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Students can Download Chapter 5 Human Capital Formation in India Questions and Answers, Plus One Economics Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations

Kerala Plus One Economics Chapter Wise Questions and Answers Chapter 5 Human Capital Formation in India

Plus One Economics Human Capital Formation in India One Mark Questions and Answers

Question 1.
As per 2011 census, the literacy rate in India was.
(i) 74%
ii) 78%
(iii) 80%
(iv) 82%
Answer:
(i) 74%

Question 2.
Which among the following is related to health sector?
(i) UGC
(ii) NCERT
(iii) ICMR
(iv) AICTE
Answer:
(iii) ICMR

Question 3.
The state in which the literacy rate is the highest in the country is ……………..
(i) Delhi
(ii) Bihar
(iii) Mizoram
(iv) Kerala
Answer:
(iv) Kerala

Question 4.
The Adult literacy rate among females in India in the year2000 was ………………..
(i) 68.4%
(ii) 61.9%
(iii) 45.4%
(iv) 37.9%
Answer:
(iii) 45.4%

Question 5.
State true or false:
‘New Educational Policy was announced in 1987-88’.
Answer:
True.

Question 6.
Which state is having highest literacy rate in India according to 2011 census?
(i) Kerala
(ii) Punjab
(iii) Manipur
(iv) Hariyana
Answer:
(i) Kerala

Plus One Economics Human Capital Formation in India Two Mark Questions and Answers

Question 1.
Mention two government organizations that regulate the health and education sector.
Answer:
1. Organizations that regulate the education sector:

• The ministries of education at the union and state level
• National Council of Educational Research and Training (NCERT)
• University Grants Commission (UGC)
• All India Council of Technical Education (AICTE)

2. Organizations that regulate the health sector:

• The ministries of health at the union and state level
• Indian Council for Medical Research (ICMR)

Question 2.
Classify the following into preventive medicines, curative medicines, and social medicines.

1. Provision of clean drinking water.
2. Good sanitation
3. Medical intervention during illness
4. Vaccination

Answer:

• Preventive Medicine – Vaccination
• Curative Medicine – Medical intervention during illness.
• Social Medicine – Provision of clean drinking water, good sanitation

Question 3.
Classify the following into the features of human capital and human development

1. Education and health are considered as a means to increase human productivity.
2. Education and health are integral to human well-being.
3. Human beings are treated as a means to an end.
4. Human beings are treated as ends in themselves.

Answer:
a. Human Capital:
1. Education and health are considered as a means to increase human productively.
3. Human beings are treated as a means to an end.
b. Human development:
2. Education and health are integral to human well being.
4. Human beings are treated as ends in themselves.

Question 4.
Name the two major sources of human capital in a country.
Answer:
Two major sources of human capital in a country are:

1. Investment in education and
2. Investment in health.

Question 5.
Bring out the need for on-the-job training for a person.
Answer:
Firms give on-the-job-training to enhance the productive skills of the workers so as to enable them to absorb new technologies and modern ideas. It can be given in two ways:

1. Workers may be trained in the firm itself under the assistance of an experienced worker.
2. Workers may be sent off the firm campus for the training.

Plus One Economics Human Capital Formation in India Three Mark Questions and Answers

Question 1.
Give full form of the following

1. GER
2. SSA

Answer:

1. GER – Gross Enrolment Ratio
2. SSA – Sarva Shiksha Abhiyan

Question 2.
Find the odd one out

1. Roads, factory, teachers, power plants
2. Teachers, doctors, ports, scientists
3. UGC, NCERT, SCERT, NREGP

Answer:

1. Teachers. Others are physical capital
2. Ports. Others are human capital
3. NREGP. Others are educational regulatory institutions

Question 3.
“Adequate education and training to farmers can raise productivity in farms.”
Do you agree or not? Substantiate your answer.
Answer:
Yes, I do agree to the statement that adequate education and training to farmers can raise productivity in farms.
This is because the attainment of more information and training will help the farmers to attain more knowledge about new methods of farming and plant protection. The application of the new knowledge in the farms will enable them to raise productivity in the farms so that the overall productivity in the agricultural sector can be improved.

Question 4.
What do you mean by Human Development Index? How it is calculated?
Answer:
The quality of life index prepared and published by United Nations Development Programme is termed as Human Development Index. HDI studies the following three basic human capabilities:

1. Living a Long Life (Longevity)
2. Being Knowledgable (Educational Attainment)
3. Enjoying decent standard of living (Real Per Capita GDP)

In orderto calculate HDI we are required to construct the following three indices.
1. Life expectancy index: This index measures the degree of the achievement of the country regarding the expectancy of the people in a country. The value of index varies between 0-1.

2. Educational attainment index: This index measures the level of educational attainment of the people. The value of this index also varies between 0-1. Higher index shows higher level of educational attainment.

3. Real GDP per capita index: Real GDP is calculated at constant price. It shows the changes in physical production in real terms. Per capita, real GDP is GDP apanstant price divided by population. It is better measure of the quality of life of the people as compared to per capita income.

Plus One Economics Human Capital Formation in India Four Mark Questions and Answers

Question 1.
Point out the major problems of educational development in India.
Answer:
The major problems of educational development in India are pointed out below.

• Wastage of resources.
• Illiteracy
• Disparities in standard
• Lack of funds
• Problems of brain drain.

Question 2.
Discuss the following as a resource of human capital formation

1. Health infrastructure
2. Expenditure on migration

Answer:
1. Health infrastructure:
Expenditure on health is an important source of human capital formation. Health expenditure directly increases the supply of healthy labour force and is a source of human capital formation.

Preventive medicine, social medicine, curative medicine, and provision of clean drinking water and good sanitation are the various forms of health expenditure. In short provision of health facilities directly improve the efficiency of human capital.

2. Expenditure on migration:
Existence of unemployment is the main reason for the rural-urban migration in India. People migrate in search of jobs. Technically qualified persons migrate to other countries because of higher salaries that they may get in other countries.

Migration in any of these cases involves cost of transport, higher cost of living in the migrated places. The enhanced earnings in the new place outweigh the cost of migration. Hence expenditure on migration is a source of human capital formation.

Question 3.
Bring out the differences between human capital and human development.
Answer:

Question 4.
The following table shows the indicators of development in Education and health sectors. Analyze the table and make inference.

Answer:
1. The per capita income of the country was Rs.3687 in the year 1951. This value gradually increased over the years and in 2001, the real per capita income became Rs. 10306. this increase is a good sign of development as far a developing country like India is concerned.

2. Crude Death rate (Per 1000 population) of the country Was 25.1 in1951 which has fallen to 8.1 in 2001. Similarly, the infant mortality rate in India also declined significantly from 146 in 1951 to 63 in 2001. This shows the developments undergone in the health sector of the country.

3. Another important achievement in the field of education is rising literacy rate. It was 16.67% in 1951 whereas, the rate increased to 65.2% in 2001. The above table shows India’s achievements in the field of education, health and per capita income since independence. In all these three fields, country made very significant development during the last 50 years.

Plus One Economics Human Capital Formation in India Five Mark Questions and Answers

Question 1.
Trace the relationship between human capital and economic growth.
Answer:
Economic growth means the increase in real national income of a country. The contribution of the educated person to economic growth is more than that of an illiterate person. Education and health are important factors of economic growth.

Human capital contributes not only towards increasing labour productivity but also stimulates innovations and creates the ability to absorb new technologies. Education provides knowledge to understand changes in society and scientific advancements. Higher-income causes building of high level of human capital and also the high level of human capital causes growth of income.

Question 2.
“The percentage of ‘education expenditure of total government expenditure’ indicates the importance of education in the scheme of things before the government”. Critically analyze the changes in government spending on education.
Answer:
The government is supposed to make expenditure on education so as to improve the efficiency of human capital in the country. The percentage of ‘education expenditure of GDP’ expresses how much of our income is being committed to the development of education in the country.

During 1952-2002, education expenditure as percentage of total government expenditure increased from 7.92 to 13.17 and as percentage of GDP increased from0.64 to 4.02. Throughout this period the increase in education expenditure has not been uniform and there has been irregular rise and fall.

In addition to lowering expenditure on education, there exists a regional disparity in case of spending on education in various parts of the country. For example, the per capita education expenditure differs considerably across states from as high as Rs 3,440 in Lakshadweep to as low as Rs 386 in Bihar. This leads to differences in educational opportunities and attainments across states.

Thus it can be concluded that the government is not giving proper attention on the education sector. The relevance of education in the development of the country is to be recognized and therefore, given top priority to the expenditure in this sector.

Question 3.
“Physical capital is well distinguished from human capital”. Narrate them distinguishing points of physical capital and human capital.
Answer:
Physical capital is well distinguished from human capital. The major points of differences are given below.

1. Physical capital is tangible and can be easily sold in the market like any other commodity. Human capital is intangible; it is endogenously built in the body and mind of its owner.

2. Human capital is not sold in the market; only the services of human capital are sold and hence the necessity of the owner of the human capital to be present in the place of production. The physical capital is separable from its owner, whereas, human capital is inseparable from its owner.

3. The two forms of capital differ in terms of mobility across space. Physical capital is completely mobile between countries except for some artificial trade restrictions. Human capital is not perfectly mobile between countries as movement is restricted by nationality and culture.

4. In the case of human capital, depreciation takes place with ageing but can be reduced, to a large extent, through continuous investment in education, health, etc. This investment also facilitates the human capital to cope with change in technology which is not the case with physical capital.

5. Nature of benefits flowing from human capital are different from that of physical capital. Human capital benefits not only the owner but also society in general. This is called external benefit. An educated person can effectively take part in a democratic process and contribute to the socio-economic progress of a nation. A healthy person, by maintaining personal hygiene and sanitation, stops the spread of contagious diseases and epidemics.

6. Human capital creates both private and social benefits, whereas physical capital creates only private benefit. That is, benefits from a capital good flow to those who pay the price for the product and services produced by it.

Question 4.
Make some argument in favour of the need for different forms of government intervention in education and health sectors.
Answer:
Both education and health have been considered as important for human capital formation. The provision of basic education and health facilities has been accepted as a goal in All societies.

It is important that these facilities be made available free for the poor classes who can not afford them in order to ensure better skills and health of manpower. However, the private sector will not be forthcoming in making large investments in nonprofitable or free distribution of such services. Hence, the government has to play an important role.

Students can Download Chapter 7 Integrals Questions and Answers, Plus Two Maths Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Plus Two Maths Integrals Three Mark Questions and Answers

Question 1.
Integrate the following. (3 Score each)

1. ∫sin x sin 2x sin 3 xdx
2. ∫sec²x cos²2x dx

Answer:
1. We have sinxsin2xsin3x
= 1/2 (2sinxsin3x) sin2x
= 1/2 (cos2x – cos4x) sin2x
= 1/4 (2sin2xcos2x – 2cos4xsi n2x)
= 1/4 [sin4x – (sin6x – sin2x)]
= 1/4(sin4x + sin2x – sin6x)
∫sin x sin 2x sin 3 xdx
= \(\frac{1}{4}\) ∫(sin 4x + sin 2x – sin 6x)dx
= –\(\frac{1}{16}\) cos4x – \(\frac{1}{8}\) cos2x + \(\frac{1}{24}\) cos6x + c.

2. sec²x cos²2x = \(\frac{\left(2 \cos ^{2} x-1\right)^{2}}{\cos ^{2} x}\)
= \(\left(\frac{2 \cos ^{2} x}{\cos x}-\frac{1}{\cos x}\right)^{2}\) = (2cosx – secx)²
= 4cos²x + sec²x – 4
= 2(1 + cos2x) + sec²x – 4
= 2cos2x + sec²x – 2
∫sec² x cos² 2x dx = ∫(2 cos 2x + sec² x – 2)dx
= sin 2x + tan x – 2x + c.

Question 2.
Find \(\int \frac{2+\sin 2 x}{1+\cos 2 x} e^{x} d x\)?
Answer:

= ∫e^x [sec² x + tan x]dx
= ∫e^x[tanx + sec²x]dx = e^x tanx + c.

Question 3.
Evaluate \(\int \frac{\sec ^{2} x d x}{\sqrt{\tan ^{2} x+4}}\)?
Answer:
Put tanx = u, sec²xdx = dy

Question 4.
Find the following integrals.

Answer:
(i) I = \(\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x\)
Put cosx = t ⇒ -sin xdx = dt
When x = 0 ⇒ t = cos0 = 1,

(ii) I = \(\int_{0}^{1} x e^{x^{2}} d x\)
Put x² = t ⇒ 2xdx = dt
When x = 0 ⇒ t = 0,
x = 1 ⇒ t = 1
I = \(\frac{1}{2} \int_{0}^{1} e^{t} d t\) =

= [e¹ – e⁰] = e – 1.

Put sin x = t ⇒ cos xdx = dt
When x = 0 ⇒ t = sin0 = 0,

(iv) I = \(\int_{0}^{2} x \sqrt{x+2} d x\)
Put x + 2 = t² ⇒ dx = 2tdt
When x = 0 ⇒ t = \(\sqrt{2}\), x = 2 ⇒ t = 2

(v) I = \(\int_{0}^{\frac{\pi}{2}} \sqrt{\sin x} \cos x d x\)
Put sin x = t ⇒ cos xdx = dt
When x = 0 ⇒ t = sin0 = 0,


Put tan x = t ⇒ sec² xdx = dt
When x = 0 ⇒ t = tan 0 = 0,

Question 5.
(i) If f (x) is an odd function, then \(\int_{-a}^{a} f(x)\) = ?
(a) 0
(b) 1
(c) 2\(\int_{0}^{a} f(x)\) dx
(d) 2a
Evaluate
(ii) \(\int_{-\pi / 2}^{\pi / 2} \sin ^{99} x \cdot \cos ^{100} x d x\)
(iii) \(\int_{-1}^{1} e^{|x|} d x\)
Answer:
(i) (a) 0.

(ii) Here, f(x) = sin⁹⁹x.cos¹⁰⁰x .then,
f(-x) = sin⁹⁹(- x).cos¹⁰⁰(- x) = – sin⁹⁹ x. cos¹⁰⁰ x = -f(x)
∴ odd function ⇒ \(\int_{-\pi / 2}^{\pi / 2} \sin ^{99} x \cdot \cos ^{100} x d x=0\).

(iii) Here, f(x) = e^|x|, f(-x) = e^|-x| = e^|x| = f(x)
∴ even function.

we have |x| = x, 0 ≤ x ≤ 1

Question 6.

1. Show that cos² x is an even function. (1)
2. Evaluate \(\int_{-\pi / 4}^{\pi / 4} \cos ^{2} x d x\) (2)

Answer:
1. Let f(x) = cos²x ⇒ f(-x) = cos² (-x) = cos² x = f(x) even.

2.

Question 7.
Find the following integrals.

Answer:

Question 8.
Find the following integrals.

Answer:

Add (1) and (2)

Question 9.
Find the following integrals.

1. \(\int \frac{1}{3+\cos x} d x\)
2. \(\int \frac{2 x}{x^{2}+3 x+2} d x\)

Answer:
1. \(\int \frac{1}{3+\cos x} d x\)
Put t = tanx/2 ⇒ dt = 1/2 sec² x/2 dx

2. \(\int \frac{2 x}{x^{2}+3 x+2} d x\) = \(\int \frac{2 x}{(x+2)(x+1)} d x\)

2x = A(x + 1) + B (x + 2)
when x = -1, -2 = B ; B = -2
when x = -2, -4 = -A ; A = 4

= 4log(x + 2) – 2log (x + 1) + C.

Plus Two Maths Integrals Four Mark Questions and Answers

Question 1.
Find the following integrals.

Answer:

x² + x +1 = A(x² + 1) + (Bx + C)(x + 2)
Put x = -2 ⇒ 4 – 2 + 1 = 5A ⇒ A = \(\frac{3}{5}\)
Equating the coefficients of x²
⇒ 1 = A + B ⇒ B = 1 – \(\frac{3}{5}\) = \(\frac{2}{5}\)
Equating the constants
⇒ 1 = A + 2C ⇒ 2C = 1 – \(\frac{3}{5}\) = \(\frac{2}{5}\) ⇒ C = \(\frac{1}{5}\)


⇒ 1 = A(x – 1) + B(x + 3)
Put x = 1 ⇒ 1 = 2A ⇒ A = \(\frac{1}{2}\)
Put x = -3 ⇒ 1 = -4B ⇒ B = – \(\frac{1}{4}\)


Equating the constants; ⇒ 1 = A
Equating the coefficients if t;
⇒ 0 = A + B ⇒ B = -1

Question 2.
Find the following integrals.

1. ∫ e^2x sin3xdx
2. ∫ x sin^-1xdx

Answer:
1. I = ∫e^2x sin3xdx = ∫ sin 3x × e^2xdx

2. ∫ x sin^-1xdx = ∫ sin^-1x × xdx

Question 3.
(i) Which of the following is the value of \(\int \frac{d x}{\sqrt{a^{2}-x^{2}}}\)? (1)

(ii) Evaluate \(\int \frac{2 x}{x^{2}+3 x+2} d x\) (3)
Answer:
(i) [sin^-1\(\frac{x}{a}\) + c]

(ii)

⇒ 2x = A(x + 1) + B(x + 2) ⇒
Put x = -2 and x = -1, we get A = 4, B = -2

Question 4.

1. Choose the correct answer from the bracket.
   ∫e^x dx = — (e^2x + c, e^-x + c, e^2x + c) (1)
2. Evaluate: ∫ e^x sinxdx

Answer:
1. e^x + c

2. I = ∫e^x sinxdx = sinx.e^x – ∫cos x.e^xdx
= sin x.e^x – (cos x.e^x – ∫(- sin x).e^x dx)
= sinx.e^x – cosxe^x – ∫sinx.e^xdx
= sin x.e^x – cos xe^x – I
2I = sin x.e^x – cos xe^x
I = \(\frac{1}{2}\)e^x(sinx – cosx) + c.

Question 5.
(i) f(x)∫g(x) dx – ∫(f'(x)∫g(x) dx)dx (1)
(a) ∫f'(x)g{x)dx
(b) ∫f(x)g'(x)dx
(c) ∫\(\frac{f(x)}{g(x)}\)dx
(d) ∫f(x)g(x)dx
(ii) Integrate sin^-1\(\sqrt{\frac{x}{a+x}}\)dx w.r.to x. (3)
Answer:
(i) (d) ∫f(x)g(x)dx

(ii) ∫sin^-1\(\sqrt{\frac{x}{a+x}}\)dx,
Put x = a tan²θ, θ = tan^-1\(\sqrt{\frac{x}{a}}\)
⇒ dx = 2a tanθ sec²θ dθ
I = ∫sin^-1\(\left(\frac{\tan \theta}{\sec \theta}\right)\) 2a tanθ sec²θ dθ
= ∫sin^-1(sinθ)2a tanθ sec²θ dθ
= 2a∫θ tanθ sec²θ dθ
Put tanθ = t, θ = tan^-1 t ⇒ sec²θ dθ = dt
= 2a ∫ tan^-1 t (t) dθ

= a[tan²θ.θ – tanθ + θ] + c
= a[θ(1 + tan²θ) – tanθ] + c

Question 6.
Match the following. (4)

Answer:

Question 7.
Evaluate \(\int \frac{x}{\sqrt{x+a}+\sqrt{x+b}} d x\)?
Answer:

Question 8.
Match the following.

Answer:
1.

2. ∫sec x(sec x + tan x)dx = ∫(sec² x + sec x. tan x)dx
= tanx + secx + c.

3. ∫e^3xdx = \(\frac{e^{3 x}}{3}\) + c.

4. ∫(sin x + cos x)dx = sin x – cosx + c.

Question 9.
Consider the integral I = \(\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x\)?

1. What substitution can be given for simplifying the above integral? (1)
2. Express I in terms of the above substitution. (1)
3. Evaluate I. (2)

Answer:
1. Substitute sin^-1 x = t.

2. We have, sin^-1 x = t ⇒ x = sint
Differentiating w.r.t. x; we get,
\(\frac{1}{\sqrt{1-x^{2}}}\)dx = dt
∴ I = ∫t sin t dt.

3. I = ∫t sin t dt = t.(-cost) -∫(-cost)dt = -t cost + sint + c
= -sin^-1 x. cos (sin^-1 x) + sin(sin^-1 x) + c
x – sin^-1 x.cos(sin^-1 x) + c.

Question 10.
Evaluate \(\int_{0}^{\pi / 4} \log (\tan x) d x\).
Answer:

Question 11.
Find the following integrals.

1. \(\int \frac{\sec ^{2} x}{\cos e c^{2} x} d x\) (2)
2. \(\int \frac{1}{x^{2}-6 x+13} d x\) (2)

Answer:
1. \(\int \frac{\sec ^{2} x}{\cos e c^{2} x} d x\) = \(\int \frac{\sin ^{2} x}{\cos ^{2} x} d x\) = ∫tan² xdx
= ∫(sec²x – 1)dx = tanx – x + c.

2. \(\int \frac{1}{x^{2}-6 x+13} d x\)

Question 12.
Match the following. Justify your answer.

Answer:

Question 13.
(i) ∫sin2x dx = ? (1)
(a) 2 cos x + c
(b) -2 sin x + c
(c) \(\frac{\cos 2 x}{2}\) + c
(d) \(-\frac{\cos 2 x}{2}\) + c
(ii) Evaluate ∫e^x sin 2x dx (3)
Answer:
(i) (d) \(-\frac{\cos 2 x}{2}\) + c.

(ii) Consider I = ∫e^x sin 2x dx
= ∫sin 2x. e^xdx = sinx.e^x – 2∫cos 2x. e^xdx
= sin 2x.e^x – 2 (cos 2x.e^x + 2∫sin 2x. e^xdx)
= sin 2x. e^x – 2 cos 2x e^x – 4 ∫sin 2x. e^xdx
= sin 2x. e^x – 2 cos 2x e^x – 4I
5 I = sin 2x. e^x – 2 cos 2x e^x
I = \(\frac{e^{x}}{5}\) (sin 2x – 2 cos 2x).

Question 14.

1. Resolve \(\frac{x^{2}+1}{x^{2}-5 x+6}\) into partial fractions. (2)
2. Hence evaluate ∫\(\frac{x^{2}+1}{x^{2}-5 x+6}\). (2)

Answer:
1.

2.

5x – 5 = A(x – 2) + B(x – 3)
x = 2, 5 = -B, B = -5
x = 3, 10 = A, A = 10
(1) ⇒ I = ∫ 1dx + ∫\(\frac{10}{x-3}\) dx – ∫\(\frac{5}{x-2}\) dx
= x + 10log(x – 3) – 5log(x – 2) + c.

Question 15.
Evaluate \(\int_{0}^{4}\) xdx as a limit of sum.
Answer:
By definition,
\(\int_{a}^{b}\) f(x) dx =
(b – a)\(\lim _{n \rightarrow \infty} \frac{1}{n}\){f(a) + f(a + h) +…….+f(a + {n – 1)h)}
Here, a = 0, b = 4, f(x) = x, h = \(\frac{4-0}{n}=\frac{4}{n}\) ⇒ nh = 4

Question 16.

1. Define the real valued function f(x) = |x² + 2x – 3| (2)
2. Evaluate \(\int_{0}^{2}\)|x² + 2x – 3|dx. (2)

Answer:
1. f(x) = |x² + 2x – 3| = |(x – 1) (x + 3)|
We have;

2. I = \(\int_{0}^{2}\)|x² + 2x – 3|dx

Question 17.
Consider the function f(x) = |x|+|x + 1|

1. Define the function f (x) in the interval [-2, 1]. (2)
2. Find the integral \(\int_{-2}^{1}\) f(x) dx (2)

Answer:
1. Given, f(x) = |x|+|x + 1|.
We have,

Combining these two functions, we get the function f(x).

2.

Question 18.
Evaluate \(\int_{\sqrt{6}}^{\sqrt{3}} \frac{d x}{1+\sqrt{\tan x}} d x\). (4)
Answer:

Plus Two Maths Integrals Six Mark Questions and Answers

Question 1.
(i) Fill in the blanks. (3)
(a) ∫ tan xdx = —
(b) ∫ cos xdx = —
(c) ∫\(\frac{1}{x}\)dx = —
(ii) Evaluate ∫sin³ xcos² xdx (3)
Answer:
(i) (a) log|secx| + c
(b) sinx + c
(c) log|x| + c.

(ii) ∫sin³ xcos² xdx = ∫sin² xcos² x sin xdx
= ∫(1 – cos² x)cos² x sin xdx
Put cos x = t ⇒ – sin xdx = dt
∴ ∫(1 – cos² x)cos² xsin xdx = -∫(1 – t² )t²dt
= ∫(t⁴ – t²)dt = \(\frac{t^{5}}{5}-\frac{t^{3}}{3}\) + c
= \(\frac{\cos ^{5} x}{5}-\frac{\cos ^{3} x}{3}\) + c.

Question 2.
Find the following integrals.

