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General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
(iv) Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
(v) Questions 13-23 in Section C are long-answer I type questions carrying 4 marks each.
(vi) Questions 24-29 in Section D are long-answer II type questions carrying 6 marks each.
Question 1
• Q1

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

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• Q2

Evaluate :

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• Q3

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.

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• Q4

Determine the value of the constant 'k' so that function  is continuous at x = 0.

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• Q5

Prove that if E and F are independent events, then the events E and F' are also independent.

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• Q6

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced.

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• Q7

Find $\int \frac{\mathrm{d}x}{{x}^{2}+4x+8}$

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• Q8

Show that all the diagonal elements of a skew symmetric matrix are zero.

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• Q9
• Q10

Show that the function $f\left(x\right)=4{x}^{3}-18{x}^{2}+27x-7$ is always increasing on $\mathrm{ℝ}$.

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• Q11

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

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• Q12

For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then fine the rate of change of the slope of the curve when x = 3.

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• Q13

Evaluate :

OR

Evaluate :

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• Q14

Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy, where C is parameter.

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• Q15

Let $\stackrel{\to }{\mathrm{a}}=\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}},\text{\hspace{0.17em}}\stackrel{\to }{\mathrm{b}}=\stackrel{^}{\mathrm{i}}$ and $\stackrel{\to }{\mathrm{c}}={\mathrm{c}}_{1}\stackrel{^}{\mathrm{i}}+{\mathrm{c}}_{2}\stackrel{^}{\mathrm{j}}+{\mathrm{c}}_{3}\stackrel{^}{\mathrm{k}},$ then

(a) Let c1 = 1 and c2 = 2, find c3 which makes $\stackrel{\to }{\mathrm{a}},\text{\hspace{0.17em}}\stackrel{\to }{\mathrm{b}}$ and $\stackrel{\to }{\mathrm{c}}$ coplanar.

(b) If c2 = –1 and c3 = 1, show that no value of c1 can make $\stackrel{\to }{\mathrm{a}},\text{\hspace{0.17em}}\stackrel{\to }{\mathrm{b}}$ and $\stackrel{\to }{\mathrm{c}}$ coplanar.

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• Q16

Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society?

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• Q17

Prove that

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• Q18

Using properties of determinants, prove that $\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|=9{y}^{2}\left(x+y\right).$

OR

Let , find a matrix D such that CD − AB = O.

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• Q19

Differentiate the function with respect to x.

OR

If ${x}^{m}{y}^{n}={\left(x+y\right)}^{m+n}$, prove that $\frac{{d}^{2}y}{d{x}^{2}}=0$.

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• Q20

The random variable X can take only the values 0, 1, 2, 3. Given that P(2) = P(3) = p and P(0) = 2P(1). If $\mathrm{\Sigma }{p}_{\mathit{i}}{x}_{i}^{2}=2\mathrm{\Sigma }{p}_{\mathit{i}}{x}_{\mathit{i}}$, find the value of p.

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• Q21

Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

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• Q22

Solve the following L.P.P. graphically

 Maximise Z = 4x + y Subject to following constraints x + y ≤ 50, 3x + y ≤ 90, x ≥ 10 x, y ≥ 0

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• Q23
• Q24

Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).

OR

Find the area bounded by the circle x2 + y2 = 16 and the line $\sqrt{3}\mathrm{y}=x$ in the first quadrant, using integration.

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• Q25

Solve the differential equation given that y = 1 when $x=\frac{\mathrm{\pi }}{2}$

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• Q26

Find the equation of the plane through the line of intersection of $\underset{r}{\to }·\left(2\stackrel{\mathit{^}}{i}-3\stackrel{\mathit{^}}{j}+4\stackrel{\mathit{^}}{k}\right)=1$ and $\underset{r}{\to }·\left(\stackrel{^}{i}-\stackrel{^}{j}\right)+4=0$ and perpendicular to the plane $\underset{r}{\to }·\left(2\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)+8=0$. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.

OR

Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines $\frac{x-8}{3}=\frac{\mathrm{y}+19}{-16}=\frac{\mathrm{z}-10}{7}$ and $\frac{x-15}{3}=\frac{\mathrm{y}-29}{8}=\frac{\mathrm{z}-5}{-5}$.

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• Q27

Consider f : R+ → [−5, ∞), given by f(x) = 9x2 + 6x − 5. Show that f is invertible with ${\mathrm{f}}^{-1}\left(y\right)\left(\frac{\sqrt{y+6}-1}{3}\right)$.

Hence Find
(i) f−1(10)
(ii) y if ${\mathrm{f}}^{-1}\left(y\right)=\frac{4}{3},$

where R+ is the set of all non-negative real numbers.

OR

Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = ab + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.

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• Q28

A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box.

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• Q29

If $\mathrm{A}=\left(\begin{array}{crr}2& 3& 10\\ 4& -6& 5\\ 6& 9& -20\end{array}\right)$, find A–1. Using A–1 solve the system of equations

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