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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 $\mathrm{A}=\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right)$ is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.

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• Q2

If are unit vectors such that  then write the value of  $\stackrel{\to }{\mathrm{a}}·\stackrel{\to }{\mathrm{b}}+\stackrel{\to }{\mathrm{b}}·\stackrel{\to }{\mathrm{c}}+\stackrel{\to }{\mathrm{c}}·\stackrel{\to }{\mathrm{a}}.$

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• Q3

If ${\left|\stackrel{\to }{\mathrm{a}}×\stackrel{\to }{\mathrm{b}}\right|}^{2}+{\left|\stackrel{\to }{\mathrm{a}}·\stackrel{\to }{\mathrm{b}}\right|}^{2}=400$ and $\left|\stackrel{\to }{\mathrm{a}}\right|=5,$ then write the value of $\left|\stackrel{\to }{\mathrm{b}}\right|.$

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• Q4

Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes.

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• Q5

If , then write the order of matrix A.

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• Q6

If , write the value of x.

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• Q7

Find the values of a and b, if the function f defined by $f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}+3x+a& ,& x⩽1\\ bx+2& ,& x>1\end{array}\right\$ is differentiable at x = 1.

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• Q8

Differentiate if x ∈ (–1, 1)

OR

If x = sin t and y = sin pt, prove that $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{p}^{2}y=0$

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• Q9

Find the angle of intersection of the curves .

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• Q10

Evaluate :

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• Q11

Find : $\int \left(2x+5\right)\sqrt{10-4x-3{x}^{2}}dx$

OR

Find : $\int \frac{\left({x}^{2}+1\right)\left({x}^{2}+4\right)}{\left({x}^{2}+3\right)\left({x}^{2}-5\right)}dx$

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• Q12
• Q13

Solve the following differential equation :

${y}^{2}dx+\left({x}^{2}-xy+{y}^{2}\right)dy=0$

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• Q14

Solve the following differential equation:

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• Q15

If , show that $\stackrel{\to }{a}-\stackrel{\to }{d}$  is parallel to $\stackrel{\to }{b}-\stackrel{\to }{c}$, where .

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• Q16

Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(–4, 4, 4).

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• Q17

A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.

OR

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

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• Q18

Prove that :

OR

Solve for x :

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• Q19

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?

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• Q20

Using integration find the area of the region bounded by the curves and the x-axis.

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• Q21

Find the equation of the plane which contains the line of intersection of the planes and whose x-intercept is twice its z-intercept.

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• Q22

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B.

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• Q23

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :

 Tablets Iron Calcium Vitamin X 6 3 2 Y 2 3 4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

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• Q24

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| –$x,\forall x\in \mathrm{R}$. Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

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• Q25

If a, b and c are all non-zero and $\left|\begin{array}{ccc}1+\mathrm{a}& 1& 1\\ 1& 1+\mathrm{b}& 1\\ 1& 1& 1+\mathrm{c}\end{array}\right|=0,$ then prove that $\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+1=0$

OR

If $\mathrm{A}=\left(\begin{array}{ccc}\mathrm{cos}\alpha & -\mathrm{sin}\alpha & 0\\ \mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 1\end{array}\right),$ find adj·A and verify that A(adj·A) = (adj·A)A = |A| I3.

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• Q26

The sum of the surface areas of a cuboid with sides x, 2x and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of  the sum of their volumes.

OR

Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.

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