Q1
mcq
1 mark
A relation R in a set A is called Reflexive relation if
- A. (a, a) ∈ R for all a ∈ A
- B. (a, a) ∈ R for atleast one a ∈ A
- C. (a, b) ∈ R implies (b, a) ∈ R
- D. (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R
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Q2
mcq
1 mark
The principal value of $\sin^{-1}(\frac{1}{\sqrt{2}})$ is
- A. $\frac{\pi}{2}$
- B. $\frac{\pi}{3}$
- C. $\frac{\pi}{4}$
- D. $\frac{\pi}{6}$
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Q3
mcq
1 mark
Match List - I with List - II.
List - I
A) Domain of $\sin^{-1} x$
B) Range of $\tan^{-1} x$
C) Range of $\cos^{-1} x$
List - II
i) $(-\frac{\pi}{2}, \frac{\pi}{2})$
ii) $[0, \pi]$
iii) $[-1, 1]$
Choose the correct answer from the options given below :
- A. A-i, B-ii, C-iii
- B. A-iii, B-ii, C-i
- C. A-ii, B-i, C-iii
- D. A-iii, B-i, C-ii
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Q4
mcq
1 mark
For a $2 \times 2$ matrix $A=[a_{ij}]$ whose elements are given by $a_{ij}=2i-j$ then A is equal to
- A. $\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$
- B. $\begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix}$
- C. $\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}$
- D. $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
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Q5
mcq
1 mark
Let A be a nonsingular matrix of order $3 \times 3$, then $|adj A|$ is equal to
- A. $|A|$
- B. $3|A|$
- C. $|A|^3$
- D. $|A|^2$
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Q6
mcq
1 mark
If $f(x) = \cos 2x$, then $f'(\frac{\pi}{4})$ is
- A. 2
- B. –2
- C. $\sqrt{2}$
- D. $-\sqrt{2}$
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Q7
mcq
1 mark
For the given figure consider the following statements 1 and 2 :
Statement 1 : Left hand derivative of $y=f(x)$ at $x=1$ is –1.
Statement 2 : The function $y=f(x)$ is differentiable at $x=1$.
Then which of the following are true?
- A. Statement 1 is true, Statement 2 is false
- B. Statement 1 is false, Statement 2 is true
- C. Both Statements 1 and 2 are true
- D. Both Statements 1 and 2 are false
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Q7
mcq
If Statement 1 : Left hand derivative of $f(x) = |x|$ at x = 0 is –1. Statement 2 : The derivative of $f(x) = |x|$ exists at x = 0. Then which of the following is true?
- A. Statement 1 is true, Statement 2 is false
- B. Statement 1 is false, Statement 2 is true
- C. Statement 1 and 2 both are true
- D. Statement 1 and 2 both are false
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Q8
mcq
1 mark
The absolute maximum value of the function f given by $f(x)=x^3$, $x \in [-2, 2]$ is
- A. 2
- C. –2
- D. 8
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Q9
mcq
1 mark
$\int e^x(\sin x - \cos x)dx$ is
- A. $-e^x \cos x$
- B. $e^x \cos x$
- C. $e^x \sin x$
- D. $e^x \sin^2 x$
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Q10
mcq
1 mark
The degree of differential equation $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + e^{\frac{dy}{dx}} = 0$ is
- A. 1
- B. 3
- C. 2
- D. not defined
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Q11
mcq
1 mark
The direction cosines of the vector $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ are
- A. $\frac{1}{\sqrt{5}}, \frac{-1}{\sqrt{5}}, \frac{2}{\sqrt{5}}$
- B. $\frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}}, \frac{2}{\sqrt{6}}$
- C. $\frac{1}{6}, \frac{-1}{6}, \frac{2}{6}$
- D. $\frac{-1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}$
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Q12
mcq
1 mark
The angle between two vectors $\vec{a}$ and $\vec{b}$ with $|\vec{a}| = \sqrt{3}$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = \sqrt{6}$ is
- A. $\frac{\pi}{6}$
- B. $\frac{\pi}{3}$
- C. $\frac{\pi}{4}$
- D. $\frac{\pi}{2}$
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Q13
mcq
1 mark
The equation of y-axis in space is
- A. x = 0, y = 0
- B. x = 0, z = 0
- C. y = 0, z = 0
- D. y = 0
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Q14
mcq
If $P(A) = \frac{1}{2}, P(B|A) = \frac{2}{3}$ then $P(A \cap B) = $ ___________
- A. $\frac{1}{3}$
- B. $\frac{1}{2}$
- C. 1
- D. $\frac{3}{5}$
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Q15
mcq
1 mark
Assertion [A] : For two events E and F if $P(E) = \frac{1}{5}, P(F) = \frac{1}{2}$ and $P(E | F) = \frac{1}{5}$ then E and F are independent events.
Reason [R] : If E and F are two independent events then $P(E | F) = P(E)$
Then which of the following are true?
