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Q5
mcq
In the determinant $$\begin{vmatrix} 7 & 5 & 1 \\ 4 & 0 & 6 \\ 5 & 3 & 2 \end{vmatrix},\ M_{23}$ is : (where $M_{ij}$ denotes the minor of element $a_{ij}$)
Correct Answer: A —
Solution
For $M_{23}$, delete the 2nd row and 3rd column of the determinant
$$\begin{vmatrix} 7 & 5 & 1 \\ 4 & 0 & 6 \\ 5 & 3 & 2 \end{vmatrix}.$$
The remaining $2\times 2$ determinant is
$$\begin{vmatrix} 7 & 5 \\ 5 & 3 \end{vmatrix} = 7\cdot 3 - 5\cdot 5 = 21 - 25 = -4.$$
So the minor $M_{23}=-4$. Since the given options do not include $-4$, the question/options appear misprinted. Mathematically, the correct value is $-4$.
Q6
mcq
If $y = \sec(\tan^{-1} x)$, then $\dfrac{dy}{dx}$ at $x = 1$ is equal to :
-
A.
2
-
B.
\dfrac{1}{2}
-
C.
1
-
D.
\dfrac{1}{2}
Correct Answer: A —
Solution
Let $y=\sec(\tan^{-1}x)$. Put $\theta=\tan^{-1}x$, so $\tan\theta=x$ and
$$\sec\theta=\sqrt{1+\tan^2\theta}=\sqrt{1+x^2}.$$
Hence
$$y=\sqrt{1+x^2}.$$
Differentiate:
$$\frac{dy}{dx}=\frac{x}{\sqrt{1+x^2}}.$$
At $x=1$,
$$\frac{dy}{dx}=\frac{1}{\sqrt{2}}.$$
This value is not present in the options, so the MCQ appears misprinted. The correct derivative at $x=1$ is $\frac{1}{\sqrt{2}}$.
Q7
mcq
The value of $k$ for which the function $f$ given by
$$f(x)=\begin{cases}2x, & x<5 \\ 2x^2, & x>5 \\ k\cos x, & x=5 \end{cases}$$ is continuous at $x=5$, is :
-
A.
6
-
B.
5
-
C.
\dfrac{5}{2}
-
D.
10
Correct Answer: D —
Solution
For continuity at $x=5$,
$$\lim_{x\to 5^-} f(x)=\lim_{x\to 5^+} f(x)=f(5).$$
Now,
$$\lim_{x\to 5^-} f(x)=\lim_{x\to 5^-} 2x=10,$$
and
$$\lim_{x\to 5^+} f(x)=\lim_{x\to 5^+} 2x^2=50.$$
Since these two one-sided limits are not equal, no value of $k$ can make the function continuous at $x=5$. Thus the question as written is inconsistent. If only the value at $x=5$ were to match the left-hand limit, then $k\cos 5=10$, but continuity still fails because the right-hand limit is $50$.
Q8
mcq
$\int x^2\cos x^2\sin x\,dx$ equals :
-
A.
\dfrac{1}{2}[\tan 2x + \cot 2x] + C
-
B.
\tan 2x\cot 2x + C
-
C.
\dfrac{1}{2}[\tan 2x\cot 2x] + C
-
D.
\dfrac{1}{2}[\cot 2x\tan 2x] + C
Correct Answer: A —
Solution
The integrand shown, $\int x^2\cos x^2\sin x\,dx$, does not match any standard simplification leading to the listed options, so the question appears misprinted. The options clearly correspond to the identity
$$\int (\tan x+\cot x)\,dx = \int \frac{2}{\sin 2x}\,dx?$$
But the given choices reduce to expressions involving $\tan 2x$ and $\cot 2x$, not a valid antiderivative of the stated integrand. Therefore, with the expression exactly as printed, none of the options is correct.
Q9
mcq
$\int_{4/3}^{4/3} x\sin x\,dx$ equals :
-
A.
$2\int_{0}^{4/3} x\sin x\,dx$
-
C.
1
-
D.
$\int_{0}^{4/3} x\sin x\,dx$
Correct Answer: B —
Solution
A definite integral with equal upper and lower limits is always zero:
$$\int_a^a f(x)\,dx=0.$$
Therefore,
$$\int_{4/3}^{4/3} x\sin x\,dx=0.$$
So the correct option is (b).