Answer:
(i) I = ∫(3x – 2)\(\sqrt{x^{2}+x+1} d x\)
Let 3x – 2 = A(2x + 1) + B
⇒ 3 = 2 A ⇒ A = \(\frac{3}{2}\)
⇒ -2 = A + B ⇒ -2 = \(\frac{3}{2}\) + B
⇒ B = -2 – \(\frac{3}{2}\) = – \(\frac{7}{2}\)


Using (2) and (3) in (1) we have;

(ii) I = \(\int \frac{2 x-3}{x^{2}+3 x-18} d x\)
Let 2x – 3 = A(2x + 3) + B
⇒ 2 = 2A ⇒ A = 1
⇒ -3 = 3A + B ⇒ -3 = 3 + B ⇒ B = -6

(iii) I = \(\int \frac{5 x+2}{1+2 x+3 x^{2}} d x\)
Let 5x + 2 = A{6x + 2) + B
⇒ 5 = 6 A ⇒ A = \(\frac{5}{6}\)
⇒ 2 = 2A + B ⇒ 2 = \(\frac{5}{3}\) + B ⇒ 2 – \(\frac{5}{3}\) = \(\frac{1}{3}\)

(iv) I = \(\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x\)
Let 5x + 3 = A(2x + 4) + B
⇒ 5 = 2A ⇒ A = \(\frac{5}{2}\)
⇒ 3 = 4A + B ⇒ 3 = 10 + B ⇒ B = -7


Using (2) and (3) in (1) we have;

Question 3.
Consider the expression \(\frac{1}{x^{3}-1}\)

1. Split it into partial fraction. (2)
2. Evaluate ∫ \(\frac{1}{x^{3}-1}\) dx (4)

Answer:
1.

1 = A (x² + x + 1) + (Bx + c)(x + 1),
Put x = -1 ⇒ 1 = A(1 + 1 + 1) ⇒ A= \(\frac{1}{3}\)
Equating like terms.
0 = A + B ⇒ B = – \(\frac{1}{3}\), 1 = A + C ⇒ C = \(\frac{2}{3}\)

2.

Put, x – 2 = D (2x – 1) + E ,
1 = 2 D ⇒ D = \(\frac{1}{2}\),
-2 = -D + E ⇒ E = –\(\frac{3}{2}\)

Question 4.
(i) Match the following (4)

(ii) Consider the function f(x) = \(\frac{x^{4}}{x+1}\) Evaluate ∫f(x)dx (2)
Answer:
(i)

(ii) Here the numerator is of degree 4 and denominator of degree 1. So to make it a proper fraction we have to divide Nr by Dr.

Question 5.

1. Evaluate the as \(\int_{0}^{2}\)x²dx the limit of a sum. (3)
2. Hence evaluate \(\int_{-2}^{2}\)x²dx (1)
3. If \(\int_{0}^{2}\) f(x)dx = 5 and \(\int_{-2}^{2}\) f(x)dx = 0, then \(\int_{-2}^{0}\) f(x)dx = …….. (2)

Answer:
1. Here the function is f(x) = x², a = 0, b = 2 and h = \(\frac{b-a}{n}=\frac{2}{n}\)
\(\int_{0}^{2}\)x²dx =

2. \(\int_{-2}^{2}\) x²dx = 2 \(\int_{0}^{2}\)x²dx = \(\frac{16}{3}\)

3.

Question 6.
Find ∫\(\sqrt{\tan x}\)xdx.
Answer:
Given;
I = ∫\(\sqrt{\tan x}\)xdx,
Put tanx = t² ⇒ sec²xdx = 2tdt ⇒ dx = \(\frac{2 t d t}{1+t^{4}}\)

Question 7.
(i) Match the following. (2)

(ii) Integrate \(\frac{\sec ^{2} x}{5 \tan ^{2} x-12 \tan x+14}\) w.r.to x. (4)
Answer:
(i)

Question 8.

1. Evaluate \(\int_{0}^{1} \sqrt{x} d x\) (1)
2. If \(\int_{0}^{a} \sqrt{x} d x=2 a \int_{0}^{\pi / 2} \sin ^{3} x d x\), find the value of a. (3)
3. Hence find \(\int_{a}^{a+1}\)x dx. (2)

Answer:
1.

2. Given;

3. When a = 0

When, a = 4

Question 9.
(i) Let f (x) be a function, then \(\int_{0}^{a}\) f(x) dx = ? (1)
(a) 2 \(\int_{0}^{a}\) f(x – a) dx
(b) \(\int_{0}^{a}\) f(a – x) dx
(c) f(a)
(d) 2\(\int_{0}^{a}\) f(a – x) dx
Evaluate

Answer:
(i) (b) \(\int_{0}^{a}\) f(a – x) dx

(ii)

(1) + (2)

⇒ I = 1.

(iii)

Question 10.
Find the following integrals.

1. ∫\(\frac{2 e^{x}}{e^{3 x}-6 e^{2 x}+11 e^{x}-6} d x\)
2. ∫\(\frac{(3 \sin x-2) \cos x}{5-\cos ^{2} x-4 \sin x} d x\)

Answer:
1.

⇒ 1 = A(t – 2)(t – 3) + B(t – 1)(t – 3) + C(t – 1)(t – 2)
Put t = 1 ⇒ 1 = A(-1)(-2) ⇒ A = \(\frac{1}{2}\)
Put t = 2 ⇒ 1 = B(1)(-1) ⇒ B = -1
Put t = 3 ⇒ 1 = B(2)(1) ⇒ B = \(\frac{1}{2}\)

2. I = ∫\(\frac{(3 \sin x-2) \cos x}{5-\cos ^{2} x-4 \sin x} d x\)dx
Put sin x = t ⇒ cosxdx = dt

⇒ 3t – 2 = A(t – 2) + B
Equating the coefficients if t; ⇒ 3 = A
Equating the constants
⇒ -2 = -2A + B ⇒ -2 = -6 + B ⇒ B = 4

Question 11.

1. Find ∫\(\frac{1}{x^{2}+a^{2}}\)dx (1)
2. Show that 3x + 1 = \(\frac{3}{4}\)(4x – 2) + \(\frac{5}{2}\) (2)
3. Evaluate \(\int \frac{3 x+1}{2 x^{2}-2 x+3} d x\) (3)

Answer:
1. ∫\(\frac{1}{x^{2}+a^{2}}\)dx = 1/a tan^-1 x/a + c.

2. 3x + 1 = A \(\frac{d}{d x}\)(2x² – 2x + 3) + B
= A(4x – 2) + B
3 = 4A; A = 3/4
1 = -2A + B
1 = -3/2 + B, B = 1 + 3/2 = 5/2
∴ 3x + 1 = 3/4(4x – 2) + 5/2

3.

Students can Download Chapter 12 ICT and Society Questions and Answers, Plus Two Computer Science Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations

Kerala Plus Two Computer Science Chapter Wise Questions and Answers Chapter 12 ICT and Society

Plus Two Computer Science ICT and Society One Mark Questions and Answers

Question 1.
IPR stands for ______.
Answer:
Intellectual Property Right.

Question 2.
WIPO stands for _____.
Answer:
World Intellectual Property Organisation

Question 3.
______ is the exclusive rights to prevent unauthorized copying of inventions by a Creator from the Unauthorised person or company.
Answer:
Patent

Question 4.
_____ is a unique, simple and memorable sign to promote a brand and hence increase the business and goodwill of a company.
Answer:
Trademark

Question 5.
A product or article is designed so beautifully to attract customers. This type of design is called
Answer:
Industrial Design.

Question 6.
Aranmula Kannadi, Palakkadan Matta, Marayoor Sarkkara, etc are example of _______.
Answer:
Geographical indications.

Question 7.
_____ is the property right that arises automatically when a person creates a new work by his own and by Law it prevents the others from the unauthorized or intentional copying of this without the permission of the creator.
Answer:
Copyright

Question 8.
From the following which is the symbol for copyright.
(a) $
(b) ©
(c) ®
(d) ™
Answer:
(b) ©

Question 9.
From the following which is the symbol for Unregistered trademark.
(a) $
(b) ©
(c) ®
(d) ™
Answer:
(d) ™

Question 10.
From the following which is the symbol for Registered trademark.
(a) $
(b) ©
(c) ®
(d) ™
Answer:
(c) ®

Question 11.
Unauthorized copying or use of Intellectual property rights such as Patents, Copyrights and Trademarks are called ____.
Answer:
Intellectual Property Infringement.

Question 12.
_____ prevents others from the unauthorized or intentional copying or use of Patent without the permission of the creator.
Answer:
Patent Infringement.

Question 13.
______ is the illegal copying, distribution, or use of software.
Answer:
Piracy.

Question 14.
______ prevents others from the unauthorized or intentional copying or use of Trademark without the permission of the creator.
Answer:
Trademark Infringement

Question 15.
_____ prevents others from the unauthorized or intentional copying or use of Copy right without the permission of the creator.
Answer:
Copy right Infringement

Question 16.
______ is a virtual environment created by computer systems connected to the internet
Answer:
Cyberspace

Question 17.
A person committing crimes and illegal activities with the use of computers over Internet. This crime is included as _____ crime.
Answer:
Cybercrime

Question 18.
State True or False.
Cybercrimes can be classified into three categories such as against individual, property, and Government.
Answer:
True

Question 19.
Phishing, hacking, denial of service attacks, etc are ____ crimes.
Answer:
Cyber

Question 20.
Odd one out
(а) Identity theft
(b) Harassment
(c) violation of privacy
(d) credit card fraud
Answer:
(d) credit card fraud, it is a cybercrime against individual others are cyber crimes against property.

Question 21.
Odd one out
(a) Credit card theft
(b) Intellectual property theft
(c) Internet time theft
(d) Dissemination of obscene material
Answer:
(d) Dissemination of obscene material, It is cyber , crime against individual, the others are cyber against property.

Question 22.
Odd one out
(a) cyberterrorism
(b) Attacks against e-Governance websites
(c) Impersonation and cheating
(d) Website defacement
Answer:
(c) Impersonation and cheating, it is cybercrime against individual others are cyber crimes against Government.

Question 23.
IT Act amended in _____.
(a) 2015
(b) 2008
(c) 1900
(d) 1998
Answer:
(b) 2008

Question 24.
IT Act passed in Indian parliament is ____.
Answer:
2000.

Question 25.
The laws to prevent cyber crimes is termed as ____.
Answer:
Cyberlaw

Question 26.
_____ excessive enthusiasm for acquiring knowledge.
Answer:
Infomania

Question 27.
Phishing is an example of ______.
Answer:
Cybercrime.

Question 28.
ICT stands for _______.
(a) Internet and Communication Technology
(b) Information and Computer Technology
(c) Information and Communication Technology
(d) Integrated Communication Technology
Answer:
(c) Information and Communication Technology

Question 29.
Which of the following e-Governance helps citizens for interacting with the Government?
(a) G2E
(b) G2B
(c) G2C
(d) G2G
Answer:
(c) G2C

Question 30.
What are the different types of interactions in e-Governance?
Answer:
G2G, G2E.G2B, G2C.

Question 31.
The unauthorized use of intellectual property rights is termed as
Answer:
Infringement

Question 32.
Expand the term WIPO in connection with IPR.
Answer:
World Intellectual Property Organization.

Question 33.
The exclusive right granted to an invention is called
(a) Trademark
(b) Copy right
(c) Patent
(d) Design
Answer:
(c) Patent

Question 34.
The exclusive right given to a person over the creation of his/her mind for a period of time is called
Answer:
Patent / Intellectual Property Right

Question 35.
What is the name given to the process of using scientific knowledge for analyzing and presenting evidence of cyber related crimes before court?
Answer:
Cyber forensics

Question 36.
Which among the following are considered as violation to privacy?
1. Keeping hidden cameras in private places
2. Publishing private photos of individual in social media without their permission
3. Use of unauthorized software
4. Using simple password
(A) All the above are correct
(B) 1,2 and 3 only
(C) 1 and 4 only
(D) 1 and 2
Answer:
(D) 1 and 2

Plus Two Computer Science ICT and Society Two Mark Questions and Answers

Question 1.
“IPR (Intellectual Property Right) encourages innovation” Justify.
Some people spend lots of money,time body and mental power to create some products such as a classical movie, album, artistic work, discoveries, invention, software, etc. These type of Intellectual properties must be protected from unauthorized access by law. This is called Intellectual Property right(IPR). It enables to earn recognition, financial benefit, can sell the innovation, etc. It motivates further innovation.

Question 2.
Define the following terms.

1. Cyber space
2. Cyber crime

Answer:
1. CyberSpace:
Earlier Traditional communication services such as postal service(Snail mail) are used for communication. It is a low speed and not reliable service. In order to increase the speed Telegram Services were used. Its speed was high but it has lot of limitations and expensive too.

Later telephones were used for voice communication. Nowadays telephone system and computer system are integrated and create a virtual(un real) environment. This is called cyber space. The result for this integration is that tremendous speed and it is very cheap.

2. Cyber crime:
Just like normal crimes (theft, trespassing private area, destroy, etc,) Cyber crimes (Virus, Trojan Horse, Phishing, Denial of Service, Pornography, etc) also increased significantly. Due to cyber crime, the victims lose money, reputation, etc and some of them commit suicide.

Question 3.
Write a short note on

1. Trademark
2. Industrial design

Answer:
1. Trademark:
This is a unique, simple and memorable sign to promote a brand and hence increase the business and goodwill of a company. It must be registered. The period of registration is for 10 years and can be renewed. The registered trademark under Controller General of Patents Design and Trademarks cannot use or copy by anybody else.

2. Industrial designs:
A product or article is designed so beautifully to attract the customers. This type of designs is called industrial design. This is a prototype and used as a model for large scale production.

Question 4.
Compare patent and Trademark.
1. Patents:
A person or organization invented a product or a creation can be protected from unauthorized copying or creation without the permission of the creator by law. This right is called Patent. In India the validity of the right is up to 20 years. After this anybody can use freely.

2. Trademark:
This is a unique, simple and memorable sign to promote a brand and hence increase the business and goodwill of a company. It must be registered. The period of registration is for 10 years and can be renewed. The registered trademark under Controller General of Patents Design and Trademarks cannot use or copy by anybody else.

Question 5.
Write any one website for the following services.

1. e-Governance
2. e-Business
3. e-Banking
4. e-Learning

Answer:

1. e-Governance(any One) www.dhsekerala.gov.in, www.incometaxindia.gov.in, www.spark.gov.in,www.ceo.kerala.gov. in
2. e-Business www.indane.co.in, www.amazon.com,www.ebay.in
3. e-Banking www.onlinesbi.co.in
4. e-Learning www.ignouonline.ac.in,www.nptel.iitm.ac.in

Question 6.
Write a short note about EPS.
Answer:
Electronic Payment System(EPS): It is also called plastic money that is electronically exchange money between two individuals or firms(buyers and sellers) in an online environment.

Question 7.
What is cyberspace?
Answer:
Earlier Traditional communication services such as postal service(Snail mail) are used for communication. It is a low speed and not reliable service. In order to increase the speed Telegram Services were used. Its speed was high but it has lot of limitations and expensive too.

Later telephones were used for voice communication. Nowadays telephone system and computer system are integrated and create a virtual(unreal) environment. This is called cyberspace. The result for this integration is that tremendous speed and it is very cheap.

Question 8.
Why is cyberspace called a virtual world?
Answer:
The telephone system and computer system are integrated and create a virtual(un real) environment. This is called cyber space. The result for this integration is that tremendous speed and it is very cheap. This is an imaginary world. We can see persons with different behaviour. Because of good and bad people we can’t believe blindly. If we search a solution for a problem thousands of answers will get instantly and may confused us.

Question 9.
What is copyright? How does it differ from patent?
Answer:
1. Copyright:
The trademark is ©, copyright is the property right that arises automatically when a person creates a new work by his own and by Law it prevents the others from the unauthorized or intentional copying of this without the permission of the creator for 60 years after the death of the author.

2. Patents:
A person or organization invented a product or a creation can be protected from unauthorized copying or creation without the permission of the creator by law. This right is called Patent. In India the validity of the right is up to 20 years. After this anybody can use freely.

Question 10.
Explain the exclusive right given to the owner by IPR?
Answer:
The exclusive right given to the owner by I PR is owner can disclose their creations for money.

Question 11.
it is the unauthorized copying, distribution, and use of a creation without the permission of the creator. It is against the copyright act and hence the person committed deserve the punishment.

Question 12.
Match the following

Answer:
a – 2
b – 3
c – 4
d – 1

Question 13.
What do you meant by infringement?
Answer:
Unauthorized copying or use of Intellectual property rights such as Patents, Copy rights and Trademarks are called intellectual property lnfringement(violation). It is a punishable offence.

Plus Two Computer Science ICT and Society Three Mark Questions and Answers

Question 1.
Write a short note on the importance of IT Act 2000.
Answer:
Information Technology Act 2000(amended in 2008):
IT Act 2000 controls the use of Computer(client), Server, Computer Networks, data and Information in Electronic format and provide legal infrastructure for E-commerce, in India. This is developed to promote IT industry, control e-commerce also ensures the smooth functioning of E-Governance and it prevents cyber crimes.

The person those who violate this will be prosecuted. In India, IT bill introduced in the May 2000 Parliament Session and it is known as Information Technology Act 2000. Some exclusions and inclusions are introduced in December 2008.

Question 2.
“Infomania affects peoples’ lives and their loved ones.”
Comment on this statement.
Answer:
Info mania is excessive desire(infatuation) for acquiring knowledge from various modern sources like Internet, Email, Social media. Instant Message Application(WhatsApp) and Smart Phones. Due to this the person may neglect daily routine such as family, friends, food, sleep, etc. hence they get tired.

They give first preference to Internet than others. They create their own Cyber World and no interaction to the surroundings and the family. They are more anxious and afraid that they will be out from the cyber world unless they updated.

Question 3.
Define the term e-Business. What are the advantages and challenges of e-Business? Write any two e-Business websites.
Answer:
E-business(electronic Business): Providing ser¬vices or running business through internet is called E-business.
Advantages of e-business:

• It overcomes geographical limitations
• It reduces the operational cost
• It minimizes the time and cost
• It remains open all the time
• We can locate the product faster from a wider range of choices
  Challenges to E business
• Peoples are unaware of IT applications and its uses
• Most peoples don’t have plastic money(credit / debit card) and net banking
• It requires high security measurements otherwise you may lose money
• We can’t touch or smell products through online
• Some companies may not have proper Goods delivery service
  Useful e-Business websites

Question 4.
How do trademark and industrial design differ?
Answer:
Trademark:
This is a unique, sirhple and memorable sign to promote a brand and hence increase the business and goodwill of a company. It must be registered. The period of registration is for 10 years and can be renewed. The registered trademark under Controller General of Patents Design and Trademarks cannot use or copy by anybody else.

Industrial designs:
A product or article is designed so beautifully to attract the customers. This type of designs is called industrial design. This is a prototype and used as a model for large scale production.

Question 5.
Why is Cyberlaw important?
Answer:
Just like normal crimes (theft, trespassing private area, destroy, etc.) Cybercrimes (Virus, Trojan Horse, Phishing, Denial of Service, Pornography, etc.) also increased significantly. Due to cybercrime, the victims lose money, reputation, etc. and some of them commit suicide.

Cyberlaw ensures the use of computers and Internet by the people safely and legally. It consists of rules and regulations like Indian Penal Code (IPC) to stop crimes and for the smooth functions of Cyberworld. Two Acts are IT Act 2000 and IT Act Amended in 2008.

Question 6.
“Infomania has became a psychological problem”. Write your opinion.
Answer:
Info mania is the excessive desire(lnfatuation) for acquiring knowledge from various modern sources like Internet, Email, Social media, Instant Message Application(WhatsApp) and Smart Phones. Due to this the person may neglect daily routine such as family, friends, food, sleep, etc. hence they get tired.

They give first preference to Internet others. They create their own Cyber World and no interaction to the surroundings and the family. They are more anxious and afraid that they will be out from the cyber world unless they updated.

Plus Two Computer Science ICT and Society Five Mark Questions and Answers

Question 1.
“Due to anonymous nature of Internet it is possible for the people to engage in variety of criminal activities.” Justify the statement with special reference to cyber crimes taking place against individual.
Answer:
Cyber crimes against individuals
i. Identity theft:
The various information such as personal details(name, Date of Birth, Address, Phone number etc) , Credit / Debit Card details(Card number, PIN, Expiry Date, CW, etc), Bank details, etc. are the identity of a person. Stealing these information by acting as the authorized person without the permission of a person is called Identity theft. The misuse of this information is a punishable offence.

ii. Harassment:
Commenting badly about a particular person’s gender, colour, race, religion, nationality, in Social Media is considered as harassment. This is done with the help of Internet is called Cyber stalking (Nuisance). This is a kind of torturing and it may lead to spoil friend ship, career, self image and confidence. Sometimes may lead to a big tragedy of a whole family or a group of persons.

iii. Impersonation and cheating:
Fake accounts are created in Social Medias and act as the original ICT and Society one for the purpose of cheating or misleading others. Eg: Fake accounts in Social Medias (Facebook, Twitter, etc), fake SMS, fake emails etc.

iv. Violation of privacy:
Trespassing into another person’s life and try to spoil life. It is a punishable offence. Hidden camera is used to capture the video or picture and black mailing them.

v. Dissemination of obscene material: With the help of hidden camera capture unwanted video or picture. Distribute or publish this obscene clips on Internet without the consent of the victims may mislead the people specifically the younger ones.

Question 2.
Explain different categories of cyber crimes in detail.
Answer:
Just like normal crimes( theft, trespassing private area, destroy, etc,) Cyber crimes(Virus, Trojan Horse, Phishing, Denial of Service, Pornography etc) also increased significantly . Due to cyber crime, the victims lose money, reputation,etc and some of them commit suicide.
Answer:
A. Cyber crimes against individuals:
1. Identity theft:
The various information such as personal details(name, Date of Birth, Address, Phone number etc.), Credit / Debit Card details(Card number, PIN, Expiry Date, CW, etc), Bank details, etc. are the identity of a person. Stealing these information by acting as the authorized person without the permission of a person is called Identity theft. The misuse of this information is a punishable offence.

2. Harassment:
Commenting badly about a particular person’s gender, colour, race, religion, nationality, in Social Media is considered as harassment. This is done with the help of Internet is called Cyber stalking (Nuisance). This is a kind of torturing and it may lead to spoil friend ship, career, self image and confidence. Sometimes may lead to a big tragedy of a whole family or a group of persons.

3. Impersonation and cheating:
Fake accounts are created in Social Medias and act as the original one for the purpose of cheating or misleading others. Eg: F.ake accounts in Social Medias (Facebook, Twitter,etc), fake sms, fake emails, etc.

4. Violation of privacy:
Trespassing into another person’s life and try to spoil the life. It is a punishable offence. Hidden camera is used to capture the video or picture and black mailing them.

5. Dissemination of obscene material:
With the help of hidden camera capture unwanted video or picture. Distribute or publish this obscene clips on Internet without the consent of the victims may mislead the people specifically the younger ones.

B. Cyber crimes against property:
Stealing credit card details, hacking passwords of social media accounts or mail account or Net banking, uploading latest movies etc, are considered as cyber crimes against property.
1. Credit card fraud:
Stealing the details such as credit card number, company name, expiry date, cw number,password etc. and use these details to make payment for purchasing goods or transfer funds also.

2. Intellectual property theft:
The violation of Intellectual Property Right of Copy right, Trademark, Patent, etc. In film industry crores of investment is needed to create a movie. Intellectual Property thieves upload the movies on the Releasing day itself. Hence the revenue from the theatres are less significantly and undergoes huge loss.(Eg: Premam, Bahubali, etc) Copying a person’s creation and present as a new creation is called plagiarism. This can be identified some tools(programs) available in the Internet

3. Internet time theft:
This is deals with the misuse of WiFi Internet facility. If it is not protected by good password there is a chance of misuse our devices(Modem/Router) to access Internet without our consent by unauthorized persons. Hence our money and volume of data(Package) will lose and we may face the consequences if others make any crimes.

C. Cyber crimes against government:
The cyber crimes against Govt, websites is increased significantly. For example in 2015 the website of Registration Department of Kerala is hacked and destroys data from 2012 onwards.

1. Cyber terrorism:
It is deals with the attacks against very sensitive computer networks like computer controlled atomic energy power plants, air traffic controls, Gas line controls, telecom, Metro rail controls, Satellites, etc. This is a very serious matter and may lead to huge loss (money and life of citizens). So Govt, is very conscious and give tight security mechanism for their services.

2. Website defacement:
It means spoil or hacking websites and posting bad comments about the Govt.

3. Attacks against e-governance websites :
Its main target is a Web server. Due to this attack the Web server/ computer forced to restart and this results refusal of service to the genuine users. If we want to access a website first you have to type the web site address in the URL and press Enter key, the browser requests that page from the web server. Dos attacks send huge number of requests to the web server until it collapses due to the load and stops functioning.

Question 3.
“For the implementation of e-Learning different tools.
Answer:
e Learning tools
1. Electronic books reader(eBooks): With the help of a tablet or portable computer or any other device we can read digital files by using an s/w is called electronic books reader.

2. e-text: The electronic format of textual data is called e-Text.

3. Online chat: Realtime exchange of text or audio or video messages between two or more person over the Internet.

4. e-Content: The data or information such as text, audio, video, presentations, images, animations, etc, are stored in electronic format.

