Prasanna

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

Kerala Plus Two Computer Science 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.
• You 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 the 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 wrapping up of data end functions into a single unit is called ______
Answer:
Data Encapsulation

Question 2.
In a linked list, the linked part of the last node contains ________ data.
Answer:
Null Pointer

Question 3.
Give the full form of VPS.
Answer:
Virtual Private Server

Question 4.
The number of rows in a relation is called ________
Answer:
Cardinality

Question 5.
In PHP, arrays that use string keys are called ________
Answer:
Associative array

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

Question 6.
Define a structure named ‘Time’ with elements hour, minute, and second.
Answer:
struct time
{
int h, m, s;
};

Question 7.
Read the following C++code:
int a[5] = {10, 15, 20, 25, 30};
int *p = a;
Write the output of the following statements:
(a) cout < < * (p + 2); .
(b) cout << * p + 3;
Answer:
(a) 20(Third element)
(b) *p is 10(First element)
So *p+3 = 10 + 3 = 13.

Question 8.
List the different operations performed on data structures.
Answer:
The operations are Traversing, Searching, Inserting(Push), Deleting(Pop), Sorting and Merging

Question 9.
Write HTML tag for the following:

1. Hyperlink to the website http://WWW.dhsekerala.gov.in
2. [email protected]

Answer:

1. <a href=”http://www.dhsekerala.gov.in”>
2. <a href=mailto:”[email protected]”>

Question 10.
Describe the use of ‘action’ and ‘method’ attributes of <FQRM> tag.
Answer:
<Form> attributes

1. Action – Here we give the name of the program (including the path) stored in the Webserver.
2. Method – There are 2 types of methods get and post.

Question 11.
Explain the two purposes of the ‘+’ operator used in JavaScript.
Answer:
‘+’ if the operands are numbers then it adds and if one of the operands is a string then it concatenates the string.
eg. 5+3=8
“BVM”+8172=”BVM8172”

Question 12.
The Javascript function given below is used to display the sum of digits of a given number. Fill in the blanks to complete the function.
<Script language = “JavaScript”>
………. sum digit( )
{var n, s;
n = document.frm.txt1. …………;
for (s = 0; ……….; n = n/10)
s = s + ………..;
document.frm.txt2.value = s;
}
</script>
Answer:
function sum digit()
n=document.frm.txt1.value;
for(s=0; n>0: n=n/10)
s=s+n%10;

Question 13.
Explain any two constraints used in SQL.
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.

Question 14.
List the core data types in PHP.
Answer:
Data types are Integer, Float/Double, Boolean, and String.

Question 15.
What is meant by GIS? Give an example.
Answer:
Geographic Information System (GIS) technology is developed from the digital cartography and Computer-Aided Design (CAD) database management system. GIS as the name implies capturing, storing for future reference, checking, and displaying data related to various positions on the earth’s surface. GIS can be applied in many areas such as soil mapping, agricultural mapping, forest mapping, e-Governance, etc.

Question 16.
Define Infringement.
Answer:
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
• Trademark Infringement
• Copy right Infringement.

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

Question 17.
What are the different memory allocations used in C++? Explain.
Answer:
The main memory can be allocated in two methods.

1. Static memory allocation
2. Dynamic memory allocation

When the amount of memory to be allocated is known in advance and memory is allocated during compilation itself, it is referred to as static memory allocation.

When the amount of memory to be allocated is not known in advance and it is required to allocate memory as and when required during run time, it is known as dynamic memory allocation.

‘new’ operator is used for dynamic allocation of memory syntax,
datatype*pointer variable = new datatype;
eg: int *ptr = new int;

Question 18.
What is polymorphism? Give an example.
Answer:
Polymorphism is the ability for a message or data to be processed in more than one form. This is achieved by function overloading, operator overloading, and dynamic binding.
There are two types of polymorphism.

a) Compile time (early binding/static) polymorphism. It is the ability of the compiler to relate or bind a function call with the function definition during compilation time itself.
Examples are Function overloading and operator overloading.
Function overloading: Functions with the same name and different signatures (the number of parameters or data types are different).

Operator overloading: It gives new meaning to an existing C++ operator.
Eg: we know that + is used to add two numbers, Operator overloading assigns + to a new job such as it concatenates two strings into one string.

b) Run time (late binding/dynamic) polymorphism. It is the ability of the compiler to relate or bind a function call with the function definition during run time. It uses the concept of pointers and inheritance.