Q10
mcq
The general solution of the differential equation $y e^x\,dx = e^y\,dy$ is :
-
A.
$e^x + e^y = C$
-
B.
$e^x e^y = C$
-
C.
$e^x e^y = C$
-
D.
$e^x e^y = C$
Q11
mcq
The sum of the order and the degree of the differential equation $$x\log x\,\frac{d^2y}{dx^2}+x^3\,\frac{d^4y}{dx^4}=0$$ is :
Q12
mcq
Let $\vec a$ and $\vec b$ be two unit vectors and $\theta$ is the angle between them. $\vec a+\vec b$ is a unit vector, if :
-
A.
$\theta = 3$
-
B.
$\theta = 4$
-
C.
$\theta = 2$
-
D.
$\theta = \dfrac{3}{2}$
Q13
mcq
If $(2\hat i + 6\hat j - 2\hat k)(\hat i + \hat j + \hat k)=0$, then ____ is equal to :
Q14
mcq
If the position vectors of two points $A$ and $B$ are $\hat i + 2\hat j - 3\hat k$ and $\hat i - 2\hat j + \hat k$ respectively, then the direction cosines of the vector $\overrightarrow{BA}$ are :
-
A.
\dfrac{2}{\sqrt 6},\ \dfrac{4}{\sqrt 6},\ \dfrac{4}{\sqrt 6}
-
B.
\dfrac{1}{\sqrt 3},\ \dfrac{2}{\sqrt 3},\ \dfrac{2}{\sqrt 3}
-
C.
\dfrac{1}{\sqrt 3},\ \dfrac{2}{\sqrt 3},\ \dfrac{2}{\sqrt 3}
-
D.
\dfrac{1}{\sqrt 3},\ \dfrac{2}{\sqrt 3},\ \dfrac{2}{\sqrt 3}
Q15
mcq
The value of ____ for which the lines $\dfrac{x-1}{1}=\dfrac{y-2}{5}=\dfrac{z}{7}$ and $\dfrac{x-3}{1}=\dfrac{y}{2}=\dfrac{z-1}{3}$ are at right angles, is :
Q15
short answer
2 marks
Check the injectivity and surjectivity of the function $f:\mathbb N\to\mathbb N$, given by $f(x)=x^3$.
Q16
mcq
The solution set of the inequation $2x+y\le 5$ is :
-
A.
half plane that contains the origin.
-
B.
open half plane not containing the origin and not containing the points on the line $2x+y=5$.
-
C.
whole $xy$-plane except the points lying on the line $2x+y=5$.
-
D.
open half plane not containing the origin, but containing the points on the line $2x+y=5$.
Q17
mcq
The minimum value of $z = 3x + 8y$ subject to the constraints $x\le 20$, $y\le 10$ and $x\ge 0$, $y\ge 0$ is :
Q18
mcq
Two events $A$ and $B$ will be independent, if :
-
A.
$A$ and $B$ are mutually exclusive
-
B.
$P(A)=P(B)$
-
C.
$P(AB)=[1-P(A)][1-P(B)]$
-
D.
$P(A)+P(B)=1$
Q18
long answer
In a parallelogram ABCD, the sides AB and AD are represented by the vectors $2\hat i-4\hat j+5\hat k$ and $\hat i-2\hat j-3\hat k$ respectively. Find the unit vector parallel to its diagonal AC.
Q19
long answer
Find the angle between the pair of lines given by $\vec r=\hat i+2\hat j-2\hat k+(\hat i-2\hat j-2\hat k)$ and $\vec r=3\hat i-5\hat j+\hat k+(3\hat i+2\hat j-6\hat k)$.
Q22
short answer
2 marks
If $\cos y=x\cos(a+y)$, and $\cos a\ne 1$, prove that $\frac{dy}{dx}=\frac{a\sin y}{a\cos y}$.
Q22
long answer
3 marks
Evaluate: $\int \frac{2}{x}\,dx\,x\cos^{-1}x\sin^{-1}e$
Q23
short answer
2 marks
Find the projection of the vector $7\hat i-\hat j+8\hat k$ on the vector $\hat i+2\hat j+2\hat k$.
Q23
long answer
3 marks
Evaluate: $\int_0^4 dx\,x\tan^{-1}(\log x)$
Q25
long answer
The radius of an air bubble is increasing at the rate of 0.5 cm/s. At what rate is the surface area of the bubble increasing when the radius is 1.5 cm?