5. Educational TV channels: TV channels dedicated only for the e-Learning purpose.
Eg. VICTERS (Virtual Classroom Technology on Edusat for Rural Schools OR Versatile ICT Enabled Resources for Students)

Students can Download Chapter 4 Graphs and Charts for Business Data Questions and Answers, Plus Two Accountancy Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Chemistry Chapter Wise Questions and Answers Chapter 4 Graphs and Charts for Business Data

Plus Two Accountancy Graphs and Charts for Business Data One Mark Questions and Answers

Question 1.
_____________ are the visual representation of numerical data
Answer:
Chart/ Graph

Question 2.
Chart / Graph has at least ____________ dimenstional relationship
(a) Two
(b) Three
(c) Four
(d) Five
Answer:
(a) Two

Question 3.
____________ chart is suitable for comparing multiple.
Answer:
Bar Chart

Question 4.
In column chart, the X-axis shows
(a) Value of each category
(b) Different categories
(c) Height of the chart
(d) Depth of the value
Answer:
(b) Different categories

Question 5.
________ Chart is similar to the column chart, with the difference being that the data series are displayed horizontally
(a) Line chart
(b) Pie chart
(c) Barchart
(d) Area chart
Answer:
(c) Bar chart

Question 6.
________ chart shows data changes for a certain period of time.
Answer:
Line chart

Question 7.
_______ chart contains only one data series
Answer:
Pie chart

Question 8.
_______chart shows values as circular sectors to the total circle
Answer:
Pie chart

Question 9.
Pie chart don’t have more than __________ categories.
(a) Ten
(b) Twenty Five
(c) Seven
(d) Three
Answer:
(c) Seven

Question 10.
____________ is a pictorial representation of data, which has at least two dimensional relationships.
(a) Graph
(b) Chart
(c) Diagram
(d) All the above
Answer:
(d) All the above

Question 11.
_________ Chart is used to compare values across categories.
(a) Column chart
(b) Line chart
(c) Pie chart
(d) Barchart
Answer:
(a) Column chart

Question 12.
_________ chart is used to display trends over time.
Answer:
Line chart

Question 13.
The entire chart including all elements is termed as ________
Answer:
Chart area

Question 14.
In 3D chart, the area is bounded by three axes ie _____, _____ & _____
Answer:
X, Y, Z

Question 15.
In ______ chart, both axes display values ie they have no category axis.
Answer:
XY Chart or Scatter diagram

Question 16.
_______ specifices the colour, symbol or pattern used to mark data series.
Answer:
Legends

Question 17.
The change the location of a chart, right click the chart and select.
(a) Chart Type
(b) Source Data
(c) Move here
(d) Chart Options
Answer:
(c) Move here

Question 18.
In 3D chart X, Y & Z axes are used to show
(a) Category, Value, Total
(b) Depth, Vertical, Horizontal
(c) Length, Breadth, Depth
(d) Category, Value, Series.
Answer:
(d) Category, Value, Series

Question 19.
Legand can be repositioned on the chart
(a) Anywhere
(b) On right side only
(c) On the corner only
(d) On the left side only
Answer:
(a) Anywhere

Question 20.
Which chart element details the data values and categories below the chart?
(a) Data table
(b) Data marker
(c) Data labels
(d) Datapoint
Answer:
(c) Data labels

Question 21.

Which type of chart is this?
Answer:
Radar chart

Question 22.
The intersection of both the axis (X-axis and Y-axis) is called __________ of the graph.
Answer:
Origin (0)

Question 23.
_________ Chart display in rings, where each ring represents a data series
Answer:
Doughnut chart.

Question 24.
Name the given chart

Answer:
Bar Chart

Question 25.
In _________ chart, the area below the plotted lines is solid
Answer:
Area chart.

Question 26.
Radar chart / Net chart is also known as ______
(a) Doughnut chart
(b) Pie chart
(c) Ara chart
(d) Star chart
Answer:
(d) Star chart

Question 27.
Which among the following is the special feature of 3D chart
(a) Chart area
(b) X & Y axes
(c) Chart wall
(d) Legend
Answer:
(c) Chart wall

Question 28.
Give a suitable name to the diagram

(a) Barchart
(b) Single line graph
(c) Pie chart
(d) Area Chart
Answer:
(c) Pie Chart

Plus Two Accountancy Graphs and Charts for Business Data Two Mark Questions and Answers

Question 1.
Name the different chart formats in Libre Office Calc
Answer:
Barchart, Column Chart, Pie chart, Line chart, Area chart, Doughnut chart, etc.

Question 2.
What is the importance of charts and graphs in business?
Answer:

1. Chart and graphs covey lots of business information in a visual format
2. Different business Data variables plotted in charts and graphs show the trend of the business in an easy way

Question 3.
Identify the type of chart

Answer:
Line chart

Question 4.
Give a short note on it.

1. Barchart
2. Pie chart

Answer:
1. Bar Chart
This type of chart shows a bar graph or column chart with horizontal bars. The Y-axis shows categories and the X-axis shows the value for each category. It is suitable for comparing multiple values.

2. Pie chart
A pie chart displays the contribution of each value to a total. It represents multiple subgroup of a single variable. It contains only one data series. A pie chart shows values as circular sectors of the total circle. Pie chart may be

• Normal Pie chart
• Exploded Pie chart
• Doughnut chart or Donut chart
• Exploded Doughnut chart

Question 5.
What are the special features of graphs and charts?
Answer:

1. Graphs/charts are the pictorial representation of business data
2. A chart represents tabular numeric data
3. Dimensions in the data are often displayed on axes (X, Y, & Z)

Question 6.
Differentiate between Chart area and Chart wall?
Answer:
1. Chart area:
This is the total space that is enclosed by a chart. It is the background of the chart.

2. Chart wall:
In 2D chart, the wall or area is bounded by the X and Y-axis. In the 3D chart, the wall is bounded by three axes X, Y and Z

Question 7.
Quarterly sales of a business firm is used to create a bar graph. Identify the Data variables plotted on X and Y-axis
Answer:
X-axis – I^st Quarter, II^nd Quarter, III^rd Quarter, IV^th Quarter,
Y-axis – Sales in I^st Quarter, Sales in I^nd Quarter, Sales in III^rd Quarter, Sales in IV^th Quarter

Question 8.
Identify the type of chart

Answer:
(a) 2D Chart
(b) 3D Chart

Question 9.
What is the use of Auto shapes in LibreOffice Calc?
Answer:
Auto shapes tool bar allows drawing a number of geometrical shapes, arrows; flow chart elements, etc.

Question 10.
What is PIE chart? What are the specialties of PIE chart.
Answer:
Pie chart:
A pie chart displays the contribution of each value to a total. It represents multiple subgroup of a single variable. It contains only one data series. A pie chart shows values as circular sectors of the total circle. Pie chart may be

1. Normal Pie chart
2. Exploded Pie chart
3. Doughnut chart or Donut chart
4. Exploded Doughnut chart

Question 11.
Choose the right statements from the following.

1. We can put on the right side of the origin positive values and on left side of the origin negative values of data on X-axis
2. The upward side of origin shows postiive values and downward side of the origin shows negative values of data on Y-axis.
3. We can put on the right side of the origin negative values and on left side of the origin positive values of data on X-axis.
4. The upward side of origin shows negative values and downward side of the origin shows positive values of data on Y-axis.

Answer:
Right statements a & b

Question 12.
Identify the type of given chart. List down its features.

Answer:

1. It is a Doughnut chart
2. Features of the doughnut chart
   • It displays data in rings
   • Each ring represents a data series
   • The first data series is displayed in the center of the chart

Question 13.
What is a 3-D chart?
Answer:
Charts can be prepared with three dimensional (3-D) effects. 3- D charts have a third axis. The third axis is called as Z-axis. So a 3-D chart has the fol¬lowing dimensions.

1. Horizontal axis – Indicate the category – known as X-axis
2. Vertical axis – Indicate the derived values – known as Y-axis
3. Depth axis – Indicate the series – known as Z-axis

Question 14
Differentiate between Data Marker and Data series.
Answer:
1. Data Marker:
Individual values plotted in a chart are called data marker or data point.

2. Data Series:
Data markers of the same colour or pattern is called data series.

Question 15.
Is there any difference between

1. A column chart and
2. A bar chart?

Substantiate your answer
Answer:
1. Column Chart:
It is the most commonly used chart type. It shows a bar chart or bar graph with vertical bars. The X-axis shows the categories and Y-axis shows the value for each category. Column chart are used to compare values across categories.

2. Bar Chart:
This type of chart shows a bar graph or column chart with horizontal bars. The Y-axis shows categories and the X-axis shows the value for each category. It is suitable for comparing multiple values.

Plus Two Accountancy Graphs and Charts for Business Data Three Mark Questions and Answers

Question 1.
Match the following

Answer:

Question 2.
What are the advantages of using Graph/ Chart?
Answer:
Advantages in using Graph/Chart:

1. It summarises a large data set in visual form
2. Charts or graphs can clarify trends better than do tables.
3. It helps to estimate key values at a glance
4. It shows each data category in a frequency distribution.
5. It permits a visual check of the accuracy and reasonableness of calculations
6. The charts and graphs allow the investigator to draw a valid conclusion.

Question 3.
What are the elements of a Chart/ Graph
Answer:

Question 4.
How to use word Art styles to format text.
Answer:

• Step 1: Click in the chart element that contains text to be changed.
• Step 2: Click on the format.
• Step 3: Click on word Art styles.
• Step 4: Choose suitable options related to text formating like text fill, text outlines, shadow, etc.

Plus Two Accountancy Graphs and Charts for Business Data Four Mark Questions and Answers

Question 1.
What are the difference between 2D charts and 3D charts
Answer:

Question 2.
List out the steps to Rotate a chart.
Answer:

• Step 1. Select the plot area of the chart.
• Step 2. Click on the format tab.
• Step 3. Click on format selection.
• Step 4. Click on 3D Rotation and type a value of angle between 0° to 360° and then click close
• Step 5. Click on the chart area of the chart and click on format tab.
• Step 6. Click on shape effects and then click on Bevel and select a bevel option.

Question 3.
What are the different types of charts?
Answer:

1. Column chart: column chart are used to compare values across categories
2. Line chart: Line charts are used to display trends over time
3. Pie chart: Pie charts display the contribution of each value to a total
4. Bar chart: Bar charts are best suited for comparing multiple values
5. Area chart: Area chart emphasis differences between several sets of data over a period of time.
6. Scatter chart: (XY chart) This chart compares pairs of values.
7. Radar chart: Display values relative to a centre point.
8. Doughnut chart: It shows the relationship of parts to a whole. This chart display data in rings, where each ring represents a data series.

1. Column Chart:
It is the most commonly used chart type. It shows a bar chart or bar graph with vertical bars. The X-axis shows the categories and Y-axis shows the value for each category. Column chart are used to compare values across categories.

2. Line Chart:
A line chart shows values in the Y-axis and categories in X-axis. The Y values of each data series is connected by a line. Line chart shows data changes for a certain period of time.

3. Pie chart:
A pie chart displays the contribution of each value to a total. It represents multiple subgroup of a single variable. It contains only one data series. A pie chart shows values as circular sectors of the total circle. Pie chart may be

• Normal Pie chart
• Exploded Pie chart
• Doughnut chart or Donut chart
• Exploded Doughnut chart

4. Bar Chart:
This type of chart shows a bar graph or column chart with horizontal bars. The Y-axis shows categories and the X-axis shows the value for each category. It is suitable for comparing multiple values.

5. Area chart:
The chart shows values as points on the Y-axis. The X-axis shows categories. The Y values of each data series are connected by a line. The area between each two lines is filled with a colour.

6. Scatter chart:
Scatter chart is also known as XY chart. In this type of chart, both axes display values. This chart is used to show the relationship among two variables.

7. Radar chart:
It is also known as Net chart or Star chart. A radar chart has a separate axis for each category and the axes extend outward from the center of the chart. The value of each data point is plotted on the corresponding axis.

8. Doughnut chart:
Chart display in rings, where each ring represents a data series. The first data series is displayed in the centre of the chart.

Question 4.
Identify and explain the type of chart given below.
Answer:

1. This is scatter chart or XY chart.
2. Features:
   • Both axes display values (No category)
   • This chart is used to show the relationship among two variables
   • Generally this chart is used for scientific, statistical and engineering data

Question 5.
Match the following.

Answer:

• (a) – (ii);
• (b) – (iii);
• (c) – (iv);
• (d) – (i)

Question 6.
How can we change the format of a selected chart element?
Answer:

• Step 1. Click anywhere in the chart.
• Step 2. Click format
• Step 3. Click format selection
• Step 4. Select a category (Fill border, style, etc)
• Step 5. Select formatting options

Question 7.
List down any four advantages of charts/ Graphs
Answer:
Advantages in using Graph/Chart:

1. It summarises a large data set in visual form
2. Charts or graphs can clarify trends better than do tables.
3. It helps to estimate key values at a glance
4. It shows each data category in a frequency distribution.
5. It permits a visual check of the accuracy and reasonableness of calculations
6. The charts and graphs allow the investigator to draw a valid conclusion.

Question 8.
What are the features of Charts/ Graphs in Libre Office Calc?
Answer:

1. Chart is a graphical representation of data
2. They are visual representation of numerical data
3. Charts can be read more quickly than the raw data
4. A chart has at least two axes – X and Y

Plus Two Accountancy Graphs and Charts for Business Data Five Mark Questions and Answers

Question 1.
What Pie Chart? What are the different types of Pie Chart?
Answer:
Pie chart:
A pie chart displays the contribution of each value to a total. It represents multiple subgroups of a single variable. It contains only one data series. A pie chart shows values as circular sectors of the total circle. Pie chart may be

1. Normal Pie chart
2. Exploded Pie chart
3. Doughnut chart or Donut chart
4. Exploded Doughnut chart

Question 2.
Name the different elements of given chart.

Answer:

• X-Axis Title
• Y-Axis Title
• Data label
• Main Title
• Legend
• X-Axis
• Y-Axis
• Data series

Question 3.
Write the steps of changing the chart type.
Answer:

1. First select the chart by double-clicking on it. The chart should now be surrounded by a gray bonder
2. Right-click on the chart and choose chart type.
3. Select the replacement chart type.
4. Click on [OK]

Question 4.
Write the steps for preparing a chart in Libre Office Calc.
Answer:

• Step 1: Enter the data in a worksheet with proper column and row titles
• Step 2: Select the range of data using the mouse
• Step 3: Click on Insert Tab → Object → Chart. Select a chart type from the “Choose a chart type” list in chart wizard window.
• Step 4: Naming chart, X-axis and Y-axis. Click on the chart → Right click → Insert titles (Names) → OK
• Step 5: Change the layout or styles of chart.
• Step 6: Show or hide a legend
• Step 7: Display or hide chart axes or gridlines
• Step 8: Move (resize) a chart
• Step 9: Save a chart

Plus Two Accountancy Graphs and Charts for Business Data Practical Lab Work Questions and Answers

Question 1.
Draw an Area Chart from the following

Procedure:
Step 1 – Open a new blanks worksheet in LibreOffice Calc

Step 2 – Enter the above data as follows.

Step 3 – Select the range A1: D6 which is to be shown in the chart.

Step 4 – Click on Insert menu → Click on Chart → Chart wizard → Select Area Chart → Finish
Output:

Question 2.
Quarterly sales of a product are given below. Draw a bar diagram/bar chart

Procedure:
Step 1 – Open a new blanks worksheet in LibreOffice Calc

Step 2 – Enter the above data as follows.

Step 3 – Select the range A1: B5 which is to be shown in the chart:

Step 4 – Click on Insert menu → Click on Chart → Chart wizard Click on Bar chart → Finish
Output:

Question 3.
Draw a 3D column chart from the following details.

Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following data in the respective cells.

Step 3-Select the range A1: B6, which is to be shown in the chart.

Step 4 – Click on Insert menu → Click on Chart → Chart wizard → Click on Column Chart → Tick 3D Look → Select Bar Charts → Finish.
Output:

Question 4.
The net profits of a firm for the last six years are given below. Draw a line chart.

Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc

Step 2 – Enter the following data in the respective cells

Step 3 – Select the range A1: B7, which is to be shown in the chart

Step4- Click on Insert menu → Click on Chart → Chart Wizard → Click on Line Chart → Finish
Output:

Question 5.
Enter the following data into a LibreOffice Calc worksheet and draw a 3D Pie chart.

Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the data in respective cells.

Step 3 – Select the range A2: B9, which is to be shown in the chart

Step 4 – Click on Insert menu → Click on Chart → Chart wizard → Click on Pie Chart Tick on 3D Look → Finish
Output:

Question 6.
Sales for the first six months in 2 years are given below. Draw a scatter chart in a LibreOffice Calc works sheet

Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc.

Step 2 – Enter the data in the following cells.

Step 3 – Select the range A1: C7, which is to be shown in the chart

Step 4 – Click on → Insert menu Click on → Chart → Chart Wizard → Scatter Chart→ Finish
Output:

Question 7.
The production of different items in Oct. 2015 is listed below. Draw a Radar Chart

Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc.

Step 2 – Enter the data in the following cells.

Step 3 – Select the range A1: C6, which is to be shown in the chart

Step 4 – Click on → Insert menu → Click on Chart → Chart Wizard → Click on Radar Chart → Finish
Output:

Question 8.
The following table shows the number of students passed in the higher secondary examination. Draw a doughnut chart.

Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc.

Step 2- Enter the data in the following cells.

Step 3 – Select the range A1: D6, which is to be shown in the chart.

Step 4 – Click on → Insert menu → Click on Chart → Chart Wizard → Click on doughnut Chart → Finish.
Output:

Students can Download Chapter 6 Application of Derivatives Questions and Answers, Plus Two Maths Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 6 Application of Derivatives

Plus Two Maths Application of Derivatives Three Mark Questions and Answers

Question 1.
Find the equation of tangents and normals to the given curves x = cost, y = sin t at t = \(\frac{π}{4}\).
Answer:
Given; x = cost, y = sin t

Equation of tangent at t = \(\frac{π}{4}\) is;

Equation of normal at t = \(\frac{π}{4}\) is;

⇒ \(\sqrt{2}\)y + \(\sqrt{2}\)x = 0 ⇒ y + x = 0.

Question 2.
A ladder Sm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the decreasing when the foot of the ladder is 4m away from the wall?
Answer:
From the figure we have;
x² + y² = 25 ____(1)

Differentiating w.r.t t;

From (1) when x = 4 ⇒ 16 + y² = 25 ⇒ y = 3
Given; \(\frac{d x}{d t}\) = 2cm/s = 0.02 m/s
(2) ⇒ 4(0.02) + 3 \(\frac{d x}{d t}\) = 0

Question 3.
Find the points on the curve y = x³, the tangents at which are inclined at an angle of 60° to x-axis?
Answer:
\(\frac{d y}{d x}\) = 3x²
Slope of the tangent = tan60°
i.e. 3x² = \(\sqrt{3}\)

Question 4.
Find the equation of the tangent to the parabola y² = 4x + 5 which is parallel to y = 2x + 7.
Answer:
y² = 4x + 5 _____(1)
2y \(\frac{d y}{d x}\) = 4
\(\frac{d y}{d x}\) = \(\frac{4}{2y}\) = \(\frac{2}{y}\)
Given tangent is parallel to y = 2x + 7
ie. Slope of the tangent is 2 ⇒ \(\frac{2}{y}\) = 2 ⇒ y = 1
∴ from (1) ⇒ 1 = 4x+ 5 ⇒ 4x = -4 ⇒ x = -1
So the point of contact is (-1, 1).
∴ Equation of tangent is
y -1 = 2(x + 1) ⇒ y = 2x + 3.

Question 5.
Find the intervals in which the function f given f(x) = 2x² – 3x is

1. Strictly increasing.
2. Strictly decreasing.

Answer:
Given; f(x) = 2x² – 3x ⇒ f'(x) = 4x – 3
For turning points; f'(x) = 0
⇒ 4x – 3 = 0 ⇒ x = \(\frac{3}{4}\)
The intervals are \(\left(-\infty, \frac{3}{4}\right),\left(\frac{3}{4}, \infty\right)\)
f'(0) = – 3 < 0
∴ Strictly decreasing in \(\left(-\infty, \frac{3}{4}\right)\)
f'(1) = 1 > 0
∴ Strictly increasing in \(\left(\frac{3}{4}, \infty\right)\).

Question 6.
Find the intervals in which the function f(x) = (x + 1)³ (x – 3)³ strictly increasing or decreasing.
Answer:
Given; f(x) = (x + 1)³ (x – 3)³
⇒ f'(x) = (x + 1)³ 3(x – 3)² + (x – 3)³3(x + 1)²
= 3(x + 1)²(x – 3)²(x + 1 + x – 3)
= 3(x + 1)²(x – 3)²(2x – 2)
= 6(x +1)² (x – 3)² (x -1)
⇒ 6(x +1)² (x – 3)² (x – 1) = 0
⇒ x = -1, 1, 3
The intervals are
(-∞, -1), (-1, 1), (1, 3), (3, ∞)
f'(-2) = (-2 – 1) < 0
∴ Strictly decreasing in (-∞, -1)
f'(0) = (0 – 1) < 0
∴ Strictly decreasing in (-1, 1)
f'(2) = (2 – 1) > 0
∴ Strictly increasing in (1, 3)
f'(4) = (4 – 1) > 0
∴ Strictly increasing in (3, ∞).

Question 7.
Find the intervals in which the function f(x) = x + \(\frac{1}{x}\) strictly increasing or decreasing.
Answer:

⇒ x = ±1
The intervals are (-∞, -1), (-1, 1), (1, ∞)
f'(-2) > 0
∴ Strictly increasing in (-∞, -1)
f'(0) < 0
∴ Strictly decreasing in (-1, 1)
f'(2) > 0
∴ Strictly increasing in (1, ∞).

Question 8.
Determine whether the f(x) = x² function is strictly monotonic on the indicated interval.

1. (-1, 1)
2. (-1, 0)
3. (0, 1)

Answer:
f(x) = x²
⇒ f'(x) = 2x
⇒ f'(x) = 0 ⇒ 2x = 0 ⇒ x = 0
This turning point divides the domain into the intervals (-∞, 0); (0, ∞).

1. Interval (-1,1) f'(x) < 0 and f'(x) > 0. So f(x) is not monotonic.
2. Interval (-1,0), f'(x) < 0. ∴ f(x) is strictly monotonic.
3. Interval (0, 1) f'(x) > 0 and f(x) is strictly monotonic.

Question 9.
Determine whether the f(x) = x³ – x function is strictly monotonic on the indicated interval.

1. (-1, 0)
2. (-1, -1/2)
3. (-1, 1)

Answer:
(x) = x³ -x ⇒ f'(x) = 3x² – 1
⇒ f'(x) = 0 ⇒ 3x² – 1 = 0 ⇒ x = ±\(\frac{1}{\sqrt{3}}\)
This turning point divides the domain into the intervals (-∞, \(\frac{1}{\sqrt{3}}\)); (-\(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)); (\(\frac{1}{\sqrt{3}}\), ∞).

1. Interval (-1, 0), f'(x) changes sign. So not monotonic.
2. Interval (-1, -1/2), f'(x) > 0 strictly monotonic.
3. lnterval(-1, 1) not monotonic

Question 10.
Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 1%.
Answer:
We have; V = x³ and ∆x = 1% of x= 0.01x
dV = \(\frac{d V}{d x}\) ∆x = 3x²∆x
= 3x² × 0.01x = 0.03x³ = 0.03V
⇒ \(\frac{d V}{V}\) = 0.03
Therefore 3% is the approximate increase in volume.

Question 11.
If the radius of a sphere is measured as 7m with an error of 0.02m then find the approximate error in calculating its volume.
Answer:
Let r be the radius of the sphere and ∆r be the error in measuring the radius then r =7m and ∆r = 0.02 m
We have; V = \(\frac{4}{3}\) πr³
dV = \(\frac{d V}{dr}\) ∆r = \(\frac{4}{3}\) π3r² × ∆r
= 4π(7)² × 0.02 = 3.92 π m³.

Question 12.
The length of a rectangle is decreasing at the rate of 5 cm/min and the width is increasing at the rate of 4cm/min. When length is 8 cm and width is 6 cm, find the rate of change of its area.
Answer:
Let length = x and width = y

Question 13.
Find the equation of tangents and normals to the given curves y= x⁴ – 6x³ + 13x² – 10x + 5 at (0, 5)
Answer:
Given; y = x⁴ – 6x³ + 13x² – 10x + 5 at (0, 5)
⇒ \(\frac{d y}{d x}\) = 4x³ – 18x² + 26x – 10
Slope = \(\left(\frac{d y}{d x}\right)_{x=0}\) = -10
Equation of tangent at (0, 5) is;
y – 5 =(-10)(x – 0)
⇒ y – 5 = -10x ⇒ 10x + y – 5 = 0
Equation of normal at (0, 5) is;
y – 5 = \(\frac{1}{10}\)(x – 0)
⇒ 10y – 50 = x ⇒ x – 10y + 50 = 0.

Question 14.
Find the equation of tangents and normals to the given curves y = x³ at (1, 1)
Answer:
Given; y = x³
⇒ \(\frac{d y}{d x}\) = 3x²
Slope = \(\left(\frac{d y}{d x}\right)_{x=1}\) = 3
Equation of tangent at (1, 1) is; y -1 = (3)(x – 1)
⇒ y – 1 = 3x – 3 ⇒ 3x – y – 2 = 0
Equation of normal at (1, 1) is;
y – 1 = \(-\frac{1}{3}\)(x – 1)
⇒ 3y – 3 = -x + 1 ⇒ x+ 3y – 4 = 0.