Question 19.
Write an algorithm to insert a new item into a Queue.
Answer:
The algorithm is given below:
Step 1: If front = 1 and rear=N or front = rear + 1.
Then print “OVERFLOW” and return
Step 2: If front = Null then
Set front = 1 and rear = 1
Else if rear = N then set rear = 1
Else
Set rear = rear+1
End if
Step 3: Set Queue[rear] = item
Step 4: stop

Question 20.
Differentiate between a static web page and a dynamic web page.
Answer:

Question 21.
Briefly explain the different ways in which a JavaScript code can be inserted into a web page.
Answer:
Ways to add scripts to a web page.
1. Inside <BODY> section
Scripts can be placed inside the <BODY> section.

2. Inside<HEAD> section: Scripts can be placed inside the <HEAD> section. This method is a widely accepted method.

3. External (another) JavaScript file: We can write scripts in a file and save it as a separate file with the extension js. The advantage is that this file can be used across multiple HTML files and can be enhanced the speed of the page loading.

Question 22.
Distinguish between shared hosting and dedicated hosting.
Answer:
1. Shared Hosting: This type of hosting shares resources, like memory, disk space, and CPU hence the name shared. Several websites share the same server. This is suitable for small websites that have less traffic and it is not suitable for large websites that have large bandwidth, large storage space, and have a large volume of traffic.

E.g.: Shared hosting is very similar to living in an Apartment (Villas) complex. All residents are in the same location and must share the available resources (Car parking area, Swimming pool, Gymnasium, playground, etc) with everyone.

2. Dedicated Hosting: A web server and its resources are exclusively for one website that has a large volume of traffic mean a large volume of requests by the visitors. Some Govt, departments, or large organizations require uninterrupted services for that round a clock power supply is needed. It is too expensive but it is more reliable and provides good service to the public.

E.g.: It is similar to living in an Our own house. All the resources in your house is only for you. No one else’s account resides on the computer and would not be capable of tapping into your resources.

Question 23.
Explain different levels of data abstraction in DBMS.
Answer:
Database Abstraction: Abstraction means hiding, it hides certain details of how data is stored and maintained.
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.

Question 24.
Describe the ‘union’ and ‘intersection’ operations in relational algebra with suitable examples.
Answer:
Union Operation (∪) – All tuples appearing in either or both of two relations.
Intersection operation (∩) – All tuples appearing in both relations.

Question 25.
Write PHP code to display all even numbers below 100.
Answer:
<?php
echo”All even numbers below 100″;
echo”<br>=================”;
for($i=2; $i<=100; $i=$i+2)
{
echo”<br>”.$i; ‘
}

Question 26.
Explain the cloud service models.
Answer:
Cloud service models (3 major services)
1. Software as a Service (SaaS): A SaaS provider company provides more services on demand such as they allow to access both resources and applications.
Examples are Google Docs, Adobe creative cloud, Microsoft Office 365, Facebook.com, etc.

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

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

Question 27.
List and explain any three e-learning tools.
Answer:
e-Learning tools
a) Electronic books reader (eBooks): With the help of a tablet or portable computer or arty another device we can read digital files by using an s/w is called electronic books reader.

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

c) Onlinechat: Real-time exchange of text or audio or video messages between two or more people over the Internet.

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

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

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

Question 28.
Explain the various attributes of the <BODY> tag.
Answer:
Attributes of <body> tag
1. BGCOLOR – Specifies the background color for the document body
Eg. <BODY BGCOLOR = ”RED”>

2. BACKGROUND – Sets the image as the background for the document body
Eg. <BODY BACKGROUNG= “C:\result.jpg”>.

3. TEXT-Specifies the color of the text content of the page.
Eg. <BODY TEXT= “Red”>

4. Link – Specifies colour of the hyperlinks that are not visited by the user.

5. ALINK – Specifies the colour of hyperlinks.

6. VLINK – Specifies the color of hyperlinks which are already visited by the viewer. ‘
Eg. < BODY ALINK= “Cyan” LINK-’ Magenta”
VLINK= “Orange”>.

