Q1
mcq
1 mark
If $A=\begin{bmatrix}2 & -3 & 0\\ 1 & 2 & 1\\ 1 & -1 & 0\end{bmatrix}$, then the value of $|A\,\operatorname{adj}(A)|$ is:
- A. -1
- B. 1
- C. 2
- D. 3
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Q2
mcq
1 mark
For two matrices $A$ and $B$, given that $A^{-1}=\frac{1}{4}B$, then inverse of $(4A)$ is:
- A. 4B
- B. B
- C. $\frac{1}{4}B$
- D. $\frac{1}{16}B$
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Q4
mcq
1 mark
The number of discontinuities of the function $f$ given by
$$f(x)=\begin{cases}-x^2, & x<0\\ e^x, & 0\le x\le 1\\ 2x, & x>1\end{cases}$$
is:
- B. 1
- C. 2
- D. 3
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Q5
mcq
1 mark
Let $y=f\!\left(\frac{1}{x}\right)$ and $f'(x)=x^3$. What is the value of $\frac{dy}{dx}$ at $x=\frac{1}{2}$?
- A. $-\frac{1}{64}$
- B. $-\frac{1}{32}$
- C. $-32$
- D. $-64$
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Q6
mcq
1 mark
If $y=x\sec\log x$, then the value of $\frac{dy}{dx}$ at $x=\frac{2}{16\pi}$ is:
- A. $\frac{1}{\pi}$
- B. $\pi$
- C. $\frac{1}{2}$
- D. $\frac{1}{4}$
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Q7
mcq
1 mark
If $x=3\cos\theta$ and $y=5\sin\theta$, then $\frac{dx}{dy}$ is equal to:
- A. $-\frac{5}{3}\tan\theta$
- B. $-\frac{3}{5}\cot\theta$
- C. $-\frac{3}{5}\tan\theta$
- D. $-\frac{5}{3}\cot\theta$
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Q8
mcq
1 mark
For the function $f(x)=x^3$, $x=0$ is a point of:
- A. local maxima
- B. local minima
- C. non-differentiability
- D. inflexion
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Q9
mcq
1 mark
The greatest integer function defined by $f(x)=[x]$, $1<x<3$ is not differentiable at $x=$
- B. 1
- C. 2
- D. $\frac{2}{3}$
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Q10
mcq
1 mark
If the radius of a circle is increasing at the rate of $0.5$ cm/s, then the rate of increase of its circumference is:
- A. $\frac{3}{2}\pi$ cm/s
- B. $\pi$ cm/s
- C. $\frac{3}{4}\pi$ cm/s
- D. $2\pi$ cm/s
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Q11
mcq
1 mark
$\int_{-\pi/4}^{3\pi/4} x\cos x\,dx$ is equal to:
- B. $-1$
- C. 1
- D. 2
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Q12
mcq
1 mark
$\int x e^{3-x}(1-x)^3\,dx$ is equal to:
- A. $x(1-x)e^{2x/3}+C$
- B. $2x(1-x)e^{2-x}+C$
- C. $(1-x)e^x+C$
- D. $2x(1-x)e^x+C$
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Q13
mcq
1 mark
The area (in sq. units) of the region bounded by the curve $y=x$, $x$-axis, $x=0$ and $x=2$ is:
- A. $\frac{2}{3}$
- B. $\frac{1}{2}\log 2$
- C. 2
- D. 4
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Q14
mcq
1 mark
The number of arbitrary constants in the general solution of the differential equation $\frac{dy}{dx}+y=0$ is:
- B. 1
- C. 2
- D. 3
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Q15
mcq
1 mark
What is the value of the ratio of projection of $\vec a$ on $\vec b$ to projection of $\vec b$ on $\vec a$ for vectors $\vec a=2\hat i-3\hat j-6\hat k$ and $\vec b=2\hat i-2\hat j+\hat k$?
- A. $\frac{7}{3}$
- B. $\frac{3}{7}$
- C. $\frac{3}{4}$
- D. $\frac{7}{4}$
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Q15
mcq
What is the value of $\frac{\text{projection of }\vec a\text{ on }\vec b}{\text{projection of }\vec b\text{ on }\vec a}$ for vectors $\vec a = 2\hat i - 3\hat j - 6\hat k$ and $\vec b = 2\hat i - 2\hat j + \hat k$?
