Plus Two

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 6 Application of Derivatives.

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

Plus Two Maths Application of Derivatives 4 Marks Important Questions

Question 1.
(a) Find the equation of the tangent to the curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=2\) at (1,1).
(b) Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. (May – 2015)
Answer:

Question 2.
(a) The slope of the tangent to the curve given
\(x=1-\cos \theta, y=\theta-\sin \theta \text { by at } \theta=\frac{\pi}{2}\)
(i) 0
(ii) – 1
(iii) 1
(iv) Not defined.

(b) Find the intervals in which the function f(x) = x² – 4x + 6 is strictly decreasing.
(C) Find the minimum and maximum value, if any, of the function f(x) = (2x – 1)² + 3 (March – 2016)
Answer:
(a) (iii) 1
(b) Given; f(x) = x² – 4x + 6 ⇒ f’(x) = 2x – 4
For turning points; f’(x) = 2x – 4 0 ⇒ x = 2
So volurn.,e is niaxirnum when h = 2r
The intervals are (- ∞, 2); (2, ∞)
f’(0) = 2 x 0 – 4 = -4
Therefore f(x) is decreasing in (- ∞, 2)
(c) f(x) = (2 x 1)² + 3
⇒ f’(x) 2(2x – 1) x 2 f”(x) = 8
For tuming points; f’(x) = 8x – 4 = 0 ⇒ x = 1/2
f(x) has minimum value at x = 1/2 minimum value is \(f\left(\frac{1}{2}\right)=3\)
2)

Question 3.
(a) Which of the following function has neither local maxima nor local minima?
(i) f(x) = x² + x
(ii) f(x) = logx
(iii) f(x) = x³ – 3x + 3
(iv) f(x) = 3 + |x|
(b) Find the equation of the tangent to the curve y = 3x² at (1,1). (March – 2016)
Answer:

Question 4.
(i) The slope of the normal to the curve, y = x³ – x² at (1, -1) is
(a) 1
(b) – 1
(c) 2
(d)0

(ii) Find the intervals in which the function f(x) = 2x³ – 24x + 25 is increasing or decreasing. (May – 2016)
Answer:
(i) (b) – 1
(ii) f(x) = 2x³ – 24x + 25
f’(x) = 6x² – 24
f’(x) = O
⇒ 6x² – 24 = 0 ⇒ x² = 4 ⇒ x = – 2,2
Therefore the intervals are (-∞, -2); (-2, 2); (2, ∞)
f(x) is increasing in the intervals (-∞, -2); (2, ∞)
f(x) is decreasing in the intervals (-2, 2)

Question 5.
(i) The slope of the normal to the curve, y² – 4x at (1,2) is
(a) 1
(b) 1/2
(c) 2
(d) – 1

(ii) Find the intervals in which the function 2x³ + 9x² + 12x – 1 is strictly increasing. (March – 2017)
Answer:
(i) (b) – 1
(ii) f(x) = 2x³ + 9x² + 12x – 1
f’(x) = 6x² + 18x + 12
= 6(x² + 3x + 2) = 6(x + 1) (x + 2)
f’(x) = O
⇒ 6(x + 1)(x + 2) = 0 ⇒ x = – 1 – 2
Therefore the intervals are
(- ∞, – 2); (- 2, – 1); (- 1, ∞)
In the ¡nterval (- ∞, – 2)
f’( – 3) = 6(- 3 + 1) (- 3 + 2) > 0
Therefore increasing In the interval (- 2, – 1)
f’(- 1.5) = 6(- 1.5 + 1)(- 1.5 + 2) < 0
Therefore decreasing In the interval (- 1, ∞)
f’(0) = 6(0 + 1)(0 + 2) > 0
Therefore increasing

Question 6.
Find two positive numbers whose sum is 16 and sum of whose cubes is minimum. (March – 2017)
Answer:
Let the numbers be x and 16 – x. Then,
S = x³ + (16 – x)³
= S’ = 3x² + 3(16 – x)²(- 1)
⇒ S” = 6x + 6(16 – x)………..(1)
For turning points S’ = 0 ⇒ 3 x² – 3(16 – x)² = 0
⇒ x² – 16 + 32x – x² =0
⇒ – 16² + 32x = 0 = x² = \(\frac{16 \times 16}{32}\) =8
(1) ⇒ S” = 6(8) + 6(16 – 8) > 0
TherefocemrnimumM x = 8
Thusthe numbers are8 and 16 – 8 = 8

Plus Two Maths Application of Derivatives 6 Marks Important Questions

Question 1.
(i) Show that the function x³ – 6x² + 15x + 4 is strictly increasing in R.
(ii) Find the approximate change in volume of a cube of side x meters caused by an increase in the side by 3%.
(iii) Find the equation of the tangent and normal at the point (1,2) on the parabola y² = 4x. (March – 2010)
Answer:
(i) Given; f(x) = x³ – 6x² + 15x + 4
f’(x) = 3x² – 12x + 15 = 3(x² – 4x +5)
= 3(x² – 4x + 4 + 1) = 3(x – 2)² + 1) > 0
For any value of x, f(x) is a strKly ¡ncreasing.

(ii) We have; V = x³ and Δx = 3% of x = 0.03x
\(d V=\frac{d V}{d x} \Delta x=3 x^{2} \Delta x\)
= 3x² x 0.03x = 0.09x³ = 0.09V
\(\Rightarrow \frac{d V}{V}=0.09\)

Therefore 9% is the approximate increase In volume.

(iii) Given; y²….4x ⇒ 2y \(\frac{d y}{d x}\) = 4 ⇒ \(\frac{d y}{d x}=\frac{2}{y}\)
Slope at (1,2) = \(\frac{2}{2}\) = 1
Equation of tangent at (1,2) is; y – 2 = 1(x – 1)
⇒ x – y + 1 = 0
Equation of normal at (1,2) is; y – 2 = – 1(x – 1)
⇒ x + y – 3 = 0

Question 2.
Consider the parametric forms
x = 1 + \(\frac{1}{t}\) – and y = t – \(\frac{1}{t}\) ofa curve
(i) Find \(\frac{d y}{d x}\)
(ii) Find the equation of the tangent at t = 2.
(iii) Find the equation of the normal at t = 2. (May – 2010)
Answer:

Question 3.
(i) The radius of a circle is increasing at the rate of 2cmls. Find the rate at which area of the circle is increasing when radius is
6cm.
(ii) Prove that the function f(x) = log sin x is strictly increasing in \(\left(0, \frac{\pi}{2}\right)\) and strictly decreasing in \(\left(\frac{\pi}{2}, \pi\right)\)
(iii) Find the maximum and minimum value of the function f(x) = x³ – 6x² + 9x + 15. (March – 2011)
Answer:

Question 4.
(i) Find the approximate value of (82)^1/4 up to three places of decimals using differentiation.
(ii) Find two positive numbers such that Their sum is 8 and the sum of their squares is minimum. (May – 2011)
Answer:

(ii) Let the numbers be x and 8 – x. Then,
S = x² + (8 – x)²
⇒ S’ = 2x + 2(8 – x)( – 1)
⇒ S” = 2 + 2 = 4 ………..(1)
For turning points S’ = 0 = 2x – 2(8 – x) = 0
⇒ 4x – 16 = 0 ⇒ x = 4
(1) ⇒ S” = 4 > 0
Therefore minimum at x = 4
Thus the numbers are 4 and 8 – 4 = 4.

Question 5.
(i) The slope of the tangent to the curve y = x³ – 1 at x = 2 is ……….
(ii) Use differentiation to approximate \(\sqrt{36.6}\)
(iii) Find two numbers whose sum is 24 and whose product as large as possible. (March – 2012, March – 2016)
Answer:

Therefore minimum at x =12
Thus the numbers are 12 and 24 – 12 = 12.

Question 6.
(i) Show that the function x³ – 3x² + 6x – 5 is strictly increasing on R.
(ii) Find the interval in which the function f(x) = sin x + cosx; 0 < x < 2π is strictly increasing or strictly decreasing. (May – 2012)
Answer:
(i) Given; f(x) = x³ – 3x² + 6x – 5
f’(x) = 3x² – 6x + 6 = 3(x² – 2x +2)
= 3(x² – 2x + 1 + 1) 3(x – 1)² + 1) > 0
For any value cit x, f(x) is a strictly increasing.

Question 7.
(i) Find the slope of the normal to the curve y = sinθ at θ = π/4
(ii) Show that the function f(x) = x³ – 6x² + 15x + 4 is strictly increasing in R.
(iii) Show that all rectangles with a given perimeter, the square has the maximum area. (March – 2013)
Answer:

(ii) f(x) = x³ – 6x² + 15x + 4
Differentiating w.r.t x;
f(x) = 3x² – 12x + 15 = 3(x² – 4x + 5)
= 3 (x² – 4x + 4 + 1)
= 3 ((x – 2)² + 1) > 0, ∀x∈R
Therefore fis strictly increasing in R.

(iii) Let x and ybe the length and breadth of a rectangle with area A and perimeter P.

Question 8.
A right circular cylinder is inscribed in a given cone of radius R cm and height H cm as shown in the figure.

(i) Find the Surface Area S of the circular cylinder as a function of x.
(ii) Find a relation connecting x and R when S is a maximum. (May – 2013)
Answer:
(i) There are two similar triangles ΔDJB and ΔDHF

Question 9.
(i) Which of the following function is Increasing for all values of x in its domain?
(a) sin x
(b) log x
(c) x²
(d) |x|

(ii) Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2,0) and (4,4).
(iii) Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x – 6x². (March – 2014)
Answer:
(i) (b) log x
(ii) Given; y = (x – 2)² ⇒ \(\frac{d y}{d x}\) = 2(x – 2)
Slope of the chord = \(\frac{4-0}{4-2}=2\)
\(\Rightarrow 2=2(x-2) \Rightarrow x=3 \Rightarrow y=(3-2)^{2}=1\)
Therefore the required point is (3, 1)

(iii) Given; p(x) = 41 – 24x – 6x²
p’(x) = – 24 – 12x
p”(x) = – 12
For turning points p’(x) = – 24 – 12x = 0
⇒ x = -2
Since p”(x) = – 12 always maximum Therefore maximum value p(- 2) = 41 – 24(- 2) 6(- 2)² = 65

Question 10.
(a) Find the slope of the tangent to the parabola y² = 4ax at (at², 2at).
(b) Find the intervals in which the function x² – 2x + 5 is strictly increasing.
(c) A spherical bubble volume at the rate of which the diminishing when the is decreasing in 2cm³/sec. Find the surface area is radius is 3cm. (May – 2014)
Answer:

Question 11.
(a) Which of the following function is always increasing?
(i) x + sin 2x
(ii) x – sin 2x
(ill) 2x + sin 3x
(iv) 2x – sin 2x
(b) The radius of a cylinder is increasing at a rate of 1cm/s and its height decreasing at a rate of 1cm/s. Find the rate of change of its volume when the radius is 5cm and the height is 5cm.
(c) Write the equation of tangent at (1,1) on the curve 2x² + 3y² = 5. (March – 2015)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 5 Continuity and Differentiability.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 5 Continuity and Differentiability

Plus Two Maths Continuity and Differentiability 3 Marks Important Questions

Question 1.
Consider \(f(x)=\left\{\begin{array}{ll}
\frac{x^{2}-x-6}{x+2}, & x \neq-2 \\
-5, & x=-2
\end{array}\right.\)

(i) Find f(-2)
(ii) Check whether the function f(x) is continuous at x= -2. (March – 2009)
Answer:

Question 2.
If f(x) = sin(Log x), prove that x² y₂ + xy₁ + y = 0 (May -2009)
Answer:
Given; y sin(Iogx)
Differentiating with respect to X;
\(y_{1}=\cos (\log x) \frac{1}{x} \Rightarrow x y_{1}=\cos (\log x)\)
Again differentiating with respect to x
\(\begin{array}{l}
\Rightarrow x y_{2}+y_{1}=-\sin (\log x) \frac{1}{x} \\
\Rightarrow x^{2} y_{2}+x y_{1}=-y \Rightarrow x^{2} y_{2}+x y_{1}+y=0
\end{array}\)

Question 3.
(i) Establish that g(x) =1 – x + |x| is continuous at origin.
(ii) Check whether h(x) = |l – x + |x|| is continuous at origin. (March – 2010)
Answer:
(i) Given; g(x) = 1 – x + |x| ⇒ g(x) (1 – x) + |x|
Here g(x) is the sum of two functions continuous functions hence continuous.
(ii) We have;
\(\begin{array}{l}
f o g(x)=f(g(x)) \\
=\quad f(1-x+|x|)=|1-x+| x \mid=h(x)
\end{array}\)
The composition of two continuous functions is again continuous. Therefore h(x) continuous.

Question 4.
Find \(\frac{d y}{d x}\) of the following
\(\begin{array}{l}
\text { (i) } x=\sqrt{a^{\sin ^{4} 4}} \quad y=\sqrt{a^{\cos ^{-1} t}} \\
\text { (ii) } y=\cos ^{-1} \frac{\left(1-x^{2}\right)}{\left(1+x^{2}\right)}, 0<x<1 \\
\text { (iii) } y=\sin ^{-1} 2 x \sqrt{1-x^{2}}, y_{\sqrt{2}}<x<y_{\sqrt{2}}
\end{array}\)
Answer:

Question 5.
Find \(\frac{d y}{d x} \text { if } x^{3}+2 x^{2} y+3 x y^{2}+4 y^{3}=5\) (March – 2015)
Answer:

Question 6.
Find all points of discontinuity of f where f is defined by \(f(x)=\left\{\begin{array}{ll}
2 x+3, & x \leq 2 \\
2 x-3, & x>2
\end{array}\right.\) (March – 2016)
Answer:
In both the intervals x \(\leq[latex] 2 and x > 2 the function f(x) is a polynomial so continuous. So we havetocheckthe continuity at x = 2.

Question 7.
If e^x-y = x^y, then prove that [latex]\frac{d y}{d x}=\frac{\log x}{[\log \operatorname{ex}]}\) (May – 2014; March – 2016)
Answer:

Plus Two Maths Continuity and Differentiability 4 Marks Important Questions

Question 1.
Find \(\frac{d y}{d x}\) of the following (March – 2009)
\(\begin{array}{l}
\text { (i) } y=\sin ^{-1}\left(3 x-4 x^{3}\right)+\cos ^{-1}\left(4 x^{3}-3 x\right) \\
\text { (ii) } y=\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)
\end{array}\)
Answer:

Question 2.
Consider the function f(x) = |x| x ∈ R
(i) Draw the graph of f(x) =|x|
(ii) Show that the function is continuous at x = 0. (March – 2010)
Answer:

f(0^–) f(0^–) = 0. therefore continuous at x = 0.
AIso from the figure we can see that the graph does not have a break or jump.

Question 3.
(i) Find the derivative of y = x^a + a^x with respect to x.
(ii) If e^y (x + 1) = 1 , showthat \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) (May – 2011)
Answer:

Question 4.
(i) Check the continuity of the function given by f(x) \(f(x)=\left\{\begin{array}{ll}
x \sin \frac{1}{x}, & x \neq 0 \\
1, & x=0
\end{array}\right.\)

(ii) Verify Mean Value Theorem for the function f(x) = x + 1/x in the interval [1,3]. (May – 2011)
Answer:


Hence Mean Value Theorem ¡s verified.

