This paper is divided into four sections – A, B, C and D.

Answer all questions.

Section A consists of one question having sub-parts of one mark each.

Section B consists of seven questions of two marks each.

Section C consists of nine questions of three marks each, and

Section D consists of three questions of five marks each

.
Internal choices have been provided in two questions each in Section B,
Section C and Section D.

The intended marks for questions are given in brackets [ ].

All working, including rough work, should be done on the same sheet as and
adjacent to the rest of the answer.

Answers to sub parts of the same question must be given in one place only.

A list of useful physical constants is given at the end of this paper.

A simple scientific calculator without a programmable memory may be used for
calculations.

#### Section-A

This section comprises multiple choice questions (MCQs) of 1 mark each

__Question
1__

####
A function f: R→R defined as f(x) = x^{2}-4x+5 is:

(A) injective but not surjective.

(B) surjective but not injective.

(C) both injective and surjective.

(D) neither injective nor surjective.

#### Solution

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__Question
2__

####
If A =
is a skew-symmetric matrix, then the value of 2a - (b + c)

#### Solution

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__Question
3__

####
If A is a square matrix of order 3 the value of |adj A| = 8 then the value of |A^{T}| is :

(A) √2

(B) -√2

(C) 8

(D) 2√2

#### Solution

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__Question
4__

####
If inverse of matrix is the matrix then value of λ is :

(A) -4

(B) 1

(C) 3

(D) 4

#### Solution

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__Question
5__

####
If [x 2 0] then value of x is :

(A) -1

(B) 0

(C) 1

(D) 2

#### Solution

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__Question
6__

__Question
7__

####
If xe^{y}= 1, then the value of dy/dx at x = 1 is :

(A) -1

(B) 1

(C) -e

(D) -1/e

#### Solution

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__Question
8__

####
Derivative of e^{sin2x} with respect to cos x is :

(A) sin x e^{sin2x}

(B) cos x e^{sin2x}

(C) -2 cos x e^{sin2x}

(D) -2 sin^{2}x cos x e^{sin2x}

#### Solution

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__Question
9__

####
The function f(x) = x/2 + 2/x has a local minima at x equal to

(A) 2

(B) 1

(C) 0

(D) -2

#### Solution

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__Question
10__

####
Given a curve y = 7x - x^{3} and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing when x = 5 is

(A) -60 units/sec

(B) 60 units/sec

(C) -70 units/sec

(D) -140 units/sec

#### Solution

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__Question
11__

####
dx is equal to :

(A) 2 log (log x)+c

(B) -1/log x + c

(C) (logx)^{3}/3 +c

(D) 3/(logx)^{3} + c

#### Solution

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__Question
12__

####
The value of dx is

(A) 1/6

(B) 1/3

(C) -1/6

(D) 0

#### Solution

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__Question
13__

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Area of the region bounded by curve y^{2} = 4x and the X-axis between x = 0
and x=1 is :

(A) 2/3

(B) 8/3

(C) 3

(D) 4/3

#### Solution

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__Question
14__

####
The order of the differential equation = 5 is :

(A) -4

(B) 3

(C) 2

(D) not defined

#### Solution

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__Question
15__

####
The position vectors of points P and Q are p^{→} and q^{→} respectively. The point
R divides line segment PQ in the ratio 3 : 1 and S is the mid-point of line
segment PR. The position vector of S is :

#### Solution

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__Question
16__

####
The angle which the line x/1 = y/-1 = z/0 makes with the positive direction of Y-axis is

(A) 5π/6

(B) 3π/4

(C) 5π/4

(D) 7π/4

#### Solution

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__Question
17__

__Question
18__

####
If A and B are events such that P(A/B) = P(B/A) ≠ 0, then :

(A) A⊂B,butA≠B

(B) A=B

(C) A⋂B≠φ

(D) P(A)=P(B)

#### Solution

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####
Questions number 19 and 20 are Assertion and Reason based questions carrying
1 mark each. Two statements are given, one labelled as Assertion (A) and the
other is labelled as Reason (R). Select the correct answer to these questons from the codes (a), (b),
(c) and (d) as given below.

(a) both Assertion (A) and Reason (R) are true and Reason (R) is the
correct explanation of the Assertion (A).

(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not
the correct explanation of the Assertion (A).

(c) Assertion (A) is true, but Reason (R) is false.

(d) Assertion (A) is false, but Reason (R) is true.

__Question
19__

####
Assertion (A) : Domain of y = cos^{-1}(x) is [-1, 1],
Reason (R) : The range of the principal value branch of y = cos^{-1}(x) is [0,π]-{π/2}

#### Solution

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__Question
20__

####
Assertion (A) : The vectors

represent the sides of a right angled triangle.

Reason (R) : Three non-zero vectors of which none of two are
collinear forms a triangle if their resultant is zero vector
or sum of any two vectors is equal to the
third.