7. Leftmargin and Right margin-Sets margin from the left and top of the document window.

Question 29.
Write HTML code to display the following table in a web page:
Result of ABC school

Answer:

<html>
<head>
<title>Table
</title>
<body bgcolor=”cyan”>
<h1 align=”center”>Result of ABC school</h1>
<tableborder=”1"align=”center">
<tralign=”center”> •
<th rowspan=”2">Year</th>
<th colspan=”2">Students</th>
<th rowspan=”2">Pass<br>Percentage</th>
</tr>
<tr align=”center”>
<th>Registered</th>
<th>Passed</th>
</tr>
<tr align="center”>
<td>2014</td>
<td>200</td>
<td> 130</td >
<td>65</td>
</tr>
<tralign=”center”>
<td>2015</td>
<td>200</td>
<td>150</td>
<td>75</td>
</tr><tralign=”center”>
<td>2016</td>
<td>200</td>
<td > 160</td >
<td>80</td>
</tr>
</table>
</body>
</html>

Question 30.
A table named ‘student’ with fields Roll no, Name, Batch, Mark, Grade is given. Write SQL statements for the following:

1. To display the details of all students in the ‘Science’ batch.
2. To display the details of these students having grade A or A+.
3. To count the number of students in each batch.
4. To change the grade of the student to A+ whose Roll no. is 50.
5. Remove the details of the student whose Roll no. is 10.

Answer:

1. select * from student where Batch=”Science”;
2. select * from student where Grade in(“A”,”A+”);
3. select Batch,count(*) from student group by Batch;
4. update student set Grade =”A+”; where Roll_no=50;
5. delete from student where Roll_no =10;

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 10 Vector Algebra.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

Plus Two Maths Vector Algebra 3 Marks Important Questions

Question 1.
(i) With help of a suitable figure for any three vectorsa,bandc show that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\vec{b}+\bar{c})\)
(ii) If \(\bar{a}\) = i – j + k and \(\bar{b}\) = 2i – 2j – k. What is the projection of a on b? (March – 2011)
Answer:
(i) Answered in previous years questions
No. 1(ii) (6 Mark question)
(ii) Projection of \(\bar{a} \text { on } \bar{b}=\frac{\bar{a} \cdot \bar{b}}{|\bar{b}|}=\frac{2+2-1}{\sqrt{4+4+1}}=1\)

Question 2.
(i) If \(\bar{a}\) = 3i – j – 5k and \(\bar{b}\) = i – 5j + 3k Show that \(\bar{a}\) + \(\bar{b}\) and a bare perpendicular.
(ii) Given the position vectors of three points as A(i – j + k); B(4i + 5j + 7k) C(3i + 3j + 5k)
(a)Find \(\bar{AB}\) and \(\bar{BC}\)
(b) Prove that A,B and C are collinear points. (March – 2011)
Answer:

Question 3.
(i) Write the unit vector in direction of i + 2j – 3k.
(ii) If \(\overline{P Q}\) = 31 + 2j — k and the coordinate of P are(1, -1,2) , find the coordinates of Q. (May – 2012)
Answer:

Question 4.
(a) The angle between the vectors \(\bar{a}\) and \(\bar{b}\) such that \(|\vec{a}|=|\bar{b}|=\sqrt{2}\)
\(\bar{a}\).\(\bar{b}\) = 1 is
\(\begin{array}{lll}
\text { (i) } \frac{\pi}{2} & \text { (ii) } \frac{\pi}{3} & \text { (iii) } \frac{\pi}{4} & \text { (iv) } 0
\end{array}\)
(b) Find the unit vector along \(\bar{a}-\bar{b}\) where \(\bar{a}\) = i + 3j – k and \(\bar{b}\) 3i + 2j + k (March -2016)
Answer:

Plus Two Maths Vector Algebra 4 Marks Important Questions

Question 1.
Consider the vectors \(\bar{a}\) = 21+ j – 2k and \(\bar{b}\) = 6i – 3j + 2k.
(i) Find \(\bar{a} \bar{b}\) and \(\bar{a} \times \bar{b}\).
(ii) Verity that \(|\bar{a} \times \bar{b}|=|\vec{a}|^{2}|\bar{b}|^{2}-(\bar{a} \cdot \bar{b})^{2}\) (March – 2012)
Answer:

Question 2.
(i) For any three vectors \(\bar{a}, \bar{b}, \bar{c}\), show that \(\bar{a} \times(\bar{b}+\bar{c})+\bar{b} \times(\bar{c}+\bar{a})+\bar{c} \times(\bar{a}+\bar{b})=0\)
(ii) Given A (1, 1, 1), B (1, 2, 3), C (2, 3, 1) are the vertices of MBCa triangle. Find the area of the ∆ABC (May – 2012)
Answer:

Question 3.
Consider A (2, 3, 4) , B (4, 3, 2) and C (5, 2, -1) be any three points
(i) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(ii) Find the area of triangle ABC (March – 2013)
Answer:

Question 4.
(i) Find the angle between the vectors \(\bar{a}\) =3i + 4j + k and \(\bar{b}\) = 2i + 3j – k
(ii) The adjacent sides of a parallelogram are \(\bar{a}\) = 3i + λj + 4k and \(\bar{b}\) = i – λj + k
(a) Find \(\bar{a} \times \bar{b}\)
(b) If the area of the parallelogram is square units, find the value of A (May – 2013)
Answer:

Question 5.
Let \(\bar{a}\) = 2i – j + 2k and \(\bar{b}\) = 6i + 2j + 3k
(i) Find a unit vector in the direction of \(\bar{a}\) + \(\bar{b}\)
(ii) Find the angle between a and b (March – 2014)
Answer:

Question 6.
Consider the triangle ABC with vertices A(1, 1, 1) , B (1, 2, 3) and C (2, 3, 1)
(i) Find \(\overline{A B}\) and \(\overline{A C}\)
(ii) Find \(\overline{A B}\) x \(\overline{A C}\)
(iii) Hence find the area of the triangle (March – 2014)
Answer:

Question 7.
Consider the vectors \(\bar{a}\) = i – 7j + 7k; \(\bar{b}\) = 3i – 2j + 2k
(a) Find \(\bar{a b}\).
(b) Find the angle between \(\bar{a}\) and \(\bar{b}\).
(c) Find the area of parallelogram with adjacent sides \(\bar{a}\) and \(\bar{b}\). (May – 2014)
Answer:

Question 8.
(a) If the points A and B are (1, 2, -1) and (2, 1, -1) respectively, then is
(i) i + J
(ii) i – J
(iii) 2i + j – k
(iv) i + j + k
(b) Find the value of for which the vectors 2i – 4j + 5k, i – λj + k and 3i + 2j – 5k are coplanar.
(c) Find the angle between the vectors a = 2i + j – k and b = i – j + k (March – 2016)
Answer:

Question 9.
(i) \((\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})\) is equaito
\(\begin{array}{lll}
\text { (a) } \bar{a} & \text { (b) }|\bar{a}|^{2}-|\bar{b}|^{2} & \text { (c) } \bar{a} \times \bar{b} \text { (d) } 2(\bar{a} \times \bar{b})
\end{array}\)

(ii) If \(\bar{a}\) and \(\bar{b}\) are any two vectors, then
\((\bar{a} \times \bar{b})^{2}=\left|\begin{array}{ll}
\bar{a} \cdot \bar{a} & \bar{a} \cdot \bar{b} \\
\bar{a} \cdot \bar{b} & \bar{b} \bar{b}
\end{array}\right|\)

(iii) Using vectors, show that the points A(1, 2, 7), B(2, 6, 3), C(3, 10, -i) are collinear. (May – 2016)
Answer:

Plus Two Maths Vector Algebra 6 Marks Important Questions

Question 1.
(i) Find a vector in the direction of \(\bar{r}\) = 3E – 4j that has a magnitude of 9.
(ii) For any three vectors \(\bar{a,b}\) and \(\bar{c}\), and Prove that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\bar{b}+\bar{c})\).
(iii) Find a unit vector perpendicular to \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}\) = i – 3j + 3k and \(\bar{b}\) and \(/bar{c}\) = 3E—3j+2k. (March – 2010)
Answer:
(i) Unit vector of magnitude 9

Question 2.
Let A(2, 3, 4), B(4, 3, 2) and C(5, 2, -1) be three points
(i) Find \(\overline{A B}\) and \(\overline{B C}\)
(ii) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(iii) Fiñd the area of the triangle ABC. (May – 2010)
Answer:

Question 3.
ABCD s a parallelogram with A as the origin, \(\bar{b}\) and \(\bar{d}\) are the position vectors of B and D respectively.