- A. $\frac{7}{3}$
- B. $\frac{3}{7}$
- C. $\frac{3}{4}$
- D. $\frac{7}{4}$
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Q16
mcq
1 mark
The direction ratios of the line $\frac{1-x}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ are:
- A. 3, 1, 2
- B. 4, 3, 2
- C. 9, $-3$, 2
- D. 9, 3, 2
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Q17
mcq
1 mark
The Cartesian equation of the line passing through the point $(1,-3,2)$ and parallel to the line $\vec r=2\hat i-\hat k+\lambda(\hat i+\hat j+2\hat k)$ is:
- A. $\frac{1-x}{2}=\frac{y+3}{0}=\frac{2-z}{1}$
- B. $\frac{x+1}{1}=\frac{3-y}{1}=\frac{z+2}{2}$
- C. $\frac{x-1}{2}=\frac{3-y}{0}=\frac{z+2}{1}$
- D. $\frac{1-x}{1}=\frac{y+3}{1}=\frac{z-2}{2}$
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Q18
mcq
If $\vec a$ and $\vec b$ are two vectors such that $\vec a\cdot\vec b>0$ and $|\vec a\cdot\vec b|=|\vec a\times\vec b|$, then the angle between $\vec a$ and $\vec b$ is:
- A. $\frac{\pi}{4}$
- B. $\frac{\pi}{3}$
- C. $\frac{3\pi}{2}$
- D. $\frac{4\pi}{3}$
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Q19
mcq
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. Assertion (A): $\cos^{-1}(\cos\frac{\pi}{6})$ is equal to $\frac{\pi}{6}$. Reason (R): The range of the principal value branch of the function $y=\cos^{-1}x$ is $[0,\pi]$.
- 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, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (R) is true.
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Q20
mcq
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. Assertion (A): If $R$ and $S$ are two events such that $P(R\mid S)=1$ and $P(S)>0$, then $S\subset R$. Reason (R): If two events $A$ and $B$ are such that $P(A\cap B)=P(B)$, then $A\subset B$.
- 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, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (R) is true.
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Q21
short answer
2 marks
Find the value of $\cos^{-1}\left(\frac{1}{2}\right)-\tan^{-1}\left(-\frac{1}{3}\right)+\cosec^{-1}(-2)$.
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Q22
short answer
2 marks
(a) If $y=(\sin^{-1}x)^2$, then find $(1-x^2)^2\frac{d^2y}{dx^2}-x\frac{dy}{dx}$. OR (b) If $y^x=x^y$, then find $\frac{dy}{dx}$.
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Q23
short answer
2 marks
Given that $f(x)=x\log x$, find the point of local maximum of $f(x)$.
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Q24
short answer
3 marks
(a) Find: $\int \frac{3}{x^3-1}\,dx$ OR (b) Evaluate: $\int_0^4 |2x-4|\,dx$
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Q25
short answer
3 marks
Find the angle between the lines $\frac{x-5}{7}=\frac{2y+5}{5}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$.
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Q26
short answer
3 marks
(a) Find a matrix $A$ such that $A\begin{bmatrix}2&-1&0&4\end{bmatrix}=\begin{bmatrix}16&0&10&17\end{bmatrix}$. Also, find $A^{-1}$. OR (b) Given a square matrix $A$ of order 3 such that $A^2=\begin{bmatrix}3&-2&2\\0&1&0\\4&4&-3\end{bmatrix}$, show that $A^3=A^{-1}$.
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Q27
short answer
If $x\sin(a+y)-\sin y=0$, prove that $$\frac{dy}{dx}=\frac{\sin(ax)}{\sin(ay)}.$$ OR Find $\frac{dy}{dx}$, if $y=(\cos x)^x+\cos^{-1}x$.
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Q28
short answer
Find: $$\int \cos x\cos 2x\,dx$$ OR Find: $$\int \frac{dx}{(1+4x)^{2x-5}}$$
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Q29
short answer
Find the general solution of the differential equation $$y\,dx-x\,dy+(x\log x)\,dx=0.$$
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Q30
short answer
If the vectors $\vec a$, $\vec b$ and $\vec c$ represent the three sides of a triangle, then show that $\vec a\times\vec b=\vec b\times\vec c=\vec c\times\vec a$.
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Q31
short answer
It is known that 20% of the students in a school have above 90% attendance and 80% of the students are irregular. Past year results show that 80% of students who have above 90% attendance and 20% of irregular students get ‘A’ grade in their annual examination. At the end of a year, a student is chosen at random from the school and is found to have an ‘A’ grade. What is the probability that the student is irregular?
