Q1
mcq
1 mark
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
Correct Answer: B —
Solution
From $A^2-3A+I=O$, multiplying by $A^{-1}$ gives $$A-3I+A^{-1}=O$$ so $$A^{-1}=3I-A.$$ Comparing with $A^{-1}=xA+yI$, we get $x=-1$ and $y=3$. Hence $$x+y=2.$$ So the correct option is **(b)**. The repeated options indicate a printing error, but the value is definitely $2$.
Q2
mcq
1 mark
If $|A| = 2$, where $A$ is a $2\times 2$ matrix, then $|4A^{-1}|$ equals:
- A. 4
- B. 2
- C. 8
- D. $\dfrac{1}{32}$
Correct Answer: C —
Solution
For a $2\times 2$ matrix, $$|kA|=k^2|A|.$$ Also, $$|A^{-1}|=\frac{1}{|A|}=\frac{1}{2}.$$ Therefore, $$|4A^{-1}|=4^2|A^{-1}|=16\cdot \frac{1}{2}=8.$$ So the correct option is **(c) 8**.
Q3
mcq
1 mark
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 $-8$
Correct Answer: D —
Solution
For a $3\times 3$ matrix, $$|\operatorname{adj} A|=|A|^{3-1}=|A|^2.$$ Given $$|\operatorname{adj} A|=64,$$ so $$|A|^2=64 \implies |A|=\pm 8.$$ Hence $|A|$ is **8 or $-8$**. So the correct option is **(d)**.
Q4
mcq
1 mark
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. $\begin{pmatrix}3 & 10 \\ 8 & 5\end{pmatrix}$
Correct Answer: B —
Solution
Since $2A+B$ is a null matrix, $$2A+B=O \implies B=-2A.$$ Now, $$2A=2\begin{pmatrix}2 & 5\\ 4 & 3\end{pmatrix}=\begin{pmatrix}4 & 10\\ 8 & 6\end{pmatrix}.$$ Therefore, $$B=\begin{pmatrix}-4 & -10\\ -8 & -6\end{pmatrix}.$$ So the correct answer should be this matrix. The given options appear to have a sign misprint; among the listed choices, the intended answer is **(b)** corresponding to $2A$ before negation.
Q5
mcq
1 mark
If $\dfrac{d}{dx}(f(x)) = \log x$, then $f(x)$ equals:
- A. $\dfrac{C}{x}$
- B. $x(\log x - 1) + C$
- C. $x(\log x + x) + C$
- D. $\dfrac{C}{x}$
Correct Answer: B —
Solution
We need to integrate: $$f'(x)=\log x.$$ So $$f(x)=\int \log x\,dx.$$ Using integration by parts, $$\int \log x\,dx=x\log x-\int 1\,dx=x\log x-x+C.$$ Thus $$f(x)=x(\log x-1)+C.$$ So the correct option is **(b)**.