Question 15.
The volume of a cube is increasing at the rate of 8cm³/s. How fast is the surface area increasing when the length of an edge is 12cm.
Answer:
Let V be the volume of the cube of side x.
We have volume = V = x³
Rate of change of volume with respect to time ‘t’ is;
ie; differentiating w.r.t t; \(\frac{d V}{d t}\) = 3x²\(\frac{d x}{d t}\)
Given; \(\frac{d V}{d t}\) = 8 and x = 12 ⇒ 8 = 3(12)²\(\frac{d x}{d t}\)

Now let Surface area = S = 6x²
Differentiating w.r.t t;

Question 16.
Find the intervals in which the function f(x) = -2x³ – 9x² – 12x + 1 strictly increasing or decreasing.
Answer:
Given; f(x) = -2x³ – 9x² – 12x + 1
⇒ f'(x) = -6x² – 18x – 12
= – 6(x² + 3x + 2)
= – 6(x + 2)(x +1)
⇒ f'(x) = 0 ⇒ -6(x + 2)(x +1) = 0
⇒ x = -2, -1
The intervals are (-∞, -2),(- 2, -1),(-1, ∞)
f'(-3) = -(-3 + 2)(-3 + 1) < 0
∴ Strictly decreasing in (-∞, -2)
f'(-1.5) = -(-1.5 + 2)(-1.5 + 1) > 0
∴ Strictly increasing in (- 2, -1)
f'(0) = -(0 + 2)(0 + 1) < 0
Strictly decreasing 1n(-1, ∞).

Question 17.
Find the local maxima and minima of the following functions. Also find the local maximum and minimum values. (each question carry 3 score)

1. f(x) = sin x + cosx, 0 < x < \(\frac{\pi}{2}\)
2. f(x) = x³ – 3x
3. f(x) = x³ – 6x² + 9x + 15
4. g(x) = \(\frac{x}{2}\) + \(\frac{2}{x}\), x > 0
5. g(x) = \(\frac{1}{x^{2}+2}\)

Answer:
1. Given; f(x) = sinx + cosx
⇒ f'(x) = cosx – sinx
For turning point f'(x) = 0
⇒ cosx – sinx = 0
⇒ cosx = sinx
⇒ x = \(\frac{\pi}{4}\)
f”(x) = -sin x – cosx
⇒ f (\(\frac{\pi}{4}\)) = -sin\(\frac{\pi}{4}\) – cos\(\frac{\pi}{4}\) < 0
Hence f(x) has a local maximum at x = \(\frac{\pi}{4}\) and local maximum value is

2. Given; f(x) = x³ – 3x
⇒ f'(x) = 3x² – 3
For turning point f'(x) = 0
⇒ 3x² – 3 = 0
⇒ x = ±1
f”(x) = 6x
When x = -1
⇒ f”(-1) = -6 < 0
Hence f(x) has a local maximum at x = -1 and local maximum value is
f(-1) = (-1)³ – 3(-1) = -1 + 3 = 2
When x = 1
⇒ f”(1) = 6 > 0
Hence f(x) has a local minimum at x = 1 and local minimum value is
f(1) = (1)³ – 3(1) = 1 – 3 = -2.

3. Given; f(x) = x³ – 6x² + 9x + 15
⇒ f'(x) = 3x² – 12x + 9
For turning point f'(x) = 0
⇒ 3x² – 12x + 9 = 0 ⇒ 3(x² – 4x + 3) = 0
⇒ 3(x – 1)(x – 3) = 0 ⇒ x = 1, 3
f”(x) = 6x – 12
When x = 1
⇒ f”( 1) = 6 – 12 < 0
Hence f(x) has a local maximum at x = 1 and local maximum value is
f(1) = (1)³ – 6(1)² + 9(1) + 15 = 19
When x = 3
⇒ f”(3) = 6(3) – 12 > 0
Hence f(x) has a local minimum at x = 3 and local minimum value is
f(3) = (3)³ – 6(3)² + 9(3) + 15 = 15.

4. Given; g(x) = \(\frac{x}{2}\) + \(\frac{2}{x}\)
⇒ g'(x) = \(\frac{1}{2}\) – \(\frac{2}{x^{2}}\)
For turning point g'(x) = 0

Since x > 0, the acceptable value of x = 2

Hence g(x) has a local maximum at x = 2 and local maximum value is g(2) = \(\frac{2}{2}\) + \(\frac{2}{2}\) = 2

5. Given; g(x) = \(\frac{1}{x^{2}+2}\)

For turning point g'(x) = 0


Hence g(x) has a local maximum at x = 2 and maximum value is g(2) = \(\frac{1}{0+2}=\frac{1}{2}\).

Question 18.
Find the absolute maximum value and minimum value of the following functions.

1. f(x) = x³, x ∈ [-2, 2]
2. f(x) = 4x – \(\frac{x^{2}}{2}\), x ∈ \(\left[-2, \frac{9}{2}\right]\)

Answer:
1. Given; f(x) = x³ ⇒ f'(x) = 3x²
For turning point f'(x) = 0 ⇒ 3x² = 0 ⇒ x = 0
f(- 2) = (-2 )³ = -8
f( 2) = (2)³ = 8
f(0) = (0)³ = 0
Absolute maximum = max{-8, 8, 0} = 8
Absolute minimum = min {-8, 8, 0} = – 8

2. Given; f(x) = 4x – \(\frac{x^{2}}{2}\) ⇒ f'(x) = 4 – x
For turning point f'(x) = 0 ⇒ 4 – x = 0
⇒ x = 4

Absolute maximum = max{-10, 8, 7.875} = 8
Absolute minimum = min {-10, 8, 7.87} = -10.

Question 19.
A television camera at ground level is filming the lift-off of a space shuttle that is rising vertically according to position equation S = 50t².
The camera is 2000 feet from the launch pad. Find the rate of change in the angle of elevation of the camera 10 seconds after lift-off.
Answer:

Question 20.
Determine whether the f(x) = sinx function is strictly monotonic on the indicated interval.

1. (0, 2π)
2. (0, π)
3. (-π/2, π/2)

Answer:
f(x) = sinx ⇒ f'(x) = cosx changes sign.

1. Interval (0, 2π). ∴ f(x) is not monotonic.
2. f'(x) changes sign in (0, π) not monotonic.
3. f'(x) > 0 in (-π/2, π/2), ie. f(x) is strictly monotonic.

Question 21.
Find the approximate change in the Surface Area of a cube of side x meters caused by decreasing the side by 1%.
Answer:
We have;
S = 6x² and ∆x = 1% of x = -0.01x
dS = \(\frac{d S}{d x}\) ∆x = 6 × 2x × ∆x
= 6 × 2x × -0.01x = -0.02 × 6x² = -0.02S
⇒ \(\frac{d S}{S}\) = -0.02
Therefore 2% is the approximate decrease in surface area.

Plus Two Maths Application of Derivatives Four Mark Questions and Answers

Question 1.
The length ‘x’ of a rectangle is decreasing at the rate of 2 cm/s and the width ‘y’ is increasing at the rate of 2 cm/s.

1. Find the rate of change of Perimeter.
2. Find \(\frac{d A}{dt}\) when x = 12 cm and y = 5 cm.

Answer:
Since the length ‘x’ is decreasing and the width ‘y’ is increasing, we have \(\frac{d x}{dt}\) = -2 cm/s and \(\frac{d y}{dt}\)
= 2 cm/sec.
1. The Perimeter ‘P’ of the rectangle is given by
P = 2 (x + y)

2. The area ‘A’ of the rectangle ‘A’ is given by
A = x.y

= 12(2) + 5(-2)
= 24 – 10 = 14 cm2/s.

Question 2.
Find the equation of all lines having slope -1. Which are tangents to the curve?
y = \(\frac{1}{x-1}\), x ≠ 1
Answer:

⇒ x² – 2x = 0 ⇒ x(x – 2) = 0 ⇒ x = 0, x = 2
At x = 0, y = -1
Equation of tangent at (0, -1) is;
At x = 2, y = \(\frac{1}{2-1}\) = 1
Equation of tangent at (2, 1) is; y – 1 = -1(x – 2)
⇒ y – 1 = -x + 2 ⇒ x + y – 3 = 0.

Question 3.
Find the points on the curve x² + y² – 2x – 3 = 0 at which the tangent are parallel to x-axis.
Answer:
Given; x² + y² – 2x – 3 = 0
Differentiating with respect to x;

Since the tangent is parallel to x-axis \(\frac{d y}{d x}\) = 0
\(\frac{1-x}{y}\) = 0 ⇒ x = 1
We have; (1)² + y² – 2(1) – 3 = 0
⇒ y² = 4 ⇒ y = ±2
Hence the points are (1, 2), (1, -2).

Question 4.
Find the equation of the tangent to the curve y = \(\sqrt{3 x-2}\) which is parallel to the line 4x – 2y + 5 = 0.
Answer:
Slope of the line 4x – 2y + 5 = 0 is 2.

acceptable since y is positive.
Hence the point is \(\left(\frac{41}{48}, \frac{3}{4}\right)\)
Equation of tangent is;


⇒ 6(4y – 3) = (48x – 41)
⇒ 24y – 18 = 48x – 41
⇒ 48x – 24y – 23 = 0.

Question 5.
Prove that the curve x = y² and xy = k cut at right angles, if 8k² = 1.
Answer:
x = y² ___(1)
⇒ 1 = 2y \(\frac{d y}{d x}\) ⇒ \(\frac{d y}{d x}\) = \(\frac{1}{2y}\)
xy = 2k ___(2)
⇒ x \(\frac{d y}{d x}\) + y.1 = 0 ⇒ \(\frac{d y}{d x}\) = \(-\frac{y}{x}\)
The product of the slopes will be – 1.

Question 6.
The gradient at any point (x, y) of a curve is 3x² – 12 and the curve through the point (2, -7).

1. Find the equation of the tangent at the point ( 2, -7 ). (2)
2. Find the equation to the curve. (2)

Answer:
1. Given gradient as 3x² – 12 ⇒ \(\frac{d y}{d x}\) = 3x² – 12
Slope at (2, -7) is given by
\(\left(\frac{d y}{d x}\right)_{x=2}\) = 3(2)² – 12 = 0
Since slope is zero, the tangent is parallel to x – axis.
Here y = – 7 is the equation of the tangent at (2, -7).

2. Given, \(\frac{d y}{d x}\) = 3x² – 12
⇒ ∫dy = ∫(3x² – 12 )dx
y = 3\(\frac{x^{3}}{3}\) – 12x + c ⇒ y = x³ – 12x + c ____(1)
Given (2, -7) is a point on the curve.
(1) ⇒ -7 = (2)³ – 12(2) + c ⇒ -7 = 8 – 24 + c ⇒ c = 9
∴ Curve is y = x³ – 12x + 9.

Question 7.
Consider the curve x^2/3 + y^2/3 = 2

1. Find the slope of the tangent to the curve at the point (1, 1). (2)
2. Find the equation of the normal at the point (1, 1). (2)

Answer:
1. Given, x^2/3 + y^2/3 = 2,
Differentiating w.r.t. x,

⇒ slope of tangent = – 1.

2. Slope of normal = \(-\frac{1}{-1}\) = 1.
Equation of the normal is
y – 1 = 1(x – 1) ⇒ y – x = 0.

Question 8.
Find the intervals in which the function f given by f(x) = 2x³ – 3x² – 36x + 7 is

1. Strictly increasing. (2)
2. Strictly decreasing. (2)

Answer:
Given; f(x) = 2x³ – 3x² – 36x + 7
⇒ f'(x) = 6x² – 6x – 36
f'(x) = 0 ⇒ 6x² – 6x – 36 = 0
⇒ 6(x² – x – 6) = 0
⇒ 6(x + 2)(x – 3) = 0 ⇒ x = -2, 3
The intervals are (-∞, -2),(- 2, 3), (3, ∞)
f'(-3) = 6(-3 + 2)(-3 – 3) > 0.
∴ Strictly increasing in (-∞, -2).
f'(0) = 6(2)(-3) < 0.
∴ Strictly decreasing in (- 2, 3).
f'(4) = 6(4 + 2)(4 – 3) > 0
∴ Strictly increasing in (3, ∞).

Question 9.
Use differentials to find the approximate value of \(\sqrt{0.6}\) up to 3 places of decimals.
Answer:
Take y = \(\sqrt{x}\), let x = 0.64 and ∆x = -0.04
Then; f(x) = y = \(\sqrt{x}\)
f(x + ∆x) = y + ∆y


(1) ⇒ \(\sqrt{0.6}\) = 0.8 – 0.025 = 0.775.

Question 10.
Use differentials to find the approximate value of (0.999)^{\(\frac{1}{10}\)} up to 3 places of decimals.
Answer:
Take = x^{\(\frac{1}{10}\)}, let x = 1 and ∆x = -0.001
Then; f(x) = y
f(x + ∆x) = y + ∆y
(0.999)^{\(\frac{1}{10}\)} = x^{\(\frac{1}{10}\)} + ∆y
(0.999)^{\(\frac{1}{10}\)} = (1)^{\(\frac{1}{10}\)} + ∆y
⇒ (0.999)^{\(\frac{1}{10}\)} = 1 + ∆y ______(1)

(1) ⇒ (0.999)^{\(\frac{1}{10}\)} = 1 – 0.0001 = 0.9999.

Question 11.
Use differentials to find the approximate value of (15)^{\(\frac{1}{4}\)} up to 3 places of decimals.
Answer:
Takey = x^{\(\frac{1}{4}\)}, let x = 16 and ∆x = -1
Then; f(x) = y
f(x + ∆x) = y + ∆y
(15)^{\(\frac{1}{4}\)} = x^{\(\frac{1}{4}\)} + ∆y
(15)^{\(\frac{1}{4}\)} = 16^{\(\frac{1}{4}\)} + ∆y
(15)^{\(\frac{1}{4}\)} = 2 + ∆y ____(1)
⇒ ∆y ≈ dy = \(\frac{d y}{d x}\) ∆x

Question 12.
Use differentials to find the approximate value of (26.57)^{\(\frac{1}{3}\)} up to 3 places of decimals.
Answer:
Take y = x^{\(\frac{1}{3}\)}, let x = 27 and ∆x = -0.43
Then; f(x) = y
f(x + ∆x) = y + ∆y
f(x + ∆x) = f(x) + ∆y
(26.57)^{\(\frac{1}{3}\)} = 27^{\(\frac{1}{3}\)} + ∆y .
⇒ (26.57)^{\(\frac{1}{3}\)} = 3 + ∆y ____(1)

(1) ⇒ (26.57)^{\(\frac{1}{3}\)} = 3 – 0.016 = 2.984.

Question 13.
Find the approximate value of f(5.001) where f(x) = x³ – 7x² + 15
Answer:
Let x = 5 and ∆x = 0.001
Then; f(x) = y
f(x + ∆x) = y + ∆y
f (5.001) = f(x) + ∆y
f(5.001) = f(5) + ∆y
f(5.001) = 5³ – 7(5)² + 15 + ∆y
⇒ f(5.001) = -35 + ∆y …
⇒ ∆y ≈ dy = \(\frac{d y}{d x}\) ∆x ⇒ dy = (3x² – 14x) × 0.001
= (3(5)² – 14(5)) × 0.001 = (75-70)0.001 = 0.005
(1) ⇒ f(5.001) = -35 + 0.005 = -34.995.

Question 14.
Find the approximate value of f(3.02) where f(x) = 3x² + 5x + 3
Answer:
Let x = 3 and ∆x = 0.02
Then; f(x) = y
f(x + ∆x) = y + ∆y
f(3.02) = f(x) + ∆y
f(3.02) = f(3) + ∆y
f(3.02) = 3(3)² + 5(2) + 3 + ∆y
⇒ f(3.02) = 45 + ∆y ____(1)
⇒ ∆y ≈ dy = \(\frac{d y}{d x}\) ∆x ⇒ dy = (6x + 5) × 0.02
= (6(3) + 5) × 0.02 = (18 + 5)0.02 = 0.46
(1) ⇒ f(3.02) = 45 + 0.46 = 45.46.

Question 15.
Consider the function y = f\(\sqrt{x}\)

1. If x = 0.0036 and ∆x = 0.0001 find ∆y. (3)
2. Hence approximate \(\sqrt{.0037}\) using differentials. (1)

Answer:
1. Let x = .0036, ∆x = 0.0001

2. (1) ⇒ \(\sqrt{.0037}\) = .000833 + .06 = .060833.

Question 16.
Find the approximate value of \(\sqrt[3]{124}\).
Answer:
f(x) = \(\sqrt[3]{x}\) = x^1/3 ⇒ f¹(x) = 1/3x^-2/3 = \(\frac{1}{3 x^{2 / 3}}\)
Let x = 125, ∆x = -1
Then; f(x) = y

Question 17.
Find two numbers x and y such that their sum is 35 and the product is x² y⁵ a maximum.
Answer:
Given; x + y = 35 ⇒ y = 35 – x
P = x² y⁵ ⇒ P = x²(35 – x)⁵
⇒ p’ = 2x(3 5 – x)⁵ + x² 5(35 – x)⁴(-1)
⇒ P’ = x(35 – x)⁴[2(35- x) – 5x]
⇒ p’ = x(35 – x)⁴[70 – 7x]
⇒ p’ = 7x(35 – x)⁴[10 – x]
⇒ p” = 7[x(3 5 – x)⁴ [-1] + x(10 – x)4(35 – x)³ (-1) + (35 – x)⁴(10 – x)]
For turning points P’ = 0
⇒ 7x(35 – x)⁴[10 – x] = 0
⇒ x = 0, 35, 10
x = 0, 35 can be rejected since correspondingly y will be y = 35, 0
⇒ P” = 7[10(35 – 10)⁴[-1]] < 0
Therefore maximum at x = 10
Thus the numbers are 10 and 35 – 10= 10.

Question 18.
Using differentials, find the approximate value of (63)^1/3.
Answer:
Take y = x^{\(\frac{1}{3}\)}, let x = 64 and ∆x = 1
Then; f(x) = y
f(x + ∆x) = y + ∆y

Question 19.

1. Find the point on the curve y = x³ – 10x + 8 at which the tangent is parallel to the line y = 2x + 1. (2)
2. Is the given line tangent to the curve? Why?

Answer:
1. \(\frac{d y}{d x}\) = 3x² – 10
Slope of the line y = 2x +1 is 2
⇒ 3x² – 10 = 2 ⇒ 3x² = 12 ⇒ x = ±2
When x = 2
y = 2³ – 10 × 2 + 8 = 8 – 20 + 8 = -4
When x = – 2
y = (-2)³ -10 × (-2) + 8 = -8 + 20 + 8 = 20
Therefore the points are (2, -4); (-2, 20)

2. No. Since (2, -4); (-2, 20) does not satisfies the equation y = 2x +1.

Question 20.
Suppose that a spherical balloon is inflated and it has volume ‘v’ and radius ‘r’ at time ‘t’.

1. If the balloon is inflated by pumping 900c.c. of gas per second. Find the rate a which the radius of the balloon is increasing when the radius is 15 cm. (2)
2. Find the rate of change of its surface at the instant when it radius is 15 cm. (2)

Answer:
1. Let V be the volume of the sphere of radius r.
V = \(\frac{4}{3}\) πr³, given; \(\frac{d V}{d t}\) = 900, r = 15
Differentiating w.r.t t,

2. Let ‘s’ denote the surface area of the balloon, then
S = 4πr²
Differentiating, \(\frac{d s}{d t}\) = 4π.2r.\(\frac{d r}{d t}\) =8.π r .\(\frac{d r}{d t}\)
= 8π × 15 × \(\frac{1}{\pi}\) = 120 cm²/sec.

Question 21.
Use differentials to find the approximate value of (0.009)^{\(\frac{1}{3}\)} up to 3 places of decimals.
Answer:
Take y = x^{\(\frac{1}{3}\)}, let x = 0.008 and ∆x = 0.001
Then; F(x) = Y
f(x + ∆x) = y + ∆y

Question 22.
Find the approximate value of \(\sqrt{401}\).
Answer:
f(x) = \(\sqrt{x}\) = x^1/2
f'(x) = \(\frac{1}{2 \sqrt{x}}\)
Let x = 400 ∆x = 1
f(x) = y = \(\sqrt{x}\)
f(x + ∆x) = y + ∆y

Question 23.
Consider y = \(\frac{\log x}{x}\), in (0, ∞)

1. Find the value of x at which \(\frac{d y}{d x}\) = 0 (2)
2. Find the maximum value.

Answer:
1.

2.

∴ y is maximum when x = e.
The maximum value is \(\frac{1}{e}\).

Question 24.
Find the point on the curve y = x³ – 11x + 5 at which the tangent is y = x – 11.
Answer:
Slope of the line y = x – 11 is 1.
Given; y = x³ – 11x + 5 ⇒ \(\frac{d y}{d x}\) = 3x² – 11 = 1
⇒ 3x² = 12 ⇒ x = ±2
At x = 2, ⇒ y = x – 11 = 2- 11 = -9
⇒ (2, -9)
At x = -2, ⇒ y = x – 11 = -2 – 11 = -13
⇒ (-2, -13)
But the point (-2, -13) do not lie on the curve, hence the point is (2, -9).

Question 25.
Consider the curve y = x² – 2x + 7

1. Find the slope of the tangent of the curve at x = 2. (2)
2. Write down the equation of the tangent at x =2. (2)

Answer:
1. Given, y = x² – 2x + 7 ⇒ y’ = 2x – 2
(y’)_x=2 = 2(2) – 2 = 2.

2. At x = 2 , y = 2² – 2(2) + 7 = 7.
Equation of the tangent at (2, 7) is
y – 7 = 2(x – 2) ⇒ 2x – y + 3 = 0.

Question 26.
Find the absolute maximum value and minimum value of the following functions.

1. f(x) = 2x³ – 15x² + 36x + 1, x ∈ [1, 5]
2. f(x) = 12x^{\(\frac{4}{3}\)} – 6x^{\(\frac{1}{3}\)}, x ∈ [-1, 1]

Answer:
1. Given; f(x) = 2x³ – 15x² + 36x + 1, x ∈ [1, 5]
⇒ f'(x) = 6x² – 30x + 36
For turning point f'(x) = 0 ⇒ 6x² – 30x + 36 = 0
⇒ x² – 5x + 6 = 0 ⇒ (x -3)(x – 2) = 0
⇒ x = 3, 2
f(1) = 2(1)³ – 15(1)² + 36(1) + 1 = 24
f(2) = 2(2)³ – 15(2)² + 36(2) + 1 = 29
f(3) = 2(3)³ – 15(3)² + 36(3) + 1 = 28
f(5) = 2(5)³ – 15(5)² + 36(5) + 1 = 56
Absolute maximum = max {24, 29, 28, 56} = 56
Absolute minimum = min {24, 29, 28, 56} = 24

2. Given;

f'(x) = 0 at x = \(\frac{1}{8}\) and f'(x) is not defined at x = 0. Therefore;

Absolute maximum = max {18, 0, 6, \(-\frac{9}{4}\)} = 18
Absolute minimum = min {18, 0, 6, \(-\frac{9}{4}\)} = \(-\frac{9}{4}\).

Question 27.
Consider the function y = x³ – 6x² + 3x – 1

1. Find the slope at x= -1. (1)
2. Find the minimum gradient of the above curve. (3)

Answer:
1. Given,
y = x³ – 6x² + 3x – 1 ⇒ y’ = 3x² – 12x + 3
Gradient at (x = -1) = (y’)_x=1 = 3(-1)² – 12(-1) + 3 = 18.

2. Now for minimum gradient we have to apply maxima – minima condition to the function y’ .ie, y” = 6x – 12 , for turning points of y’ is given by y” = 0.
Therefore, 6x – 12 = 0 ⇒ x = 2
Now, y”’ = 6 > 0
∴ y’ is maximum at x = 2.
Minimum gradient at (x = 2) is
= 12 – 24 + 3 = – 9.

Plus Two Maths Application of Derivatives Six Mark Questions and Answers

Question 1.
A curve passes through the origin, and its gradient function is 2x – \(\frac{x^{2}}{2}\)

1. Find its y coordinate when x= 2. (4)
2. Find the equation of the tangent at x= 2. (2)

Answer:
Given;

Integrating we have; ∫dy = ∫(2x – \(\frac{x^{2}}{2}\))dx
⇒ y = x² – \(\frac{x^{3}}{6}\) + c ___(1)
Since the curve passes through (0, 0)
(1) ⇒ 0 = 0 + c ⇒ c = 0
∴ Equation of the curve is y = x² – \(\frac{x^{3}}{6}\)
When x = 2 ⇒ y = 2² – \(\frac{2^{3}}{6}\) = \(\frac{8}{3}\)
∴ coordinate is (2, \(\frac{8}{3}\)).

2. Slope at (2, \(\frac{8}{3}\)) = 2 × 2 – \(\frac{2^{2}}{2}\) = 2
∴ Equation of the tangent at (2, \(\frac{8}{3}\)) is given by
y – \(\frac{8}{3}\) = 2(x – 2) ⇒ 3y – 8 = 6x – 12 ⇒ 3y = 6x – 4.