- A. [A] is true but [R] is false
- B. Both [A] and [R] are false
- C. Both [A] and [R] are true
- D. [A] is false but [R] is true
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Q16
short answer
1 mark
The value of $\cos(\sec^{-1}(2) - \sin^{-1}(\frac{\sqrt{3}}{2}))$ is _________.
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Q17
short answer
1 mark
If $y = \sin^{-1}(\cos x)$ then $\frac{dy}{dx} = $ __________.
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Q18
short answer
1 mark
The value of $\int_{7}^{13} 1 dx = $ __________.
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Q19
short answer
1 mark
The projection of vector $\hat{i} + \hat{j}$ along the vector $\hat{i} - \hat{j}$ is __________.
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Q20
short answer
1 mark
If $P(A \cap B) = \frac{4}{13}$ and $P(B) = \frac{9}{13}$ then $P(A' | B) = $ __________.
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Q21
short answer
2 marks
Find the equation of the line through the points (1, 2) and (3, 6) using determinants.
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Q22
short answer
2 marks
If $\sqrt{x} + \sqrt{y} = \sqrt{10}$ then show that $\frac{dy}{dx} + \sqrt{\frac{y}{x}} = 0$.
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Q23
short answer
2 marks
A balloon which is always remains spherical has a variable radius. Find the rate at which its volume is increasing with radius when the radius is 10 cms.
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Q24
short answer
2 marks
Find the interval in which the function given by $f(x) = 4x^3 - 6x^2 - 72x + 30$ is decreasing.
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Q25
short answer
2 marks
Find $\int \cot x \cdot \log(\sin x) dx$.
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Q26
short answer
2 marks
Verify that the function $y = a \sin x + b \cos x$ is a solution of differential equation $\frac{d^2y}{dx^2} + y = 0$.
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Q27
short answer
2 marks
If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$ then find unit vector parallel to the vector $2\vec{a} - \vec{b} + 3\vec{c}$.
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Q28
short answer
2 marks
If the lines $\frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2}$ and $\frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5}$ are perpendicular to each other, then find the value of k.
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Q29
short answer
2 marks
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls black?
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Q30
long answer
3 marks
Check whether the relation R in $\mathbb{R}$ defined by $R = \{(a, b) : a \leq b^3\}$ is reflexive, symmetric and transitive.
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Q31
long answer
3 marks
Prove that $\tan^{-1}(\frac{63}{16}) = \sin^{-1}(\frac{5}{13}) + \cos^{-1}(\frac{3}{5})$.
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Q32
long answer
3 marks
Express $\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
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Q33
long answer
3 marks
Find $\frac{dy}{dx}$ if $x = a(\cos t + \log(\tan(\frac{t}{2})))$ and $y = a \sin t$.
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Q34
long answer
3 marks
Find the two positive numbers x and y such that $x + y = 60$ and $xy^3$ is maximum.
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Q35
long answer
3 marks
Evaluate $\int \frac{2x}{x^2 + 3x + 2} dx$.
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Q36
long answer
3 marks
Find the area of triangle ABC where position vectors of A, B, C are $\hat{i} - \hat{j} + 2\hat{k}$, $2\hat{j} + \hat{k}$, $\hat{j} + 3\hat{k}$ respectively.
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Q37
long answer
3 marks
Derive the equation of a line in space through a given point and parallel to a given vector $\vec{b}$ in the vector form.
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Q38
long answer
3 marks
In two identical boxes, box I contains 2 gold coins, while box II contains one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also a gold?
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Q39
long answer
5 marks
If $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$ and $f: A \to B$ is a function defined by $f(x) = \frac{x-2}{x-3}$. Is f one-one and onto? Justify your answer.
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Q40
long answer
5 marks
If $A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}$, verify that $(AB)' = B'A'$.
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Q41
long answer
5 marks
Solve the following system of linear equations by matrix method: $4x + 3y + 2z = 60, 2x + 4y + 6z = 90, 6x + 2y + 3z = 70$.
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Q42
long answer
5 marks
If $y = (\tan^{-1} x)^2$ then show that $(x^2 + 1)^2 y_2 + 2x(x^2 + 1)y_1 = 2$.
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Q43
long answer
5 marks
Find the integral of $\frac{1}{x^2 + a^2}$ with respect to ‘x’ and hence find $\int \frac{1}{x^2 - 6x + 13} dx$.
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Q44
long answer
5 marks
Find the area of circle $x^2 + y^2 = a^2$ by method of integration.
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Q45
long answer
5 marks
Solve the differential equation $\cos^2 x \frac{dy}{dx} + y = \tan x$ $(0 \leq x < \frac{\pi}{2})$.
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Q46
long answer
6 marks
Prove that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$ and hence evaluate $\int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) dx$.
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Q46
long answer
6 marks
Solve the following Linear Programming Problem graphically : Minimise and Maximise $Z = 5x + 10y$ Subject to $x + 2y \leq 120, x + y \geq 60, x - 2y \geq 0, x \geq 0, y \geq 0$.
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Q47
long answer
4 marks
If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = O$ and hence find $A^{-1}$.
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Q47
long answer
4 marks
Determine the value of k if $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x} & x \neq \frac{\pi}{2} \\ 3 & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.
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