Question 5.
(i) Determine the value of k so that the function (May – 2012)

Answer:

Question 6.
Consider a fUnction f: R → R defined by
\(f(x)=\left\{\begin{array}{cc}
a+x, & x \leq 2 \\
b-x, & x>2
\end{array}\right.\)

(i) Find a relation between a and b if f is continuous at x = 2.
(ii) Find a and b, if f is continuous at x2 and a + b = 2. (May – 2013)
Answer:
(i) Since fis continuous at x = 2, we have;

(ii) Given a = 2 …(2) Solving (1) and (2) we have;
⇒ 2a = – 2 ⇒ a = – 1
⇒ b = 2 – a = 2 + 1 = 3

Question 7.
(i) Find if x = a(t – sin t) y = a(1 + cos t)
(ii) Verify Rolles theorem for the function f(x) = x2 + 2 in the interval [-2, 2] (March – 2014)
Answer:

Question 8.
(a) Find the relationship between a and b so that the function f defined by
\(f(x)=\left\{\begin{array}{ll}
a x^{2}-1, & x \leq 2 \\
b x+3, & x>2
\end{array}\right.\) is continuous.

(b) Verify mean value theorem for the function f(x) = x2 – 4x -3 ¡n the interval [1, 4]. (May – 2014)
Answer:
(a) Since fis continuous

Hence mean value theorem satisfies for the funcion.

Question 9.
(a) Find ‘a’ and ‘b’ if the function
\(f(x)=\left\{\begin{array}{ll}
\frac{\sin x}{x}, & -2 \leq x \leq 0 \\
a \times 2^{x}, & 0 \leq x \leq 1 \\
b+x, & 1<x \leq 2
\end{array}\right.\) is continous on [-2, 2]

(b) How many of the functions
f(x) = |x|, g(x) = |x|², h(x) = |x|³ are not differentiable at x = 0?
(i) 0
(ii) 1
(iii) 2
(iv) 3 (March – 2015)
Answer:
(a) Since f(x) is continuous on [-2, 2]

Question 10.
(a) Find the relation between ‘a’ and ‘b’ if the function f defined by
\(f(x)=\left\{\begin{array}{l}
a x+1, x \leq 3 \\
b x+3, x>3
\end{array}\right.\) is continuous.
lbx+3.x>3
(b) If e^y (x + 1) = 1, show thats \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) (May -2015)
Answer:

Question 11.
Find the value of a and b such that the function \(f(x)=\left\{\begin{array}{cc}
5 a & x \leq 0 \\
a \sin x+\cos x & 0<x<\frac{\pi}{2} \\
b-\frac{\pi}{2} & x \geq \frac{\pi}{2}
\end{array}\right.\) is continuous. (March – 2010)
Answer:

Question 12.
(i) Find \(\frac{d y}{d x}, \text { if } x=a \cos ^{2} \theta ; y=b \sin ^{2} \theta\)
(ii) Find the second derivative of the function y = e^x sinx. (May – 2017)
Answer:

Question 13.
Find \(\frac{d y}{d x}\) of the following (4 score each)
(i) y^x = x^y (May – 2015)
(ii) (COSx)^y = (cosy)^x (March – 2017)
Answer:

Plus Two Maths Continuity and Differentiability 6 Marks Important Questions

Question 1.
Find \(\frac{d y}{d x}\) if
(i) sinx + cosy = xy
(ii) x = acos³t, y = asin³t
(iii) y = x^x + (logx)^x (May -2009; May -2011; March -2015)
Answer:
(i) Given; sinx + cosy = xy
Differentiating with respect to x;

Question 2.
(i) Let y =3 cos(log x) + 4 sin (log x)
(a) Find \(\frac{d y}{d x}\)
(b) Prove that x² y₂ + xy₁ + y = 0

(ii) (a) Find the derivative of y = e^2x+logx
(b) Find \(\frac{d y}{d x}\)
if x = a (θ – sinθ), y = a(1 – cosθ) (March – 2009)
Answer:

Question 3.
(i) Show that the function f (x) defined by f(x) = sin (cosx) is a continuous function.
(ii) If \(\frac{d y}{d x}=\frac{1}{\frac{d x}{d y}}\), Show that \(\frac{d^{2} y}{d x^{2}}=\frac{-\frac{d^{2} x}{d y^{2}}}{\left(\frac{d x}{d y}\right)^{3}}\) (May -2010)
Answer:
Given; f(x) = sin(cos x)
Let g(x) = sin(x) and h(x) = cos x
Both these function are trigonometric functions hence continuous.
goh(x) = g(h(x)) = g(cos x) = sin(cos x) = f(x)

Since f(x) is the composition of two continuous functions, hence continuous.

Question 4.
(i) Let y = x^{sin x} + (sinx)^x. Find \(\frac{d y}{d x}\)
(ii) Given; \(y=\sqrt{\tan ^{-1} x}\)
(a) \(2\left(1+x^{2}\right) y \frac{d y}{d x}=1\)
(b) \(\left(1+x^{2}\right) y \frac{d^{2} y}{d x^{2}}+\left(1+x^{2}\right)\left(\frac{d y}{d x}\right)^{2}+2 x y \frac{d y}{d x}=0\) (May – 2010; Onam – 2017)
Answer:

Question 5.
(i) The function \(f(x)=\left\{\begin{array}{ll}
5, & x \leq 2 \\
a x+b, 2< & x<10 \text { is } \\
21, & x \geq 10
\end{array}\right.\) continuous. Find a and b
(ii) Find \(\frac{d y}{d x}\) (a) if y = Sin (xsinx)
(iii) If y = ae” + be’; show that \(\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0\) (March – 2011)
Answer:

Question 6.
(i) Match the following.

(ii) If y = sin^-1 x, prove that (1 – x²) y₂ – xy₁ = 0 (March – 2012; May -2017)
Answer:

Question 7.
(i) Consider \(f(x)=\left\{\begin{array}{ll}
3 x-8, & x \leq 5 \\
2 k, & x>5
\end{array}\right.\) Find the value of k if f(x) is continuous at x = 5.
(ii) Find \(\frac{d y}{d x}, \text { if } y=(\sin x)^{\log x}, \sin x>0\)
(iii) If y = (sin-1 x)2, then show that \(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=2\). (March -2013)
Answer:

Question 8.
(i) Find, if y = 1ogx, x>0
(ii) Is f(x) = |x| differentiable at x = 0?
(iii) Find if x = sin θ – cos θ and y= sinθ + cosθ (May – 2013)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 4 Determinants.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 4 Determinants

Plus Two Maths Determinants 3 Marks Important Questions

Question 1.
Prove that \(\begin{array}{|lll|}
1 ! & 2 ! & 3 ! \\
2 ! & 3 ! & 4 ! \\
3 ! & 4 ! & 5 !
\end{array}\) (March – 2015)
Answer:

Question 2.
Using properties of determinants prove the following. (March – 2010; Christmas -2017)
Answer:

Plus Two Maths Determinants 4 Marks Important Questions

Question 1.
Consider the matrix \(A=\left[\begin{array}{ll}
2 & 5 \\
3 & 2
\end{array}\right]\)

(i) Find adj (A)
(ii) Find A1
(iii) Using A^-1 solve the system of linear equations 2x + 5y = 13x + 2y = 7 (March – 2010)
Answer:

Plus Two Maths Determinants 6 Marks Important Questions

Question 1.
Consider the matrix \(A=\left[\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right]\)

(i) Using the column operat,on
C₁ → C₁ + C₂ + C₃,
show that \(|A|=(a+b+c)\left|\begin{array}{ccc}
1 & b & c \\
1 & c & a \\
1 & a & b
\end{array}\right|\)
(ii) Show that |A| = – (a3 + b3 + e3 — 3abc)
(iii) Find A x adj(A) if a = 1,b = 10,c = 100 (May – 2010)
Answer:

Question 2.
(i) (a) If \(A=\left[\begin{array}{ccc}
1 & 1 & 5 \\
0 & 1 & 3 \\
0 & -1 & -2
\end{array}\right]\)

What is the value of |3A|?
(b) Find the equation of the line joining the points (1,2) and (-3,-2) using determinants.
(ii) Show that \(\left|\begin{array}{lll}
1 & a & a^{2} \\
1 & b & b^{2} \\
1 & c & c^{2}
\end{array}\right|=(a-b)(b-c)(c-a)\)
Answer:

(b) Let (x,y) be the coordinate of any point on The line, then (1,2), (-3, -2) and (x, y) are collinear.

Hence the area formed will be zero.

Question 3.
Consider the following system of linear equations; x + y + z = 6, x – y + z = 2, 2x + y + z = 1
(i) Express this system of equations in the Standard form AXB
(ii) Prove that A is non-singular.
(iii) Find the value of x, y and z satisfying the above equation. (May – 2011)
Answer:

Question 4.
(i) lf \(\left|\begin{array}{ll}
x & 3 \\
5 & 2
\end{array}\right|=5\), then x = ………..
(ii) Prove that
\(\left|\begin{array}{ccc}
y+k & y & y \\
y & y+k & y \\
y & y & y+k
\end{array}\right|=k^{2}(3 y+k)\)
(iii) Solve the following system of linear Equations, using matrix method; 5x + 2y = 3, 3x + 2y = 5 (March – 2012)
Answer:

Question 5.
(i) Let B is a square matrix of order 5, then |kB| is equal to ………..
(a) |B|
(b) k|B|
(c) k⁵|B|
(d) 5|B|

(ii) Prove that \(\left|\begin{array}{lll}
1 & x & x^{2} \\
1 & y & y^{2} \\
1 & z & z^{2}
\end{array}\right|=(x-y)(y-z)(z-x)\)
(iii) Check the consistency of the following equations; 2x + 3y + z = 6, x + 2y – z = 2, 7x + y + 2z =10 (May – 2012)
Answer:

Therefore the system is consistent and has unique solutions.

Question 6.
(i) Find the values of x in which \(\left|\begin{array}{ll}
3 & x \\
x & 1
\end{array}\right|=\left|\begin{array}{ll}
3 & 2 \\
4 & 1
\end{array}\right|\)

(ii) Using the property of determinants, show that the points A(a,b + c), B(b,c + a), C(c,a + b) are collinear.
(iii) Examine the consistency of system of following equations: 5x – 6y + 4z = 15, 7x + y – 3z = 19, 2x + y + 6z = 46 (EDUMATE – 2017; March – 2013)
Answer:

Since, the system is consistent and has unique solutions.

Question 7.
Consider a system of linear equations which is given below;
\(\begin{array}{l}
\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4 ; \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 \\
\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2
\end{array}\)

(i) Express the above equation in the matrix form AX = B.
(ii) Find A^-1, the inverse of A.
(iii) Find x,y and z. (May – 2013)
Answer:

Question 8.
Consider the matrices \(A=\left[\begin{array}{ll}
2 & 3 \\
4 & 5
\end{array}\right]\)

(i) Find A² – 7A – 21 = 0
(ii) Hence find A-1
(iii) Solve the following system of equations using matrix method 2x + 3y = 4; 4x + 5y = 6 (March – 2014)
Answer:

(iii) The given system of equations can be converted into matrix form AX = B

Question 9.
(i) Let A be a square matrix of order 2 x 2 then |KA| is equal to
(a) K|A|
(b) K²|A|
(c) K³|A|
(d) 2K|A|

(ii) Prove that
\(\left|\begin{array}{ccc}
\mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & 2 \mathbf{a} \\
2 \mathrm{~b} & \mathrm{~b}-\mathrm{c}-\mathrm{a} & 2 \mathrm{~b} \\
2 \mathrm{c} & 2 \mathrm{c} & \mathrm{c}-\mathrm{a}-\mathrm{b}
\end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})^{3}\)

(iii) Examine the consistency of the system of Equations. 5x + 3y = 5; 2x + 6y = 8 (May- 2014)
Answer:

(iii) The given system of equation can be written in matrix form as

solution exist and hence it is consistent.

Question 10.
(a) Choose the correct statement related to the matnces \(A=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right], B=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)
\(\begin{array}{l}
\text { (i) } A^{3}=A, B^{3} \neq B \\
\text { (ii) } A^{3} \neq A, B^{3}=B \\
\text { (iii) } A^{3}=A, B^{3}=B \\
\text { (iv) } A^{3} \neq A, B^{3} \neq B
\end{array}\)

(b) lf \(M=\left[\begin{array}{ll}
7 & 5 \\
2 & 3
\end{array}\right]\) then verity the equation M² – 10M + 11 I₂ = O

(c) Inverse of the matrix \(\left[\begin{array}{lll}
0 & 1 & 2 \\
0 & 1 & 1 \\
1 & 0 & 2
\end{array}\right]\) (March – 2015)
Answer:

Question 11.
Solve the system of Linear equations x + 2y + z = 8; 2x + y – z = 1; x – y + z = 2 (March – 2015)
Answer:

Question 12.
(a) If \(\left|\begin{array}{ll}
x & 1 \\
1 & x
\end{array}\right|=15\) then find the value of X.

(b)Solve the following system of equations 3x – 2y + 3z = ?, 2x + y – z = 1 4x – 3y + 2z = 4 (May – 2015)
Answer:

Question 13.
(i)The value of the determinant \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & -1 \\
1 & 1 & -1
\end{array}\right|\) is
(a) -4
(b) 0
(c) 1
(d) 4

(ii) Using matrix method, solve the system of linear equations x + y + 2z = 4; 2x – y + 3z = 9; 3x – y – z = 2 (May – 2016)
Answer:
(i) (d) 4
(ii) Express the given equation in the matrix form as AX = B

Question 14.
(i) If \(A=\left[\begin{array}{ll}
a & 1 \\
1 & 0
\end{array}\right]\) is such that A² = I then a equals
(a) 1
(b) -1
(c) 0
(d) 2

(ii)Solve the system of equations x – y + z = 4; 2x + y – 3z = 0; x + y + z = 2 Using matrix method. (March – 2017)
Answer:

Question 15.
(i) IfA is a 2 x 2 matrix with |A| = 5, then |adjA| is
(a) 5
(b) 25
(c) 1/5
(d) 1/25

(ii) Solve the system of equations using matrix method.
x + y + z = 1; 2x + 3y – z = 6; x – y + z = -1 (May – 2017)
Answer:
(i) (a) 5
(ii) LetA X=B,

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 3 Matrices.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 3 Matrices

Plus Two Maths Matrices 3 Marks Important Questions

Question 1.
Write A as the sum of a symmetric and a skew-symmetric matrix. \(A=\left[\begin{array}{ccc}
1 & 4 & -1 \\
2 & 5 & 4 \\
-1 & -6 & 3
\end{array}\right]\) (March – 2010)
Answer:

Question 2.
Consider the matrices
\(A=\left[\begin{array}{lll}
2 & 1 & 3 \\
2 & 3 & 1 \\
1 & 1 & 1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
-1 & 2 & 3 \\
-2 & 3 & 1 \\
-1 & 1 & 1
\end{array}\right]\)
(i) Find A+B
(ii) Find (A + B) (A-B) (May -2010)
Answer:

Question 3.
Given \(P=\left[\begin{array}{cc}
2 & -3 \\
-1 & 2
\end{array}\right]\) Find the inverse of P by elementary row operation. (March 2011)
Answer:

Question 4.
Let \(A=\left[\begin{array}{lll}
3 & 6 & 5 \\
6 & 7 & 8
\end{array}\right] \text { and } C=\left[\begin{array}{ccc}
1 & 2 & -3 \\
4 & 5 & 6
\end{array}\right]\)

(i) Find 2A
(ii) Find the matrix B such that 2A + B = 3C (May 2011)
Answer:

Question 5.
Let \(A=\left[\begin{array}{cc}
2 & 4 \\
-1 & 1
\end{array}\right]\)
(i) Apply elementary transformation R → R R1/2 in the matrix A.
(ii) Find the inverse of A by the elementary transformation. (May 2011)
Answer:

Question 6.
Consider the matrix \(A=\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\)
(i) Find A²
(ii) Find ksothat A2 = kA – 7I (March – 2012)
Answer:

Question 7.
Consider a 2×2 matrix
\(A=\left[a_{i j}\right]$ where $a_{i j}=|2 i-3 j|\)
(i) Write A
(ii) Find A + A^T (March – 2012)
Answer:

Question 8.
If \(A=\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\) then
(i) Find A²
(ii) Hence show that A² – 5A + 7I = 0. (March 2013)
Answer:

Question 9.
If a matrix \(A=\left[\begin{array}{ll}3 x & x \\ -x & 2 x\end{array}\right]\) is a solution of the equation x² – 5x + 7 = 0, find any one value of X. (May 2013)
Answer:

Question 10.
Consider the matrices \(A=\left[\begin{array}{cc}1 & -2 \\ -1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$\) \(A B=\left[\begin{array}{ll}2 & 9 \\ 5 & 6\end{array}\right]\), find the values of a,b,c,d (March – 2014)
Answer:

Question 11.
Consider a 2 x 2 matrix A=[a_ij] Where \(a_{i j}=\frac{(i+2 j)^{2}}{2}\)
(i) Write A
(ii) Find A + A^T (March – 2014)
Answer:

Question 12.
If X + Y = \(\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]\) and X – Y = \(\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]\) then
(i) Find X and Y.
(ii) Find 2X + Y. (May – 2014)
Answer:

Question 13.
i) If A, B are symmetric matrices of same order then AB – BA is always a ………….
A) Skew-Symmetric matrix
B) Symmetric matrix
C) Identity matrix
D) Zero matrix
(ii) For the matrix \(A=\left[\begin{array}{ll}2 & 4 \\ 5 & 6\end{array}\right]\), verify that A + A^T is a symmetric matrix. (March – 2015)
Answer:

Question 14.
Consider the matrix \(A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\)
(i) Find A²
(ii) Find k so that A² = kA – 21 (May – 2015)
Answer:

Plus Two Maths Matrices 4 Marks Important Questions

Question 1.
(i) Find the value of x and y from the equations \(a\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{cc}3 & -4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 15 & 14\end{array}\right]\)
(ii) Given \(A=\left[\begin{array}{cc}1 & 2 \\ 3 & -1 \\ 4 & 2\end{array}\right], B=\left[\begin{array}{ccc}-1 & 4 & -5 \\ 2 & 1 & 0\end{array}\right]\) Show that AB ≠ BA (March – 2011)
Answer:

Question 2.
(i) Find a, b matrix \(\left[\begin{array}{ccc}0 & 3 & a \\ b & 0 & -2 \\ 5 & 2 & 0\end{array}\right]\) is skew symmetric matrix.
(ii) Express \(A=\left[\begin{array}{ccc}7 & 3 & -5 \\ 0 & 1 & 5 \\ -2 & 7 & 3\end{array}\right]\) sum of a symmetric and a skew symmetric matrix. (May – 2012)
Answer:

Question 3.
Consider the matrices \(A=\left[\begin{array}{cc}2 & -6 \\ 1 & 2\end{array}\right]$ and $A+3 B=\left[\begin{array}{cc}5 & -3 \\ -2 & -1\end{array}\right]\)
(i) Find matrix B
(il) Find matrix AB.
(iii) Find the transpose of B. (May – 2013)
Answer:

Question 4.
(i) The value of k such that matrix \(\left[\begin{array}{cc} 1 & k \\ -k & 1 \end{array}\right]\) is symmetric if
(a) 0
(b) 1
(c) – 1
(d) 2

(ii) If \(A=\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\) then prove that \(A^{2}=\left[\begin{array}{cc}
\cos 2 \theta & \sin 2 \theta \\
-\sin 2 \theta & \cos 2 \theta
\end{array}\right]\)
\(\text { (iii) if } A=\left[\begin{array}{ll}
1 & 3 \\
4 & 1
\end{array}\right], \text { then find }\left|3 A^{T}\right|\) (March – 2017)
Answer:

Plus Two Maths Matrices 6 Marks Important Questions

Question 1.
Let A be a matrix of order 3 x 3 whose elements are given by a_ij = 21 – j
(i) Obtain the matrix A.
(ii) Find AT Also express A as the sum of symmetric and skew-symmetric matrix. (March – 2010)
Answer:

Question 2.
Consider a 2 x 2 matrix \(A=\left[a_{\theta}\right]\) with a_ij = 2ⁱ + j
(i) Construct A.
(ii) Find A + A^T, A – A^T
(iii) Express A as sum of a symmetric and skew-symmetric matrix. (May -2015)
Answer:

Question 3.
(i) \(A=\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right], B=\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right]\) then BA = _____
\(\begin{array}{l}
\text { (a) }\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] & \text { (b) }\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right] \\
\text { (c) }\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right] & \text { (d) }\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]
\end{array}\)

(ii) Write \(A=\left[\begin{array}{cc}
3 & 5 \\
1 & -1
\end{array}\right]\) as the sum of a symmetric and a skew symmetric matrix.
(iii) Find the inverse of \(A=\left[\begin{array}{ll}
2 & -6 \\
1 & -2
\end{array}\right]\) (March 2016)
Answer:

Question 4.
(i) If the matrix A is both symmetric and skew-symmetric, then A is a
(a) diagonal matrix
(b) zero matrix
(c) square matrix
(d) scalar matrix

(ii) If \(A=\left[\begin{array}{cc}
1 & 3 \\
-2 & 4
\end{array}\right]\), then show that

(iii) Hence find A^-1 (May 2016)
Answer:

Question 5.
(i) The number of all possible 2 x 2 matrices with entries O or 1 is
(a) 8
(b) 9
(c) 16
(d) 25

(ii) If the area of a triangle whose vertices are (k,0), (5,0), (0,1) is 10 square units the find k.
(iii) Using elementary transformations find the inverse of the matrix \(\left[\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right]\) (May 2017)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 2 Inverse Trigonometric Functions.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 2 Inverse Trigonometric Functions

Plus Two Maths Inverse Trigonometric Functions 3 Marks Important Questions

Question 1.
(i) Find the principal value of \(\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
(ii) Prove that \(2 \sin ^{-1}\left(\frac{3}{5}\right)=\tan ^{-1}\left(\frac{24}{7}\right)\) (March – 2010)
Answer:

Question 2.
Prove \(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}\) (March – 2015; March – 2016)
Answer:

Plus Two Maths Inverse Trigonometric Functions 4 Marks Important Questions

Question 1.
(i) Find the principal value of \(\cos ^{-1}\left(-\frac{1}{2}\right)\)
(ii) Show that \(\left(\frac{\cos x}{1-\sin x}\right)=\frac{\pi}{4}+\frac{x}{2}\) (March-2011)
Answer:

Question 2.
(i) The principal value of \(\cos ^{-1}\left(-\frac{1}{2}\right)\)
(ii)Expresstan \(\tan ^{-1}\left(\frac{\cos x}{1-\sin x}\right)\) in the simplest Form. (May – 2012)
Answer:

Question 3.
(i) Write the principal value of \(\sin ^{-1}\left(\frac{1}{2}\right)\)
(ii) Show that \(\sin ^{-1}\left(\frac{3}{5}\right)-\sin ^{-1}\left(\frac{8}{17}\right)=\cos ^{-1}\left(\frac{84}{85}\right)\) (March – 2013, Onam – 2017)
Answer:

Question 4.
(a) The principal value of tan’1 (-1) is \(\left[\frac{\pi}{4},-\frac{\pi}{4}, \quad \pi-\frac{\pi}{4}, \quad \pi+\frac{\pi}{4}\right]\)
(b) If \(\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\) then find the value of x. (May 2014)
Answer:

Plus Two Maths Inverse Trigonometric Functions 6 Marks Important Questions

Question 1.
Match the following. (1 + 1 + 1 + 3 = 6) (May 2010)

Answer:

Question 2.
(i) Give an expression for tan(x + y)
(ii) Prove that xy < 1, \(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)\)
(iii) Using the above result prove that \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=\frac{\pi}{4}\) (May 2011)
Answer:

Question 3.
(i) Show that \(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{8}=\frac{\pi}{4}\)
(ii) Given that \(\cot 3 \theta=\frac{3 \cot ^{2} \theta-1}{\cot ^{3} \theta-3 \cot \theta}\), Show that \(\cot ^{-1} \frac{3 x^{2}-1}{x^{3}-3 x}, \quad|x|<\sqrt{3}$ is $3 \cot ^{1} x\) (May 2013 )
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 1 Relations and Functions.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 1 Relations and Functions

Plus Two Maths Relations and Functions 3 Marks Important Questions

Question 1.
Consider the set A = {1, 2, 3, 4, 5}, and B = {1, 4, 9, 16, 25} and a function ƒ : A → B defined by f(1) = 1, f(2) = 4, f(3) = 9, f(4) = 16 and f(5) = 25
(i) Show that f is one-to-one
(ii) Show that f is onto.
(iii) Does ƒ^-1 exists? Explain (May – 2013)
Answer:
(i) ƒ = {(1,1), (2,4), (3,9), (4,16), (5,25)}
Every element in A is mapped to different elements in B. Therefore one-to-one.
(ii) R (ƒ) = {1, 4, 9, 16, 25} = B. Therefore onto.
(iii) Since f is one-to-one and onto function, ƒ^-1 exists.
ƒ^-1 = {(1,1), (4,2), (9,3), (16,4), (25,5)}

Question 2.
a) When a relation R on a set A is said to be reflexive
b) Show that ƒ : [-1, 1] → R given by \(f(x)=\frac{x}{x+2}\) is one-one (May – 2015)
Answer:

Question 3.
a) The function ƒ :N → N given by ƒ(x) = 2x
i) one-one and onto
ii) one-one and not onto
iii) not one-one and not onto
iv) onto but not one-one
b) Find goƒ(x), if ƒ(x) = 8x³ and g(x) = x^1/2
c) Let * be an operation such that a*b= LCM of a and b defined on the set A = {1,2,3,4,5}. Is * a binary operation? Justify your answer. (March-2016)
Answer:
a) ii) one-one and not onto
b) Answered in previous years questions No. 2(ii) (6 Mark question)
c) LCM of 2 and 3 is 6 ∉ A, therefore not a binary operation.

Plus Two Maths Relations and Functions 4 Marks Important Questions

Question 1.
(i) ƒ : {1,2,3,4} → {5} defined by ƒ = {(1,5), (2,5), (3,5), (4,5)} Does the function is invertible?
(ii)
(iii) Let A = Nx N, N-set of natural numbers and * 1be a binary operation on A defined by (a,b) * (c,d) = (ac—bd,ud +bc). Show that* is commutative on A. (March -2011)
Answer:
(i) Inverse does not exists because fis not one-one.

Hence cummutative.

Question 2.
Let N be the set of Natural numbers. Consider the function ƒ: N → N defined by ƒ(x) = x + l, x ∈ N
(i) Prove that f is not onto
(ii) \(If g(x)=\left\{\begin{array}{ll}x-1, & x>1 \\ 1, & x=1\end{array}\right. then find g o f\)
(iii) Check whether goƒ is an onto function. (May 2011)
Answer:

(iii) Since f is not onto goƒ is also not onto.

Question 3.
(i) Give a relation on a set A = {1,2,3,4} which is reflexive , symmetric and not transitive.
(ii) Show that ƒ : [-1,1] → R given by \(f(x)=\frac{x}{x+2}\) is one-one.
(iii) Let ‘*’ be a binary operation on Q⁺ defined by a*b = \(a * b=\frac{a b}{6}\) ’.Find the inverse of 9 with respect to ’ * ’. (March -2012)
Answer:
(i) Given A = {1,2,3,4}
R = {(1,1)(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1)}

Question 4.
(i) *:R x R → Ris given by a * b = 3a2 – b
Find the value of 2 * 3. Is ‘*’ commutative? Justify your answer.
(ii) ƒ :R → R is defined by ƒ(x) = x² – 3x + 2 Find ƒoƒ (x) and ƒoƒ. (May 2012)
Answer:

Question 5.
(i) Consider ƒ : R → R given by ƒ(x) = 5x + 2
(a) Show that f is one-one.
(b) Is f invertible? Justify your answer.

(ii) Let * be a binary operation on N defined by a * b = HCF of a and b
(a) Is * commutative?
(b) Is * associative? (March-2013)
Answer:
(i) (a) Let x₁, x₂, ∈ R
ƒ(x₁) = ƒ(x₂) ⇒ 5x₁ + 2 = 5x₂ + 2
⇒ 5x₂ = 5x₂ ⇒ x₁ = x₂
Therefore fis one-one.

(b) Yes.
Let y e range of ƒ
⇒ ƒ(x) = y ⇒ 5x + 2 = y
\(\Rightarrow x=\frac{y-2}{5} \in R\)
Therefore corresponding to every y ∈ R there existsa real number \(\frac{y-2}{5}\) Therefore f is onto.
Hence bijective, so invertible.

(ii) (a) Yes.
a * b = HCF (a,b) = HCF (b,a) = b * a
Hence commutative.

(b) Yes.
a * (b * c) = a* HF(b,c) = HCF(a,b,c)
(a*b) * c =HCF(a,b) * c HCF(a,b,c)
a * (b * c) = (a * b) * c
Hence associative.

Question 6.
(a) Let f: R → R be given by ƒ (x) = \(\frac{2 x+1}{3}\) find ƒoƒ and show that f is invertible.
(b) Find the identity element of the binary operation * on N defined by a * b = ab². (May 2014)
Answer:

Therefore f is onto.
Hence f is bijective and invertible.

(b) let ‘e’ be the identity element, then

Since e is not unique, this operation has no identity element.

Question 7.
a) What is the minimum number of pairs to form a non-zero reflexive relation on a set of n elements?
b) On the set R of real numbers, S is a relation defined as S = {(x,y)/X∈R, y ∈ R, x + y = xy}. Find a ∈ R such that ‘a’ is never the first element of an ordered pair in S. Also find b ∈ R such that ‘b’ is never the second element of an ordered pair in S.
c) Consider the function \(f(x)=\frac{3 x+4}{x-2}, x \neq 2\) Find a function on a suitable domain such that goƒ(x) = x = ƒog(x). (March 2015)
Answer:

Question 8.
(i) If ƒ: R → R and g: R → R defined by ƒ(x) = x² and g(x) = x + 1, then goƒ (x) is
(a) (x + 1)²
(b) x³ + l
(c) x² + l
(d) x + l

(ii) Consider the function ƒ: N → N, given by ƒ(x) = x³. Show that the function ‘ƒ’ is injective but not surjective.
(iii) The given table shows an operation on A = {p,q}

(a) Is * a binary operation?
(b) * commutative? Give reason. (May 2016)
Answer:
(i) (C) x² + 1
(ii) ƒ : N → N , given by ƒ(x) = x³
for x,y ∈ N ⇒ ƒ(x) = ƒ(y)
x³ = y³ ⇒ x = y

There fore f is injective.
Now 2 ∈ N, but there does not exists any element x in domain N such that ƒ(x) = x³ = 2 their fore f is not surjective.

(iii) (a) Yes
(b) No, because p*q = q; q*p = p
⇒ p*q ≠ q*p

Question 9.
(i) Let R be a relation defined on A{1,2,3} by R = {(13),(3,1),(2,2)} is
(a) Reflexive
(b) Symmetric
(C) Transitive
(d) Reflexive but not transitive.
(ii) Find fog and gof if ƒ(x) = |x+1| and g(x) = 2x – 1
(iii) Let * be a binary operation defined on N x N by (a,b) * (c,d.) = (a + c, b + d)
Find the identity element for * if it exists. (March – 2017)
Answer:
(i) (b) Symmetric

(ii) ƒog(x) = |g(x) + 1| = |2x – 1 + 1| = |2x|
goƒ(x) = 2 ƒ(x) – 1 = 2 |x + 1| – 1

(iii) Let e =(e₁, e₂) be the identity element of the operation in ? N x N then, (a,b)*(e₁, e₂) = (a + e₁, b + e₂) ≠ (a,b) Since, e₁ ≠ 0, e₂ ≠ 0

Therefore identity element does not existš.

