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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

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

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• Q4

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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• 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

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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• Q9

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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• Q10

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm.

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• Q11

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

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• Q12

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

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• Q13

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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• Q14

If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to .  Also, find the angle which makes with .

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• Q15

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

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• 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

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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• Q18

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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• Q19

Evaluate :

OR

Evaluate :

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• Q20

Solve the following L.P.P graphically:

 Maximise Z = 20x + 10y Subject to the following constraints x + 2y ≤ 28, 3x + y ≤ 24, x ≥ 2, x, y ≥ 0

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• Q21

Show that the family of curves for which $\frac{dy}{dx}=\frac{{x}^{2}+{y}^{2}}{2xy}$, is given by x2y2 = cx.

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• Q22
• Q23

Solve the following equation for x:

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• 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

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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• Q26

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 âˆŠ A. Also find the identity element of * in A and hence find the invertible elements of A.

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• Q27

If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is $\frac{\mathrm{\pi }}{3}.$

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• Q28

If , find A–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2yz = 8.

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• Q29

Find the particular solution of the differential equation

given that y = 0 when $x=\frac{\pi }{2}$.

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