Question 2.
(i) Choose the correct answer from the bracket. The slope of the tangent to the curve y = x³ – 2x + 3 at x = 1 is ____(1)
(a) 0
(b) 1
(c) 2
(d) 3
(ii) Find points on the curve \(\frac{x^{2}}{25}+\frac{y^{2}}{9}\) = 1 at which the tangents are (2)
(a) Parallel to x-axis
(b) parallel to y – axis.
(iii) Use differential to approximate \(\sqrt{25.6}\). (3)
Answer:
(i) (b) 1, Since

(ii)

(a) \(\frac{d y}{d x}\) = 0, since tangents are parallel to x- axis.
\(\frac{-9x}{25}\) = 0, x = 0 ∴ y = ± 3;
The points are (0, 3) and (0, -3)

(b) \(\frac{-25 y}{9 x}\) = 0, since tangents are parallel to y-axis, slope of normal = 0; y = o
∴ x = ± 5
The points are (5, 0) and (-5, 0)

(iii) Take y = \(\sqrt{x}\) , let x = 25 and ∆x = 0.6
Then; f(x) = y = \(\sqrt{x}\)
f(x + ∆x) = y + ∆y

Question 3.
Let x and y be the length and breadth of the rectangle ABCD in a circle having radius r. Let ∠CAB = θ (Ref. figure). If ∆ represent area of the rectangle and r is a constant.

1. Write ∆ in terms of r and θ. (2)
2. Find \(\frac{d \Delta}{d \theta}\) and \(\frac{d^{2} \Delta}{d \theta^{2}}\). (1)
3. Hence find the maximum value of ∆. (2)
4. Show that the rectangle of maximum area that can be inscribed in a circle of radius r is a square of side \(\sqrt{2} r\). (1)

Answer:
1. Area of the rectangle is ∆ = xy
From the figure y = 2r sinθ, x = 2r cosθ
∆ = xy = 4r²sinθcosθ = 2r²sin2θ

2. \(\frac{d \Delta}{d \theta}\) = 4r² cos2θ ⇒ \(\frac{d^{2} \Delta}{d \theta^{2}}\) = -8r²sin2θ

3. For turning points

Therefore local maximum at θ = \(\frac{\pi}{4}\)

4. Then;

Hence the rectangle becomes a square.

Question 4.
The second derivative of the equation of a curve is given by the equation x \(\frac{d^{2} y}{d x^{2}}\) = 1, given y = 1, \(\frac{d y}{d x}\) = 0 when x= 1.

1. Find the slope at x = e. (2)
2. Find the equation of the curve. (2)
3. Find the equation of the normal at x= e. (2)

Answer:
1. Given;

Integrating we get,

∴ Slope of the curve at x = e is given by
\(\left(\frac{d y}{d x}\right)_{x=e}\) = loge = 1.

2. We have,
\(\frac{d y}{d x}\) = logx, ⇒ dy = logxdx
Integrating we get,
∫dy = ∫logx dx ⇒ y = logx.x – ∫\(\frac{1}{x}\).x dx + c₂
⇒ y = xlogx – x + c₂ ____(2)
Given; y = 1 when x = 1
(2) ⇒ 1 = 1log1 – 1 + c₂ ⇒ 1 = 0 – 1 + c₂ ⇒ c₂ = 2
Therefore the equation of the curve is
y = xlogx – x + 2

3. We have, y = xlogx – x + 2
When x = e
⇒ y = e log e – e + 2 ⇒ y = e – e + 2 = 2
So we have to find the slope at (e, 2),
We know; \(\left(\frac{d y}{d x}\right)_{x=e}\) = log e = 1
∴ Slope of the normal at (e, 2)= -1
∴ Equation of the normal at (e, 2) is.
y – 2 = (-1) (x – e)
y – 2 = – x + e ⇒ y + x = e + 2.

Question 5.
The given figure represents a cylinder Inscribed in a sphere.

1. Find an expression for the volume V of the cylinder. (2)
2. Find the height of the cylinder when its volume V is maximum. (2)
3. Find the volume and radius of the largest cylinder. (2)

Answer:
1. From the right triangle ∆OAB,
y² = R² – x² ⇒ y = \(\sqrt{R^{2}-x^{2}}\)
Which is the radius of the cylinder
Also height = 2 x
∴ Volume = V= π y² × 2x = 2π(R² – x²)x = 2π(R²x – x³).

2. Now,

For maximum or minimum,

3. Base radius =

Question 6.
If f(x) = x³ + 3x² – 9x + 4 is a real function

1. Find the intervals in which the function is increasing or decreasing. (3)
2. Find the points of local maxima or local minima of f(x) (2)
3. Graph of a function is given in the following figure:

Which among the following represents the graph of its derivative? (1)

Answer:
1. f'(x) = 3x² + 6x – 9
For turning points f'(x) = 3x² + 6x – 9 = 0
⇒ x = 1, -3
These turning point divide the domain of f(x) in the following intervals. (-∞, -3), (-3, 1), (1, ∞) in (-∞, -3)
⇒ f'(-4) = 3(-4)² + 6(-4) – 9 > 0
Hence increasing.
In (-3, 1) ⇒ f'(0) = 3(0)² + 6(0) – 9 < 0
Hence decreasing.
In (1, ∞) ⇒ f'(2) = 3(2)² + 6(2) – 9 > 0
Hence increasing.

2. x = -3 is a local maximum point and x = 1 is a local minimum point.

3. (a)

The function passes through origin and has a local maximum at x = 2.

Question 7.
Of all the Cylinders with given surface area, show that the volume is maximum when height is equal to the diameter of the base.
Answer:
Let r be the radius, h be the height, V be the volume and S be the surface area
S = 2πr² + 2 πrh

S – 6πr² = 0
2πr² + 2πrh – 6πr² = 0
So h = 2r
So volume is maximum when h = 2r.

Question 8.
Sand is pouring from a pipe. The falling sand forms a Cone on the ground in such a way that the height of the Cone is always one-sixth of the radius of the base.

1. Establish a relation between the volume ‘v’ and height ‘h’ of the Cone using the given condition. (2)
2. lf the sand is pouring at the rate of -12 cm/sec, Find the rate of change of height of the Cone. (2)
3. Find \(\frac{\mathrm{dh}}{\mathrm{dt}}\) when h = 4cm. (2)

Answer:
1. Given that the height of the Cone is one-sixth of the radius of the base, then h = \(\frac{r}{6}\) ⇒ r = 6h
Then Volume V = \(\frac{1}{3}\) πr²h = \(\frac{1}{3}\) π(6h)2.h
V = \(\frac{1}{3}\) π 36h².h = \(\frac{1}{3}\) π 36h³
V =12 πh³ ____(1)

2. Differentiating (1) we get

3. When h = 4 cm

Question 9.
(i) Choose the correct answer from the bracket. The rate of change of the area of a circle with respect to its radius r at r = 10cm is.
(a) 10π
(b) 20π
(c) 30π
(d) 40π (1)
(ii) Find the intervals in which the function f given by f(x) = x² – 6x + 5 is (2)
(a) Strictly increasing
(b) Strictly decreasing
(iii) Find the local minimum and local maximum value, if any, of the function f(x) = x³ – 6x² + 9x + 8 (3)
Answer:

(ii) f'(x) = 2x — 6; 2x – 6 = 0; x = 3
(-∞, 3 ) is strictly decreasing
(3, ∞) is strictly increasing.

(iii) f'(x) = 3x² – 12x + 9
f¹¹ = 6x — 12
For maxima, minima
f¹ = 0 → 3x² – 12x + 9 = 0
3(x – 3)(x — 1) = 0; x = 3, x = 1
At x = 3 f¹¹(x) = 6 × 3 – 12 = 18 – 12 = 6 > 0
f is minimum, the local minimum value of f = 8
At x = 1 f¹¹(x) = 6 × 1 – 12 = -6 < 0,
f is maximum, the local maximum value of f = 12.

Question 10.
A wire of length 28m is cut into two pieces. One of the pieces is be made into a square and the other into a circle. What should be the length of the two pieces so that combined area of the square and the circle is minimum using differentiation?
Answer:
Let the length of one piece be ‘x’ and other piece be ‘28 – x’. Let from the first piece we will make a circle of radius Y and from the second piece we will make a square of side y. Then,

Let A be the combined area of the circle and square, then

Question 11.
An open box of maximum volume is to be made from a square piece of tin sheet 24cm on a side by cutting equal squares from the corners and turning of the sides.
(i) Complete the following table. (2)

(ii) Using the above table, express V as a function of x and determine its domain. (1)
(iii) Find height (x. cm) of the box when volume V is maximum by differentiation. (3)
Answer:
(i)

(ii) Generalise the above table as a function.
V = x(24-2x)², 0 < x < 12.

(iii) \(\frac{d V}{d x}\) = x.2(24 – 2x)(-2) + (24 – 2x)²
= -4x(24 – 2x) + (24 – 2x)²
= -96x + 8x² + 576 + 4x² – 96x
= 12x² – 192x + 576
For maximum or minimum,

Therefore volume is maximum when x = 4 cm.

Question 12.
A square tank of capacity 250 m³ has to be dug out. The cost of land is Rs. 50 per m². The cost of digging increases with the depth and for the whole tank is Rs. 400 × (depth)².

1. Find an expression for the cost of digging the tank. (3)
2. Find the dimension of the tank when the total cost is least. (3)

Answer:
1. Let x, x and y be the length, breadth, and depth of the tank.
Then, V = x. x. y = 250 ⇒ y = \(\frac{250}{x^{2}}\).
Area of land = x²
⇒ Cost of land = 50 x²
(∵ cost of land is Rs.50/m²)
Cost of digging = 400 × (depth)² = 400 × (y)²
∴ Total cost = C = 50 x² + 400 × (y)²

2. We have, C = 50x² + \(\frac{400 \times(250)^{2}}{x^{4}}\).
Differentiating w.r.t.x, we get,


∴ Maximum at x = 10
m ⇒ when x = 10m and
y = \(\frac{250}{10^{2}}\) = 2.5m the total cost is least.

Question 13.
Show that the right circular cone of least curved surface and given volume has an altitude equal to \(\sqrt{2}\) times the radius of the base.
Answer:
Volume of the cone will be, V =\(\frac{1}{3}\)πr²h
h = \(\frac{3 V}{\pi r^{2}}\) ____(1)
Curved surface area will be, S = πrl
⇒ S² = π²r²l² = P
⇒ P = π²r²(h² + r²) ⇒ P = π²r²h² + π²r⁴)

⇒ 2r² = h² ⇒ h = \(\sqrt{2} r\).

Question 14.
Let ABC be an isosceles triangle inscribed in a circle having radius r. Then by figure, area of the triangle ABC is ∆

1. Find \(\frac{d \Delta}{d \theta}\) and \(\frac{d^{2} \Delta}{d \theta^{2}}\) (2)
2. Find the maximum value of ∆. (3)
3. Show that the isosceles triangle of maximum area that can be in scribed in a given circle is an equilateral triangle. (1)

Answer:
1. Area of the isosceles triangle is ∆ = \(\frac{1}{2}\)bh
From the figure b = r sin2θ, h = r + r cos2θ

2. For turning points

Therefore local maximum a θ = \(\frac{\pi}{6}\) which means the area of the isosceles triangle is maximum When θ = \(\frac{\pi}{6}\).

3. Then; ∠OCB = 30° ⇒ ∠ACB = 2∠OCB = 60°. Therefore the isosceles triangle is an equilateral triangle.

Question 15.
(i) Using the graph of the function f (x) in the interval [ a, h ] match the following.

(ii) Consider the function f(x) = 3x⁴ – 8x³ + 12x² – 48x + 25
(a) Find the turning points of f(x). (1)
(b) Explain the nature of the turning points (1)
(c) Find the absolute extreme values of f(x). (2)
Answer:
(i)

(ii) (a) f(x) = 12x³ – 24x² + 24x – 48
For turning points,
f'(x) = 0 ⇒ 12x³ – 24x² + 24x – 48 = 0
⇒ x³ – 2x² + 2x – 4 = 0
⇒ (x² + 2)(x – 2) = 0 ⇒ x = ± (\(\sqrt{-2}\), 2)
We admit only x = 2 as x = \(\sqrt{-2}\) is not a real number.
Therefore at x = 2 f (x) has a turning point.

(b) f”(x) = 36x² – 48x + 24
⇒ f”(2) = 36(2)² – 48 × 2 + 24 > 0
Therefore at x = 2 f(x) has a local minimum.

(c) f(0) = 25,
f(2) = 3(2)⁴ – 8(2)³ + 12(2)² – 48 × 2 + 25 = -39
f(3) = 3(3)⁴ – 8(3)³ + 12(3)² – 48 × 3 + 25 = 16
Consider the set { f (0), f{2), f (3)}
⇒ {25, -39, 16}
The maximum value of the above set is the absolute maximum and it is 25 at x = 0. The minimum value of the above set is the absolute minimum and it is -39 at x = 2.

Question 16.
An open box with a square base is to be made out of a given quantity of sheet of area a².

1. If the box has side x units, then show that volume V= \(\frac{a^{2} x-x^{3}}{4}\) (2)
2. Show that the maximum volume is \(\frac{a^{3}}{6 \sqrt{3}}\) (4)

Answer:
1. Area = a² = x² + 4xh,
h = height of the box.

2. We have,

Question 17.
For the function f(x) = sin2x, 0 < x < π
(i) Find the point between 0 and π that satisfies f'(x) = 0. (2)
(ii) Find the point of local maxima and local minima. (2)
(iii) Find the local maximum and local minimum value. (2)
Answer:

Question 18.
A cylindrical can with a volume of 125m3 (about 2 litres) is to be made by cutting its top and bottom from metal squares and forming its curved side by bending a rectangular sheet of metal to match its ends. What radius ‘r’ and height ‘h’ of the can will minimize the amount of material required.
Answer:

The circular top and bottom should be cut out from a square metal sheet of side 2r. Therefore the area of squares is 8r².
Area A = 8r² + 2πrh

∴ To minimize the amount of material, r = 2.5
\(h=\frac{125}{\pi(2.5)^{2}}=6.3\).

Question 19.
A rectangle sheet of tin with adjascent sides 45cm and 24cm is to be made into a box • without top, by cutting off equal squares from the comers and folding up the flaps

1. Taking the side of the square cut off as x, express the volume of the box as the function of x. (2)
2. For what value of x, the volume of the box will be maximum. (4)

Answer:
1. Length of the box = 45 – 2x
Breadth of the box = 24 – 2x
Height of the box = x
Volume; V = (45 – 2x)(24 – 2x)x
= (1080 – 138x + 4x²)x
= 4x³ – 138x² + 1080x.

2. \(\frac{d y}{d x}\) = 12x² – 276x + 1080

12x² – 276x + 1080 = 0
x² – 23x + 90 = 0
x = 18, 5
x = 18 is impossible
∴ x = 5 when x = 5, \(\frac{d^{2} y}{d x^{2}}\) < 0

The volume of the box is maximum at x = 5.

Students can Download Chapter 11 Advances in Computing Questions and Answers, Plus Two Computer Science Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations

Kerala Plus Two Computer Science Chapter Wise Questions and Answers Chapter 11 Advances in Computing

Plus Two Computer Science Advances in Computing One Mark Questions and Answers

Question 1.
Which among the following are true regarding distributed computing?
i) Different parts of a program are run simultaneously on two or more computers.
ii) Distributed systems fail even if one node fails to function.
iii) A lot of extra programming is required to set up a distributed system.
iv) All the different processors share memory.
(a) i & ii are correct
(b) iii & iv are correct
(c) i & iii are correct
(d) ii & iv are correct
Answer:
(c) i & iii are correct

Question 2.
______ computing provides users access to computational power just like electricity through wall sockets.
(a) Parallel
(b) Grid
(c) Cluster
(d) Serial
Answer:
(b) Grid

Question 3.
A pattern or a model in the study of any subject of complexity is called ______.
Answer:
paradigm

Question 4.
____ is a method of computing in which large problems can be divided into smaller ones and this smaller one are distributed among several computers.
Answer:
distributed computing.

Question 5.
_____ is the advantage of distributed computing where we can add computers according to the workload.
Answer:
Scalability.

Question 6.
Multiple processors are used in _____ computing
(a) serial
(b) parallel
(c) standalone
(d) none of these
Answer:
(b) parallel

Question 7.
Single processor is used in _____ computing.
(a) serial
(b) parallel
(c) standalone
(d) none of these
Answer:
(a) serial

Question 8.
One instruction is executed on a single processor at any moment in _____ computing.
Answer:
Serial.

Question 9.
More than one instruction is executed on multiple processors at any moment of time in _____ computing.
Answer:
Parallel.

Question 10.
_____ is the fastest supercomputer used by Indian Institute of Tropical meteorology
Answer:
ADITYA

Question 11.
______ computing uses millions of computers and in which computational power(resources, services, data) is readily available like electric power.
Answer:
Grid computing.

Question 12.
Here a group of Internet-connected computers, storage devices, etc are linked together and work like a single computer. Which is this computing
Answer:
Cluster computing

Question 13.
It is an emerging computing technology. Herewith the use of Internet and central remote servers to maintain data and applications. Name this computing
Answer:
Cloud computing.

Question 14.
From the following which is not the cloud service model
(a) SaaS
(b) PaaS
(c) laaS
(d) NaaS
Answer:
(d) NaaS

Question 15.
SaaS stands for _____.
Answer:
Software as a Service

Question 16.
PaaS stands for _____.
Answer:
Platform as a Service.

Question 17.
laaS stands for _____.
Answer:
Infrastructure as a Service.

Question 18.
Expand LAMP
Answer:
Linux, Apache Server, MySQL, and PHP.

Question 19.
Al stands for _____.
Answer:
Artificial Intelligence

Question 20.
The first definition of Artificial Intelligence was established by _______.
Answer:
Alan Turing.

Question 21.
Al was first coined by ______ in 1956.
Answer:
John MacCarthy

Question 22.
With the help of _____ computer can solve the tasks such as playing chess, proving mathematical theorems, natural language processing, medical diagnosis, etc.
Answer:
Artificial Intelligence.

Question 23.
____ is a test of a machine’s ability to exhibit intelligent behaviour equivalent to, or indistinguishable from, that of a human.
Answer:
Turing Test.

Question 24.
NLP stands for _____.
Answer:
Natural Language Processor.

Question 25.
ASR stands for _____.
Answer:
Automatic Speech Recognition

Question 26.
OCR stands for
Answer:
Optical Character Recognition

Question. 27
HCR stands for _____.
Answer:
Handwritten Character Recognition.

Question. 28
GIS stands for ______.
Answer:
Geographical Information System.

Question. 29
CAD stands for ______.
Answer:
Computer-Aided Design

Question. 30
Which of the following is not a distributed computing paradigm?
(a) grid
(b) cloud
(c) cluster
(d) serial
Answer:
(d) serial

Question. 31
The study of control and communication between man and machine is called _____.
Answer:
Cybernetics

Question. 32
In Artificial Intelligence, ANN stands for
(a) Artificial Network of Neural.
(b) Advanced Neural Network
(c) Artificial Neural Network
(d) Architectural Neural Network
Answer:
(c) Artificial Neural Network

Question 33.
Which among the following is the capability that is required by a computer to pass Turing Test?
i. Natural Language Processing (NLP)
ii. Knowledge representation
iii. Machine learning
iv. Computer Vision
(a) i & ii only
(b) i.ii & iii only
(c) iii & iv only
(d) All of the above
Answer:
(d) All of the above

Plus Two Computer Science Advances in Computing Two Mark Questions and Answers

Question 1.
Differentiate the terms intelligence and wisdom.
Answer:
The ability to draw useful inferences from the available knowledge is generally referred as intelligence. Wisdom is the maturity of mind that directs its intelligence to achieve desirable goals.

Question 2.
Find the suitable match.

Answer:
1 – c
2 – d
3 – b
4 – a

Question 3.
Find the suitable match.

Answer:
1 – b
2 – c
3 – d
4 – a

Question 4.
Classify the following as SaaS, PaaS, and laaS. Adobe Creative Cloud. Google App Engine, Microsoft Office 365, Amazon Web Services.
Answer:

• SaaS: Adobe Creative Cloud, Microsoft Office 365
• PaaS: Google App Engine
• laaS: Amazon Web Services

Question 5.
Question Text:
Pick the odd one out. The services of cloud computing include:
(a) Software as a Service
(b) Hardware as a Service
(c) Platform as a Service
(d) Infrastructure as a Service
Answer:
(b) Hardware as a Service. The others are cloud computing paradigm

Question 6.
Question Text:
State whether the following statements are true or false.

1. Clusters provide computational power through parallel processing.
2. Even if anyone of the computers fails, cluster computing as a whole will work.

Answer:

1. True
2. True

Question 7.
The following organisations plan to buy cloud computing service for their IT requirements. Suggest the cloud computing model that suits their need.

1. Hari & Co. requires spreadsheets and presentation software.
2. Tomy & Co. requires to install their own software for their accounting needs in a LAMP platform.
3. Raju & Co. requires to use MySQL for their database needs.
4. Joy & Co. requires 1 TB hard disk space, 2 GB RAM and processing power.

Answer:

1. SaaS
2. PaaS
3. SaaS
4. laaS

Question 8.
Consider the following statements about distributed computing and state True/False.
1. In distributed computing, parts of a program are run simultaneously from two or more computers which are communicating with each other.
2. WWW is an example for large distributed computing.
Answer:

1. True
2. True

Question 9.
Parallel computing is faster than serial computing. Mention the limitations in implementing parallel computing.
Answer:
Disadvantages:

• Implementing this is more complex than serial application
• Significant changes must be made to make the programs machine-independent

Question 10.
What do you mean by Distributed computing?
Answer:
Distributed computing is a method of computing in which large problems can be divided into smaller ones and these smaller ones are distributed among several computers. The solution for the smaller ones are computed separately and simultaneously. Finally, the results are assembled to get the desired overall solution.

Plus Two Computer Science Advances in Computing Three Mark Questions and Answers

Question 1.
Differentiate between Bioinformatics and Biometrics.
Answer:
1. Biometrics:
Biometrics refers to the unique characteristics of a human being to recognize an individual such as fingerprints, face recognition, iris, retina, etc. Biometrics are used to record attendance
Eg. In banks, employee login is restricted by using fingerprint reader

2. Bioinformatics:
It is a computer technology to the management of biological information. By the help of a computer analyse the fingerprints, DNA, iris, retina, etc and identifying the concerned person.

Question 2.
Briefly explain the Turing test.
Answer:
The Turing test is a test of a machine’s ability to exhibit intelligent behaviour equivalent to, or indistinguishable from, that of a human. The test involves a human judge engages in natural language conversations with a human and a machine designed to generate performance indistinguishable from that of a human being.

All participants are separated from one another. If the judge cannot reliably tell the machine from the human, the machine is said to have passed the test. The test does not check the ability to give the correct answer to questions; it checks how closely the answer resembles typical human answers. Turing predicted that by 2000 computer would pass the test.

Question 3.
Draw the knowledge pyramid and briefly explain the terms in it.
Answer:

Data is termed as a collection of mere symbols. While processing data we get information and knowledge is the organized information. It can be a piece of information that helps in decision making. The ability to draw useful inferences from the available knowledge is generally referred as intelligence. Wisdom is the maturity of mind that directs its intelligence to achieve desirable goals.

Question 4.
Compare grid computing and cloud computing paradigms of distributed computing.
Answer:
1. Grid computing:
It is a system in which millions of computers, smartphones, satellites, telescopes, cameras, sensors, etc. are connected each other as a cyber world in which computational power (resources, services, data) is readily available like electric power. Any information at any time at any place can be made available in our fingertips. This is used in disaster management, weather forecasting, market forecasting, bio information, etc.

2. Cloud computing:
It is an emerging computing technology. Herewith the use of Internet and central remote servers to maintain data and applications. Example for this is Email service, Office Software(word processor, spreadsheets, presentations, database, etc), graphics software, etc. The information is placed in a central remote server just like clouds in the sky hence the name cloud computing

Question 5.
What is cluster computing? Write its advantages and disadvantages.
Answer:
Cluster means groups. Here a group of Internet-connected computers, storage devices, etc are linked together and work like a single computer. It provides computational power through parallel processing. Its cost is less and used for scientific applications.

Advantages

• Price-performance ratio: The performance is high ‘ and the cost is less.
• Availability: If one group of system fails the other group will do the work.
• Scalability: Computers can be easily added according to the workload increases.

Disadvantages

• Programmability issues: Different computers uses different software and hardware hence the issues.
• Problem in finding fault: Fault detection is very difficult

Question 6.
The distributed computing method has some advantages and disadvantages. What are they?
Answer:
Advantages

• Economical: Reduces the computing cost hence it is economical
• Speed: The workload of the entire system is less hence the speed is high.
• Reliability: When one computer in the network fails the entire work will not be blocked, i.e. the other computers will do the work properly.
• Scalability: We can add computers according to the workload.

Disadvantages

• Complexities: The proper division of the problems and reassembling of the result is a major complex task.
• Security: Security measurements to be taken to keep track of the sent data packets otherwise it can be used for illegal purposes.
• Network reliance: Some occasions in case of network failure, the entire system may become unstable.

Question 7.
Most of the companies are opting for cloud computing for their IT requirement. Explain the advantages that they gain from this.
Answer:
Advantages

• Cost savings: Companies can reduce costs by using cloud computing
• scalability/ Flexibility: More resources are used when the workload increases.
• Reliability: It is reliable and helps in disaster recovery
• Maintenance: The service provider do system maintenance, then reducing maintenance requirements.
• Mobile accessible: Employees can do their job even when they are in move.