Question 10.
(i) If R = {(x,y) : x, y ∈ Z, x – y ∈ Z}, then the relation R is
(a) Reflexive but not transitive
(b) Reflexive but not symmetric
(C) Symmetric but not transitive
(d) An equivalence relation.

(ii) Let * be a binary operation on the set Q of rational numbers by a*b = 2a + b. Find 2 * (3 * 4) and (2 * 3) * 4.
(iii) Let ƒ : R → R, g : R → R be two one-one funçtions. Check whether gof is one-one or not. (May- 2017)
Answer:
(i) (d) An equivalance relation.
(ii) 2* (3 * 4) = 2 * 10 = 14
(2 * 3)* 4 = 7 * 4 = 18
(iii) ƒ : R → R, g : R → R
Let x₁, x₂, ∈ R
goƒ(x₁) = g(ƒ(x₁)) = g(ƒ(x₂)) = g(ƒ(x₂))
⇒ x₁ = x₂

Plus Two Maths Relations and Functions 6 Marks Important Questions

Question 1.
(i) (a) A function ƒ : X → Y is onto if range of ƒ = ………….
(b) Let ƒ : {1, 3, 4} {3, 4, 5} and
g: {3, 4, 5} → {6, 8, 10} be functions defined by
ƒ (1) = 3, ƒ (3) = 4, ƒ (4) = 5;
g (3) = 6, g(4) = 8, g(5) = 8 ,then (goƒ) (3) = …………..

(ii) Let Q be the set of Rational numbers and ‘*’ be the binary operation on Q defined by \(a * b=\frac{a b}{4}\) for all a,b in Q
(a) What is the identity element of ‘ * ’on Q?
(b) Find the inverse element of * ’ on Q.
(c) Show that a * (b * c) = (a * b) * c, ∀a,b,c ∈ Q.
Answer:

Question 2.
(i) Let R be the relation on the set N of natural numbers given by
R = {(a,b): a – b > 2, b>3}
Choose the correct answer
(a) (4, 1) ∈ R
(b) (5, 8) ∈ R
(c) (8, 7) ∈ R
(d) (10, 6) ∈ R

(ii) If ƒ(x) = 8x³ and g(x) = x^1/3, findg(ƒ(x)) and ƒ(g(x))
(iii) Let * be a binary operation on the set Q of rational numbers defined by a*b = \(\frac{a}{b}\). Check whether * is commutative and associative? (March – 2014, May – 2015, March – 2016)
Answre:

Question 3.
Let \(f(x)=\frac{x-1}{x-3}, x \neq 3\) and \(g(x)=\frac{x-3}{x-1}, x \neq 1\) be two functions defined on R.

(i) Find ƒog(x), x ≠ 0
(ii) Find ƒ^-1 (x) and g^-1 (x), x ≠ 1
(iii) Find (goƒ)^-1 (x) (May-2010)
Answer:

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper Say 2018 with Answers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

Question 1 to 7 carry 3 scores each. Answer any 7 questions.

Question 1.
a) Construct a 2 × 2 matrix whose elements are given a_ij = 2i + j
b) Find A².
Answer:

Question 2.
a) If \(\int \frac{f(x)}{x^{2}+1} d x\) = log | x² + 1 | + C, then f(x) = …………
b) Find ∫ xe^x dx
Answer:
a) f(x) = 2x
b) ∫ xe^x dx = x∫e^x dx – ∫1 × e^x dx
= xe^x – e^x + c = e^x (x – 1) + c

Question 3.
Form the differential equation of the family of all circles touching the y-axis at origin.
Answer:
Equation of the family of circle which touches the y-axis at origin is of the form.
(x – a)² + y² = a² ……… (1)

Question 4.
Consider the relation in the set N of Natural numbers defined as R = { (a,b): ab is a factor of 6}. Determine whether the relation is reflexive, symmetric or transitive.
Answer:
(2, 2) ∉ R, Not reflexive
(x, y) ∈ R ⇒ (y, x) ∈ R ⇒ xy = yx, Symmetric
(3, 2) ∈ R, (2, 3) ∈ R ⇒ (3, 3) ∉ R,
Since 3 × 3 = 9, not transitive.

Question 5.
Find the area bounded by the curve y= cos x and x axis between x = 0 and x = π.
Answer:
Area = 2 \(\int_{0}^{\pi / 2}\) cos xdx
= 2 \([\sin x]_{0}^{\pi / 2}\) = 2[1 + 0] = 2

Question 6.

A rectangular plot is to be fenced using a rope of length 20 meters with one of its sides is a wall as shown in the figure. Find the maximum area of such rectangle.

Length = 20 – 2x, breadth = x
A = x(20 – 2x) = 20x – 2x²
A'(x) = 20 – 4x ⇒ A'(x) = 0 ⇒ x = 5
A”(x) = -4 < 0
Hence A is maximum at x = 5
Maximum area = 5 × 10 = 50

Question 7.
A manufacture produces nuts and bolts. The time required to produce one packet of nuts and one packet of bolts on machines A and B is given in the following table

He earns a profit of Rs. 25 per packet of nuts and Rs. 12 per packet of bolts. He operates his machines for almost 15 hours a day. Formulate a linear programming problem to maximise his profit.
Answer:
Maximise: Z = 25x + 12y
Subject to
2x + 3y ≤ l5; 3x + y ≤ 5; x, y ≥ 0

Questions 8 to 17 carry 4 scores each. Answer any 8.

Question 8.
Consider the curve y = x³ + 8x + 3
a) Find the point on the curve at which the slope of the tangent is 20.
b) Does there exist a tangent to the curve with negative slope? Justify your answer.
Answer:
a) \(\frac{d y}{d x}\) = 3x² + 8
Slope is given as 20
20 = 3x² + 8 ⇒ x = ±2
Therefore points (2, 27), (-2, -21)
b) No. 3x² + 8 ≥ 0 (Always positive for any value of x.)

Question 9.
a) Which of the following functions is not continuous at zero?
i) f(x) = sin x

b) Find the Values of a and b such that the function defined by
f(x) = \(\left\{\begin{array}{cc}
10, & x \leq 3 \\
a x+b, & 3<x<4 \\
20, & x \geq 4
\end{array}\right.\)
[Here f(x) is oscillating between -1 and 1 as x approaches to 0. In other cases the limit value and function value are same. So continuous.]
b) \(\lim _{x \rightarrow 3^{+}}\)f(x) = 3a + b ⇒ 3a + b = 10
\(\lim _{x \rightarrow 4^{-}}\) f(x) = 4a + b ⇒ 4a + b = 20
Solving both equations we get a = 10, b = -20

Question 10.
Consider the plane 2x – 3y + z = 5
a) Find the equation of the plane passing through the point (1, 1, 3) and parallel to the above plane.
b) Find the distance between above planes
Answer:
a) Equation of a plane parallel to the
2x – 3y + z = 5 is of the form 2x – 3y + z = k .
Since it passes through the point (1, 1, 3),
we have 2 – 3 + 3 = k ⇒ k = 2
⇒ 2x – 3y + z = 2

b) Distance between the planes
= \(\left|\frac{5-2}{\sqrt{4+9+1}}\right|=\frac{3}{\sqrt{14}}\)

Question 11.
Consider the vectors
\(\vec{a}\) = 2i + j + 3k; \(\vec{b}\) = i + 4j – k
a) Find the projection of \(\vec{a}\) on \(\vec{b}\)
b) If \(\vec{a}\) is perpendicular to a vector \(\vec{c}\) then projection of \(\vec{a}\) on \(\vec{c}\)
c) Write a vector \(\vec{d}\) such that the projection of \(\vec{a}\) on \(\vec{d}\) = |\(\vec{a}\)|
Answer:

b) Projection will be zero.
c) Projection of \(\vec{a}\) on \(\vec{d}\) = |\(\vec{a}\)|, means the angle between \(\vec{a}\) and \(\vec{d}\) is zero. Hence both are parallel. So any vector parallel to \(\vec{a}\) is \(\vec{d}\).

Question 12.
a)

In the figure ABCD is a Parallelogram. If
\(\overrightarrow{A B}\) = 3i – j + 2k; \(\overrightarrow{A D}\) = i + j + 2k, find \(\overrightarrow{A C}\) and \(\overrightarrow{D B}\)
b) If \(\vec{a}\) and \(\vec{b}\) are adjacent sides of any parallelogram \(\vec{c}\) and \(\vec{d}\) are diagonals,
then show that |\(\vec{c}\) × \(\vec{d}\)| = 2|\(\vec{a}\) × \(\vec{b}\)|
Answer:
a) \(\overrightarrow{A C}\) = \(\overrightarrow{A B}\) + \(\overrightarrow{A D}\) = 4i + 4j
\(\overrightarrow{B D}\) = \(\overrightarrow{A B}\) – \(\overrightarrow{A D}\) = 2i – 2j

b) Let \(\vec{c}\) = \(\vec{a}\) + \(\vec{b}\) and d = \(\vec{a}\) – \(\vec{b}\)
\(\vec{c}\) × \(\vec{d}\) = (\(\vec{a}\) + \(\vec{b}\)) × (\(\vec{a}\) – \(\vec{b}\))
= \(\vec{a}\) × \(\vec{a}\) – \(\vec{a}\) × \(\vec{b}\) + \(\vec{b}\) × \(\vec{a}\) – \(\vec{b}\) × \(\vec{b}\)
= 0 – \(\vec{a}\) × \(\vec{b}\) – \(\vec{a}\) × \(\vec{b}\) – 0 = -2(\(\vec{a}\) × \(\vec{b}\))

Question 13.
Find ∫(4x + 7)\(\sqrt{x^{2}+4 x+13}\)dx
Answer:
4x + 7 = A(2x + 4) + B ⇒ A = 2, B = -1
I = 2∫\(\sqrt{x^{2}+4 x+13}\)dx – ∫\(\sqrt{x^{2}+4 x+13}\)dx
I = 2I₁ – I₂

Question 14.
a) Write integrating factor of the linear differential equation \(\frac{d y}{d x}+\frac{y}{x}\) = sin x
b) Slope of the tangent to a curve at any point is twice the x coordinate of the point. If the curve passes through the point (1, 4), find its equation.
Answer:
IF = \(e^{\int P d x}\) = \(e^{\log x}\) = x
b) \(\frac{d y}{d x}\) = 2x ⇒ dy = 2x dx
Integrating we get;
∫dy = ∫ 2x dx + C ⇒ y = x² + C
Since passes through (1, 4) we have;
4 = 1 + C ⇒ C = 3
Hence equation is y = x² +3

Question 15.
Solve the linear programming problem graphically Maximise Z = 3x + 5y
Subject to the constraints
x + 3y ≤ 3
x + y ≤ 2
x, y ≥ 0

Maximum 7 at B = \(\left(\frac{3}{2}, \frac{1}{2}\right)\)

Question 16.
a) If cos^-1 \(\frac{12}{13}\) = tan^-1 x then find x.
b) Show that cos^-1 \(\frac{4}{5}\) + cos^-1 \(\frac{12}{13}\) = tan^-1 \(\frac{14}{33}\)
Answer:

Question 17.
Consider the binary operation * on the set R of real numbers, defined by a*b = \(\frac{a b}{4}\)
a) Show that * is commutative and associative.
b) Find the identity element for * o R.
c) Find the inverse of 5.
Answer:

Questions from 18 to 24 carry 6 scores each. Answer any 5.

Question 18.
Consider the matrix A = \(\left[\begin{array}{lll}
1 & 0 & 2 \\
0 & 1 & 2 \\
0 & 4 & 9
\end{array}\right]\)
a) Find A^-1 using elementary row operations.
b) Find the solution of the system of equations given below:
(A^-1 obtained above may be used)
x + 2z + 2; y + 2z + 1; 4y + 9z = 3
Answer:
a) A = IA
\(\left[\begin{array}{lll}
1 & 0 & 2 \\
0 & 1 & 2 \\
0 & 4 & 9
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] A\)

Question 19.
a) Show that
\(\left|\begin{array}{lll}
1 & a & b c \\
1 & b & a c \\
1 & c & a b
\end{array}\right|\) = (a – b)(b – c)(c – a)
b) If A = \(\left[\begin{array}{cc}
2 & 3 \\
4 & -1
\end{array}\right]\) verift that A × adj A = |A|I
Answer:

Question 20.
a) If f is a function such that f(-x) = f(x), then \(\int_{-a}^{a}\) f(x) dx = ………
b) Evaluate \(\int_{-\pi / 2}^{\pi / 2}\) cos x dx
c) Evaluate \(\int_{0}^{1}\) (x² + 1)dx as the limit of a sum.
Answer:

Question 21.
a) Verify mean value theorem for the function f(x) = x² – 4x – 3 in the interval [1, 4].
b) Consider the function
f(x) = sin^-1 2x \(\sqrt{1-x^{2}}\), \(\frac{-1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}\)
i) Show that f(x) = 2sin^-1 x
ii) Find f'(x)
Answer:
a) f(x) is a continuous function in [1, 4], since it is a polynomial.
f'(x) = 2x – 4, f is differentiable in (1, 4).
f(4) = 16 – 16 – 3 = -3, f(1) = 1 – 4 – 3 = -6

Hence mean value theorem is verified.

b) i) Put x = sinθ
f(x) = sin^-1 (2 sin θ\(\sqrt{1-\sin ^{2} \theta}\))
= sin^-1 (2 sin θ cos θ)
= sin^-1 (sin 2θ) = 2θ = 2 sin^-1 x
ii) f'(x) = \(\frac{2}{\sqrt{1-x^{2}}}\)

Question 22.
a) Show that the lines
\(\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-3}{1} ; \frac{x-3}{2}=\frac{y-1}{1}=\frac{z-4}{2}\) are coplanar.
b) Find the equation of the plane that contains the above lines.
c) Show that the above lines intersect at the point (3, 1, 4).
Answer:
a) Points (2, -1, 3) and (3, 1, 4)

Normal direction ratios are 3, 0, -3.
Therefore equation of the plane is
3(x – 2) -3 (z – 3) = 0
3x – 6 – 3z + 9 = 0
x – z + 1 = 0

c) (3, 1, 4) is a point on the second line. Substitute the point in the first line
\(\frac{3-2}{1}=\frac{1+1}{2}=\frac{4-3}{1} \Rightarrow \frac{1}{1}=\frac{2}{2}=\frac{1}{1}\)
Therefore the point (3, 1, 4) satisfies the first line. Hence both interest at (3, 1, 4).

Question 23.
a) A coin is tossed 3 times. Find the probability distribution of the number of heads.
b) A bag contains 5 black and 6 white balls, 4 balls of the same colour (Black or white) are added to the bag, shuffled well and one ball is drawn. If the ball obtained is white. What is the probability that the balls added are black?
Answer:
a) Let X be the random variable denoting the
number of heads appears. Then X = {0, 1, 2, 3}
P(X = x) = ⁿC_x P^xq^n-x; p = \(\frac{1}{2}\), q = \(\frac{1}{2}\), n = 3

E₁ = Balls added are black.
E₂ = Balls added are white.
A = Ball drawn is white.

Question 24.