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Q32
long answer
5 marks
Check whether the relation $S$ in the set of all real numbers $(\mathbb{R})$ defined by $S=\{(a,b):a\le b^3\}$ is reflexive, symmetric or transitive.
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Q33
long answer
5 marks
Using integration, find the area of the region bounded by the curve $y=2^{-4x}$, the lines $x=-2$ and $x=3$ and the x-axis.
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Q33
long answer
5 marks
(a) Using integration, find the area of the region bounded by the curve $y = 2 - 4x$, the lines $x = -2$ and $x = 3$ and the x-axis.
OR
(b) Using integration, evaluate the area of the region bounded by the curve $y = x^2$, the lines $y = 1$ and $y = 3$ and the y-axis.
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Q34
long answer
5 marks
Find the shortest distance between the lines $$\frac{3x}{8}=\frac{9y-16}{10}=\frac{7z}{1}$$ and $$\frac{3x}{15}=\frac{8y}{29}=\frac{5-5z}{1}.$$
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Q34
long answer
5 marks
Find the point of intersection of the lines $$\vec r=\hat i-\hat j+6\hat k+l(3\hat i-\hat k),$$ and $$\vec r=-3\hat j+3\hat k+m(\hat i+2\hat j-\hat k).$$ Also, find the vector equation of the line passing through the point of intersection of the given lines and perpendicular to both the lines.
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Q34
long answer
5 marks
(a) Find the shortest distance between the lines
$\dfrac{x-3}{8} = \dfrac{y+16}{9} = \dfrac{z-7}{10}$
and
$\dfrac{x-3}{15} = \dfrac{y-8}{29} = \dfrac{z+5}{5}$.
OR
(b) Find the point of intersection of the lines
$\vec r = \hat i - \hat j + 6\hat k + \lambda (3\hat i - \hat k)$, and
$\vec r = -3\hat j + 3\hat k + \mu (\hat i + 2\hat j - \hat k)$.
Also, find the vector equation of the line passing through the point of intersection of the given lines and perpendicular to both the lines.
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Q35
long answer
5 marks
Solve the following linear programming problem graphically: Minimise $Z=6x+7y$ subject to
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Q35
long answer
5 marks
Solve the following linear programming problem graphically:
Minimise $Z = 6x + 7y$
subject to constraints
$x + 2y \geq 240$
$3x + 4y \leq 620$
$2x + y \geq 180$
$x, y \geq 0$.
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Q36
short answer
A sandbag is dropped from a balloon at a height of 60 metres.
When the angle of elevation of the sun is $30^\circ$, the position of the sandbag is given by the equation $y = 60 - 4.9t^2$, where $y$ is the height of the sandbag above the ground and $t$ is the time in seconds.
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Q37
short answer
A salesman receives a commission for each sale he makes together with a fixed daily income. The number of sales he makes in a day along with their probabilities are given in the table below:
$X : 0\ 1\ 2\ 3\ 4\ 5$
$P(X) : 0.42\ 3k\ 0.3\ 0.05\ 2k\ 0.03$
His daily income $Y$ (in ₹) is given by:
$Y = 800X + 50$
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A sandbag is dropped from a balloon at a height of 60 metres. When the angle of elevation of the sun is $30^\circ$, the position of the sandbag is given by the equation $y = 60 - 4.9t^2$, where $y$ is the height of the sandbag above the ground and $t$ is the time in seconds.
Q41
short answer
Find the relation between $x$ and $y$, where $x$ is the distance of the shadow at $P$ from the point $Q$ and $y$ is the height of the sandbag above the ground.
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Q42
short answer
After how much time will the sandbag be 35 metres above the ground?
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Q43
short answer
Find the rate at which the shadow of the sandbag is travelling along the ground when the sandbag is at a height of 35 metres.
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Q44
short answer
How fast is the height of the sandbag decreasing when 2 seconds have elapsed?
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A salesman receives a commission for each sale he makes together with a fixed daily income. The number of sales he makes in a day along with their probabilities are given in the table below: $X : 0\ 1\ 2\ 3\ 4\ 5$ $P(X) : 0.42\ 3k\ 0.3\ 0.05\ 2k\ 0.03$ His daily income $Y$ (in ₹) is given by: $Y = 800X + 50$
Q48
short answer
Calculate the expected weekly income of the salesman assuming he works five days per week.
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Q49
short answer
Formulate the linear equations in $x$ and $y$ to represent the given information.
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Q50
long answer
Find the dimensions of the plot of land by matrix method.
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