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General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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Question 1
• Q1

If , then write the cofactor of the element a21 of its 2nd row.

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• Q2

Write the sum of the order and degree of the differential equation

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• Q3

Write the solution of the differential equation

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• Q4

Find the unit vector in the direction of the sum of the vectors

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• Q5

Find the area of a parallelogram whose adjacent sides are represented by the vectors

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• Q6

Find the sum of the intercepts cut off by the plane $2x+y-z=5,$ on the coordinate axes.

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• Q7

Evaluate:
$\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\mathrm{cos}x}{1+{e}^{x}}dx$

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• Q8

Three machines E1, E2 and E3 in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E1 and E2 are defective and that 5% of those produced by machine E3 are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.

OR

Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.

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• Q9

The two vectors represent the two sides vectors respectively of triangle ABC. Find the length of the median through A.

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• Q10

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$.

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• Q11

If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.

OR

If , then find the value of θ.

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• Q12

If and I is the identity matrix of order 2, then show that A2= 4 A − 3 I. Hence find A−1.

OR

If , then find the values of a and b.

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• Q13

Using properties of determinants, prove the following :
$\left|\begin{array}{ccc}1& a& {a}^{2}\\ {a}^{2}& 1& a\\ a& {a}^{2}& 1\end{array}\right|={\left(1-{a}^{3}\right)}^{2}$

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• Q14

Evaluate :

OR

Evaluate :

$\int \frac{{x}^{2}}{\left({x}^{2}+4\right)\left({x}^{2}+9\right)}dx$

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• Q15

Find whether the following function is differentiable at x = 1 and x = 2 or not :

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• Q16

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as

The number of contacts of each type made in two cities X and Y is given in the matrix B as

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity or their social activities ?

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• Q17

Evaluate :

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• Q18

Find the point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts on the axes.

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• Q19
• Q20

Find the minimum value of (ax + by), where xy = c2.

OR

Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3.

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• Q21

Maximise z = 8x + 9y subject to the constraints given below :
2x + 3y ≤ 6
3x − 2y ≤6
y ≤ 1
x, y ≥ 0

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• Q22

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6.

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• Q23

Let f : N → ℝ be a function defined as f(x) = 4x2 + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f.

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• Q24

Using integration, find the area of the region bounded by the line x y + 2 = 0, the curve x = $\sqrt{y}$ and y-axis.

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• Q25

Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.

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• Q26

Solve the following differential equation :

OR

Solve the following differential equation :

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