In a circle of radius 2 a square is inscribed as shown in the figure. Using integration, find the area of the shaded region (Area of a square may be calculated using any convenient method)
Answer:

Equation of the circle is x² + y² = 4
Area of the sector in the first quadrant

Area of the triangle = \(\frac{1}{2}\) (2)(2) = 2
Area of the shaded region in the first quadrant = π – 2
Hence the area of the required region = 4(π – 2) = 4π – 8

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper March 2019 with Answers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1 to 7 carry 3 scores each. Answer any 6 questions. (6 × 3 = 18)

Question 1.
a) If f(x) = sinx, g(x) = x², x∈R, them find (fog)(x)
b) Let u and v be two functions defined on R as u(x) = 2x – 3 and v(x) = \(\frac{3+x}{2}\). Prove that u and v are inverse to each other.
Answer:
a) f(x) = sinx, g(x) = x², x∈R
fog(x) = f(g(x)) = f(x²) = sin(x²)

b) uov(x) = u(v(x))
= \(u\left(\frac{3+x}{2}\right)=\frac{2(3+x)}{2}-3\) = X
vou(x) = v(u(x))
v(2x – 3) = \(\frac{3+2 x-3}{2}\) = x

Question 2.
a) For the symmetric matrix
A = \(\left[\begin{array}{lll}
2 & x & 4 \\
5 & 3 & 8 \\
4 & y & 9
\end{array}\right]\)
Find the values of x and y.
b) From Part (a), verify AA’ and A + A’ are symmetric matrices.
Answer:
x = 5, y = 8
b)

Question 3.
a) Find the slope of tangent line to the curve y = x² – 2x + 1
b) Find the equation to the above curve which is parallel to the line 2x – y + 9 = 0.
Answer:
a) y = x² – 2x + 1 ⇒ \(\frac{d y}{d x}\) = 2x – 2
⇒ slope = 2x – 2

b) Since the tangent is parallel to the line 2x – y + 9 = 0 , both have same slope.
Slope of the line 2x – y + 9 = 0 is 2.
⇒ 2x – 2 = 2 ⇒ X = 2 ⇒ y = 1
Therefore the point is (2, 1)
Hence the equation of the tangent line is
y – 1 = 2 (x – 2) ⇒ y – 2x + 3 = 0

Question 4.
a) If ∫ f(x) dx = log |tan x| + C . Find f(x).
b) Evaluate ∫ \(\frac{1}{\sqrt{1-4 x^{2}}} d x\)
Answer:

Question 5.
a) Area bounded by the curve y = f(x) and the lines x = a, x = b and the x axis = ………..

b) Find area of the shaded region using integration.

Answer:
a) i) \(\int_{a}^{b} x d y\)
b) Here the slope of the line is 3 and passes through the origin. So its equation is y = 3x.

Question 6.
a) The order of the differential equation formed by y = A sin x + B cos x + c, where A and B are arbitrary constants is
i) 1   ii) 2   iii) 0   iv) 3
b) Solve the differential equation
sec²x tan ydx + sec²y tan xdy = 0
Answer:
a) ii) 2
b) sec²x tan ydx + sec²y tan xdy = 0

⇒ log |tan x| + log |tan y| = log c
⇒ log |tan x tan y| = log c
⇒ tan x tan y = c

Question 7.
A factory produces three items P, Q and R at two plants A and B. The number of items produced and operating cost per hour is as follows:

It is desired to produce at least 500 items of type P, at least 400 items of type Q and 300 items of type R per day.
a) Is it a maximisation case or a minimisation case? Why?
b) Write the objective function and constraints.
Answer:
a) Cost of operation should be minimum for a factory.
Hence this is a minimisation problem.

b) Maximise : Z = 1000 x + 800 y
Subject to
20x + 30y ≥ 500; 15x +12y ≥ 400;
25x + 23y ≥ 300; x, y ≥ 0

Questions 8 to 17 carry 4 scores each. Answer any 8. (8 × 4 = 32)

Question 8.
a) The function P is defined as “to each person on the earth is assigned a date of birth.” Is this a function one-one? Give reason.
b) Consider the function f: \(\left[0, \frac{\pi}{2}\right]\) → R given by f(x) = sin x and g: \(\left[0, \frac{\pi}{2}\right]\) → R given by g(x) = cos x.
i) Show that f and g are one-one functions.
ii) Is f + g one-one? Why?
c) The number of one-one functions from a set containing 2 elements to a set containing 3 elements is ………..
i) 2   ii) 3   iii) 6   iv) 8
Answer:
a) Not one-one. Since different persons have same birthdays.

b) i) f(x) = sin x and g(x) = cos x are one-one in the domain value in the domainone \(\left[0, \frac{\pi}{2}\right]\). Since for each value in domain \(\left[0, \frac{\pi}{2}\right]\) both have only one image.

ii) (f + g)(x) = sin x + cos x
(f + g)(0) = sin0 + cos0 = 0 + 1 = 1

Hence not one-one.

c) ³P₂ = 3 × 2 = 6

Question 9.
If A = sin^-1 \(\frac{2 x}{1+x^{2}}\) ,B = cos^-1 \(\frac{1-x^{2}}{1+x^{2}}\), C = tan^-1 \(\frac{2 x}{1+x^{2}}\) satisfies the condition 3A – 4B + 2C = \(\frac{\pi}{3}\). Find the value of x.
Answer:
a) 3A – 4B + 2C = \(\frac{\pi}{3}\)
3sin-1 \(\frac{2 x}{1+x^{2}}\) – 4cos-1 \(\frac{1-x^{2}}{1+x^{2}}\) + 2tan-1 \(\frac{2 x}{1+x^{2}}\)
3 × 2 tan^-1 x – 4 × 2 tan^-1 x + 2 × 2 tan^-1 x = \(\frac{\pi}{3}\)
⇒ 6tan^-1 x – 8tan^-1 x + 4tan^-1 x = \(\frac{\pi}{3}\)
⇒ 2tan^-1 x = \(\frac{\pi}{3}\) ⇒ tan^-1 x = \(\frac{\pi}{6}\)
⇒ x = \(\frac{1}{\sqrt{3}}\)

Question 10.
a) Write the function whose graph is shown below.

b) Discuss the continuity of the function obtained in part (a).
c) Discuss the differentiability of the function obtained in part (a).
Answer:
a) f(x) = \(\left\{\begin{array}{ll}
x^{2}, & x \leq 0 \\
x, & x>0
\end{array}\right.\)

b)

For x > 0, f(x) = x which is a polynomial, hence continuous.
For x < 0, f(x) = x² which is a polynomial, hence continuous. Therefore the function is continuous.

c) Since the function has a sharp corner at x = 0.
The function is not differentiable at x = 0.
Hence the function is not differentiable.

Therefore left derivative is not equal to right derivative. Hence not differentiable at x = 0.

Question 11.
A cuboid with a square base and given volume ‘V’ is shown in the figure:

a) Express surface area ‘S’ as a function of x.
b) Show that the surface area is minimum when it is a cube.
Answer:

Question 12.
a) If 2x + 4 = A(2x + 3) + B, find A and B.
b) Using part (a) evaluate ∫\(\frac{2 x+4}{x^{2}+3 x+1} d x\)
Answer:
a) A = 1, B = 1


Question 13.
Consider the Differential equation cos² x \(\frac{d y}{d x}\) + y = tan x. Find
a) its degree
b) the integrating factor
c) the general solution.
Answer:
a) One.

Question 14.
The position vectors of three points A, B, C are given to be i + 3j + 3k, 4i + 4k, -2i + 4j + 2k respectively
a) Find \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\)
b) Find the angle between \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\)
c) Find a vector which is perpendicular to both \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\) having magnitude 9 units.
Answer:
a) \(\overrightarrow{A B}\) = 3i – 3j + k, \(\overrightarrow{A C}\) = -3i + j – k


Question 15.
a) If \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar vectors, write the vector perpendicular to \(\bar{a}\)
b) If \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar, prove that [\(\bar{a}\) + \(\bar{b}\) \(\bar{b}\) + \(\bar{c}\) \(\bar{c}\) + \(\bar{a}\)] are coplanar.
Answer:
a) Cross product of \(\bar{a}\) with any of the vectors \(\bar{b}\) or \(\bar{c}\).

b) Given,
[\(\bar{a}\), \(\bar{b}\), \(\bar{c}\)] = 0

Question 16.
a) Write all the direction cosines of x-axis.
b) If a line makes α, β, γ with x, y, z axis respectively, then prove that sin² α + sin² β + sin² γ = 2
c) If a line makes equal angles with the coordinate axes, find the direction cosines of the lines.
Answer:
a) 1, 0, 0

b) LHS = sin² α + sin² β + sin² γ
= 1 – cos² α + 1 — cos² β + 1 — cos² γ
= 3 – (cos² α + cos² β + cos² γ) = 3 – 1 = 2

c) Given α, β, γ are equal. Then
⇒ cos² α + cos² α + cos² α = 1
⇒ 3 cos² α = 1
⇒ cos α = \(\frac{1}{\sqrt{3}}\)
⇒ α = cos^-1 \(\frac{1}{\sqrt{3}}\)

Question 17.
The activities of a factory are given in the following table:

Solve the linear programming problem graphically and find the maximum profit subject to the above constraints.
Answer:
Maximise: Z= 5x + 8y
x + 4y ≤ 24; 3x + y ≤ 21; x + y ≤ 9; x, y ≥ 0

Maximum is at (4, 5); Z = 60

Questions from 18 to 24 carry 6 scores each. Answer any 5. (5 × 6 = 30)

Question 18.
If A = \(\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\). Show that A² – 5A + 7I = 0. Hence find A⁴ and A^-1
Answer:

A² – 5A + 7I = 0
Multiplying by A^-1 we have;
A^-1 (A² – 5A + 7I) = 0
⇒ A – 5I + 7A^-1 = 0

Question 19.
If A = \(\left[\begin{array}{ccc}
2 & -3 & 5 \\
3 & 2 & -4 \\
1 & 1 & -2
\end{array}\right]\), then
a) Find A^-1
b) Using A^-1 from part (a) solve the system of equations.
Answer:
|A| = \(\left|\begin{array}{ccc}
2 & -3 & 5 \\
3 & 2 & -4 \\
1 & 1 & -2
\end{array}\right|\) = -1
C₁₁ = 0, C₁₂ = 2, C₁₃ = 1
C₂₁ = -1, C₂₂ = -9, C₂₃ = -5
C₃₁ = 2, C₃₂ = 23, C₃₃ = 13

b) X = A^-1 B

Question 20.
Find for the following:
a) sin² x + cos² y = 1
b) y = x^x
c) x = a(t – sin t), y = a(1 + cos t)
Answer:
a) sin² x + cos² y = 1
Differentiating w.r.to x we have;
2 sinx cosx + 2 cos y(-sin y) \(\frac{d y}{d x}\) = 0
sinx cosx = cosy siny \(\frac{d y}{d x}\)
\(\frac{d y}{d x}=\frac{\sin x \cos x}{\cos y \sin y}=\frac{\sin 2 x}{\sin 2 y}\)

b) y = x^x Take log on both sides;
logy = x log x
Differentiating w.r to x

Question 21.
Evaluate the following:

Answer:
i)



Question 22.
a) Find the area bounded by the curve y = sin x and the lines x = 0, x = 2π and x axis.
b) Two fences are made in a grass field as shown in the figure. A cow is tied at the point O with a rope of length 3m.

i) Using integration, find the maximum area of grass that cow graze within the fences. Choose D as origin.
ii) If there is no fences find the maximum area of grass that cow can graze.
Answer:

b) i) The Area the cow grazes in the sector of the circle with radius 3 and centered at origin.
x² + y² = 9 ⇒ y = \(\sqrt{9-x^{2}}\)
Area = \(\int_{a}^{b}\) ydx = \(\int_{0}^{3} \sqrt{9-x^{2}} d x\)

ii) The required area is area inside the full circle = 4 × \(\frac{9 \pi}{4}\) = 9π

Question 23.
a) Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).
b) The Cartesian equation of two lines are given by \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}, \frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\). Write the vector equation of these two lines.
c) Find the shortest distance between the lines mentioned in part (b).
Answer:
a) (3x – y + 2z – 4) + k (x + y + z – 2) = 0
It passes through the point (2, 2, 1)
(3(2) – 2 + 2(1) -4)+ k (2 + 2 + 1 – 2) = 0
⇒ 2 + k(3) = 0 ⇒ k = \(-\frac{2}{3}\)
(3x – y + 2z – 4) –\(\frac{2}{3}\) (x + y + z – 2) = 0
⇒ 9x -3y + 6z – 12 – 2x – 2y – 2z + 4 = 0
⇒ 7x – 5y + 4z – 8 = 0

b) \(\vec{r}\) = (-i – j – k) + λ (7i – 6j + k)
\(\vec{r}\) = (3i + 5j + 7k) + μ (i – 2j + k)

c) \(\overline{a_{1}}\) = -i -j -k; \(\overline{b_{1}}\) =7i – 6j + 2k
\(\overline{a_{2}}\) = 3i + 5j + 7k; \(\overline{b_{2}}\) =i – 2j + k
\(\overline{a_{2}}\) – \(\overline{a_{1}}\) = 4i + 4j + 8k

Question 24.
a) A bag contains 4 red and 4 black balls. Another bag contains 2 red and 5 black balls. One of the two bags is selected at random and a ball is drawn from the bag and which is found to be red. Find the probability that the ball is drawn from first bag.
b) A random variable X has the following distribution function:

i) Find k.
ii) Find the mean and the variance of the random variable.
Answer:
E₁ = Event of choosing bag I
E₂ = Event of choosing bag II
A = Event of drawing a red ball.
P(E₁) = P(E₂) = \(\frac{1}{2}\)

b) i) ΣP_i = 1
⇒ k + 3k + 5k + 7k + 4k = 1
⇒ 20k = 1 ⇒ k = \(\frac{1}{20}\)

ii)

Kerala State Board New Syllabus Plus Two Computer Application Previous Year Question Papers and Answers.

Kerala Plus Two Computer Application Previous Year Question Paper Say 2018 with Answers

Time: 2 Hours
Cool off time : 15 Minutes

General Instructions to candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• Your are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Part – A

Answer all questions from 1 to 5. Each question carries 1 score. (5 × 1 = 5)

Question 1.
Write the name of the built-in function of C++ to convert the given character into its lower case.
Answer:
tolower()

Question 2.
Which is the tag used to create a line break in an HTML page?
Answer:
<br>

Question 3.
A candidate key that is not a primary key is called the _______ key.
Answer:
alternate key

Question 4.
Which is the keyword used in the SQL SELECT command to eliminate duplicate values in the selection.
Answer:
distinct

Question 5.
Expand the term CDMA.
Answer:
Code Division Multiple Access

Part – B

Answer any 9 questions from 6 to 16. Each question carries 2 scores. (9 × 2 = 18)

Question 6.
Write the function prototype for the following function:

1. A function sum( ) takes two integer arguments and returns integer value.
2. A function print() has no argements and nonreturn value.

Answer:

1. int sum(int, int);
2. void print();

Question 7.
Differentiate actual arguments and formal arguments in C++.
Answer:
The argument that present in the called function is called the formal argument and it is present in the calling function is called an actual argument. The data type of both is the same.

Question 8.
Write the output of the following HTML code:
<OL Type = “I” Start = “10”>
<LI> keyboard </LI>
<LI> mouse </LI>
<LI> light pen </LI>
Answer:
10. keyboard
11. mouse
12. light pen

Question 9.
Describe any four values of Type attributes of the <INPUT> Tag in HTML.
Answer:
<Input> It is used to create input controls. Its type of attribute determines the control type.
Main values of the type attribute are given below.

1. Text – To create a text box.
2. Password – To create a password text box.
3. Checkbox – To create a check box.
4. Radio – To create a radio button.
5. Reset – To create a Reset button.
6. Submit-To creates a submit button.
7. Button – To create a button

Question 10.
Write a short note on a virtual private server.
Answer:
Virtual Private Server (VPS): A VPS is a virtual machine sold as a service by an Internet hosting Service. A VPS runs its own copy of an OS (Operating System) and customers have super level access to that OS instance, so they can install almost any s/w that runs on that OS. This type is suitable for websites that require more features than shared hosting but fewer features than dedicated hosting.