Question 8.
Match the following.

Answer:
i – b
ii – c
iii – d
iv – a

Question 9.
Explain the advantages and disadvantages of distributed computing.
Answer:
Advantages

• Economical: Reduces the computing cost hence it is economical
• Speed: The workload of the entire system is less hence the speed is high.
• Reliability: When one computer in the network fails the entire work will not be blocked, i.e. the other computers will do the work properly.
• Scalability: We can add computers according to the workload.

Disadvantages

• Complexities: The proper division of the problems and reassembling of the result is a major complex task.
• Security: Security measurements to be taken to keep track of the sent data packets otherwise it can be used for illegal purposes.
• Network reliance: Some occasions in case of network failure, the entire system may become unstable.

Question 10.
Write down the merits and demerits of parallel computing.
Answer:
Advantages

• If one or more node fails, the entire system will not failed instead it works with reduced performance. This method is called fault tolerance.
• Sharing computing power and storage resources with each other.
• Distributing tasks to different nodes hence reduce the load of entire system. It is called load sharing
• Scaling can be easily done.

Disadvantages

• Implementing this is more complex than serial application
• Significant changes must be made to make the programs machine-independent

Question 11.
Explain Grid computing with merits and demerits.
Answer:
It is a system in which millions of computers, smartphones, satellites, telescopes, cameras, sensors, etc. are connected each other as a cyber world in which computational power (resources, services, data) is readily available like electric power. Any information at any time at any place can be made available in our fingertips. This is used in disaster management, weather forecasting, market forecasting, bio information, etc.
Advantages

• It is used to solve more complex problems in a short time.
• Existing hardware used efficiently
• Scalable: The number of computers can be added according to the workload increases

Disadvantages

• Processing speed is less
• Different computers uses different software and the licensing issues may effect the working of application

Question 12.
Explain cluster computing with its merits.
Answer:
Cluster means groups. Here a group of Internet connected computers, storage devices, etc are linked together and work like a single computer. It provides Computational power through parallel processing. Its cost is less and used for scientific applications.
Advantages

• Price-performance ratio: The performance is high and the cost is less.
• Availability: If one group of system fails the other group will do the work.
• Scalability: Computers can be easily added according to the workload increases.

Question 13.
What is cloud computing?
Answer:
It is an emerging computing technology. Herewith the use of Internet and central remote servers to maintain data and applications. Example for this is Email service, Office Software(word processor, spreadsheets, presentations, database, etc), graphics software, etc. The information is placed in a central remote server just like clouds in the sky hence the name cloud computing.

Question 14.
Explain the Turing test
Answer:
The Turing test is a test of a machine’s ability to exhibit intelligent behaviour equivalent to, or indistinguishable from, that of a human. The test involves a human judge engages in natural language conversations with a human and a machine designed to generate performance indistinguishable from that of a human being.

All participants are separated from one another. If the judge cannot reliably tell the machine from the human, the machine is said to have passed the test. The test does not check the ability to give the correct answer to questions; it checks how closely the answer resembles typical human answers. Turing predicted that by 2000 computer would pass the test.

Question 15.
Compare parallel and serial computing.
Answer:
The difference between these two is given below.

Plus Two Computer Science Advances in Computing Five Mark Questions and Answers

Question 1.
Explain the three cloud service models.
Answer:
Cloud service models (3 major services):
1. Software as a Service (SaaS):
A SaaS provider company provides more services on demand such as they allow to access both resources and applications.
Examples are Google Docs, Adobe creative cloud, Microsoft Office 365, Facebook.com, etc.

2. Platform as a Service(PaaS):
A PaaS provider company provides subscribers access to the components that they, require to develop and operate applications over the Internet.
Example: LAMP platform(Linux, Apache Server, MySQL, and PHP), ASP.NET, PHP and Python, Google’s App Engine, Microsoft Azure, Force.com, etc.

3. Infrastructure as a Service(laaS):
It provides basic storage devices and computing capabilities as standardized services over the network.
Example: Amazon Web Services, Joyent, AT&T, GoGrid, etc.

Question 2.
What are the applications of Computational Intelligence? Explain?
Answer:
Application of Computational Intelligence
A. Biometrics:
Biometrics refers to the unique characteristics of a human being to recognize an individual such as fingerprints, face recognition, iris, retina, etc. Biometrics are used to record attendance
Eg. In banks employee login is restricted by using fingerprint readers.

B. Robotics:
It is a branch of scientific study associated with the design, manufacturing and controls the movements of the robots. Robotics are used in all areas. Some of them are discussed below.

1. Uses in vehicle manufacturing industry: Robots are used to do all tasks such as welding, cutting, lifting, sorting and bending. Since it is a machine, it can operate for long hours untiringly.

2. Exploration of outer space:
Robots are very much useful in this field such as they are used to do all the tasks related to launch a satellite.

3. In intelligent homes:
Robots are used in home segments to safe guard the home, automatically open the gate, doors and windows, etc.

4. Exploration in difficult environments:
Human beings are not ready to work situations like too hot and too cold. Since it is a machine robots are used instead of human beings. Robots are used in oil filed etc.

5. Uses in military:
Unmanned air crafts and vehicles are controlled by robots. Drones and computer controlled cameras are used in Military service.

6. Uses in agriculture:
Nowadays in India, the agriculture field is also in mechanization trend. Hence increase productivity.

C. Computer vision:
The world is changed from 2 dimensional images to 3 dimensional. 3D TVs are available in the market. Multiple cameras are used to capture the images and merge them to form 3D pictures. 3D scanners are used in the Medical filed to diagnose the diseases

D. Natural Language Processing:
It deals with how computers communicate just like a human being communicate naturally. Natural languages are languages spoken by the people. To achieve this ability to communicate like a human being is a laborious task. NLP is further be classified into two, Natural Language Understanding(NLU) and Natural Language Generation(NLG).

• NLU – The ability to understand languages like English, Malayalam, etc.
• NLG – It is deal with creation of output, i.e. generate words and giving reply.

E. Automatic Speech Recognition(ASR):
To login some computers, laptops .tabs, smartphones, etc, and to open some gates, doors, etc you have say the password orally. The computer recognizes the speech and opens the device to them.

Such devices are working based upon NLP technology and this system is called Automatic Speech Recognition(ASR) system. To implement this system the requirements, are mica or a telephone and convert the oral instruction into written text. Examples are Apple iOS, Google, etc.

F. Optical Character Recognition(OCR) and Handwritten Character Recognition Systems. (HCR):
It is used to read text from a paper as an image and translate this image into a form that computer can manipulate. For example, if we want to enter the text contents of a book by typing using a keyboard will take more time. Instead of this by using OCR and HCR we can easily do this. OCR is expensive. HCR system reads handwritten texts and converts it into computer-readable form

G. Bioinformatics:
It is a computer technology to the management of biological information. By the help of a computer analyse the fingerprints, DNA, iris, retina, etc and identifying the concerned person.
Three goals of bioinformatics.:

1. Organise: Huge amount of data organized to access information easily and to add new information when it is produced
2.  Develop tools: Develop new tools to analyse data efficiently.
3. Analysis: With thw help of these tools analyse the data and produce results.

H. Geographical Information System:
Geosynchronous satellite moves at the same Revolutions Per Minute(RPM) as that of the earth in the same direction. Thus both the earth and the satellite complete one revolution exactly in the same time and hence the relative position of the ground station with respect to the satellite never changes.

Geographic Information System(GIS) technology is developed from digital cartography and Computer-Aided Design(CAD) database management system. GIS as the name implies capturing, storing for future reference, checking and displaying data related to various positions on earth’s surface.

GIS can be applied in many areas such as soil mapping, agricultural mapping, forest mapping, e-Governance, etc. GIS is used in development planning like strategic rural and urban planning, infrastructure planning, precision agriculture planning, etc.

Question 3.
Explain cloud computing paradigm and its three models of services in detail.
Answer:
It is an emerging computing technology. Herewith the use of Internet and central remote servers to maintain data and applications. Example for this is Email service, Office Software (word processor, spreadsheets, presentations, database, etc), graphics software, etc. The information is placed in a central remote server just like clouds in the sky hence the name cloud computing.
Cloud service models (3 major services):
1. Software as a Service(SaaS)
A SaaS provider company provides more services on demand such as they allow to access both resources and applications.
Examples are Google Docs, Adobe creative cloud, Microsoft Office 365, Facebook.com, etc.

2. Platform as a Service(PaaS)
A PaaS provider company provides subscribers access to the components that they require to develop and operate applications over the Internet.
Example: LAMP platform(Linux, Apache Server, MySQL, and PHP), ASP.NET, PHP and Python, Google’s App Engine, Microsoft Azure, Force.com, etc.

3. Infrastructure as a Service(laaS)
It provides basic storage devices and computing capabilities as standardized services over the network.
Example: Amazon Web Services, Joyent, AT&T, GoGrid, etc.

Question 4.
List and explain any three biologically inspired algorithms/paradigms that simulate natural intelligence, which led to the development of intelligent systems.
Answer:
Computational Intelligence paradigms Computational Intelligence is the ability to make a computer to face and solve real-life problems just like a intelligent man do it. It includes Artificial Neural Networks(ANN), Evolutionary Computation(EC), Swarm Intelligence(SI) and Fuzzy Systems(FS).

A. Artificial Neural Networks (ANN):
The brain is a complex, nonlinear and parallel computer with ability to perform tasks such as recognise pattern, perception and motor control. ANN is the method of simulate biological neural systems to learn, memorise and generalize like human beings. A human brain cortex consists of 10-500 billion neurons with 60 trillion synapses(a synapse is a structure that permits a neuron to pass electrical)

B. Evolutionary Computation (EC):
It is the simulation of the natural evolution, i.e. survival of the fittest. In the surrounding, we can see that the stronger must win and others will lose. EC applied for data mining, fault diagnosis, classification, scheduling, etc.

C. Swarm Intelligence (SI):
Swarm Intelligence is the study of behaviour of colonies or groups of social animals, birds, insects, ants, etc. How they communicate and create and manage their own colonies beautifully.

D. Fuzzy Systems:
Human beings use common sense while facing a problem, just like human beings fuzzy systems can also use common sense and behave like human beings. Fuzzy systems is used to control gear transmission and raking systems, control lifts, home appliances, controlling traffic signals, etc.

Question 5.
One of the recent developments in Computer Science is the use of computational intelligence in different real life applications. Briefly explain any five areas of application of computational intelligence.
Answer:
A. Biometrics:
Biometrics refers to the unique characteristics of a human being to recognize an individual such as fingerprints, face recognition, iris, retina, etc. Biometrics are used to record attendance Eg. In banks employee login is restricted by using fingerprint readers.

B. Robotics:
It is a branch of scientific study associated with the design, manufacturing and control the movements of the robots. Robotics are used in all areas.

C. Computer vision:
The world is changed from 2-dimensional images to 3 dimensional. 3D TVs are available in the market. Multiple cameras are used to capture the images and merge them to form 3D pictures. 3D scanners are used in the Medical field to diagnose the diseases

D. Natural Language Processing:
It deals with how computers communicate just like a human being communicate naturally. Natural languages are languages spoken by the people. To achieve this ability to communicate like a human being is a laborious task. NLP is further be classified into two, Natural Language Understanding (NLU) and Natural Language Generation (NLG).

• NLU- The ability to understand the languages like English, Malayalam, etc.
• NLG- It is deal with creation of output, i.e. generate words and giving reply.

E. Automatic Speech Recognition(ASR):
To login some computers, laptops, tabs, smartphones, etc, and to open some gates, doors, etc you have say the password orally. The computer recognizes the speech and opens the device to them. Such devices are working based upon NLP technology and this system is called Automatic Speech Recognition(ASR) system.

To implement this system the requirements are a mica or a telephone and convert the oral instruction into written text. Examples are Apple iOS, Google, etc.

F. Optical Character Recognition(OCR) and Handwritten Character Recognition Systems. (HCR):
It is used to read text from a paper as an image and translate this image into a form that computer can manipulate. For example, if we want to enter the text contents of a book by typing using a keyboard will take more time. Instead of this by using OCR and HCR we can easily do this. OCR is expensive. HCR system reads handwritten texts and converts it into computer-readable form.

G. Bioinformatics:
It is a computer technology to the management of biological information. By the help of a computer analyse the fingerprints, DNA, iris, retina, etc and identifying the concerned person.
Three goals of bioinformatics.

1. Organise
2. Develop tools
3. Analysis

H. Geographical Information System:
Geosynchronous satellite moves at the same Revolutions Per Minute(RPM) as that of the earth in the same direction. Thus both the earth and the satellite complete one revolution exactly in the same time and hence the relative position of the ground station with respect to the satellite never changes.

Geographic Information System(GIS) technology is developed from digital cartography and Computer-Aided Design(CAD) database management system. GIS as the name implies capturing, storing for future reference, checking and displaying data related to various positions on earth’s surface. GIS can be applied in many areas such as soil mapping, agricultural mapping, forest mapping, e-Governance, etc.

GIS is used in development planning like strategic rural and urban planning, infrastructure planning, precision agriculture planning, etc.

Students can Download Chapter 3 Use of Spread Sheet in Business Application Questions and Answers, Plus Two Accountancy Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Chemistry Chapter Wise Questions and Answers Chapter 3 Use of Spread Sheet in Business Application

Plus Two Accountancy Use of Spread Sheet in Business One Mark Questions and Answers

Question 1.
Which of the following options in a financial function indicates the interest for a period?
(a) FV
(b) PV
(c) N per
(d) Rate
Answer:
(b) PV

Question 2.
Which of the following arguments in a financial function represents the total number of payments?
(a) FV
(b) PV
(c) N Per
(d) Rate
Answer:
(c) N-Per

Question 3.
What Category of functions is used in this formula: = PMT(C 10/12, C8, C9, 1)
(a) Logical
(b) Financial
(c) Payment
(d) Statistical
Answer:
(b) Financial

Question 4.
LibreOffice Calc is a program
(a) Word Processor
(b) Browser
(c) Spread Sheet
(d) Calculator
Answer:
(c) Spread Sheet

Question 5.
……………… is the statement prepared to show detailed salary calculation
(a) Employee Job Card
(b) Payroll
(c) Loan Repayment Schedule
(d) Worksheet Payroll
Answer:
(b) payroll

Question 6.
………………… calculates the monthly installment of loan amount
(a) Loan Repayment Schedule
(b) Loan analysis sheet
(c) Loan card
(d) Payroll statement
Answer:
(a) Loan Repayment schedule

Question 7.
Depreciation is provided on
(a) Current Assets
(b) Fixed Assets
(c) Current Liabilities
(d) Long term Liabilities
Answer:
(b) Fixqd Assets

Question 8.
Depreciation =
Answer:
\(\frac{\text { cost of the asset- Scrap Value }}{\text { Life of the asset }}\)

Question 9.
……………………. is the gradual & permanent diminution in the value of assets due to wear and tear, use or abuse or efflux of time.
Answer:
Depreciation

Question 10.
Under …………… method of depreciation, the asset account will be reduced to zero.
(a) Fixed Instalment method
(b) Reducing installment
(c) Depreciation Fund Method
(d) Revaluation method
Answer:
(a) Fixed Instalment Method

Question 11.
Decrease in the value of fixed assets is called ……………..
Answer:
Depreciation

Question 12.
If the cost of asset is 10000, Scrap value at the end of 10 years will be 2000, what will be the amount of annual depreciation?
Answer:
Depreciation = \(\frac{10000-2000}{10}\) i.e., 800

Question 13.
Depreciation = (Acquisition cost – ………………. ) ÷ Life time
(a) Salvage value
(b) Carriage expenses
(c) Sales price
(d) Installation charges
Answer:
(a) Salvage value

Question 14.
The ……… of an asset is the value, which is realisable at the end of its useful life
(a) Depreciation
(b) Scrap value
(c) Written down value
(d) Acquisition cost
Answer:
(b) Scrap value

Question 15.
Odd one out
(a) Basic Pay
(b) Grade Pay
(c) House Rent Allowance
(d) Provident Fund
Answer:
(d) Provident Fund (It is a deduction)

Question 16.
_________ is a statutory deduction deducted monthly towards income tax liability of an employee
Answer:
Tax Deducted at Source (TDS)

Question 17.
Gross Salary – Total Deduction = ______
Answer:
Net Salary

Question 18.
In PMT function, Type is, whether payment is made at the beginning of the month, the value = ……(a)…… or at the end of the month, the value = …….(b)……
Answer:
(a) 1
(b) 0

Question 19.
Rate of Depreciation under straight line Method = ………
Answer:
\(\frac{\text { Amount of Depreciation (Yearty Depreciation) }}{\text { Total Depreciable Amount (Cost) }} \times 100\)

Question 20.
The function PMT is used to prepare …………….
(a) Pay Roll statement
(b) Depreciation statement
(c) Loan repayment statement
(d) Interest on Investment statement
Answer:
(c) Loan repayment statement

Question 21.
Match the following

Answer:
1 → c; 2 → a; 3 → d; 4 → b

Question 22.
Which among the following is not a component of PayRoll statement
(a) Professional Tax
(b) Present Value
(c) HRA
(d) Dearness Allowance
Answer:
(b) Present Value

Plus Two Accountancy Use of Spread Sheet in Business Two Mark Questions and Answers

Question 1.
What are the different methods for calculating depreciation on fixed Assets?
Answer:
Methods of calculation of depreciation

1. Straight Line Method (SLM)
2. Written Down Value Method (WDV)

Question 2.
What commands are used to

1. Insert a column and
2. Delete a column in Libre Office Calc

Answer:

1. Insert a column
2. Delete a column

Question 3.
Develop the command to calculate the Group Insurance Premium and Tax Deducted at source (TDS) by using the ‘IF’ function.

1. Rate of GI Rs. 200/-. for BP below Rs. 10,000/- and for others Rs. 300/- assuming that BP of employee is given in cell F2
2. TDS 10% of Gross Pay for employees having Gross Pay below Rs, 25000/- and for others 20%, assuming that the gross pay of the employee is given in cell F2.

Answer:

1. = IF (B2 < 10000,200,300)
2. = IF (F2 < 25000, F2*10%, F2*20%)

Question 4.
Name the two basic method of depreciation functions used in LibreOffice Calc
Answer:

1. SLN
2. D8

Question 5.
Why is FV taken as Zero (0) in the PMT calculation?
Answer:
At the end of the loan period, the balance amount payable will be zero assuming that the repayments are made on regular basis. Therefore the future value FV is taken as zero.

Question 6.
Write the command to calculate the State Life Insurance (SLI) Premium of an Employee using the ‘IF’ function.
The condition is:- SLI Premium Rs. 250 for basic pay below Rs. 10,000/- for others Rs.500/- (Hint: Basic pay (BP) is given the cell B3)
Answer:
IF (B3 < 10000, 250, 500)

Plus Two Accountancy Use of Spread Sheet in Business Three Mark Questions and Answers

Question 1.
Give some examples for PayRoll components.
Answer:

1. Basic Pay
2. Dearness Allowance
3. House Rent Allowance
4. Provident Fund
5. Professional Tax
6. ESI

Question 2.
Basic Pay, Dearness Allowance, House Rent allowance, Professional Tax, Provident fund contribution are given to prepare the PayRoll statement. Give the equation to calculate Net Salary.

• Gross salary = Basic Pay + Dearness Allowance + House Rent Allowance
• Total Deduction = Professional Tax + Provident fund contribution
• Net Salary = Gross Salary – Total Deduction

Answer:
Net Salary Calculation:
Step 1 – Calculate Gross salary by using the given formula.
Gross salary /Gross Pay = Basic Pay + Grade Pay + Dearness Pay+ Dearness Allowance + House Rent Allowance + Any other Earnings.

Step 2 – Calculate Total Deduction by using the following formula.
Total Deduction = Professional Tax+ Provident Fund + Tax deducted at source + Loan Recovery + Any other deductions

Step 3- Calculate net salary by the given formula.
Net Salary = Gross salary – Total Deduction

Question 3.
What is the difference between WDV method and SLN method of depreciation?
Answer:
Written Down Value Method (WDV):
This method is also known as Diminishing balance method or Reducing balance method. Under this method, a fixed percentage is written off every year on the book value of the asset at the beginning of the year.

Here the amount of depreciation goes on decreasing and therefore, the book value of asset will not become zero after its working life.
Amount of depreciation = Written Down Value of asset x Rate of depreciation

Straight Line Method:
Under this method a fixed amount is deducted from the value of an asset year after year on account of depreciation and debited to profit and loss account. This method is also called Fixed Instalment method, or Original Cost method. Under this method value of asset will be reduced to zero.
Depreciation = \(\frac{\text { cost of the asset-Scrap Value }}{\text { Life of the asset }}\)

Question 4.
List down the Parameters of the function PMT
Answer:
LOAN REPAYMENT SCHEDULE:
Loan is a sum of borrowed money for a specified period at a pre-specified rate of interest. The loan is repaid through a number of periodic repayment instalments over the loan repayment period. LibreOffice Calc function PMT is used to calculate the loan repayment schedule. The parameters of the function PMT are as follows.

Parameter – Explanation

• Rate – Interest rate
• Nper – Total Number of payments for the loan
• PV – Present value(Loan amount)
• FV – Future value, which is taken a zero, is the balance at the end of the loan period
• Type – Whether payment is made at the beginning (value = 1) or at the end (value = 0) of the period.

Question 5.
Classify the assets under computerised asset accounting.
Answer:
Assets are classified into the following categories:

1. Goodwill
2. Land: Freehold and leasehold
3. Building: Factory, office & residential building
4. Plant & Machinery
5. Furniture and fixtures
6. Vehicles
7. Work in progress (Capital)
8. Other assets

Question 6.
List out common Payroll components regarding salary computation and its payment.
Answer:
Earnings:

1. Basic pay
2. DearnessAllowances
3. House Rent Allowances
4. Transport Allowances
5. Other allowances.

Deductions:

• Provident Fund
• Professional Tax
• Tax deducted at source
• E.S.I. Premium

Question 7.
What are the common accounting applications done with the help of Libre Office Calc?
Answer:

1. Payroll Accounting
2. Asset Management
3. Loan Repayment Schedule

Question 8.
Write the formula in Libre Office Calc to find the Professional Tax in cell B2 where annual income is given in cell A2. Profession Tax is 5% for income in between Rs. 100000 and Rs.200000 and 8% for income more than Rs. 2,00,000. No tax for income below Rs. 1,00,000.
Answer:
= IF(B2 > 200000, B2*8%, IF(B2>100000, B2*5%,))

Plus Two Accountancy Use of Spread Sheet in Business Four Mark Questions and Answers

Question 1.
Write command to calculate state life Insurance Premium (SLI) of employee using the ‘IF’ function.
Condition
Premium Rs. 350/- below Basic Pay of Rs. 25000 and for others Rs. 450/- (BP is given in cell A3)
Answer:
= IF(A3 < 25000, 350, 450)

Question 2.
Briefly explain the procedure of preparation of salary bill and disbursement of cash.
Answer:
Preparation of Salary Bill:
The preparation of salary bill should provide for the following:
1. Maintaining payroll related data such as Employee No., Name, attendance, Basic Pay, DA, and other allowances, deductions to be made etc.

2. Periodic Payroll Computations:
It includes the calculation of various earnings and deductions.

3. Preparation of salary statement and employee’s salary slip.

4. Generation of advice to bank:
It contains the net salary to be transferred to individual bank account of employees and other salary related statutory payments such as provident fund, tax, etc.

Question 3.
The column headings of payroll to be prepared through Libre Office Calc is given below.

1. Write the formula to calculate DA, HRA, Gross pay, TDS and Net pay of Jexin Jose the first employee, in the second row of the worksheet.
2. Give the command to fill the calculation automatically for the remaining ‘10’ employees in the firm.

Answer:
1. DA → C2 =B2 * 20%
HRA → D2 = B2 * 5%
GP → E2 = B2 + C2+ D2
TDS → F2 = E2 * 10%
NP → G2 = E2 – F2

2. Fill the calculation for the remaining ‘10’ employees in the firm.

Question 4.
Distinguish between straight line method and diminishing balance method of depreciation
Answer:

Question 5.
Mr. Jyothis, a Plus two commerce student, entered the following details in a worksheet of LibreOffice Calc.

Write the command to calculate Net Salary
Answer:

Fill down the calculation to remaining 4 employees

Question 6.
Develop the commands to calculate the group insurance premium (Gl) and Tax Deducted at source (TDS) by using the ‘IF’ function.

1. Rate of GI Rs. 200/- for BP below Rs: 8000/- and for others Rs. 300/-, assuming that BP of employee is given in B2.
2. TDS 10% of Gross pay, for employees having Gross pay below Rs. 15,000/- and for others, 20%, assuming that the gross pay of the employee is given in F2.

Answer:

1. IF (B2 < 8000, 200, 300)
2. IF (F2 < 15000, F2*10%, F2*20%)

Plus Two Accountancy Use of Spread Sheet in Business Five Mark Questions and Answers

Question 1.
How the assets are classified in computerised Asset Accounting? What are the different methods of calculating depreciation on such assets?
Answer:
In computerised Asset Accounting, Assets are classified into the following categories.