(i) What is the position vector of C?
(ii) What is the angle between \(\bar{AB}\) and \(\bar{AD}\)?
(iii) If \(|\overrightarrow{A C}|=|\overrightarrow{B D}|\), show that ABCD is a rectangle. (May – 2011)
Answer:
(i) Since ABCD is a parallelogram with A as the

Question 4.
(a) If \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) respectively are the position vectors representing the vertices A,B,C,D of a parallelogram, then write \(/bar{d}\) in terms of \(\bar{a}, \bar{b}, \bar{c}\).
(b) Find the projection vector of \(/bar{b}\) = i + 2j + k along the vector \(/bar{a}\) = 21 – i – j + 2k. Also write \(/bar{b}\) as the sum of a vector along \(/bar{a}\) and a perpendicular to \(/bar{a}\).
(C) Find the area of a parallelogram for which the vectors 21 + j, 31 + j +4k are adjacent sides. (March – 2015)
Answer:

Question 5.
(a) Write the magnitude of a vector \(/bar{a}\) in terms of dot product.
(b) If \(\bar{a}, \bar{b}, \bar{a}+\bar{b}\) are unit vectors, then prove that the angle between \(/bar{a}\) and \(/bar{b}\) is \(\frac{2 \pi}{3}\)
(c) If 2i + j – 3k and mi + 3j – k are perpendicular to each other, then find ‘m’.
Also find the area of the rectangle having these two vectors as sides. (March – 2015)
Answer:

Question 6.
Consider the triangle ABC with vertices A(1, 2, 3), B(-1, 0, 4), C(0, 1, 2)
(a) Find \(\overline{A B}\) and \(\overline{A C}\)
(b) Find \(\angle A\)
(c) Find the area of triangle ABC. (May – 2015)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 9 Differential Equations.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 9 Differential Equations

Plus Two Maths Differential Equations 3 Marks Important Questions

Question 1.
Form a differential equation of the family of circles having a centre on y-axis and radius 3 units. (May -2013)
Answer:
The equation of the circle passing through the point (O,k)and radius 3 is of the form

Question 2.
Consider the Differential equation
\(\frac{d^{2} y}{d x^{2}}+y=0\)
(i) Write the order and degree.
(ii) Verify that y = a cos x + b sin x where a,b ∈ R is a solution of the given DE. (March – 2014)
Answer:
(i) Order = 2; Degree = I
(ii) Given; y = acosx + bsin x
y₁ = – asin x + bcos x
y₂ = – acos x – bsin x
We have; y₂ = – (a cos x + b sin x)
⇒ y₂ = -y ⇒ y₂ + y = 0

Plus Two Maths Differential Equations 4 Marks Important Questions

Question 1.
If cosx\(\frac{d y}{d x}\) + y sin x = tan² x is a DE then
(i) Find its order and degree.
(ii) Find its general solution. (May -2010)
Answer:

Question 2.
(i) Write the order and degree of the DE
\(\left[\frac{d y}{d x}\right]^{2}+\frac{d y}{d x}-\sin ^{2} y=0\)
(ii) Solve the \(\frac{d y}{d x}+2 y \tan x=\sin x\) (May-2011)
Answer:

Question 3.
(i) The general solution of the DE \(\frac{d y}{d x}=e^{x-y} \text { is }\)
(a) e^y + e^x = c
(b) e^y e^x = c
(c) e^-y + e^-x = c
(d) e^-y – e^x = c
(ii) Solve the DE \(\frac{d y}{d x}=\frac{2 x y}{1+x^{2}}+x^{2}+2\) (March – 2013)
Answer:

Question 4.
(a) Consider the family of all circles having their centre at the point (1,2). Write the equation of the family. Write the corresponding differential equation.
(b) Write the integrating factor of the differential equation
\(\cos x \frac{d y}{d x}+y=\sin x, \quad 0 \leq x<\frac{\pi}{2}\) (March – 2015)
Answer:
(a) The equation of the circle ¡s

Question 5.
(a) Write the order and degree of the differential equations.
\(x y\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x\left(\frac{d y}{d x}\right)^{3}-y \frac{d y}{d x}=0\)

(b) Find the general solution of the differential equation ylog ydx – xdy = 0
(c) Find the integrating factor of the differential equation \(x \frac{d y}{d x}-y=2 x^{2}\) (May -2015)
Answer:

Question 6.
(a) y = a cosx +b sin x is the solution of the differential equation

(b) Find the solution of the differential equation \(x \frac{d y}{d x}+2 y=x^{2}, \quad(x \neq 0)\) given that y = 0 when x=1. (March – 2016)
Answer:

Plus Two Maths Differential Equations 6 Marks Important Questions

Question 1.
(i) Form the DE corresponding to the Function y = ae^x + be^2x
(ii) State the order and degree of the above DE.
(iii) Solve \(x \frac{d y}{d x}=x+y\) (March – 2009)
Answer:

Question 2.
(i) Form the DE corresponding to the function Xy = C²
(ii) Consider the DE (x² + y² ) dx = 2xydy
(a) Write the DE in the ton \(\frac{d y}{d x}=g\left[\frac{y}{x}\right]\)
(b) Solve the DE completely (May -2009, May -2013)
Answer:

Question 3.
(i) Equation of a circle touching the y-axis at origin is x² + y² – 2ax = 0. Find the DE of all such circles.
(ii) SolvetheDE \(\left(1+x^{2}\right) \frac{d y}{d x}+y=\tan ^{-1} x\) (March – 2010)
Answer:

Question 4.
(i) Solution of the DE y^‘ – y = 0 is y = ………….
(ii) Solve the DE \(\Rightarrow \frac{d y}{d x}+y \sec x=\tan x\)
(iii) Form the DE of the family of ellipse having foci on the x-axis and centre at the origin. (March-2011)
Answer:

Question 5.
Consider the DE \(x d y-y d x=\sqrt{x^{2}+y^{2}} d x\)
(i) Express it in the form \(\frac{d y}{d x}\) = F(x, y)
(ii) Find the general solution. (March -2012; Edumate -2017)
Answer:

Question 6.
(i) Prove that the DE is (3xy + y²) dx + (x² + xy) dy = 0 a homogeneous DE of degree 0.
(ii) Solve the DE (3xy + y²) dx + (x² + xy) dy = 0 (May —2012, Edumate -2017)
Answer:

Question 7.
Consider the differential equation \(\frac{d y}{d x}-3 \cot x y=\sin 2 x\)
(a) Find its integrating factors.
(b) Fînd its solution, given that y = 2 When x = \(\frac{\pi}{2}\). (May-2014)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 8 Application of Integrals.

Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 8 Application of Integrals

Plus Two Maths Application of Integrals 4 Marks Important Questions

Question 1.
(i) The area bounded by the curve y = f(x) x-axis and the line x= a and x= b is……
(ii) Find the area enclosed between the Parabola y = x² and the straight line 2x – y + 3 = 0 (March – 2010)
Answer:

Question 2.
Find the area enclosed between the curve x² = 4y and the line x = 4y – 2 (March -2011)
Answer:

Question 3.
(i) Area of the shaded portion in the figure is equal to

(ii) Consider the curves y = x², x = 0, y = 1, y = 4.
Draw a rough sketch and shade the region bounded by these curves, Find area of the shaded region.
Answer:

Question 4.
Consider the following figure:

(i) Find the point of Intersection P of the circle x² + y² = 32 and the line y = x.
(ii) Find the area of the shaded region. (EDUCATE – 2017; March – 2013; March – 2014)
Answer:

Question 5.
(a) The area bounded by the curve, above the x-axis, between x = a and x = b is

(b) Find the area of the circle x² + y² = 4 using integration. (March – 2016)
Answer:

Question 6.
(i) The area bounded by y = 2cosx , the x-axis from x = 0 to x = \(\frac{\pi}{2}\) is
(a) 0
(b) 1
(c) 2
(d) -1
(ii) Find the area of the region bounded by the y² = 4ax and x² = 4ay, a > 0 (March – 2017)
Answer:

When x = 4a, y = 4a and x = 0, y = 0.
Therefore the point is (0, 0) and (4a, 4a).

Plus Two Maths Application of Integrals 6 Marks Important Questions

Question 1.
Consider the circle x² + y² = 16 and the straight line \(y=\sqrt{3} x\) as shown ¡n the figure

(i) Find the points A and B as shown in the figure.
(ii) Find the area of the shaded region in the figure using definite integral. (May -2010)
Answer:
(i) The point of intersection of x² + y² = l6 and

Question 2.
(i) Draw the rough sketch of \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)
(ii) Find the area bounded by the above curve using integration. (May – 2011)
Answer:
(i) The curve is an ellipse.

Question 3.
(i) Find the area enclosed between the curve y² = x, x = 1, x = 4 and x-axis.
(ii) Using ntegration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1). (March-2012)
Answer:

Required area ΔABC = Area ΔAB2
+ Area 2BC3 – Area ΔAC3
Equation AC is y = 2(x – 1)
Equation BC is y = 4 – x
Equation AB is y = 1/2 (x – 1)

Question 4.