Question 11.
Define primary key and alternate key.
Answer:
Primary key – It is a set of one or more attributes used to uniquely identify a row.
Alternate key – A candidate key other than the primary key.

Question 12.
Write a short note on UNION operation in Relational algebra.
Answer:
UNION operation: This operation returns a relation consisting of all tuples appearing in either or both of the two specified relations. It is denoted by U. duplicate tuples are eliminated. Union operation can take place between compatible relations only, i.e., the number and type of attributes in both the relations should be the same and also their order.
e.g. SCIENCE U COMMERCE gives all the tuples in both COMMERCE and SCIENCE.

Question 13.
Differentiate the data type CHAR and VARCHAR in SQL.
Answer:
Char – It is used to store a fixed number of characters. It is declared as char (size).
Varchar – It is also used to store characters but it uses only enough memory.
the char data type is fixed length. It allocates maximum memory and maybe there is a chance of memory wastage. But Varchar allocates only enough memory to store the actual size.

Question 14.
Write a short note on Supply Chain Management.
Answer:
Supply Chain Management (SCM): This deals with moving raw materials from suppliers to the company as well as finished goods from the company to customers. The activities include are inventory (raw materials, work in progress, and finished goods) management, warehouse management, transportation management, etc.

Question 15.
Write a short note on the mobile operating system.
Answer:
Mobile Operating System: It is an OS used in handheld devices such as smartphones, tablets, etc. It manages the hardware, multimedia functions, Internet connectivity, etc. Popular OSs are Android from Google, iOS from Apple, BlackBerry OS from BlackBerry, and Windows Phone from Microsoft.

Question 16.
Define the following term:

1. SIM
2. MMS

Answer:

1. The network is identified using the SIM (Subscriber Identity Module).
2. Multimedia Messaging Service (MMS): It allows sending Multi-Media (text, picture, audio, and video file) content using mobile phones. It is an extension of SMS.

Part – C

Answer any 9 questions from 17 to 27. Each carries 3 scores. (9 × 3 = 27)

Question 17.
Rewrite the following C++ code using the if…else statement:

switch (choice)
{
Case 1:
cout<<"one";
break;
case 0:
cout<<"zero";
break;
default;
cout<<"End"
break;
}
Answer:
if(choice==1)
cout<<“One”;
else if (choice==0)
cout<<“Zero”;
else
cout<<“End”;

Question 18.
Write the output of the following C++ code. Justify your answer.

for(i=1; i<5; i++)
{
cout<<“\t”<<i;
if(i==3)
break;
}

Answer:
This prints 1 2 3. This is because the value of i becomes ‘3’ then the break statement executes, it terminates the loop and hence the output.

Question 19.
Consider the following C++ code :
a) char name [20];
cin>>name;
cout<<name;
b) char name [20];
gets (name);
cout<<name;
Write the output in both cases if the string entered value is “NEW DELHI”. Justify your answer.
Answer:
a) The output is New. This is because of cin operator reads up to the delimiter space. The characters after space will not be read.
b) The output is New Delhi. This is because of the gets() function reads characters upto the user press the enter key, including space.

Question 20.
Define array traversal with an example.
Answer:
Traversal: All the elements of an array is visited and processed is called traversal
Eg:

#include<iostream>
using namespace std;
int main()
{
int n[10], i, sum=0;
for(i=0; i<10; i++)
{
cout<<“Enter value for number”<<i+1<<":";
cin>>n[i];
if(n[i]%5==0)
sum+=n[i];
}
cout<<"The sum of numbers which are exact multiple of 5 is "<<sum;
}

Question 21.
Consider the following function definition in C++:

void sum (int a, int b=10, int c=20)
{
int sum = a + b + c;
cout<<sum:
}

Write the output of the above code for the following function call:
(a) sum (1, 2, 3);
(b) sum (2, 3);
(c) sum (3);
Answer:
a) 6
Here a = 1, b = 2 and c = 3
So the answer is 6

b) 25
Here a = 2, b = 3 and c = 20 (The default value)
So the answer is 25

c) 33
Here a = 3, b = 10 and c = 20 (Default values for b and c)
So the answer is 33

Question 22.
Compare client-side scripting and server-side scripting.
Answer:

Question 23.
Write the HTML code to generate the following table:

Answer:

<html>
<head>
<title>
Table creation
</title>
</head>
<body bgcolor="red">
<table border="1">
<tr align="center">
<th>Roll No</th>
<th> Name</th>
<th> Class</th>
</tr>
<tr align="center">
<td>100</td>
<td> Vishnu</td>
<td> C1</td>
</tr>
<tralign="center">
<td>101</td>
<td> Anupama</td>
<td> C2</td>
</tr>
<tralign="center">
<td> 102</td>
<td> Biju</td>
<td> A1 </td>
</tr>
</table>
</body>
</html>

Question 24.
Classify the following values in JavaScript into suitable data type:
“Hello”, False, 125.0, 148, “True”, True
Answer:
String – “Hello”, “True”
Numeric – 125.0,148
Boolean – False, True

Question 25.
What is Content Management System? Write any two popular CMS software.
Answer:
Content Management System(CMS): CMS is a collection of programs that is used to create, modify, update, and publish website contents. CMS can be downloaded freely and is useful to design and manage attractive and interactive websites with the help of templates that are available in CMS. WordPress, Joomla, etc. are examples of CMS.

Question 26.
Define the following terms:

1. Cardinality
2. Schema
3. Tuple

Answer:

1. Cardinality – The number of rows.
2. Schema – The structure of the table is called the schema.
3. Tuple means the rows.

Question 27.
Explain any three benefits of the ERP system.
Answer:
Benefits of ERP system
1. Improved resource utilization: Resources such as Men, Money, Material, and Machine are utilized maximum hence increase productivity and profit.

2. Better customer satisfaction: Without spending more money and time all the customer’s needs are considered well. Because the customer is the king of the market. Nowadays a customer can track the status of an order by using the docket number through the Internet.

3. Provides accurate information: Right information at the right time will help the company to plan and manage the future cunningly. A company can increase or reduce production based upon the right information hence increase productivity and profit.

4. Decision-making capability: Right information at the right time will help the company to take a good decisions.

5. Increased flexibility: A good ERP will help the company to adopt good things as well as avoid bad things rapidly. It denotes flexibility.

6. Information integrity: A good ERP integrates various departments into a single unit. Hence reduce the redundancy, inconsistency, etc.

Part – D

Answer any 2 questions from 28 to 30. Each question carries 5 scores. (2 × 5 = 10)

Question 28.
Consider the following HTML code and answer the following:
<EM> COMPUTER </EM> <BR>
<STRONG> APPLICATION </STRONG> <BR> <HR>
(a) Name the tag used to make the text as italics and bold in the above code. (1)
(b) What is the purpose of <HR> tag? Explain its any two attributes. (2)
(c) Write the HTML statement to scroll the text given in <EM> from top to bottom. (2)
Answer:
a) for Italics <I> or <i> is used
for bold <strong> or <b> is used
b) <HR> is used to draw a horizontal line. Its attributes are size, width, shade, and color.

c) <html>
<head>
<title>
Demo of Marquee
</title>
</head>
<body bgcolor="red">
<marquee direction="down">
<em>hi welcome to BVM</em>
</marquee>
</body>
</html>

Question 29.
Consider the following JavaScript code:

function print ()
{
var i,
for (i=1; i<=10; ++i)
{
document.write(i);
document.write("<<BR>");
}
}

(i) Write the output of the above code. (1)
(ii) Rewrite the above code using a while loop. (2)
(iii) Modify the above code to find the sum of first 10 counting numbers. (2)
Answer:
i) It prints 1 to 10 line by line

ii) function Print()
{
var i;
i=1;
while(i<= 10)
{
document.write(i);
document.write (" <BR> ");
i++;
}
}

iii) function Print()
{
var i, sum=0;
for(i=1; i<=10; i++)
sum=sum+i;
document.write("The sum of first 10 countimg numbers is "+sum);
}

Question 30.
Define constrain. Explain any four-column constraints.
Answer:
Constraints are used to ensure database integrity.

1. Not Null – It ensures that a column can never have NULL values.
2. Unique – It ensures that no two rows have the same value in a column.
3. Primary key – Similar to unique but it can be used only once in a table.
4. Default – We can set a default value.
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.

Kerala State Board New Syllabus Plus Two Computer Application Previous Year Question Papers and Answers.

Kerala Plus Two Computer Application Previous Year Question Paper March 2019 with Answers

Time: 2 Hours
Cool off time : 15 Minutes

General Instructions to candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• Your are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Part – A

Answer all questions from 1 to 5. Each carries 1 score. (5 × 1 = 5)

Question 1.
The input operator in C++ is ___________
Answer:
>> or cin >>

Question 2.
_________ character is stored at the end of the string.
Answer:
NULL or ‘\0’

Question 3.
The process of breaking large program into smaller sub-programs is called __________
Answer:
Modularization

Question 4.
Name the keyword used to declare variables in JavaScript.
Answer:
var

Question 5.
Expand MIS.
Answer:
Management Information System

Part – B

Answer any 9 questions from 6 to 16. Each carries 2 scores. (9 × 2 = 18)

Question 6.
List the type modifiers in C++.
Answer:
Type modifiers used in C++ are signed, unsigned, short and long.

Question 7.
Rewrite the following code using for loop:

int x = 1;
start:
cout<<x;
x = x + 5;
if (x < = 50)
goto start;

Answer:

for(x=1; x<=50; x+=5)
cout<<x;

Question 8.

1. Define an Array.
2. Initialize an integer array with 5 elements.

Answer:

1. Array: An array is a collection of elements with the same data type store in contiguous memory location.
2. int mark[] = {40, 42, 44, 46, 48, 50};

Question 9.
Write the port number for the following web services:

1. Simple Mail Transfer Protocol.
2. HTTP secure (HTTPS)

Answer:

1. 25
2. 443

Question 10.
What is the use of frame tag in HTML? What is its limitation?
Answer:
frame tag helps to view multiple web pages in a single window. The main limitation is that all browsers not supporting the frame tag.

Question 11.
Write the HTML code to display the following using list tag:
i) Biology Science
ii) Commerce
iii) Humanities
Answer:

<html>
<head>
<title>list demo
</title>
</head>
<body bgcolor="red">
<ol type="i">
<li> Biology Science</li>
<li> Commerce</li>
<li> Humanities</li>
</ol>
</body>
</html>

Question 12.
What is the difference between isNaN() and Number() functions in JavaScript?
Answer:
isNaN() function checks the given value is a number or not. If it is not a number it returns a true value otherwise false.
Number() function converts the data into numerical type.

Question 13.
What is CMS? Give two examples.
Answer:
CMS means Content Management System. It is a collection of programs that are used to create, modify, update, and publish website content. CMS can be downloaded freely and is useful to design and manage attractive and interactive websites with the help of templates that are available in CMS. WordPress, Joomla, etc. are examples of CMS.

Question 14.
First table containing 4 rows and 3 columns, the second table contains 5 rows and 2 columns, then the Cartesian product table contains ______ rows and ______ columns.
Answer:
The number of rows is the product of rows, i.e. 4 × 5 = 20 rows
The number of columns is the sum of columns, i.e. 3+2 = 5 columns

Question 15.
How Business Process Re-Engineering (BPR) is related to Enterprise Resource Planning (ERP)?
Answer:
ERP and BPR will not make much change if they are in stand-alone. To improve the efficiency of an enterprise integrate both ERP and BPR because they are the two sides of a coin. For better results conducting BPR before implementing ERP, will help an enterprise to avoid unnecessary modules from the software.

Question 16.
Define the terms:
i) Cyber Forensics
ii) Infomania
Answer:
i) Cyber Forensics: Critical evidence of a particular crime is available in electronic format with the help of computer forensics. It helps to identify the criminal with help of blood, skin or hair samples collected from the crime site. DNA, polygraph, finger prints are another effective tool to identify the accused person is a criminal or not.

ii) Infomania: Right information at the right time is considered as the key to success. The information must be gathered, stored, managed and processed well. Infomania is the 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 the Internet than others. They create their own Cyber World and no interaction to the surroundings and the family.

Part – C

Answer any 9 questions from 17 to 27. Each carries 3 scores. (9 × 3 = 27)

Question 17.
Compare the selection statements ‘if’ and ‘switch’.
Answer:
Following are the difference between the switch and if-else if ladder.

1. Switch can test only for equality but if can evaluate a relational or logical expression.
2. If else is more versatile.
3. If else can handle floating values but switch cannot
4. If the test expression contains more variable if else is used.
5. Testing a value against a set of constants switch is more efficient than if-else.

Question 18.
Write a program in C++ to accept a string with white space like “good morning” from the keyboard and display the same string.
Answer:

#include<iostream>
#include<cstdio>
using namespace std;
int main()
{
char str[80];
cout<<"Enter a string:";
gets(str);
puts(str);
}

Question 19.
Compare static webpage and dynamic webpage.
Answer:

Question 20.
i) What is the use of reserved characters for HTML entities? (1)
ii) List any four reserved characters and their use. (2)
Answer:
(i) HTML entities are used to print reserved characters in HTML.
(ii)

Question 21.
Write the built-in JavaScript functions used for the following situation:

1. Display warning message in the screen.
2. Character at a particular position.
3. Convert uppercase to lowercase.

Answer:

1. alert()
2. charAt()
3. toLowerCase()

Question 22.
Write the merits and demerits of free Webhosting.
Answer:
The name implies it is free of cost service and the expense is met by the advertisements. Some service providers allow limited facility such as limited storage space, do not allow multimedia (audio and video) files.

Question 23.
What is the key? Explain any two keys in a relational database management system.
Answer:
Key is used to identify or distinguish a tuple in a relation.

• Candidate key – It is used to uniquely identify the row.
• Primary key – It is a set of one or more attributes used to uniquely identify a row.
• Alternate key – A candidate key other than the primary key.
• Foreign key – A single attribute or a set of attributes, which is a candidate key in another table is called a foreign key.

Question 24.
Define the term Data independence. Explain different levels of data independence.
Answer:
Data Independence – It is the ability to modify the schema definition in one level without affecting the scheme definition at the next higher level.

• Physical Data Independence – It is the ability to modify the physical scheme without causing application programs to be rewritten.
• Logical Data Independence – It is the ability to modify the logical scheme without causing application programs to be rewritten.

Question 25.
Explain any three situations to modify the structure of a table with the help of alter command in SQL.
Answer:
We can alter the table in two ways.
We can add a new column to the existing table using the following syntax,
ALTER TABLE <tablename>ADD(<cloumnname> <type> <constraint>);
We can also change or modify the existing column in terms of type or size using the following syntax,
ALTER TABLE<tablename>MODIFY(<column> <newtype>);

Question 26.
Explain the merits of ERP system.
Answer:
Benefits of ERP system
1. Improved resource utilization: Resources such as Men, Money, Material and Machine are utilized maximum hence increase productivity and profit.

2. Better customer satisfaction: Without spending more money and time all the customer’s needs are considered well. Because the customer is the king of the market. Nowadays a customer can track the status of an order by using the docket number through the Internet.

3. Provides accurate information: Right information at the right time will help the company to plan and manage the future cunningly. A company can increase or reduce production based upon the right information hence increase productivity and profit.

4. Decision-making capability: Right information at the right time will help the company to take a good decision.

5. Increased flexibility: A good ERP will help the company to adopt good things as well as avoid bad things rapidly. It denotes flexibility.

6. Information integrity: A good ERP integrates various departments into a single unit. Hence reduce the redundancy, inconsistency, etc.

Question 27.
Compare GPRS and EDGE.
Answer:
GPRS (General Packet Radio Services): It is a packet-oriented mobile data service on the 2G on GSM. GPRS was originally standardized by European Telecommunications Standards Institute (ETSI) GPRS usage is typically^fiarged based on the volume of data transferred. Usage above the bundle cap is either charged per megabyte or disallowed.

