Q4
mcq
The value of $p$ for which the vectors $2\hat{i}+p\hat{j}+\hat{k}$ and $4\hat{i}-6\hat{j}+2\hat{k}$ are perpendicular to each other, is:
- A. 3
- B. 3
- C. $\frac{3}{17}$
- D. $\frac{3}{17}$
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Q5
mcq
If the vector $\hat{i}+b\hat{j}+\hat{k}$ is equally inclined to the coordinate axes, then the value of $b$ is:
- A. 1
- B. 1
- C. 3
- D. $\frac{1}{3}$
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Q6
mcq
If $\mathbf{a}+\mathbf{b}=\hat{i}$ and $\mathbf{a}=2\hat{i}-2\hat{j}+2\hat{k}$, then $|\mathbf{b}|$ equals:
- A. $\sqrt{14}$
- B. 3
- C. $\sqrt{12}$
- D. $\sqrt{17}$
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Q7
mcq
1 mark
Direction cosines of a line perpendicular to both x-axis and z-axis are:
- A. 1, 0, 1
- B. 1, 1, 1
- C. 0, 0, 1
- D. 0, 1, 0
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Q8
mcq
1 mark
If $P(B\mid A)=0\cdot3$, $P(A)=0\cdot4$ and $P(B)=0\cdot8$, then $P(A\mid B)$ is equal to:
- A. 0·6
- B. 0·3
- C. 0·06
- D. 0·4
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Q9
mcq
1 mark
For what value of $k$ may the function be continuous?
$ f(x)=\begin{cases} kx, & x>0 \\ 3x+5, & x<0 \\ \cos x, & x=0 \end{cases}$
- B. 1
- C. $\frac{1}{2}$
- D. No value
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Q9
mcq
For what value of $k$ may the function
$$f(x)=\begin{cases}x^2\cos x, & x\neq 0\\ k, & x=0\end{cases}$$
become continuous?
- B. 1
- C. $\frac{1}{2}$
- D. No value
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Q10
mcq
1 mark
If $A$ is a $2\times 2$ matrix and $(3I+4A)(3I-4A)=x^2I$, then the value(s) of $x$ is/are:
- A. $7$
- C. 5
- D. 25
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Q10
mcq
If $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and $(3I+4A)(3I-4A)=x^2I$, then the value(s) of $x$ is/are:
- A. 7
- C. 5
- D. 25
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Q11
mcq
The general solution of the differential equation $x\,dy+(1+x^2)\,dx=dx$ is:
- A. $y=2x+\frac{3}{x^3}+C$
- B. $y=2\log x+\frac{3}{x^3}+C$
- C. $y=\frac{2}{x^2}+C$
- D. $y=2\log x+\frac{2}{x^2}+C$
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Q12
mcq
If $f(x)=a(x\cos x)$ is strictly decreasing in
- A. {0}
- B. (0, )
- C. ( , 0)
- D. ( , )
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Q13
mcq
The corner points of the feasible region in the graphical representation of a linear programming problem are $(2,72)$, $(15,20)$ and $(40,15)$. If $z=18x+9y$ be the objective function, then:
- A. $z$ is maximum at $(2,72)$, minimum at $(15,20)$
- B. $z$ is maximum at $(15,20)$, minimum at $(40,15)$
- C. $z$ is maximum at $(40,15)$, minimum at $(15,20)$
- D. $z$ is maximum at $(40,15)$, minimum at $(2,72)$
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Q14
mcq
The number of corner points of the feasible region determined by the constraints $x\ge 0$, $y\ge 0$, $2y\le x+2$, $x\ge 0$, $y\ge 0$ is:
- A. 2
- B. 3
- C. 4
- D. 5
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Q15
mcq
If for a square matrix $A$, $A^2-3A+I=O$ and $A^{-1}=xA+yI$, then the value of $x+y$ is:
- A. 2
- B. 2
- C. 3
- D. 3
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Q16
mcq
If $A^{-1}=|A|^k$, where $A$ is a $3\times 3$ matrix, then the value of $k$ is:
- A. $\frac{1}{8}$
- B. 8
- C. 2
- D. $\frac{1}{2}$
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Q17
mcq
Let $A$ be a $3\times 3$ matrix such that $|\operatorname{adj} A|=64$. Then $|A|$ is equal to:
- A. 8 only
- B. 8 only
- C. 64
- D. 8 or 8
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Q18
mcq
If $A=\begin{pmatrix}2&5\\4&3\end{pmatrix}$ and $2A+B$ is a null matrix, then $B$ is equal to:
- A. $\begin{pmatrix}4&10\\8&6\end{pmatrix}$
- B. $\begin{pmatrix}4&10\\8&6\end{pmatrix}$
- C. $\begin{pmatrix}3&10\\8&5\end{pmatrix}$
- D. $\begin{pmatrix}3&10\\8&5\end{pmatrix}$
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Q19
mcq
1 mark
Assertion (A): Equation of a line passing through the points $(1, 2, 3)$ and $(3, 1, 3)$ is $\frac{x-1}{2} = \frac{y-2}{-1} = \frac{z-3}{0}$.
Reason (R): Equation of a line passing through points $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$ is given by $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true and Reason (R) is false.
- D. Assertion (A) is false and Reason (R) is true.
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Q20
mcq
1 mark
Assertion (A): The number of onto functions from a set $P$ containing 5 elements to a set $Q$ containing 2 elements is 30.
Reason (R): Number of onto functions from a set containing $m$ elements to a set containing $n$ elements is $n^m$.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true and Reason (R) is false.