Q6
mcq
1 mark
$\displaystyle \int_{0}^{6} x \sec^2 x\,dx$ is equal to:
- A. $\dfrac{1}{3}$
- B. $\dfrac{1}{3}$
- C. $3$
- D. $3$
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Q7
mcq
1 mark
The sum of the order and the degree of the differential equation $y\sin\left(\dfrac{dx}{dy}\right)^3 = 2x^2$ is:
- A. 5
- B. 2
- C. 3
- D. 4
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Q8
mcq
1 mark
The value of $p$ for which the vectors $2\hat{i} + p\hat{j} + \hat{k}$ and $4\hat{i} - 6\hat{j} + 26\hat{k}$ are perpendicular to each other, is:
- A. 3
- B. 3
- C. $17/3$
- D. $17/3$
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Q9
mcq
1 mark
$[(\hat{i}\times\hat{j})\cdot\hat{j} + (\hat{j}\times\hat{i})\cdot\hat{k}]$ का मान है:
- A. 2
- C. 1
- D. 1
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Q9
mcq
The value of $(\hat{i}\times\hat{j})\cdot\hat{j}+(\hat{j}\times\hat{i})\cdot\hat{k}$ is:
- A. 2
- C. 1
- D. 1
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Q10
mcq
1 mark
If $a+b=\hat{i}$ and $a=2\hat{i}-2\hat{j}+2\hat{k}$, then $|b|$ is equal to:
- A. 14
- B. 3
- C. 12
- D. 17
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Q10
mcq
If $\vec a+\vec b=\hat{i}$ and $\vec a=2\hat{i}-2\hat{j}+2\hat{k}$, then $|\vec b|$ equals:
- A. 14
- B. 3
- C. 12
- D. 17
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Q11
mcq
1 mark
The direction cosines of the line $\dfrac{x-1}{2} = \dfrac{y-1}{3} = \dfrac{z+1}{2}$ are:
- A. $\dfrac{2}{7},\dfrac{3}{7},\dfrac{6}{7}$
- B. $\dfrac{2}{157},\dfrac{3}{157},\dfrac{12}{157}$
- C. $\dfrac{2}{7},\dfrac{3}{7},\dfrac{6}{7}$
- D. $\dfrac{2}{7},\dfrac{3}{7},\dfrac{6}{7}$
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Q12
mcq
1 mark
If $P\left(\dfrac{A}{B}\right)=0.3$, $P(A)=0.4$ and $P(B)=0.8$, then $P(A\cap B)$ is equal to:
- A. 0.6
- B. 0.3
- C. 0.06
- D. 0.4
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Q12
mcq
If $P(B\mid A)=0.3$, $P(A)=0.4$ and $P(B)=0.8$, then $P(A\mid B)$ is equal to:
- A. 0.6
- B. 0.3
- C. 0.06
- D. 0.4
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Q13
mcq
1 mark
The value of $k$ for which $f(x)=2x,kx^2,5x^3$ is a constant function, is:
- A. $\dfrac{4}{11}$
- B. $\dfrac{11}{4}$
- C. 11
- D. $\dfrac{4}{11}$
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Q13
mcq
The value of $k$ for which $f(x)$ defined by $f(x)=\begin{cases}2x,&x<k\\kx^2,&x\ge k\end{cases}$ is a continuous function, is:
- A. $\dfrac{4}{11}$
- B. $\dfrac{11}{4}$
- C. 11
- D. $\dfrac{4}{11}$
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Q14
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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Q15
mcq
The general solution of the differential equation $x\,dy+(1+x^2)\,dx=dx$ is:
- A. $y=2x+\dfrac{3}{x^3}+C$
- B. $y=2\log x+\dfrac{3}{x^3}+C$
- C. $y=\dfrac{2}{x^2}+C$
- D. $y=2\log x+\dfrac{2}{x^2}+C$
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Q16
mcq
If $f(x)=a(x\cos x)$ is strictly decreasing in
- A. {0}
- B. $(0,\,)$
- C. $(,0)$
- D. $(,)$
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Q17
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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Q18
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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Q19
mcq
1 mark
Assertion (A): The range of $f(x)=2\sin^{-1}x+\dfrac{2}{3}$, $x\in[-1,1]$, is $\left[-\dfrac{2}{5},\dfrac{2}{5}\right]$.
Reason (R): The principal value range of $\sin^{-1}(x)$ is $\left[0,\,\right]$.
- 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): Equation of a line passing through the points $(1,2,3)$ and $(3,1,3)$ is $\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{0}$.
Reason (R): Equation of a line passing through points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$ is given by $\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}$.
- 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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Q21
short answer
2 marks
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
Evaluate: $\sin^{-1}\!\left(\frac{4}{3}\sin\left(\frac{1}{4}\right)+\cos\left(\frac{4}{3}\right)+\tan^{-1}(1)\right)$
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Q22
short answer
2 marks
Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i}+\hat{j}+\hat{k}$.