Q26
long answer
3 marks
Find: $\int x\cot^{-1}(1)\,dx$
Q27
short answer
Evaluate : $\int \frac{2}{x}\,dx$
Q27
short answer
Evaluate : $\int \frac{dx}{x\,\tan^{-1}(\log x)}$
Q29
short answer
Solve the differential equation $x\,dy-y\,dx=\sqrt{2xy}\,dx$.
Q31
short answer
From a lot of 10 bulbs which includes 2 defectives, a sample of 2 bulbs is drawn at random without replacement. Find the probability distribution of the number of defective bulbs. Hence, find the mean.
Q32
long answer
(a) A relation $R$ in the set $A = \{5, 6, 7, 8, 9\}$ is given by $R = \{(x, y) : xy$ is divisible by $2\}$. Write $R$ in roster form and prove that $R$ is an equivalence relation. Also, find the elements related to element $7$.
OR
(b) Let $A = \{3\}$ and $B = \{1\}$ be two sets. Prove that the function $f : A \to B$ given by $f(x) = 3^2$ is onto. Is the function $f$ one-one? Justify your answer.
Q33
short answer
Using matrices, solve the following system of linear equations: $x-y+2z=7$; $3x+4y-5z=5$; $2x-y+3z=12$.
Q34
short answer
Find the area of the region bounded by the curve $y=x$, the line $x=2y+3$ and the $x$-axis, using integration.
Q35
short answer
Find the foot of the perpendicular drawn from the point $(2,3,8)$ to the line $\frac{x-2}{3}=\frac{y-4}{6}=\frac{z-1}{2}$. Also, find the perpendicular distance of the given line from the given point.
Q35
long answer
(a) Find the shortest distance between the lines whose vector equations are:
$
\vec r = \hat i + 2\hat j - 4\hat k + \lambda(2\hat i + 3\hat j + 6\hat k)
$
and
$
\vec r = 3\hat i - 3\hat j - 5\hat k + \mu(2\hat i + 3\hat j + 8\hat k)
$.
OR
(b) Find the foot of the perpendicular drawn from the point $(2, 3, 8)$ to the line $\dfrac{x-3}{2} = \dfrac{y-1}{4} = \dfrac{z}{6}$. Also, find the perpendicular distance of the given line from the given point.
Q36
long answer
Case Study 1
A balloon is being inflated with the help of an air pump, and it remains spherical. Its radius, the surface area and the volume of air in it are all increasing.
Based on the above, answer the following questions:
(i) Are the quantities: radius, surface area and volume of the spherical balloon changing at the same rate or different rates, when air is filled in it? 1
(ii) Write the expressions for the surface area $(S)$ and the volume $(V)$. 1
(iii) (a) At the instant when the radius of the balloon is $6$ cm and the radius $(r)$ is increasing at the rate of $2$ cm/s, find at what rate the surface area $(S)$ of the balloon is increasing. 2
OR
(iii) (b) At the instant when the radius of the balloon is $6$ cm and the radius $(r)$ is increasing at the rate of $2$ cm/s, find at what rate the volume $(V)$ of the spherical balloon is increasing. 2
Q37
long answer
Case Study 2
A fighter-jet of the enemy is flying along the parabolic path $4y = x^2$. A soldier is located at the point $(0, 5)$ and is aiming to shoot down the jet when it is nearest to him.
Based on the above, answer the following questions:
(i) Let $(x, y)$ be the position of the jet at any instant. Express the distance between the soldier and the jet as the function $f(x)$. 1
(ii) Taking $S = [f(x)]^2$, find $\dfrac{dS}{dx}$. 1
(iii) (a) What will be the position of the jet when the soldier shoots it down? 2
OR
(iii) (b) What will be the distance between the soldier and the jet at the instant when he shoots it down? 2
Reading Passage
There are ten cards numbered $1$ to $10$ and they are placed in a box and then mixed up thoroughly. Then one card is drawn at random from the box.
Q38
long answer
Case Study 3
Read the following passage and answer the questions given below:
There are ten cards numbered $1$ to $10$ and they are placed in a box and then mixed up thoroughly. Then one card is drawn at random from the box.
Based on the above, answer the following questions:
(i) What is the probability that the number on the drawn card is greater than 4? 2
(ii) If it is known that the number on the drawn card is greater than 4, then what is the probability that it is an even number? 2