#### Solution

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#### Section-B

This section comprises very short answer (VSA) type questions of 2 marks each.

__Question
21__

__Question
22__

####
(a) Verify whether the function f defined by

is continuous at x = 0 or not.

OR

(b) Check for differentiability of the function f defined by f(x) = |x - 5|, at the point x = 5.

#### Solution

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__Question
23__

####
The area of the circle is increasing at a uniform rate of 2 cm^{2}/sec . How fast is the circumference of the circle increasing when the radius r = 5cm ?

#### Solution

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__Question
24__

__Question
25__

####
Find the vector equation of the line passing through the point (2, 3, -5)
and making equal angles with the co-ordinate axes.

#### Solution

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#### Section-C

This section comprises short answer (SA) type questions of 3 marks each.

__Question
26__

####
(a) Find dy/dx if (cos x)^{y} = (cos y)^{x}

OR

(b) If √1-x^{2} + √1-y^{2} = a(x-y), prove that dy/dx = √(1-y^{2})/(1-x^{2})

#### Solution

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__Question
27__

####
If x = a sin^{3}θ, y = b cos_{3}θ, then find d^{2}y/dx^{2} at θ = π/4

#### Solution

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__Question
28__

__Question
29__

####
(a) Find the particular solution of the differential equation

dy/dx - 2xy = 3x^{2}e^{x2} ; y(0) = 5.

OR

(b) Solve the following differential equation :
x^{2}dy + y(x+y)dx =0

#### Solution

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__Question
30__

####
Find a vector of magnitude 4 units perpendicular to each of the vectors
and hence verify your answer.

#### Solution

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__Question
31__

####
The random variable X has the following probability distribution where a
and b are some constants:

If the mean E(X) = 3, then find values of a and b and hence determine
P(X ≥ 3).

#### Solution

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#### Section-D

This section comprises long answer (LA) type questions of 5 marks eac h.

__Question
32__

####
(a) If A = then find A^{-1} and hence solve the following system of equations :

x+2y-3z=1;
2x-3z=2;
x+2y=3

OR

(b) Find the product of the matrices

hence solve the system of linear equations :
x+2y-3z=-4;
2x+3y+2z=2;
3x-3y-4z=11

#### Solution

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__Question
33__

####
Find the area of the region bounded by the curve 4x^{2} + y^{2} = 36 using
integration.

#### Solution

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__Question
34__

####
(a) Find the co-ordinates of the foot of the perpendicular drawn from the
point (2, 3, -8) to the line 4-x/2 = y/6 = 1-z/3 Also, find the perpendicular distance of the given point from the line.

OR

(b) Find the shortest distance between the lines L_{1} & L_{2} given below :

L_{1} : The line passing through (2,-1,1) and parallel to x/1=y/1=z/3

#### Solution

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__Question
35__

####
Solve the following L.P.P. graphically :

Maximise Z = 60x + 40y

Subject to x+2y ≤ 12

2x+y ≤ 12

4x + 5y ≥ 20

x,y≥0

#### Solution

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#### Section-E

This section comprises 3 case study based questions of 4 marks each.

__Question
36__

####
(a) Students of a school are taken to a railway museum to learn about
railways heritage and itg history.

An exhibit in the museum depicted many rail lines on the track near
the railway station. Let L be the set of all rail lines on the railway
track and R be the relation on L defined by

R={(l_{1},l_{2}) : l_{1} is parallel to l_{2}}

On the basis of the above information, answer the following questions :

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation R is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the
equation y = 3x + 2, then find the set of rail lines in R related to
it.

OR

(b) Let S be the relation defined by S ={(l_{1},l_{2}) : l_{1} is perpendicular to l_{2} }
check whether the relation S is symmetric and transitive.

#### Solution

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__Question
37__

####
A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins
on the left and right are to be 1^{1/2} cm as shown below:

On the basis of the above information, answer the following questions:

(i) Write the expression for the area of the visiting card in terms of x.

(ii) Obtain the dimensions of the card of minimum area.

#### Solution

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__Question
38__

####
A departmental store sends bills to charge its customers once a month.
Past experience shows that 70% of its customers pay their first month bill
in time. The store also found that the customer who pays the bill in time
has the probability of 0.8 of paying in time next month and the customer
who doesn't pay in time has the probability of 0.4 of paying in time the
next month.

Based on the above information, answer the following questions :

(i) Let E_{1} and E_{2} respectively denote the event of customer paying or
not paying the first month bill in time.
Find P(E_{1}), P(E_{2}).

(ii) Let A denotes the event of customer paying second month's bill in
time, then find P(A|E_{1}) and P(A|E_{2}).

(iii) Find the probability of customer paying second month's bill in time.

OR

(iii) Find the probability of customer paying first month's bill in time if it
is found that customer has paid the second month's bill in time.

#### Solution

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