- D. Assertion (A) is false and Reason (R) is true.
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Q21
short answer
2 marks
(a) Position vectors of the points A, B and C as shown in the figure below are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. If $AC = \frac{4}{5}AB$, express $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
OR
(b) Check whether the lines given by equations $x = 2 + 2t$, $y = 7 + t$, $z = 3 + 3t$ and $x = 2$, $y = 2 + 8s$, $z = 4 + 5s$ are perpendicular to each other or not.
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Q22
short answer
2 marks
If $y = \left(x + \frac{1}{x^2}\right)^2$, then show that $(x^2 - 1)^2 \frac{dy}{dx} = 4y^2$.
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Q23
short answer
2 marks
Find the sub-intervals in which $f(x) = \frac{\log(2 + x)}{x^2 - x}$, $x > 2$ is increasing or decreasing.
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Q24
short answer
2 marks
(a) A function $f : A \to B$ defined as $f(x) = 2x$ is both one-one and onto. If $A = \{1, 2, 3, 4\}$, then find the set $B$.
OR
(b) Evaluate: $\sin^{-1}\!\left(\frac{1}{4}\right) + \sin^{-1}\!\left(\frac{3}{4}\right) + \cos^{-1}\!\left(\frac{1}{4}\right) + \cos^{-1}\!\left(\frac{3}{4}\right) + \tan^{-1}(1)$.
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Q25
short answer
2 marks
For two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$, if $|\mathbf{a} - \mathbf{b}| = |\mathbf{a} + \mathbf{b}|$, then find the angle between $\mathbf{a}$ and $\mathbf{b}$.
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Q26
long answer
3 marks
(a) Find the general solution of the differential equation: $\frac{dy}{dx} = \frac{y}{x} e^{\frac{y}{x}}$.
OR
(b) Find the particular solution of the differential equation $\frac{dy}{dx} + \cot x \cdot y = \cos^2 x$, given that when $x = \frac{\pi}{2}$, $y = 0$.
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Q26
short answer
26. (a) Find the general solution of the differential equation: $\dfrac{dy}{dx}=\dfrac{y}{x}e^{\,y/x}$.
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Q27
long answer
3 marks
Solve the following linear programming problem graphically: Maximize $P = 100x + 5y$ subject to the constraints $x + y \le 300$, $3x + y \le 600$, $y \ge x + 200$, $x, y \ge 0$.
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Q28
long answer
3 marks
(a) The probability distribution of a random variable $X$ is given below:
$\begin{array}{c|ccc}
X & 1 & 2 & 3 \\
\hline
P(X) & \frac{2}{k} & \frac{3}{k} & \frac{6}{k}
\end{array}$
(i) Find the value of $k$.
(ii) Find $P(1 < X < 3)$.
(iii) Find the mean $E(X)$.
OR
(b) Two independent events $A$ and $B$ are such that $P(A \cup B) = \frac{1}{4}$ and $P(\overline{A} \cap \overline{B}) = \frac{1}{6}$. Find $P(A)$ and $P(B)$.
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Q28
short answer
28. (a) The probability distribution of a random variable $X$ is given below: $X$: 1 2 3; $P(X)$: $\frac{2}{k}$ $\frac{3}{k}$ $\frac{6}{k}$. (i) Find the value of $k$. (ii) Find $P(1<X<3)$. (iii) Find $E(X)$, the mean of $X$.
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Q28
short answer
28. (b) A and B are independent events such that $P(A\cap B)=\frac{1}{4}$ and $P(A'\cap B)=\frac{1}{6}$. Find $P(A)$ and $P(B)$.
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Q29
long answer
3 marks
(a) Find the value of $\int_0^2 e^x \sin x\,dx$
OR
(b) Find $\int (a x \cos x)(b x \cos x)\,dx$.
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Q30
short answer
30. Evaluate: $\int_0^2 |\sin x\cos x|\,dx$.
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Q31
short answer
31. Find: $\int \dfrac{1}{x(x+1)(x+2)}\,dx$.
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Q33
long answer
5 marks
33. Find the area of the region $\{(x,y):x^2+y^2\le 1\le x+y\}$, using integration.
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Q34
long answer
5 marks
34. A relation $R$ is defined on a set of real numbers as $R=\{(x,y):x\cdot y\text{ is an irrational number}\}$. Check whether $R$ is reflexive, symmetric and transitive or not.
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Q35
long answer
5 marks
35. (b) Solve the following system of equations by matrix method: $x+2y+3z=6$, $2x-y+z=2$, $3x+2y-2z=3$.
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Q36
short answer
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2\text{ cm}^3/\text{s}$. The semi-vertical angle of the conical tank is $45^\circ$. On the basis of given information, answer the following questions:
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Q37
short answer
The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0.44$.
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Q38
short answer
1 mark
Find the volume of water in the tank in terms of its radius $r$.
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Q38
short answer
The equation of the path traced by a roller-coaster is given by the polynomial $f(x)=a(x+9)(x+1)(x-3)$. If the roller-coaster crosses the $y$-axis at a point $(0,1)$, answer the following:
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Q39
short answer
1 mark
Find rate of change of radius at an instant when $r = 2\sqrt{2}\text{ cm}$.
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Q40
short answer
2 marks
Find the rate at which the wet surface of the conical tank is decreasing at an instant when radius $r = 2\sqrt{2}\text{ cm}$.
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Q41
short answer
2 marks
Find the rate of change of height when height is $4\text{ cm}$.
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Q45
short answer
2 marks
Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.
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