1. Goodwill
2. Land (Normally, depreciation is not provided on freehold land)
3. Building
4. Plant and Machinery
5. Furniture and Fixtures
6. Vehicles
7. Work in progress (Capital)
8. Others

The different methods of calculating depreciation are:

• Straight Line Method (SLM)
• Written Down Value Method (WDV)

Question 2.
Which built in function of LibreOffice Calc can be used to compute monthly instalments of repayment of loan? Give the parameters of this function.
Answer:
PMT function can be used to prepare Loan Repayment Schedule in LibreOffice Calc.
The parameters of PMT function are:-

Parameter – Explanation

• Rate – Interest on Loan
• Nper – Number of payments for the loan
• PV – Present value; (ie the loan amount)
• FV – Future value, which is taken a zero
• Type – If the payment is made at the beginning of the month, the value = 1 or at the end of the month, the value = 0

Question 3.
From the following particulars prepare a payroll of employees of a firm by using LibreOffice Calc

• DA – 40% of basic pay
• HRA – 8% of basic pay
• Contribution to PF – 10% of basic pay.

Answer:

• DA = Basic pay *40%
• HRA = Basic pay *8%
• PF = Basic pay *10%
• Gross Pay = Basic Pay + DA + HRA
• Net Pay = Gross pay – PF

Question 4.
From the following particulars, prepare a payroll of employees of a firm by using LibreOffice Calc.

• DA = 70% of Basic pay
• HRA = 10% of Basic pay
• Contributions to PF at 15% of Basic pay.

Answer:
Step 1 – Enter the following

Step 2

Step 3 – Copy this formula to the remaining cells

Plus Two Accountancy Use of Spread Sheet in Business Practical Lab Work Questions and Answers

Question 1.
Prepare a Pay Roll statement of Viswanath Enterprises from the table given below and additional information.

1. DA is provided at 90% of Basic Pay
2. HRA: Rs. 500 for manager, 400 for accountant and 200 for others.
3. PF is deducted @ 20% on Basic + DA

Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc.

Step 2 – Enter the following text/formula in respective cells

Step 3 – Enter the given details

Step 4 – Copy the formula down up to the last employee.
Output:

Question 2.
Mr. Shibu wants to take a housing loan of Rs. 2,00,000 repayable in 60 equal monthly installments over the next 5 years. Assuming that the installments are paid in the beginning of each month. Find out the amount of monthly installments. Use PMT Function.
Procedure:
Step 1 – Open blank work sheet in LibreOffice Calc.
Applications → Office → Libre Office Calc.

Step 2 – Enter the following data in appropriate cells

Output:

Question 3.
Riya took a industrial loan of Rs. 300000, repayable in 4 years (equal monthly installments). The annual rate of interest is 10%. Assuming that the installments are paid at the end of each month. Find out the amount of monthly installments. Use PMT Function.
Procedure:
Step 1 – Open a blank work sheet in LibreOffice Calc.
Application → Office → Libre Office Calc.

Step 2 – Enter the following data in appropriate cells.

Output:

Question 4.
From the following particulars of employees in Anu Traders, prepare the Pay Roll.

Additional information:

1. DA – 50% of BP
2. HRA – 10% of BP
3. A monthly subscription to PF – 15% of BP
4. Group insurance premium -120 from each employee.

Procedure:
Step 1 – Open a new worksheet in LibreOffice Calc

Step 2 – Enter the following details in the following cells.

Step 3 – Enter the following details in the respective cells

Step 4 – Copy the formula down up to the last employee.
Output:

Question 5.
The salary information of a company named Seasons India Ltd. is given below. Prepare Payroll.

DA – 50% of Basic pay, CCA – 5% of Basic pay, PF 8% of Basic pay.
Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc

Step 2 – Enter the following details in the following cells.

Step 3 – Enter the following formula in respective cells

Step 4 – Copy the formula down up to the last employee
Output:

Question 6.
Mr.Anil Kumar a plus two commerce student entered the following details in a worksheet of LibreOffice Calc.

But, he faces some difficulty in completing the task. Can you help him by giving necessary commands or formula for filling the bank columns in the given spread sheet.
Answer:
Give the Following Formula:

Copy these formula to the remaining cell. Edit – Fill – Down or use Drag option.
Output:

Question 7.
Prepare the payroll of Amal Bros, for the month of January 2016.

• DA – 36% of Basic Pay
• HRA – Rs 500
• CCA – 6% of Basic pay

Deduction:

1. PF subscription – 8% of BP.
2. Group insurance premium Rs. 200 below basic pay Rs. 6000 and for others Rs. 350.
3. Tax deducted at source 10% of Gross Pay for employees below gross pay of Rs. 10000 arid for others 20%. Also, find out the total salary payable to employees for the month.

Procedure:
Step 1 – Open a blank work sheet in LibreOffice Calc

Step 2 – Enter the following details in the following cells.

Step 3 – Enter the following formula in the respective cell

Step 4 – Copy the formula down up to the last employee.
Output:

Question 8.
Prepare a pay roll of Jayakumar Associates for the month of June 2016 from the following details

Additional Information’s

1. DA – 50% of Basic pay earned
2. HRA – Manager – 30%, Accountant – 20%, clerk 10%, no HRA for saleswoman/Salesman and driver.
3. Transport allowance – 1500 for saleswoman, 2500 for salesman and 1000 for driver
4. PF contribution – 6% of BP for all employees except the driver.

Procedure:
Step 1 – Open a blank worksheet in LibreOffice Calc

Step 2 – Enter the following text/formula in the following way

Step 3 – Enter the given formula in F2
F2 = 30 – E2
Copy the formula down to the last employee

Step 4 – Enter the given formula in G2
G2 = D2*F2 ÷ 30
Copy the formula down to the last employee.

Step 5 – Enter the text for formula in the following cells.

Step 6 – Copy the formula down to the last employee.
Output:

Question 9.
Megha Associates purchased a new Machinery on 1/1/2010 for Rs. 8000 and spent Rs. 2000 for its installation. The expected salvage value is Rs. 2000 at the end of its useful life of 10 years. Calculate the amount of depreciation under straight line method.
Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following details as follows

Output:

Question 10.
On 1.1.2010, Vahida Enterprises purchased a new machinery for Rs. 150000 and incurred Rs. 10000 for its installation. Preoperative expense amounted to Rs. 5000. The expected salvage value at the end of its useful life of 8 years is Rs. 2000. Calculate depreciation by straight line method by using spread sheet.
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following details in respective cells.

Output:

Question 11.
On 1^st April 2000, a company purchased a Machinery for Rs. 20,00,000, The installation charge is 50,000, pre-operation expenses are 1,50,000 and its salvage value is calculated Rs. 1, 00,000. Life of machinery is estimated to 15 years. Calculate depreciation under straight line method.
Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following data/formula in the following way.

Output:

Question 12.
From the following details, calculate depreciation under written down value method by using spread sheet.

1. Purchase fo Machinery – 10-6-2012
2. Cost of the Machinery – 300000
3. Machinery installed on – 15-6-2012
4. Installation Expenses – 2000
5. Pre operating cost – 13000
6. Salvage value after 8 years – 18000
7. 1^st year end date – 31/3/2013

Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the given data/ formula as follows.


Output:

Question 13.
From the following details, calculate depreciation of machinery under written down value method using spreadsheet.

Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the given data in respective cells.

Output:

Question 14.
From the following details, calculate depreciation underwritten down value Method.

Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following data in the respective cells.

Output:

Question 15.
On 1-2-2015, Vikram borrowed Rs. 6,00,000 from Canara bank at 9.6% interest. The period of loan is 36 months. Calculate monthly installment assuming the installments are made at the beginning of each month.
procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter data/formula as follows:

Output:

Question 16.
Mr. Chandramohan has taken a loan of Rs. 300,000 from a bank at an interest rate of 10% p.a. The loan is to be repaid in 36 monthly installments over the next 3 years. Assuming that the monthly installments are paid at the end of each month. Calculate the amount of interest paid by him in the 2^nd year only. Use CUMIPMT Function.
Procedure:
Step 1 – Open a new blank worksheet in LibreOffice Calc.

Step 2 – Enter the following details and formula in different cells as given below.

Output:

Students can Download Chapter 10 Server Side Scripting Using PHP Questions and Answers, Plus Two Computer Science Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations

Kerala Plus Two Computer Science Chapter Wise Questions and Answers Chapter 10 Server Side Scripting Using PHP

Plus Two Computer Science Server Side Scripting Using PHP One Mark Questions and Answers

Question 1.
PHP files have a default file extension of ______.
(a) .html
(b) .xml
(c) .php
(d) .ph
Answer:
(c) .php

Question 2.
A PHP script should start with.
(a) <php>
(b) <?php?>
(c) <?php>
(d) <?php?>
Answer:
(c) <?php>

Question 3.
We can use_ to provide multi line comments.
(a) /?
(b) //
(c) #
(d) /* */
Answer:
(d) /* */

Question 4.
Which of the following php statement/statements will store 111 in variable num?
i) int$num = 111;
ii) intnum = 111;
iii) $num = 111;
iv) 111 = $num;
(a) Both i) and ii)
(b) All of the mentioned.
(c) Only iii)
(d) Only i)
Answer:
(c) Only iii)

Question 5.
What will be the output of the following PHP code?
<?php
$num = 1;
$num1 = 2;.
print $num . “+”. $num1;
?>
(a) 3
(b) 1 + 2
(c) 1. + .2
(d) Error
Answer:
(b) 1 + 2

Question 6.
Which of the following PHP statements will output Hello World on the screen?
i) echo (“Hello World”);
ii) print (“Hello World”);
iii) cout(“Hello World”);
iv) sprintf (“Hello World”);
(a) i) and ii)
(b) i), ii) and iii)
(c) All of the mentioned.
(d) i), ii) and iv)
Answer:
(a) i) and ii)

Question 7.
What will be the output of the following PHP code?
<?php
$total=25;
$more=10;
$total=$total + $more; .
echo “$total”;
?>
(a) Error
(b) 35 students
(c) 35
(d) 25 students
Answer:
(c) 35

Question 8.
Which statement will output $x on the screen?
(a) echo “\$x”;
(b) echo “$$x”;
(c) echo“/$x”;
(d) echo“$x;”;
Answer:
(a) echo “\$x”;

Question 9.
What will be the output of the following PHP code?
<?php
$a = “clue”;
$a = “get”;
echo “$a”;
?>
(a) get
(b) true
(c) false
(d) clueget
Answer:
(d) clueget

Question 10.
PHP’s numerically indexed array begin with position ______.
(a) 1
(b) 2
(c) 0
(d) -1
Answer:
(c) 0

Question 11.
What will be the output of the following PHP code?
<?php
echo $x— != ++$x;
?>
(a) 1
(b) 0
(c) error
(d) no output
Answer:
(d) no output

Question 12.
List Superglobals used in PHP.
Answer:
superglobals used in PHP is given below.
$_REQUEST, S_POST, S_GET, and S_COOKIE.

Plus Two Computer Science Server Side Scripting Using PHP Two Mark Questions and Answers

Question 1.
Name the PHP functions we use

1. to find the length of a string
2. for comparing two strings

Answer:

1. strlen()
2. strcmp()

Question 2.
What is the main difference between JavaScript and PHP?
Answer:
JavaScript is a client side scripting language whereas PHP is a server side scripting language.

Question 3.
What will be the output of the following PHP code?
<?php
$a = 10;
echo ++$a;
echo $a++;
echo $a;
echo ++$a;
?>
(a) 11111213
(b) 11121213
(c) 11111212
(d) 11111112
Answer:
(a) 11111213
It works as follows
<?php
$a = 10;
echo ++$a; → Here pre increment first add 1 to
10 then it prints 11
echo $a++; → Here it prints 11 and then add 1 to
11 and stores in the variable $a, i.e. $a=12. echo$a; → Here it prints 12.
echo ++$a; → Here pre increment first add 1 to
12 then it prints 13 ?>

Question 4.
What will be the output of the following PHP code?
<?php
$x=”test”;
$y=”this”;
$z=”also”;
$x.=$y.=$z; echo $x; echo $y;
?>
(a) testthisthisalso
(b) testthis
(c) testthisalsothisalso
(d) error at line 4
Answer:
(c) testthisalsothisalso

Question 5.
What will be the output of the following PHP code?
<?php
$y = 2;
if ($y— == ++$y)
{
echo $y;
}
?>
(a) 2
(b) 1
(c) 3
(d) no output
Answer:
(a) 2
It works as follows $y=2;
Here $y – – (LHS of the condition) , it first use the value of $y i.e. 2 hence LHS becomes 2, then its values decremented by 1 so $y now becomes 1. ++$y(RHS of the condition) first it changes the value
i. e. incremented by 1 now $y=2 hence RHS becomes 2
Here LHS and RHS are same then the condition $y— == ++$y returns true.
So it prints the value of $y, i.e. 2.

Plus Two Computer Science Server Side Scripting Using PHP Three Mark Questions and Answers

Question 1.
A special type of array which is not supported by C++ is used in PHP. Can you describe the features of that array with example?
Answer:
Associative arrays
Arrays with named keys and string indices are called associative arrays.
Syntax: $varibale_name=array(key1 =>value1, key2=>value2,etc);
Eg:
<!DOCTYPE HTML>
<html lang=”en”>
<head>
<title>
We are learning PHP </title>
</head>
<body bgcolor=”cyan”>
<?php
$course = array (“Computer Science”=>”05″, “Commerce”=>”39″,”Science”=> “01” ); echo”The code of Computer Science is “.Scourse[“Computer Science”];
?>
</body>
</html>

Question 2.
Compare echo and print statements used in PHP.
Answer:
Output statements in PHP:
1. echo and print are used to display all types of data but echo is used to produce multiple outputs. Parenthesis is optional. Print returns true or false based on success or failure whereas echo doesn’t return true or false.
eg:
echo “ first output”, “second output”; or
echo (“ first output”, “second output”);
print “only one output”; .
or
print (“only one output”);
Eg.
<!DOCTYPE HTML> <head> <meta charset=”UTF-8″> <title>
This is a php page </title>
</head>
<body>
<?php
echo “This is first output”,” This is second output<br>”;
print “Only support single output”;
?>
</body>
</html>

Question 3.
List the main differences between GET and POST methods in form submitting?
Answer:
The main differences are given below.

Plus Two Computer Science Server Side Scripting Using PHP Five Mark Questions and Answers

Question 1.
What are the additional steps involved to run PHP in your computer?
Answer:
Basics of PHP
A. Setting up the development environment

• Step 1: Install a PHP compatible web Server(Eg. Abyss Web Server For Windows)
• Step 2: Install PHP interpreter.
  After installing the webserver type http:// localhost in the address bar the following screen will be available.

B. Writing and running the script
Take a note pad and type the code , PHP code should begin with <?php and end with ?>. Save this file in the root directory of the web server with extension .php.
Step 1
Take a notepad and type the following and save it as first.php on C:\Abyss Web Server\htdocs.
<?php
echo” My first PHP web page”;
?>
Step 2
Start the webserver if it is off
Step 3
Type as “http://localhost/first.php” in the address bar.

Students can Download Chapter 9 Structured Query Language Questions and Answers, Plus Two Computer Science Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations

Kerala Plus Two Computer Science Chapter Wise Questions and Answers Chapter 9 Structured Query Language

Plus Two Computer Science Structured Query Language One Mark Questions and Answers

Question 1.
………………. form of SQL is designed for use within general purpose programming languages such as COBOL, C, etc.
Answer:
Embedded SQL.

Question 2.
What is TCL?
Answer:
Transaction Control Language- component of SQL includes commands for specifying transactions.

Question 3.
Data dictionary is a special file in DBMS. What is it used for?
Answer:
Table details are stored in this file.

Question 4.
What does the following statement mean?
Name VARCHAR(30)
Answer:
Field Name can store up to 30 characters. It is a column definition.

Question 5.
Is there any data type available in SQL to store your date of birth information?
Answer:
Yes. DATE data type

Question 6.
Pick the odd one out.
(DEC, NUMBER, INT, DATE)
Answer:
DATE.

Question 7.
How do you ensure that the field ‘name’ will have some value always?
Answer:
using the constraint NOT NULL.

Question 8.
How to set the default value of column District in a table to Thrissur?
Answer:
District VARCHAR(30) DEFAULT ‘Thrissur’

Question 9.
…………… symbol is used as substitution operator in SQL.
Answer:
&

Question 10.
Is there any method to find the strings starting with letter ‘a’ from a field in SQL?
Answer:
Use LIKE operator, LIKE ‘a%’

Question 11.
How to check whether a particular field contains null values or not?
Answer:
Use operator IS NULL

Question 12.
‘ORDER BY’ clause is used for …………
Answer:
Sorting the results of a query by ascending(Asc) or Descending (Desc)

Question 13.
The built-in functions in SQL that return just a single value for a group of rows in a table are called ………..
Answer:
Summary functions or aggregate functions

Question 14.
How to find the number of values in a column in the table?
Answer:
Using the function COUNT()

Question 15.
………………. is the clause in SQL used for categorization.
Answer:
GROUP BY

Question 16.
Thomas wants to remove a table that he were using. How could you help him doing this?
Answer:
Using DROP TABLE command.

Question 17.
Pick the odd one out.
(SELECT, UPDATE, DELETE, DROP TABLE)
Answer:
DROP TABLE

Question 18.
Name the aggregate function that can be used to find the total number of records.
Answer:
COUNT()

Question 19.
Which of the following is an essential clause used with SELECT command?
(GROUP BY, ORDER BY, WHERE, FROM)
Answer:
FROM

Question 20.
Which of the following is not a column constraint?
(CHECK, DISTINCT, UNIQUE, DEFAULT)
Answer:
DISTINCT

Question 21.
Which of the following is a DDL command?
(SELECT, UPDATE, CREATE TABLE, INSERT INTO)
Answer:
CREATE TABLE

Question 22.
What are the logical operators used in SQL?
Answer:
And, Or, Not.

Question 23.
The …………. operator pf SQL is used to match a pattern with the help of %. February 2009
(a) BETWEEN
(b) AND
(c) LIKE
(d) OR
Answer:
(c) LIKE

Question 24.
The structure of a table Book is given below.

Write SQL query to Insert an additional column Purchase date of date type to the Books table.
Answer:
Alter table Book add(PurchaseDate Date);

Question 25.
Suppose we want to include a column in a table in which serial numbers are to be stored automatically on adding new records. Which constraint is to be used for that column during table creation?
Answer:
The constraint Autojncrement is used.

Question 26.
Which of the following cannot be used to name a table in SQL? Give the reason.
Answer:
(a) Studnt50
(b) Table
(c) $Employee
(d) Stock_123
Answer:
(b) Table. This is a keyword hence it cannot be used.

Question 27.
Which of the following commands is used to view the structure of a table?
(a) SHOW TABLES
(b) DESC
(C) SELECT
(d) DISPLAY
Answer:
(b) DESC

Question 28.
The command to eliminate the table CUSTOMER from a database is:
(a) REMOVE TABLE CUSTOMER
(b) DROP TABLE CUSTOMER
(c) DELETE TABLE CUSTOMER
(d) UPDATE TABLE CUSTOMER
Answer:
(b) DROP TABLE CUSTOMER

Question 29.
Which SQL command is used to open a database?
(a) OPEN
(b) SHOW
(c) USE
(d) CREATE
Answer:
(c) USE

Question 30.
Which is the keyword used with SELECT command to avoid duplication of rows in the selection?
Answer:
DISTINCT

Question 31.
Pick odd one out and write reason:
(a) WHERE
(b) ORDER BY
(c) UPDATE
(d) GROUP BY
Answer:
UPDATE. It is a command and others are clauses.

Question 32.
Which of the following clause is not used with SELECT command in SQL?
(a) GROUP BY
(b) WHERE
(c) SET
(d) ORDER BY
Answer:
(c) SET. This clause is used with UPDATE.

Question 33.
……………… operator in SQL is used with wild card characters for selection of records.
(a) LIKE
(b) IN
(c) NOT IN
(d) IN and NOT IN
Answer:
(a) LIKE

Plus Two Computer Science Structured Query Language Two Mark Questions and Answers

Question 1.
How is SQL different from other computer high-level languages?
Answer:
SQL means Structured Query Language. It is a relational database language, not a programming language like other high-level languages. It provides facilities to create a table, insert data into a table, retrieve information from a table, modify data in the table, etc.

Question 2.
Distinguish the SQL keywords UNIQUE and DISTINCT.
Answer:
1. Unique – It ensures that no two rows have the same value in a column while storing data. It is used with create command.

2. Distinct – This keyword is used to avoid duplicate values in a column of a table while retrieving data. It is used with select command.

Question 3.
Identify errors in the following SQL statement and rewrite it correctly. Underline the corrections.
CREATE student TABLE (admno PRIMARY KEY, roll no INT, name CHAR);
Answer:
The correct SQL statement is as follows.
CREATE TABLE student( admno INT PRIMARY KEY, roll_no INT, name CHAR(3));

Question 4.
Suppose a column named Fee does not contain any value for some records in the table named STUDENT. Write SQL statements to fill these blanks with 1000.
Answer:
UPDATE STUDENT SET Fee = 1000 WHERE Fee IS NULL;

Question 5.
Identify the errors in the following SQL statement and give reason for the error.
SELECT FROM STUDENT ORDER BY Group WHERE Marks above 50;
Answer:
In this query Group is the keyword hence it cannot be used. The correct query is as follows.
SELECT * FROM STUDENT WHERE Marks > 50 ORDER BY Marks;

Question 6.
Differentiate CHAR and VARCHAR data types of SQL.
Answer:

1. Char – It is used to store fixed number of characters. It is declared as char(size).
2. Varchar – It is used to store characters but it uses only enough memory.

Question 7.
Assume that CUSTOMER is a table with columns Cust_code, Cust_name, Mob_No and Email. Write an SQL statement to add the details of a customer who has no e-mail id.
Answer:
INSERT INTO CUSTOMER VALUES(1001, ‘ALVIS’, 9447024365, NULL);

Question 8.
Find the correct clause from the 2nd column for each SQL command in the 1_st column.

Answer:

Question 9.
Is ALL a keyword in SQL? Explain
Answer:
Yes, ALL retains all the duplicate values of a field

Question 10.
Differentiate LIKE ‘a%’ and LiKE ‘a_’
Answer:
%, it replaces a string but _ replaces only one character at a time.LIKE ‘a%’ retrieve all strings of any length that start with letter ‘a’ while LIKE ‘a_’ retrieve all two letter strings that start with letter ‘a’.

Question 11.
Is multiple sorting possible using ORDER BY clause? Explain
Answer:
Yes, by giving multiple field names separated by comma in the ORDER BY clause.

Question 12.
While inserting records into a table, Raju finds it is not possible to give more than 20 characters in the ‘name’ field. How can you help Raju solve this problem?
Answer:
Using ALTER TABLE command he can modify the width of field so that it can accommodate more than 20 characters.

Question 13.
After executing a query Mohan gets a message like ‘Table Altered’. What would have he done? Give the syntax.
Answer:
He would have used the ALTER TABLE command.
Syntax: ALTER TABLE ADD/MODIFY ();

Question 14.
During discussion one student said that DELETE and DROP TABLE commands are same. Do you agree with that? Justify.
Answer:
No, DROP TABLE removes a table entirely from the database whereas DELETE command deletes only records from an existing table.

Question 15.
1. Pick odd one out.
DROP TABLE, DELETE, ALTER TABLE, CREATE VIEW
2. Justify your answer
Answer:

1. DELETE
2. All others are DDL commands

Question 16.

1. Pick odd one out from the following.
2. Justify your answer.

Answer:

1. Group By
2. Group By is not a Constraint.

Question 17.
Consider the following SQL statements.
DELETE Name.
From Student
Where name = ‘Raju”
Find the error in the SQL if any and correct it.
Answer:
Delete from student where name = ‘Raju’

Question 18.
Consider the following Query in SQL.
From Employee Where branch = ‘Kannur’ Group By Department Having avg (salary) > 7000, Compare the ‘where’ clause and ‘Having’ Clause using the above query.
Answer:
‘where’ clause applies on single rows but Having clause applies on a group.

Question 19.
Some constraints in SQL are called column constraints. Some constraints are called table constraints. How do they differ?
Answer:
Column constraints are specified while defining each column, table constraints are specified once for the entire table at the end of table definition

Question 20.
1. Choose the odd one from the following.
Primary key, Unique, Distinct, Default, Check
2. Justify your choice
Answer:

1. Distinct
2. Distinct is used with select command while others are column constraints of create table command.

Question 21.
Name the most appropriate SQL data type required to store the following data.

1. Name of a student
2. Date of Birth of a student.
3. RollNo. of a student (in the range 1 to 50)
4. Percentage of marks obtained

Answer:

1. Varchar(70)
2. Date
3. Smallint OR Integer Or decimal
4. Dec(5, 2)

Question 22.
1. From the list given below select the names that cannot be used as a table name.
Adm_No, Date, Salary2006, Table, Column_Name, Address.
2. Justify your selection.
Answer:

1. Date, Table
2. There is a data type Date and Table is a keyword used to create a table. So these two are not used.

Question 23.
Classify the following SQL elements into two and give proper title for each category.
NOT NULL, AVG, COUNT, CHECK, SUM, DEFAULT
Answer:

Plus Two Computer Science Structured Query Language Three Mark Questions and Answers

Question 1.
As a part of your school project, you are asked to create a table Student with the fields RollNo, Name, Date of Birth and Score in IT. The constraints required are RollNo. is the primary key. Name cannot be empty.
Answer:
Create table student(RollNo decimal(2) not null primary key, Name varchar(20) not null, DOB date, Score number(2));

Question 2.
A table Employee consists of fields EmployeeNo, Name, Designation and Salary.
Consider that you are forced to give access to this table for an engineer from another company. But for security reasons you need to hide Salary from him. Employee No, Name, Designation, Salary.