EDGE (Enhanced Data rates for GSM Evolution): It is three times faster than GPRS. It is used for voice communication as well as an internet connection.

Part – D

Answer any 2 questions from 28 to 30. Each carries 5 scores. (2 × 5 = 10)

Question 28.
Identify the built-in C++ function for the following cases:

1. to convert -25 to 25.
2. compare ‘computer’ and ‘COMPUTER’ ignoring cases.
3. to check the given character is a digit or not.
4. to convert the character from ‘B’ to ‘b’.
5. to find the square root of 64 or a number.

Answer:

1. abs()
2. strcmpi()
3. isdigit()
4. tolower()
5. sqrt()

Question 29.
(i) Write the name of the tag used to group related data in an HTML form. (1)
(ii) Write the HTML code to display the following webpage: (4)

Answer:
(i) <fieldset> tag

(ii) <html>
<head>
<title>
login page
</title>
</head>
<BODY BGCOLOR="cyan">
<FORM NAME="frmlogin">
<center>
User Name
<input type="text" name="txtname">
<br><br>
Password
<input type="password" name="txtpass">
<br><br>
<input type="Submit" value="Submit">
<input type="Reset" value="Reset">
</center>
</FORM>
</body>
</html>

Question 30.
Consider the table student with attribute admno, Name, course, percentage. Write the SQL statements to do the following:

1. Display all the student details. (1)
2. Modify the course’Commerce1 to’Science1. (1)
3. Remove the student details with a percentage below 35. (1)
4. Create a view from the above table with a percentage greater than 90. (2)

Answer:

1. select * from student;
2. update student set course=”Science” where course=”Commerce”;
3. delete from student where percentage<35;
4. create view stud view as select * from student where percentage > 60;

Kerala State Board New Syllabus Plus Two Computer Application Notes Chapter 11 Trends and Issues in ICT.

Kerala Plus Two Computer Application Notes Chapter 11 Trends and Issues in ICT

Mobile Computing
The drawbacks of Desk computers are, it is heavy and power consumption rate is high and it is not portable(not mobile).
The advancements in computing technology, lightweight and low power consumption have led to the developments of more computing power in handheld devices like laptops, tablets, smartphones, etc.

Nowadays instead of desktops, lightweight and low power consumption devices are used because they are cheap and common. Moreover people are able to connect to others through the internet even when they are in motion.

Mobile Communication
The term ‘mobile’ help people to change their lifestyles and become the backbone of society. Mobile communication networks do not require any physical connection.

Generations in mobile communication
The mobile phone was introduced in the year 1946. Early-stage it was expensive and limited services hence its growth was very slow. To solve this problem, cellular communication concept was developed in 1960’s at Bell Lab. 1990’s onwards cellular technology became a common standard in our country.
The various generations in mobile communication are
a) First Generation networks(1G):
It was developed around 1980, based on analog. system and only voice transmission were allowed.

b) Second Generation networks (2G):
This is the next-generation network that was allowed voice and data transmission. Picture message and MMS(Multimedia Messaging Service) was introduced. GSM and CDMA standards were introduced by 2G.

i) Global System for Mobile(GSM):
It is the most successful standard. It uses narrowband TDMA (Time Division Multiple Access), allows simultaneous calls on the same frequency range of 900 MHz to 1800 MHz. The network is identified using the SIM(Subscriber Identity Module).

GPRS (General Packet Radio Services): It is a packet-oriented mobile data service on the 2G on GSM. GPRS was originally standardized by European Telecommunications Standards

Institute (ETSI) GPRS usage is typically charged based on the volume of data transferred. Usage above the bundle cap is either charged per megabyte or disallowed.

EDGE (Enhanced Data rates for GSM Evolution): It is three times faster than GPRS. It is used for voice communication as well as an internet connection.

ii) Code Division Multiple Access (CDMA):
It is a channel access method used by various radio communication technologies. CDMA is an example of multiple access, which is where several transmitters can send information simultaneously over a single communication channel. This allows several users to share a band of frequencies To permit this to be achieved without undue interference between the users and provide better security.

c) Third Generation networks (3G):
It allows a high data transfer rate for mobile devices and offers high-speed wireless broadband services combining voice and data. To enjoy this service 3G enabled mobile towers and handsets required.

d) Fourth Generation networks (4G): It is also called Long Term Evolution(LTE) and also offers ultra-broadband Internet facility such as high quality streaming video. It also offers good quality image and videos than TV.

e) Fifth Generation networks (5G): This is the next-generation network and expected to come into practice in 2020. It is more faster and cost-effective than the other four generations. More connections can be provided and more energy efficient.

Mobile communication services

a) Short Message Service(SMS): It allows transferring short text messages containing up to 160 characters between mobile phones. The sent message reaches a Short Message Service Center(SMSC), that allows ‘store and forward’ systems. It uses the protocol SS7 (Signaling System No7). The first SMS message ‘Merry Christmas’ was sent on 03/12/1992 from a PC to a mobile phone on the Vodafone GSM network in the UK.

b) Multimedia Messaging Service (MMS): It allows sending Multi-Media(text, picture, audio, and video file) content using mobile phones. It is an extension of SMS.

c) Global Positioning System(GPS): It is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites. The system provides critical capabilities to military, civil, and commercial users around the world. It is maintained by the United States government and is freely accessible to anyone with a GPS receiver. GPS was created and realized by the U.S. Department of Defense (DoD) and was originally run with 24 satellites. It is used for vehicle navigation, aircraft navigation, ship navigation, oil exploration, Fishing, etc. GPS receivers are now integrated with mobile phones.

d) Smart Cards: A smart card is a plastic card with a computer chip or memory that stores and transacts data. A smart card (may be like your ATM card) reader used to store and transmit data. The advantages are it is secure, intelligent and convenient.
The smart card technology is used in SIM for GSM phones. A SIM card is used as identification proof.

Mobile operating system: It is an OS used in hand held devices such as smart phone, tablet, etc. It manages the hardware, multimedia functions, Internet connectivity,etc. Popular OSs are Android from Google, iOS from Apple, BlackBerry OS from BlackBerry and Windows Phone from Microsoft.

Android OS: It is a Linux-based OS for Touch screen devices such as smartphones and tablets.lt was developed by Android Inc. founded in Palo Alto, California in 2003 by Andy Rubin and his friends. In 2005, Google acquired this. A team led by Rubin devel¬oped a mobile device platform powered by the Linux Kernel. The interface of Android OS is based on touch inputs like swiping, tapping, pinching in, and out to manipulate on-screen objects. In 2007 onwards this OS is used in many mobile phones and tablets. An¬droid SDK(Software Development Kit) is available to create applications(apps) like Google Maps, FB, What’s App,etc. It is of open-source nature and many Apps are available for free download from the Android Play Store hence increase the popularity. Different Android Versions are shown below:

Version – Code name
4.4 – KitKat
4.1 – Jelly Bean
4.0.3 – Ice Cream Sandwich
3.1 – Honeycomb
2.3 – Gingerbread
2.2 – Froyo
2.0 – Eclair
1.6 – Donut
1.5 – Cupcake

ICT in business: Drastic developments in ICT have changed the shopping habits of people. Earlier people shops traditionally. But nowadays people buy products and services online. A study reveals that online shopping habits of people are increased. Aftersale service is also good, delivery of the products is prompt and safe. The status of the product can be tracked easily hence increase the confidence level of the online customers.

Social networks and big data analytics: Earlier before buying a product people may consult two or three shop keepers or local friends and take decisions. But nowadays before taking decisions, people search shopping sites, social network groups(Facebook, WhatsApp, Instagram, Twitter, etc), web portals, etc. for the best prices. Almost all online sites have product comparison menus. By this, we can compare the price, features, etc. Earlier a product is created and customers are forced to buy. But today customer is the King of the market, so products are created for the choices of the customers.

So companies gathering information about the customers from various sources such as social media like Internet forums, social blogs, Microblogs, etc. The volume of such data is very large and considered big data in business. With the help of an s/w analysis this big data and generate a report that contains all the information such as choices, taste, needs, status etc of a customer.

Business logistics: It is the management of the flow(transportation) of resources such as food, consumer goods, services, animals etc in a business between the point of origin(source) and the point of consumption (destination) in order to meet the needs of companies and customers. Business logistics consists of many more complexities. The effective use of hardware and software reduces the complexities faced in Business logistics.

For this the hardware used is RFID(Radio Frequency Identification) tag and the reader. It is like the bar code. The RFID tag contains all the details of a product and it consists of a combination of a transmitter and a receiver. The data stored in the RFID tag can be accessed by a special reader and to read the data no need for an RFID tag and reader in a line of site instead both are within a range.

This tag is used in Vehicles as a prepaid tag and makes the payments easier in Toll booths. Similarly, it is useful to take the Census of wild animals also.

Information Security: The most valuable to a company(An enterprise ora Bank, etc) is their data base hence it must be protected from accidental or unauthorised access by unauthorised persons

Intellectual Property Right: 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 types of Intellectual properties must be protected from unauthorized access by law. This is called Intellectual Property right(IPR).
Paris convention held in 1883 protects Industrial Property
Berne Convention held in 1886 protects Literary and Artistic work.
World Intellectual Property Organisation(WIPO) in 1960, Guided by the United Nations(UN) ensures/protects the rights of creators or owners and rewarded for their creation.

A person or an organization can register their Intellectual property such as creations, trademarks, designs, etc.

Intellectual property is divided into two categories

1. Industrial Property
2. Copyright

1) Industrial property: It ensures the protection of industrial inventions, designs, Agricultural products etc from unauthorized copying or creation or use. In India, this is done by the Controller of Patents Designs and Trademarks.

Patents: A person or organization that invented a product or 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 it freely.

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^opy by anybody else.

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

Geographical indications: Some products are well known by the place of its origin. Kozhikkodan Halwa, Marayoor Sharkkara (Jaggery), Thirupathi Ladoo, etc are examples.

B) Copyright: The trademark is ©, copyright is the property right that arises automatically when a person creates a new work on 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.

Infringement (Violation): Unauthorized copying or use of Intellectual property rights such as Patents, Copyrights, and Trademarks are called intellectual property Infringement(violation). It is a punishable offense.

Patent Infringement: It prevents others from unauthorized or intentional copying or use of Patent without the permission of the creator.

Piracy: It is the unauthorized copying, distribution, and use of a creation without the permission of the creator. It is against the copy right act and hence the person committed deserves the punishment.

Trademark Infringement: It prevents others from unauthorized or intentional copying or use of Trademark without the permission of the creator.

Copy right Infringement: It prevents others from unauthorized or intentional copying or use of Copy right without the permission of the creator.

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 systems and computer systems are integrated and create a virtual(un real) environment. This is called cyberspace. The result of this integration is that tremendous speed and it is very cheap. The various departments of Govt, are providing speed, reliable and convenient online service hence increase productivity. Online shopping, Online banking, Online debate, Online Auction etc. are the various services offered by the Internet.

Through this one can transfer funds from our account to another account, hence one can pay bills such as telephone, electricity, purchase tickets(Flight, Train, Cinema, etc). As much as CyberSpace helps us that much as it gives us troubles.

Cyber Crimes: 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.

A) Cybercrimes against individuals
i) Identity theft: The various information such as personal details(name, Date of Birth, Address, Phone number etc), Credit / Debit Card details(Cand number, PIN, Expiry Date, CW, etc), Bank details, etc. are the identity of a person. Stealing this 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 the Internet is called Cyberstalking (Nuisance). This is a kind of torturing and it may lead to spoiling friendship, 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 media and act as the original 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 offense. A hidden camera is used to capture the video or picture and blackmailing them.

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

B) Cybercrimes against property: Stealing credit card details, hacking passwords of social media accounts or mail account or Net banking, uploading the latest movies etc, are considered as cyber crimes against property.

i) Credit card fraud: Stealing the details such as credit card number, company name, expiry date, CVV number, password etc. and use these details to make payment for purchasing goods or transfer funds also.
ii) Intellectual property theft: The violation of Intellectual Property Right of Copyright, Trademark, Patent, etc. In the film industry crores of investment are needed to create a movie. Intellectual Property thieves upload the movies on the Releasing day itself. Hence the revenue from the theatres is less significant 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 as some tools(programs) available in the Internet

iii) Internet time theft: This is deals with the misuse of WiFi Internet facilities. If it is not protected by a good password there is a chance of misuse of our devices (Modem/Router) to access the 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) Cybercrimes against the government: The cyber crimes against Govt, websites is increased significantly. For example in 2015 the website of the Registration Department of Kerala is hacked and destroys data from 2012 onwards.

i) Cyber terrorism: It deals with 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 a huge loss (money and life of citizens). So Govt is very conscious and give tight security mechanism for their services.

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

iii) 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 in 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 the Enter key, the browser requests that page from the webserver. Dos attacks send a huge number of requests to the webserver until it collapses due to the load and stop functioning.

Cyberethics
Guidelines for using computers over the internet

• Emails may contain Viruses so do not open any unwanted emails
• Download files from reputed sources(sites)
• Avoid clicking on pop-up Advt.
• Most of the Viruses spread due to the use of USB drives so use cautiously.
• Use a firewall on your computer
• Use anti-virus and update regularly
• Use spam blocking software
• Take backups in regular time intervals
• Use strong passwords, i.e a mixture of characters (a-z & A-Z), numbers, and special characters.
• Do not use bad or rude language in social media and emails.
• Untick ‘Remember Me’ before login.

CyberLaws: It ensures the use of computers and the Internet by people safely and legally. It consists of rules and regulations like the 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

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 provides the legal infrastructure for E-commerce, in India.

This is developed to promote the IT industry, control e-commerce also ensures the smooth functioning of E-Governance and it prevents cyber crimes.

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

Cyber Forensics: Critical evidence of a particular crime is available in electronic format with the help of computer forensics. It helps to identify the criminal with help of blood, skin, or hair samples collected from the crime site?: DNA, polygraph, fingerprints are other effective tools to identify the accused person is the criminal or not.

Info mania: Right information at the right time is considered the key to success. The information must be gathered, stored, managed, and processed well. Infomania is excessive desire(infatuation) for acquiring knowledge from various modem sources like the Internet, Email, Social media, Instant Message applications (WhatsApp), and Smart Phones. Due to this, the person may neglect daily routines such as family, friends, food, sleep, etc. hence they get tired. They give first preference to the Internet than others. They create their own Cyber World and no interaction with the surroundings and the family.

Kerala State Board New Syllabus Plus Two Computer Application Notes Chapter 10 Enterprise Resource Planning.

Kerala Plus Two Computer Application Notes Chapter 10 Enterprise Resource Planning

The goal(aim) of the management of an enterprise(Proprietor of a Company or a Venture or an organization) is to handle the resources in a good manner and .make good profit. The resources include the employees, customers, raw materials, finished goods machinery etc… Hence an enterprise handles large amount of data(DataBase) such as employee data, customer data, raw material purchase, sales data, financial data etc. The size of data to be handle is large and hence the complexity is also high. To solve this problem .organizations use ERP packages

Overview of an enterprise
Let us consider a production unit in an enterprise. The activities involved are planning, purchasing raw material, production, storing finished goods(warehouse), sales, finance etc. These activities are performed by different departments and theirduties are interlinked. Altogether the resources are classified into four M’s, That is Man, Material, Money and Machine.

Concepts of Enterprise Resource Planning
An enterprise(organization) is considered as a system(A system is an orderly grouping of interdependent components linked together to achieve an objective, according to a plan. Human body is an example for System). All the departments of an enterprise are connected to a centralized data base. ERP consists of single database and a collection of programs to handle the database hence handle the enterprise efficiently and hence enhance the productivity.

Functional units of ERP
Different modules are given below:
Financial Module: It is the core. This is used to generate financial report such as balance sheet, general ledger, trial balances, financial statements etc.