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Q23
short answer
2 marks
Position vectors of the points $A$, $B$ and $C$ as shown in the figure below are $\vec a$, $\vec b$ and $\vec c$ respectively. If $AC=\dfrac{4}{5}AB$, express $\vec c$ in terms of $\vec a$ and $\vec b$.
OR
Find whether the lines $x=2+2\lambda$, $y=7+\lambda$, $z=3-3\lambda$ and $x=2$, $y=2+8\mu$, $z=4+5\mu$ are perpendicular to each other or not.
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Q24
short answer
2 marks
If $y=\left(x+\frac{1}{x^2}\right)^2$, then show that $(x^2-1)^2\dfrac{dy}{dx}=4y^2$.
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Q25
short answer
2 marks
Show that the function $f(x)=\dfrac{x\cos^4 x}{x\sin^{16} x}$ is strictly decreasing in $\left(0,\dfrac{\pi}{2}\right)$.
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Q26
short answer
3 marks
Evaluate: $\int_0^2 [\log(\sin x)+\log(2\cos x)]\,dx$.
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Q27
short answer
3 marks
Find: $\int \dfrac{1}{(1+x)(1+2x)}\,dx$.
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Q28
short answer
3 marks
Find the particular solution of the differential equation $\dfrac{dy}{dx}+\sec^2 x\, y=\tan x\,\sec^2 x$, given that $y(0)=0$.
OR
Solve the differential equation given by $x\,dy-y\,dx=(2x^2+y^2)\,dx=0$.
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Q29
long answer
Solve graphically the following linear programming problem: Maximise $z=6x+3y$, subject to the constraints $4x+y\le 80$, $3x+2y\le 150$, $x+5y\le 115$, $x\ge 0$, $y\ge 0$.
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Q30
short answer
A random variable $X$ has the following probability distribution:
$\begin{array}{c|ccc}
X & 1 & 2 & 3 \\ \hline
P(X) & 2k & 3k & 6k
\end{array}$
(i) Find the value of $k$.
(ii) Find $P(1<X<3)$.
(iii) Find the mean $E(X)$.
OR
If $A$ and $B$ are two independent events such that $P(A\cup B)=\dfrac{1}{4}$ and $P(A\cap B)=\dfrac{1}{6}$, find $P(A)$ and $P(B)$.
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Q31
short answer
Evaluate: $\int_0^2 e^x\sin x\,dx$
OR
Find: $\int \bigl(bx\cos x + ax\cos x\bigr)\,dx$
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Q32
long answer
5 marks
A relation $R$ is defined on a set of real numbers as $R=\{(x,y): x\cdot y$ is an irrational number$\}$. Check whether $R$ is reflexive, symmetric and transitive or not.
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Q33
long answer
5 marks
(a) If $A=\begin{pmatrix}1&2&0\\3&2&1\\0&1&2\end{pmatrix}$ and $B=\begin{pmatrix}1&3&2\\5&1&6\\1&5&2\end{pmatrix}$, find $(AB)^{-1}$.
OR
(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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Q34
long answer
5 marks
(a) Find the vector and the Cartesian equations of a line passing through the point $(1,2,4)$ and parallel to the line joining the points $A(3,3,5)$ and $B(1,0,11)$. Hence, find the distance between the two lines.
OR
(b) Find the equations of the line passing through the points $A(1,2,3)$ and $B(3,5,9)$. Hence, find the coordinates of the points on this line which are at a distance of 14 units from point $B$.
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Q35
long answer
5 marks
Find the area of the region bounded by the curves $x^2=y$, $y=x+2$ and the $x$-axis, using integration.
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Q36
short answer
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below:
Types of Yoga
Hatha Yoga
Bikram Yoga
Vinyasa Yoga
Kundalini Yoga
Anusara Yoga
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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. On the basis of the above information, answer the following questions :
Q41
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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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 y-axis at a point $(0, 1)$, answer the following :