Using the above figure
Find the equation of AB.
Findthe point P.
Find the area of the shaded region by integration. (May – 2013)
Answer:
(i) The equation of a line passing through (0,2)

Question 5.
Consider the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and the line \(\frac{x}{3}+\frac{y}{2}=1\).
(a) Find the points where the line intersects the ellipse?
(b) Shade the smaller region bounded by the ellipse and the line.
(c) Find the area of the shaded region. (May – 2014)
Answer:

Question 6.
Consider the function f(x) = |x| + 1; g(x) = 1 – |x|
(a) Sketch the graph and shade the enclosed region between them.
(b) Find the area of the shaded region. (March – 2015)
Answer:

(b) The equation of the line through AB is

Question 7.
Using the given figure answer the following questions.

Define the equation of the given curve.
Find the area of the enclosed region.
Find the area when a = lo and b = 5. (March – 2011; May – 2015; March – 2017)
Answer:
(i) The figure represents an ellipse with equation

Ellipse is symmetric w.r.t coordinate axles. Therefore the area of the enclosed region is same as four times area enclosed by the curve in first quadrant.

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 7 Integrals.

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

Plus Two Maths Application of Derivatives 3 Marks Important Questions

Question 1.
Find the following integrals. (May -2011)
\(\begin{array}{l}
\text { (i) } \int x^{2} e^{2 x} d x \\
\text { (ii) } \int e^{x} \sin x d x
\end{array}\)
Answer:

Question 2.
(i) \(\int e^{x} \sec x(1+\tan x) d x=\ldots \ldots\)
(a) ex cosx + c (b) ex sec x + c
(C) ex tanx + c (d) ex sin x + c
(ii) Find \(\int \sin 2 x \cos 3 x d x\) (March – 2014)
Answer:

Question 3.
Find the following integrals.
(i) \(\begin{array}{l}
\text { (i) } \int \frac{1}{(x+1)(x+2)} d x \\
\text { (ii) } \int \frac{2 x-1}{(x-1)(x+2)^{2}} d x
\end{array}\) (March – 2014)
Answer:

Plus Two Maths Application of Derivatives 4 Marks Important Questions

Question 1.
Consider the integral \(I=\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x\)
(i) Express \(I=\frac{\pi}{2} \int_{0}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x\)
(ii) Show that \(I=\frac{\pi^{2}}{4}\) (March – 2012)
Answer:

Question 2.
(i) Evaluate: \(\int_{2}^{3} \frac{x}{x^{2}+1} d x\)
(ii) Evaluate: \(\int_{0}^{\pi} \frac{x}{1+\sin x} d x\) (March – 2014)
Answer:

Question 3.
(a) What is the value of \(\int_{0}^{1} x(1-x)^{9} d x\) If the
\(\begin{array}{llll}
\text { (i) } \frac{1}{10} & \text { (ii) } \frac{1}{11} & \text { (iii) } \frac{1}{90} & \text { (iv) } \frac{1}{110}
\end{array}\)
(b) Find \(\int_{0}^{1}(2 x+3) d x\) of a sum. (March – 2015)
Answer:

Question 4.
Evaluate \(\int_{0}^{x} \log (1+\cos x) d x\)
Answer:

Question 5.
Find \(\int_{0}^{5}(x+1) d x \text { as limit of a sum. }\)
Answer:

Question 6.
Evaluate \(\int_{0}^{4} x^{2} d x\) as the limit of a sum. (March – 2017)
Answer:

Plus Two Maths Application of Derivatives 6 Marks Important Questions

Question 1.
(i) Fill in the blanks; \(\int \frac{1}{x} d x=\)_____
(ii) Evaluate \(\int \frac{5 x+1}{x^{2}-2 x-35} d x\)
(iii) Integrate with respect to x. \(\sqrt{x^{2}+4 x+8}\) (March – 2010)
Answer:

Question 2.
(i) Evaluate \(\int-\frac{\cos e c^{2} x}{\sqrt{\cot ^{2} x+9}} d x\)
(ii) Evaluate \(\int\left(\cos ^{-1} x\right)^{2} d x\) (May -2010)
Answer:

Question 3.
(i) Evaluate \(\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x\)
(ii) Evaluate \(\int_{0}^{2} e^{x} d x \text { as limit of a sum. }\) (May -2010)
Answer:

Question 4.
(i) Fill in the blanks \(\int \cot x d x=\)_____
(ii) Evaluate the integrals
\(\begin{array}{l}
\text { (a) } \int \sin 2 x \cos 4 x d x \\
\text { (b) } \int \frac{x}{(x+1)(x+2)} d x
\end{array}\) (March -2011)
Answer:

Question 5.
(i) Evaluate \(\int_{0}^{1} x d x\) as the limit of a sum.
(ii) Evaluate \(\int_{0}^{1} x(1-x)^{n} d x\) (March – 2011)
Answer:

Question 6.
(i) Evaluate \(\int_{1}^{2} \frac{1}{x(1+\log x)^{2}} d x\)
(ii) Evaluate \(\int_{0}^{3}\left(2 x^{2}+3\right) d x\) as the limit of a sum. (May – 2011)
Answer:

Question 7.
(i) What is \(\int \frac{1}{9+x^{2}} d x=?\)
(ii) Evaluate the integrals \(\int \frac{1}{1+x+x^{2}+x^{3}} d x\) (May – 2012)
Answer:

Question 8.
(i) Evaluate \(\int_{0}^{3} f(x) d x\)
where \(f(x)=\left\{\begin{array}{ll}
x+3, & 0 \leq x \leq 2 \\
3 x, & 2 \leq x \leq 3
\end{array}\right.\)

(ii) Prove that \(\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x=\int_{0}^{1} \log \left(\frac{1-x}{x}\right) d x\) Find the value of \(\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x\) (May – 2012)
Answer:

Question 9.
(i) Find \(\int \cot x d x=\ldots \ldots\)
(ii) Find \(\int x \log x d x\)
(iii) Find \(\int \frac{x-1}{(x-2)(x-3)} d x\) (March – 2013)
Answer:

Question 10.
Evaluate
\(\text { (i) } \int \frac{x+3}{\sqrt{5-4 x-x^{2}}} d x\)
\(\text { (ii) } \int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}\) (May – 2013)
Answer:

Question 11.
Evaluate
\(\begin{array}{l}
\text { (i) } \int x^{2} \tan ^{-1} x d x \\
\text { (ii) } \int_{-1}^{2} x^{3}-x d x
\end{array}\) (May – 2013)
Answer:

Question 12.
Evaluate \(\int_{0}^{\pi / 4} \log (1+\tan x) d x\) (March – 2013)
Answer:

Question 13.
(a) The value of \(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos x d x\) (May – 2014)
(b) Prove that \(\int_{0}^{\pi} \frac{x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} d x=\frac{\pi^{2}}{2 a b}\)
Answer:

Question 14.
(a) \(\int \frac{1}{x^{2}+a^{2}} d x=\)
(b) Find \(\int \frac{1}{9 x^{2}+6 x+5} d x\)
(c) Find \(\int \frac{x}{(x-1)^{2}(x+2)} d x\) (May – 2014)
Answer:

Question 15.
Integrate the following
\(\begin{array}{l}
\text { (a) } \frac{x-1}{x+1} \\
\text { (b) } \frac{\sin x}{\sin (x-a)} \\
\text { (c) } \frac{1}{\sqrt{3-2 x-x^{2}}}
\end{array}\) (March – 2015)
Answer:

Question 16.
(a) Prove that \(\int \cos ^{2} x d x=\frac{x}{2}+\frac{\sin 2 x}{4}+c\)
(b)Find \(\int \frac{1}{\sqrt{2 x-x^{2}}} d x\)
(c) Find \(\int x \cos x d x\) (May – 2015)
Answer:

Question 17.
Find the following:
\(\begin{array}{l}
\text { (i) } \int \frac{1}{x\left(x^{7}+1\right)} d x \\
\text { (ii) } \int_{1}^{4}|x-2| d x
\end{array}\)
Answer:

Question 18.
Find \(\int_{0}^{\frac{\pi}{2}} \log \sin x d x\)
Answer:

Question 19.
Find the following: \(\begin{array}{l}
\text { (i) } \int \cot x \log (\sin x) d x \\
\text { (ii) } \int \frac{1}{x^{2}+2 x+2} d x \\
\text { (iii) } \int x e^{9 x} d x
\end{array}\) (May – 2017)
Answer:

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;