Manufacturing Module: It provides information for the production and capable to change the methods in the manufacturing sector.

Production planning Module: This module ensures the effective use of resources and helps the enterprise to enhance productivity hence increase profit.

HR (Human Resource) Module: This model ensures the effective use of Human resources and Human capital.

Inventory control Module: This model is useful to maintain the appropriate level of stock(includes raw material, work in progress and finished goods)

Purchasing Module: This module is useful to make available the required raw materials in good condition and in the right time and price.

Marketing Module: It is used for handle the orders of customers.

Sales and distribution Module: The existence of a company is based on the income from sales. This module will help to handle the sales enquiries, order placement ans scheduling, dispatching and invoicing.

Quality (Ql & QC) management module: The quality of a product or service is very much important to a company.This module helps to maintain the quality of the product. Quality planning, inspection and control are the main activities involved in this module.

Business Process Re-engineering (BPR)
In this world, tight competition is based on price, quality, wide variety of selection and quick service. To increase the business and hence increase the profit of a Business firm various activities are involved. IT and Re-engineering play major roles to increase productivity.

In general, BPR is a series of activities such as rethinking and redesign the business process to enhance the enterprise’s performance such as reducing the cost(expenses), improve the quality, prompt, and speed(time-bound) service.

BPR enhances the productivity and profit of an enterprise.

A business process consists of three elements

1. Input – Supply data for processing
2. Processing – Series of activities to convert the input into output
3. Outcome – After processing, we will get the result as output.

The connection between ERP and BPR
ERP and BPR will not make much change if they are stand-alone. To improve the efficiency of an enterprise integrate both ERP and BPR because they are the two sides of a coin. For better results conducting BPR before implementing ERP, will help an enterprise to avoid unnecessary modules from the software.

Implementation of ERP
Wonderful changes are shown if you select and implement the correct ERP. Right ERP implemented at the right time will enhance the productivity and profit of an enterprise.

The different phases of ERP implementation are given below
Pre-evaluation screening: Many ERP packages are available in the markets. At most care should be taken before implementing an ERP. Select a few from the available ERP packages.

Package selection: The selection of the right ERP to our enterprise is a laborious task and it needs huge investment. Various factors should be kept in mind before you purchase an ERP that should meet our complete needs.

Project planning: Good planning is essential to implement an ERP. From the beginning to the end activities are depicted in this phase.

Gap analysis: A cent percent(100%) problem-solving ERP is not available in the market. Most of them solve a maximum of 70% to 80% problems. The rest (30% to 20%) of the problems and their solutions are mentioned here.

Business Process Reengineering: In general BPR is the series of activities such as rethinking and redesign of the business process to enhance the enterprise’s performance such as reducing the cost(expense), improve the quality, prompt and speed(time-bound) service.
BPR enhances the productivity and profit of an enterprise

Installation and configuration: In this phase the new system are installing, before implementing the whole system a miniature of the actual system is going to be implemented as a test dose. Then check the reactions if it is good it is the time to install the whole system completely.

Implementation team training: In this phase the company trains its employees to implement and run the system.

Testing: This phase is very important. It determines whether the system produces proper result. Errors in design and logic are identified.

Going live: Here a change over is taken place to new system from old system. It is not an easy process without the support and service from the ERP vendors.

End-user training: This phase will start familiarising the users with the procedures to be used in the new system. It is very important.

Post-implementation: Once the system is implemented maintenance and review begin. In this phase repairing or correct previously ill-defined problems and upgrade or adjust the performance according to the company needs.

ERP solution providers / ERP packages
The selection of right ERP is a difficult task. Many ERP packages are available in the market. Most of them are too expensive and cannot afford by small enterprises. The reason behind the expensiveness is that the ERP companies investing huge amount of time, money and effort in the research and development of ERP packages.

Popular ERP packages are given below
Oracle
American based company famous in database(Oracle 9i-SQL) packages situated in Redwood shores, California.
ERP package is a solution for finance and accounting problems. Their other products are

1. Customer Relationship Management (CRM)
2. Supply Chain Management (SCM) Software

SAP
SAP stands for Systems, Applications and Products for data processing.
It is a German MNC in Walldorf and founded in 1972. Earlier they developed ERP packages for large MNC. But nowadays they developed for small scale industries also.

The other software products they developed are

1. Customer Relationship Management(CRM)
2. Supply Chain Management(SCM)
3. Product Life cycle Management(PLM)

Odoo
Formerly known as OpenERP.
It is an open-source code ERP. Unlike other companies, their source code is available and can be modified as and when the need arises.

Microsoft Dynamics

• American MNC in Redmond, Washington
• ERP for midsized companies.
• This ERP is more user friendly
• Another s/w is Customer Relationship Management (CRM)

Tally ERP
Indian company situated in Bangalore.
This ERP provides a total solution for accounting, inventory and Payroll.

Benefits and risks of ERP
ERP packages have a lot of advantages as well as many drawbacks also.

Benefits of ERP system

1. Improved resource utilization: Resources such as Men, Money, Material, and Machine are utilized maximum hence increase productivity and profit.

2. Better customer satisfaction: Without spending more money and time all the customer’s needs are considered well. Because the customer is the king of the market. Nowadays a customer can track the status of an order by using the docket number through the Internet.

3. Provides accurate information: Right information at the right time will help the company to plan and manage future cunningly. A company can increase or reduce production based upon the right information hence increase productivity and profit.

4. Decision-making capability: Right information at the right time will help the company to make good decisions.

5. Increased flexibility: A good ERP will help the company to adopt good things as well as avoid bad things rapidly. It denotes flexibility.

6. Information integrity: A good ERP integrates various departments into a single unit. Hence reduce the redundancy, inconsistency, etc.

Risks of ERP implementation

1. High cost: Very huge investment is required to purchase and configure an ERP. Moreover, it requires up gradation or replacement of hardware(Man, computer, or machine) is an additional investment. So small-scale enterprises cannot afford this.

2. Time consuming: The full-fledged implementation of the ERP package needs one or two years. That is highly time-consuming.

3. Requirement of additional trained staff: The existing staff may not capable to work with ERP. To overcome this give proper training to them otherwise appoint trained and experienced employees to cop up.

4. Operational and maintenance issues: The first major problem is that the resistance from the existing employees. To overcome this give awareness to the existing employees. The second problem is that the ERP package is a cyclic process-oriented package. It is a continuous process and should be maintained well otherwise the correct output will not available.

ERP and related technologies
It is an all in one system. It integrates various functions such as raw material purchase, production planning, marketing, financial etc, into a single application.

Product Life Cycle Management (PLM): It manages the entire life cycle of a product. PLM consists of programs to increase the quality and reduce the price by the efficient use of resources.

Customer Relationship Management (CRM): As we know the customer is the king of the market. The existence of a company mainly the customers. CRM consists of programs to enhance the customer’s relationship with the company.

Management Information System (MIS): Management is the decision and policymakers. Good management can make a good decision and that will help to do the business well. A good relationship between Management and employees is a key to success. MIS will collect relevant data from inside and outside of a company. Based on this information produce reports and take appropriate decisions.

Supply Chain Management (SCM): This is deals with moving raw materials from suppliers to the company as well as finished goods from the company to customers. The activities include are inventory(raw materials, work in progress, and finished goods) management, warehouse management, transportation management, etc.

Decision Support System (DSS): It is a computer-based system that takes inputs as business data and after processing it produces good decisions as output that will make the business easier.

Kerala State Board New Syllabus Plus Two Computer Application Notes Chapter 9 Structured Query Language.

Kerala Plus Two Computer Application Notes Chapter 9 Structured Query Language

SQL – Structured Query Language developed at IBM’s San Jose Research Lab.

The result of the compilation of DDL statements is a set of tables, which are stored in a special file called a data dictionary.

Creating a database in Mysql
CREATE DATABASE <database_name>;
Eg: mysql>CREATE DATABASE BVM;

Opening a database
USE command used to use a database
USE <database_name>;
Eg: mysql>USE BVM;

SHOW command is used to list entire database in our system.
mysql>SHOW DATABASES;

Data Types

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.

3. Dec or Decimal – It is used to store numbers with decimal point. It is declared as Dec (size, scale). We can store a total of size number of digits.

4. Int or Integer – It is used to store numbers with¬out decimal point. It is declared as int. It has no argument. Eg: age int.

5. Smallint – Used to store small integers.

6. Date – It is used to store date. The format is yyyy-mm-dd.
Eg: ‘1977-05-28’.

7. Time – It is used to store time. The format is

DDL commands (3 commands)

• Create table
• Avertable
• Drop table

DML commands (4 commands)

• Select
• Insert
• Delete
• Update

DCL (Data Control Language) commands

• Grant
• Revoke

Rules for naming tables and columns

• The name may contain alphabets(A-Z, a-z), digits(0-9), underscore(_) and dollar ($) symbol
• The name must contain at least one character.
• Special characters cannot be used except _ and $
• Cannot be a keyword
• The name must be unique.

Constraints are used to ensure database integrity.

• Not Null
• Unique
• Primary key
• Default
• Auto_increment

Order By – Used to sort rows either in ascending (asc) or descending (desc) order.

Aggregate functions

• Sum() – find the total of a column.
• Avg() – find the average of 3 column.
• Min() – find the smallest value of a column.
• Max() – find the largest value of the column.
• Count() – find the number of values in a column.

Group by clause is used to group the rows. Having clause is used with Group By to give conditions.

Kerala State Board New Syllabus Plus Two Computer Application Notes Chapter 8 Database Management System.

Kerala Plus Two Computer Application Notes Chapter 8 Database Management System

DBMS means Data Base Management System. It is a tool used to store a large volume of data, retrieve and modify the data as and when required. DBMS consists of data and programs.

Advantages of DBMS

1. Data Redundancy
2. Inconsistency can be avoided
3. Data can be shared
4. Standards can be enforced
5. Security restrictions can be applied
6. Integrity can be maintained
7. Efficient data access
8. Crash recovery

Structure of DBMS

1. Fields – Smallest unit of data. Eg: Name, age, sex, …
2. Record – Collection of related fields.
3. File – Collection of records

Components of DBMS

1. Databases – It is the main component.
2. Data Definition Language (DDL) – It is used to define the structure of a table.
3. Data Manipulation Language (DML) – It is used to add, retrieve, modify and delete records in a database.
4. Users – With the help of programs users interact with the DBMS.

Database Abstraction – Abstraction means hiding, it hides certain details of how data is stored and main-tained.

Levels of Database Abstraction:

1. Physical Level (Lowest Level) – It describes how the data is actually stored in the storage medium.
2. Logical Level (Next Higher Level) – It describes what data are stored in the database.
3. View Level (Highest level) – It is closest to the users. It is concerned with the way in which the individual users view the data.

Data Independence – It is the ability to modify the scheme definition in one level without affecting the scheme definition at the next higher level.

1. Physical Data Independence – It is the ability to modify the physical scheme without causing application programs to be rewritten.
2. Logical Data Independence – It is the ability to modify the logical scheme without causing application programs to be rewritten.

Users of Database

1. Database Administrator
2. Application Programmer
3. New users

Data models – It is a collection of tools for describing data, data relationship, data semantics and consistency problem. 3 models.

1. Hierarchical model
2. Network model
3. Relational model

RDBMS – Relational DataBase Management System. It consists of a collection of relations as database.

Relation means table.

Domain – A pool of possible values from which col-umn values are drawn. ‘

Tuple means rows.

Attributes means columns.

Cardinality – The number of rows.

Degree – The number of columns

View – A view is a virtual table derived from one or more base tables.

Key is used to identify or distinguish a tuple in a relation.

Candidate key – It is used to uniquely identify the row.

Primary key – It is a set of one or more attributes used to uniquely identify a row.

Alternate key – Acandidate key other than the primary key.

Foreign key – A single attribute ora set of attributes, which is a candidate key in another table is called foreign key.

Relational Algebra – It consists of a set of opera¬tions that takes one or two relations as input and produces a new relation as a result.

1. Select operation (σ)
2. Project Operation (π)
3. Cartesian Product
4. Union Operation (∪)
5. Intersection operation (∩)
6. Set difference operation (-)

Kerala State Board New Syllabus Plus Two Computer Application Notes Chapter 7 Web Hosting.

Kerala Plus Two Computer Application Notes Chapter 7 Web Hosting

Web hosting
Buying or renting storage space to store website in a web server and provide service(made available 24×7) to all the computers connected to the Internet. This is called web hosting. Such service providing companies are called web hosts. Programming languages used are PHP, ASP.NET, JSP.NET, etc.

Types of web hosting
Various types of web hosting services are available. We can choose the web hosting services according to our needs depends upon the storage space heeded for hosting, the number of visitors expected to visit, etc.
1) Shared Hosting
2) Dedicated Hosting
3) Virtual Private Server (VPS)

Buying hosting space
We designed a website of our school and we decide our school website to be made available to all over the world, we have to place the website files on a web server for that we have to purchase hosting space(memory space) in a web server.
Following factors to be considered
1) Buying sufficient amount of memory space for storing ourwebsite files
2) If the web pages contain programming contents supporting technology must be consider
3) Based upon the programs select Windows hosting or Linux hosting

Domain Name System(DNS) Registration
Millions of websites are available over Internet so that ourwebsite must be registered with a suitable name. Domain Name registration is used to identify a website over Internet. A domain name must be unique(i.e. no two website with same name is available). So you have to check the availability of domain name before you register it, for this www.whois.net website will help. If the domain name entered is available then we can register it by paying the Annual registration fees through online.

FTP (File Transfer Protocol) client software When a client requests a website by entering website address. Then FTP client software helps to establish a connection between client computer and remote server computer. Unauthorised access is denied by using username and password hence secure our website files forthat SSH(Secure Shell) FTP simply SFTP is used. Instead of http://, it uses ftp://.
By using FTP client s/w we can transfer(upload) the files from our computer to the web server by using the ‘drag and drop’ method. The popular FTP client software are FileZilla, CuteFTP, SmartFTP, etc.

Free hosting
The name implies it is free of cost service and the expense is meet by the advertisements. Some service providers allow limited facility such as limited storage space, do not allow multimedia(audio and video) files.
A paid service website’s address is as follows
eg: www.bvmhsskalparamba.com

Usually two types of free web hosting services as follows
1) as a directory service.
Service provider’s website address/ ourwebsite address
eg: www.facebook.com / bvm hss kalparambu
2) as a Subdomain
Our website address, service providers website address
eg: bvmhsskalparamba.facebook.com

Earlier web hosting services are expensive but nowadays it is cheaper hence reduced the need for free web hosting.
Example for free web hosting.

Content Management System(CMS)
Do you heard about Data Base Management System(DBMS). DBMS is a software(collection of programs) used to create, alter, modify, delete and retrieve records of a DataBase. Similarly, CMS is a collection of programs that is used to create, modify, update and publish website contents. CMS can be downloaded freely and is useful to design and manage attractive and interactive websites with the help of templates that are available in CMS. WordPress, Joomla, etc are examples of CMS.

Responsive web design
The home page is displayed differently according to the screen size of the browser window(different screen sized devices-mobile phone, palmtop, tablet, laptop, and desktop) we used. The website is designed dynamically(flexibly) that suit the screen size of a different device introduced by Ethan Marcotte. Before this, companies have to design different websites for different screen sized devices. By responsive web design, companies have to design only one website that suitably displayed according to the screen size of the devices. It is implemented by using a flexible grid layout, images, and media queries

Flexible grid layouts: It helps to set the size of the web page to fit the screen size of the device.

Flexible image and video: It helps to set the image or video dimension to fit the screen size of the device.

Media queries: There is an option(settings) to select the size of the web page to match our device, this can be done by using media queries inside the CSS file.

A well known Malayalam daily Malayala Manorama launched their responsive website.