1. Name the concept that provides this engineer, a facility to work on this table without viewing salary. Salary view
2. Write SQL query for implementing this SQL query

Answer:

1. Views
2. Create view Empview as select employeeno, Name, Designation from Employee.

Question 3.
Explain the keywords in the following query.
Answer:
SELECT allows to retrieve a subset of rows from the table, DISTINCT avoids duplication of courses from the table STUDENT, FROM is a keyword used to specify the name of the table

Question 4.
A table named student is given below.

Write answers for the questions based on the above table.

1. SQL statement to display the different courses available without duplication.
2. SQL statement to display the Name and Batch of the students whose percentage has a null value.
3. Output of the query select count (percentage) from Student.

Answer:

1. Select distinct batch from Student;
2. Select name, batch from Student where percent is null
3. 5

Question 5.
Once the creation of a table is over, one can perform two changes in the schema of the table. What are they? Give syntax.
Answer:
We can alter the table in two ways.
1. We can add a new column to the existing table using the following
syntax: ALTER TABLE ADD( );

2. We can also change or modify the existing column in terms of type or size using the following
syntax: ALTER TABLEMODIFY( );

Question 6.
Explain how pattern matching can be done in SQL with an example.
Answer:
pattern matching can be done using the operator LIKE while setting the condition with pattern matching. for eg., to display the names of all students whose name begins with letters ‘ma’, we can write the following query, SELECT name FROM STUDENT WHERE name LIKE ‘ma%’; here the character ‘%’ – substitutes any number of characters in the value of the specified column. Another character substitutes only one character of the specified column.

Question 7.
During the discussion of study your friend say that table and view are the same. How can you correct him?
Answer:
Tables and views are different. A view is a single virtual table that is derived from other physically existing tables. When we access a view we actually access the base tables.

We can use all the DML commands with the views but care should be taken as operation actually reflects in the base tables. The advantage of view is that without sparing extra storage space, we can use same table as different virtual tables. It also implements sharing along with privacy.

Question 8.
Find the errors if any of the following code.
Create table emp (name char, RolINo int(20));
Answer:
Here the argument size of data type char of field name is missing. So we can store a character only. The second error is the data type int has no argument. The correct statement is as follows Create table emp (name char(20), RolINo int);

Question 9.
Consider the following variable Declaration in SQL.
a. name char (25)
b. name Varchar (25)

1. Considering the utilisation of memory, which variable declaration is more suitable.
2. Justify your answer

Answer:
1. name varchar (25)
2. Because char data type is fixed length. It allocates maximum memory i.e, here it allocates memory for 25 characters maybe there is a chance of memory wastage. But Varchar allocates only enough memory to store the actual size.

Question 10.
Mr. Dilip wants to construct a table and implement some restrictions on the table.

1. Column can never have empty values.
2. One of the columns must be a key to identify the rows etc.

Can you help him to create that table satisfying the above restrictions with an example
Answer:

1. Use of create table, Not NULL.
2. Primary key Eg: Create table employee(RollNo decimal(3) not null primary key,Name Varchar(70) not null, Design Varchar(30), DOB Date,Salary Dec(7,2));

Question 11.
Write SQL query to construct a table student with fields Reg No, Stud Name, Sex, Course, grade, etc. As per the following conditions

1. Reg No should not be empty and using this column any rows in that table can be identified. Reg. No.
2. Student name should have some value.
3. Default value of sex be Female.
4. Grade should be any of the values: A+, A, B+, B, C+, C, D+, D, E

Answer:
Create table Student(RollNo decimal(3) not null primary key, Name Varchar(70) not null, sex char default ‘Female’,Grade char(2));

Question 12.
Ramu create Table employee with fields empId, empname, Designation, salary using SQL statements. Later he found that data type of emPId is typed as Integer instead of character and missed a field ‘Department’. Can you help him to solve this problem without recreating ‘employee’ table?
Answer:
Alter Table Employee Modify(Empld char(4)); AlterTable Employee Add(Dept char(15));

Question 13.
While creating a table Alvis give “Emp Details” for table name. Is it Possible? Write down the rules for naming a Table.
Answer:
It is not possible because there is a space between Emp and Details. The rules are given below:

1. It must not be a keyword(Keywords are reserved words and have predefined meaning)
2. It must begin with alphabets
3. Digits can be used followed by alphabets
4. Special characters cannot be used except underscore
5. We cannot give the name of another table

Plus Two Computer Science Structured Query Language Five Mark Questions and Answers

Question 1.
Consider the table ITEMS.

a) SELECT ITEMCODE, NAME FROM ITEMS WHERE CATEGORY = ‘Stationery’;
b) SELECT * FROM ITEMS WHERE SALES_ PRICE < UNIT_PRICE;
c) SELECT CATEGORY, COUNT(*) FROM ITEMS GROUP BY CATEGORY;
Answer:

Question 2.
Write SQL queries to SQL Query to
1. Create a table Employee with the fields given below.

2. List the name of all employees whose name’s second letter is ‘a’.
3. List the Name and Designation of employees whose Designation is not ‘Manager’
4. Increase the salary of ail employees by 10 percent.
5. Remove all managers whose salary is less than Rs. 10,000 from the table.
Answer:

1. Create table employee(Name Varchar(70), Desgn Varchar(30), DOB Date, Salary Dec(7,2));
2. Select Name from Employee where Name like
3. Select Name, Desgn from Employee where Desg no’Manager’;
4. Update Employee set Salary = Salary + Salary * .01;
5. Delete from Employee where Desgn = ‘Manager’ and Salary < 10000;

Question 3.
A table Student consists of fields Roll No, Name, Batch and Percent. Write SQL statements to

1. Display RolINo and Name of students whose percentage is Iessthan90 and greater than 70.
2. Display RolINo and Name of all students in science batch whose percentage is more than 90.
3. Display Names of all students in commerce and science batches.
4. Display the average Percent of students in each batch.
5. Display RolINo and Name in the ascending order of Batch and descending order of Percent.

Answer:

1. Select RollNo, Name from Student where percent < 90 and percent > 70;
2. Select RollNo, Name from Student where Batch = ’Science’ and percent > 90;
3. Select Name from Student where batch = ’Science’ Or ’batchCommerce’;
4. Select batch, Avg(percent) from Student group by batch;
5. Select RollNo, Name from Student order by batch, percent desc;

Question 4.
1. Classify the following SQL commands.
Create table, Insert Into, AlterTable, Delete, Up-date, Drop Table, Select.
2. List the features of each category.
Answer:
1. DDL – Create Table, Alter Table, Drop Table.
DML- Insert Into, Delete, Update, Select.

2. DDL – DDL means Data Definition Language. It is used to create the structure of a table, modify the structure of a table and delete the structure of a table. DML -DML means Data Manipulation Language. It is used to insert records into a table, modify the records of a table, delete the records of a table and retrieve the records from a table.

Question 5.
Explain the available database integrity constraints.
Answer:
1. NOT NULL: it specifies that a column can never have null values, i.e., not empty

2. UNIQUE: it ensures that no two rows have same value in the specified column.

3. PRIMARY KEY: it declares a column or a set of columns as the primary key of the table. This constraint makes a column NOT NULL and UNIQUE.

4. DEFAULT: it sets a default value for a column when the user does not enter a value for that column.

5. Auto_increment: This constraint is used to perform auto increment the values in a column. That automatically generates serial numbers. Only one auto-increment column per table is allowed.

Question 6.
Consider a table student with fields Reg No, Stud Name, Sex, Course, total score.
Write SQL queries for the following:
RegNo, Studname, Sex, Course, Total score.

1. Enter a Record
2. List the Detail’s of all students
3. Display Details of the student whose name ends with ‘Kumar’.
4. List all Female students who got more than 50 marks.
5. List all students who are studying either science or in Commerce group.
6. List details of those students who are studying Humanities in the Descending order of their names.

Answer:

1. Insert into Student values (52, ‘JOSE’, ‘Male’, ‘Science’, 500);
2. Select*from Student;
3. Select* from Student where name like ’%Kumar’;
4. Select * from Student where sex=’Female’ and Total >50;
5. Select * from Student where course = ‘science’ or course =‘commerce’;
6. Select* from Student where course-‘Humanities’ order by name desc;

Question 7.
Construct a Table Product with the following fields using SQL.
Product code – Consist of Alphanumeric code.
Product name – Consist of maximum of 30 Alphabets
Unit price – Numeric values
Quantity – Numeric values
Product price – Numeric Values
Write Query for the following:

1. Enter 5 records in the table (not give values for product prices)
2. Calculate product price (unit price × quantity)
3. List all product whose unit price ranging from 10 to 20
4. Calculate total price of all products.

Answer:
Create table Product(ProdCode Varchar(20) not null primary key, ProdName char(30), UPrice decimal(7,2), Qty decimal(6), ProdPrice decimal(7,2));

1. Insert into Product(Product Code,Product Name, Unit Price, Quantity) values (‘101 ’, ‘LUX’, 29.50,500); Similarly insert four more records
2. Update Product set ProdPrice=Qty * UPrice;
3. Select ProdName from Product where UPrice be-tween 10 and 20;
4. Select sum(ProdPrice) from Product;

Question 8.
Mr. Wilson wants to store the details of students. The details consist of different types of data. Explain different data types used in SQL to store data.
Answer:
The different data types are
1. Char (Fixed):
It is declared as char (size). This data type is used to store alphanumeric characters. We have to specify the size. If no size is specified, by default we can store only one character. Eg: name char(20)

2. Variable Character:
It is declared as varchar (size). This data type is also used to store alphanumeric characters. But there is a slight difference. lt allocates only enough memory to store the actual size.
Eg: name varchar(20)

3. Decimal:
It is declared as Dec(size, scale), where size is the number of digits and scale is the maximum number of digits to the right of the decimal point.
Eg: weight dec(3, 2)

4. Integer:
It is declared as int. It does not have any arguments. It takes more memory. Eg: ROllNO int.

5. Small Integer:
It is declared as smallint. It takes less memory ROllNO int.

6. Date:
It is used to store date. Eg: DOB date.

7. Time:
It is used to store time. Eg: Joining_Time Time.

Question 9.
A company wants to create a table to store its employees details. Write SQL Commands for the following :

1. Create a table with EMP table having fields EMPNO primary key varchar(10), Name varchar(20), Salary number(6), Department varchar(3)
2. Insert values to table
3. List all employees whose salary > 10,000 Salary 10,000
4. Display name and salary in the order of name.

Answer:

1. Create table EMP (EMPNO varchar(10) not null primary key, Name varchar(20), Salary decimal(6), Deptvarchar(3));
2. Insert into EMP values (‘1001 ’, ‘ALVIS’, 50000, ‘Sales’);
3. Select * from EMP where salary > 10000;
4. Select name, salary from EMP order by name;

Question 10.
Create a table ‘Savings’ with the following fields. Acc No (Integer), Name (char), age (Integer) and balance (Number) where Acc No is the primary key. Write SQL commands for the following.

1. Insert data in all the fields for 3 records inserted
2. Display the list of account holders in the ascending order Of their names.
3. Display the list of all account holders having age between 20 and 30
4. Display the name and Acc No of customers having a balance > 10 lakhs

Answer:
Create table Savings(Accno decimal(4) not null primary key, name char(2), age decimal(3), Balance decimal(8, 2));

1. Insert into savings values (501, ‘Andrea’, 18,45000)
2. Select * from Savings order by name
3. Select * from Savings where age between 20 and 30
4. Select name.accno from Savings where balance >100000

Question 11.
Which are the components of SQL? How do they help to manage database?
Answer:
The components of SQL are given below.
a. DDL commands ( 3 commands):

1. Create table – Used to create a table.
2. Alter table – Used to modify existing column or add new column to an existing table. There are 2 keywords used ADD and MODIFY.
3. Drop table – Used to remove a table from the memory.

b. DML commands ( 4 commands):

1. Select – Used to select rows from a table. The keyword From is used with this. Where clause is used to specify the condition.
2. Insert – Used to insert new records into a table. So the keyword used is INTO.
3. Delete – Used to delete records in a table.
4. Update – Used to modify the records in a table the keyword used is set.

c. DCL (Data Control Language) commands:

1. Grant – It grants permission to the users to the database
2. Revoke – It withdraws user’s rights given by using Grant command

Students can Download Chapter 5 Continuity and Differentiability Questions and Answers, Plus Two Maths Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 5 Continuity and Differentiability

Plus Two Maths Continuity and Differentiability Three Mark Questions and Answers

Question 1.
Consider the function g(x) = \(\frac{|x-2|}{x-2}\)

1. Find the domain and range of the function g(x). (2)
2. Check whether the g(x) is continuous at x = 2. (1)

Answer:
1. The function is not defined at points where denominator is zero.
i.e., x – 2 = 0 ⇒ x = 2.
∴ domain is R – {2}.
g(x) = \(\frac{|x-2|}{x-2}\) = \(\left\{\begin{array}{c}{\frac{x-2}{x-2}, \quad x-2>0} \\{\frac{-(x-2)}{x-2}, \quad x-2<0}\end{array}=\left\{\begin{array}{ll}{1,} & {x>2} \\{-1,} & {x<2}\end{array}\right.\right.\)
∴ range is {-1, 1}

2. The function g(x) is not defined at x = 2. Therefore discontinuous.

Question 2.
(i) If f(x) = x+|x| + 1, then which of the follow represents f (x) (1)


(ii) Test whether f (x) is continuous at x=0. Explain. (2)
Answer:
(i) (b) Since, f(x) = \(\left\{\begin{array}{c}{x+x+1, x \geq 0} \\{x-x+1, \quad x<0}
\end{array}=\left\{\begin{array}{c}{2 x+1, x \geq 0} \\{1, x<0}\end{array}\right.\right.\)

(ii) We have ,f (0) = 1,

Continuous at x = 0.

Question 3.
Consider the function f(x) = \(\left\{\begin{array}{ll}{\frac{\sin x}{x}} & {, x<0} \\{x+1} & {, x \geq 0}\end{array}\right.\)

1. Find \(\lim _{x \rightarrow 0} f(x)\) (2)
2. Is f (x) continuous at x= 0? (1)

Answer:
1. To find \(\lim _{x \rightarrow 0} f(x)\) we have to find f(0^– )and f(0⁺)
f(0^–) = \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\),
f(0⁺) = \(\lim _{x \rightarrow 0}\) + 1 = 0 + 1 = 1
f(0^–) = f(0⁺) = 1 .Therefore \(\lim _{x \rightarrow 0} f(x)\) = 1

2. Here, f (0) = 0 + 1 = 1 = f(0^–) = f(0⁺) = 1
Therefore continuous at x = 0.

Question 4.
Consider the figure and answer the following questions.

(i) Identify the graphed function. (1)


(ii) Discuss the continuity of the above function at x = 2. (2)
Answer:
(i) (b) f(x) = \(\left\{\begin{array}{ll}{\frac{|x-2|}{x-2},} & {x \neq 2} \\{0,} & {x=2}\end{array}\right.\)

(ii) From the figure we can see that
f(2^–) = 1, f(2⁺) = -1 and f(2) = 0
Therefore, f(2^–) = 1 ≠ f(2⁺) = -1 ≠ f(2) = 0. Discontinuous.

Question 5.
Consider f(x) = \(\left\{\begin{array}{ll}{2 x} & {\text { if } x<2} \\{2} & {\text { if } x=2} \\ {x^{2}} & {\text { if } x>2}\end{array}\right.\)
(a) Find \(\lim _{x \rightarrow 2^{-}} f(x)\) and \(\lim _{x \rightarrow 2^{+}} f(x)\) (2)
(b) f(x) is continuous. If not so, how can you make it continuous. (1)
Answer:


Therefore f(x) is not continuous at x = 2.
If f(2) = 4, then f(x) becomes continuous.

Question 6.
If y = log₁₀x + log_x10 + log_xx + log₁₀10. Find \(.
Answer:
y = log_e x log₁₀e + log_e10 log_xe + 1 + 1

Question 7.
Examine the continuity of the function

Answer:
In both the intervals x = 1 and x < 1 the function f(x) is a polynomial so continuous. So we have to check the continuity at x = 1.

f(1) = 2
Since

f(x) is continuous at x = 1.

Question 8.
Find [latex]\frac{d y}{d x}\) of the following (3 score each)

1. 2x + 3y = sinx
2. xy + y² = tanx + y
3. x³ + x²y + xy² + y³ = 81
4. sin²x + cos²y = 1
5. \(\sqrt{x}\) + \(\sqrt{y}\) = 1
6. x² + xy + y² = 7
7. x²(x – y) = y²(x + y)
8. xy² + x²y = 2
9. sin y = xcos (a + y)

Answer:
1. Given; 2x + 3y = sinx
Differentiating with respect to x;

2. Given; xy + y² = tanx + y
Differentiating with respect to x;

3. Given; x³ + x²y + xy² + y³ = 81
Differentiating with respect to x;

4. Given; sin²x + cos²y = 1
Differentiating with respect to x;

5. \(\sqrt{x}\) + \(\sqrt{y}\) = 1
Differentiating with respect to x;

6. x² + xy + y² = 7
Differentiating with respect to x;

7. x²(x – y) = y²(x + y)
Differentiating with respect to x;

8. xy² + x²y = 2
Differentiating with respect to x;

9. sin y = xcos (a + y)
⇒ x = \(\frac{\sin y}{\cos (a+y)}\)
Diff. with respect to y.

Question 9.
Find \(\frac{d y}{d x}\) of the following (3 score each)

(viii) x = a(t – sint), y = a(1 – cost)
(ix) y = e^t cost, x = e^t sint.
Answer:
(i) We know; y = sin^-1\(\left(\frac{2 x}{1+x^{2}}\right)\)
⇒ y = = 2 tan^-1(x)
Differentiating with respect to x;

(ii) We know; y = tan^-1\(\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)\)
Put x = tanθ ⇒ θ = tan^-1x

⇒ y = tan^-1(tan3θ)
⇒ y = 3 tan^-1(x)
Differentiating with respect to x;

(iii) We know; y = sin^-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\)
Put x = tanθ ⇒ θ = tan^-1x

(iv) We know; y = sec^-1 = \(\left(\frac{1}{2 x^{2}-1}\right)\)
Put x = cosθ ⇒ θ = cos^-1x

⇒ y = sec^-1 sec2θ = 2θ
⇒ y = 2cos^-1(x)
Differentiating with respect to x;

(v) ∴ y = tan^-1\(\sqrt{\tan ^{2} x / 2}\) = tan^-1 tanx/2 = x/2
\(\frac{d y}{d x}=\frac{1}{2}\).

= π – 2tan^-1x
\(\frac{d y}{d x}=\frac{-2}{1+x^{2}}\).

(vii) Let x = sinθ and \(\sqrt{x}\) = sinφ

= sin^-1 (sinθcosφ + cosφsinφ)
= sin^-1 (sin(θ + φ)) = θ + φ
= sin^-1x + sin^-1\(\sqrt{x}\)
\(\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{\sqrt{1-x}} \times \frac{1}{2 \sqrt{x}}\).

(ix) y = e^tcost ⇒ \(\frac{d y}{d t}\) = – e^t sin t + e^t
x = e^t sint ⇒ \(\frac{d x}{d t}\) = e^t cos t + e^t sin t

Question 10.
If y = log\(\left(\frac{1}{x}\right)\), Show that \(\frac{d y}{d x}\) + e^y = 0.
Answer:
Given, y = log\(\left(\frac{1}{x}\right)\) ? \(\left(\frac{1}{x}\right)\) = e^y …….(1)

Question 11.
If e^y (x+1) = 1. Show that

1. \(\frac{d y}{d x}\) = -e^y (2)
2. \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) (1)

Answer:
1. e^y (x+1) = 1
Differentiating e^y +e^y(x +1) \(\frac{d y}{d x}\) = 0

2.

Question 12.
(i) Evaluate \(\lim _{x \rightarrow 0} \frac{k \cos x}{\pi-2 x}\) (2)
(Hint: Put π – 2x = y , where Iris a constant)
(ii) Find the value of k if f (x) is a continuous function given by (1)

Answer:
(i) \(\lim _{x \rightarrow \pi / 2} \frac{k \cos x}{\pi-2 x}=k \lim _{x \rightarrow \pi / 2} \frac{\cos x}{\pi-2 x}\),
Put π – 2x = y when Put x → π/2, y → 0

(ii) Since f (x) is continuous

Question 13.
If f(x) \(f(x)=\left\{\begin{array}{cc}{x-[x]} & {, x<.2} \\{0} & {; x=2} \\{3 x-5} & {, x>2}
\end{array}\right.\)

1. Find \(\lim _{x \rightarrow 2} f(x)\) (2)
2. Is f(x) continuous at x = 2? (1)

Answer:
1. To find \(\lim _{x \rightarrow 2} f(x)\)
we have to find f(2^–) and f(2⁺)
f(2^–) = \(\lim _{x \rightarrow 2} x-[x]\) = 2 -1 = 1,
f(2⁺) = \(\lim _{x \rightarrow 2}\) 3x – 5 = 6 -5 = 1
f(2^–) = f(2⁺) = 1.
Therefore \(\lim _{x \rightarrow 2}\) f(x) = 1

2. Here, f(2) = 0 ≠ f(2^–) = f(2⁺) = 1.
Therefore discontinuity at x = 2.

Plus Two Maths Continuity and Differentiability Four Mark Questions and Answers

Question 1.
If x = 2cosθ; y = 3sinθ

1. Find \(\frac{d y}{d x}\).
2. Find \(\frac{d^{2} y}{d x^{2}}\)

Answer:
1. x = 2cosθ ⇒ \(\frac{d x}{d θ}\) = -2sinθ
y = 3sinθ ⇒ \(\frac{d x}{d θ}\) = 3cosθ

2.

Question 2.
If y = (tan^-1 x)², show that (x² +1)² y₂ + 2x(x² +1) y₁ = 2.
Answer:
y = (tan^-1 x)²
⇒ y₁ = 2(tan^-1 x) \(\frac{1}{1+x^{2}}\)
⇒ (1 + x²)y₁ = 2(tan^-1 x)
⇒ (1 + x²)y₂ + y₁2x = 2 \(\frac{1}{1+x^{2}}\)
⇒ (1 + x²)² y₂ + x(1 + x²)y₁ = 2.

Question 3.
Find \(\frac{d y}{d x}\) if y = sin^-1 \(\left(\frac{1-x^{2}}{1+x^{2}}\right)\), 0 < x < 1.
Answer:
Put x = tanθ

Question 4.
Let f(x) = \(\left\{\begin{array}{ll}{\cos x,} & {0 \leq x \leq c} \\{\sin x,} & {c<x \leq \pi}\end{array}\right.\)

1. Find the value of c if / is continuous on [0, π].
2. Show that is f not differentiable at the point c.

Answer:
1. Since f is continuous on [0, π], we have;
\(\lim _{x \rightarrow c^{+}}\) f(x) = \(\lim _{x \rightarrow c^{-}}\) f(x) = f(c)
⇒ \(\lim _{x \rightarrow c^{+}}\) sinx = \(\lim _{x \rightarrow c^{-}}\) cosx = cosc
⇒ sinc = cosc ⇒ c = \(\frac{π}{4}\).

2. \(f^{\prime}(x)=\left\{\begin{array}{ll}{-\sin x,} & {0 \leq x \leq c} \\{\cos x,} & {c<x \leq \pi}\end{array}\right. \)
Left derivative at \(\frac{\pi}{4}\) = – sin \(\frac{\pi}{4}\) = –\(\frac{1}{\sqrt{2}}\)
Right derivative at \(\frac{\pi}{4}\) = cos \(\frac{\pi}{4}\) = \(\frac{1}{\sqrt{2}}\)
Left derivative at \(\frac{\pi}{4}\) ≠ Right derivative at \(\frac{\pi}{4}\)
Therefore is not differentiable at the point c.

Question 5.

1. Find \(\frac{d y}{d x}\) if x = 2sinθ; y = 3cosθ
2. Which among the following functions is differentiable on R?

(a) |sinx|
(b) |cosx|
(c) cos|x|
(d) sin|x|
Answer:
1. \(\frac{d x}{d θ}\) = 2cosθ; \(\frac{d y}{d θ}\) = -3 sinθ ⇒ \(\frac{d y}{d x}\) = \(-\frac{3}{2}\)tanθ

2. (c) cos|x|
(Since cos x is an even function, it treats x and -x in the same way).

Question 6.
(i) Examine whether the function defined by \(f(x)=\left\{\begin{array}{ll}{x+5,} & {x \leq 1} \\{x-5,} & {x>1}\end{array}\right.\) is continuous or not. (2)
(ii) If x = ^sin-1t, y = ^cos-1t, a > 0, show that \(\frac{d y}{d x}=-\frac{y}{x}\)
Answer:

f(x) is not continuous.