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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 find the projection of .

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• Q2

Find λ, if the vectors are coplanar.

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• Q3

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

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• Q4

Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by ${a}_{ij}=\frac{\left|i-j\right|}{2}.$

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• Q5

Find the differential equation representing the family of curves $\mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}}$+ B, where A and B are arbitrary constants.

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• Q6

Find the integrating factor of the differential equation
$\left(\frac{{{e}^{-2}}^{\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{dx}{dy}=1$.

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• Q7

If $\mathrm{A}=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)$ find ${\mathrm{A}}^{2}-5\mathrm{A}+4\mathrm{I}$ and hence find a matrix X such that ${\mathrm{A}}^{2}-5\mathrm{A}+4\mathrm{I}+\mathrm{X}=\mathrm{O}$

OR

If

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• Q8

If $f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^{2}& ax& a\end{array}\right|$, using properties of determinants find the value of f(2x) − f(x).

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• Q9

Find :

OR

Integrate the following w.r.t. x
$\frac{{x}^{2}-3x+1}{\sqrt{1-{x}^{2}}}$

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• Q10

Evaluate :

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• Q11

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

OR
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

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• Q12
• Q13

Find the distance between the point (−1, −5, −10) and the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane xy + z = 5.

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• Q14

If sin [cot−1 (x+1)] = cos(tan1x), then find x.

OR

If (tan1x)2 + (cot−1x)2 = $\frac{5{\mathrm{\pi }}^{2}}{8}$, then find x.

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• Q15

If then find $\frac{dy}{dx}$.

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• Q16

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that ${y}^{2}\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+y=0.$

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• Q17

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

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• Q18
• Q19

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:

 School Article A B C Hand-fans 40 25 35 Mats 50 40 50 Plates 20 30 40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation.

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• Q20

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

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• Q21

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

OR

Evaluate as a limit of a sum.

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• Q22

Solve the differential equation :

$\left({\mathrm{tan}}^{-1}y-x\right)dy=\left(1+{y}^{2}\right)dx.$

OR

Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{xy}{{x}^{2}+{y}^{2}}$ given that y = 1, when x = 0.

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• Q23

If lines intersect, then find the value of k and hence find the equation of the plane containing these lines.

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• Q24

If A and B are two independent events such that then find P(A) and P(B).

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• Q25

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

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• Q26

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

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