Q1
mcq
1 mark
The inverse of $\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ is :
- A. $\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$
- B. $\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$
- C. $\begin{bmatrix} -3 & 5 \\ 1 & -2 \end{bmatrix}$
- D. $\begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix}$
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Q2
mcq
1 mark
The centre of the hyperbola $\frac{(x-1)^2}{16} - \frac{(y-1)^2}{25} = 1$ is :
- A. $(\frac{1}{2}, \frac{1}{2})$
- B. $(-1, 1)$
- C. $(1, -1)$
- D. $(0, 0)$
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Q3
mcq
1 mark
The order and degree of the differential equation $\frac{dy}{dx} + \left(\frac{d^2y}{dx^2}\right)^3 = 0$ are :
- A. 2, 6
- B. 2, 3
- C. 2, 4
- D. 3, 3
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Q4
mcq
1 mark
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six sided die and 1, 2, 3, 4 of a four sided die is rolled and the sum is determined. If the random variable X denote the sum, then the number of elements in the inverse image of 7 is :
- A. 3
- B. 1
- C. 4
- D. 2
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Q5
mcq
1 mark
If $|z|=1$, then the value of $\frac{1+z}{1+\frac{1}{z}}$ is :
- A. $\frac{1}{z}$
- B. $z$
- C. 1
- D. $z$
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Q6
mcq
1 mark
The value of $\int_{-\pi/2}^{\pi/2} \sin^2 x \cos^2 x dx$ is :
- B. $\frac{3}{2}$
- C. $\frac{2}{3}$
- D. $\frac{1}{2}$
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Q7
mcq
1 mark
The function $f(x)=x^2$, in the interval $[0, \infty)$ is :
- A. cannot be determined
- B. increasing function
- C. increasing and decreasing function
- D. decreasing function
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Q8
mcq
1 mark
The volume of the parallelepiped with its edges represented by the vectors $\vec{i}+\vec{j}, \vec{i}+\vec{j}+\pi\vec{k}$ is :
- A. $\pi$
- B. $\frac{2}{\pi}$
- C. $\frac{4}{\pi}$
- D. $\frac{3}{\pi}$
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Q9
mcq
1 mark
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R ?
- A. $a * b = a$
- B. $a * b = \min(a, b)$
- C. $a * b = a^b$
- D. $a * b = \max(a, b)$
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Q10
mcq
1 mark
The position of a particle ‘s’ moving at any time t is given by $s(t)=5t^2-2t-8$. The time at which the particle is at rest, is :
- A. 1
- C. 3
- D. 1/3
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Q11
mcq
1 mark
If the function $f(x) = \frac{1}{b-a}$ for $a < x < b$, represents a probability density function of a continuous random variable X, then which of the following cannot be the values of a and b ?
- A. 7 and 19
- B. 0 and 12
- C. 16 and 24
- D. 5 and 17
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Q12
mcq
1 mark
If $P(x, y)$ be any point on $16x^2 - 125y^2 = 400$ with foci $F_1(3, 0)$ and $F_2(-3, 0)$, then $|PF_1 - PF_2|$ is :
- A. 10
- B. 8
- C. 12
- D. 6
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Q13
mcq
1 mark
If the planes $\vec{r} \cdot (2\vec{i} - \lambda\vec{j} + \vec{k}) = 0$ and $\vec{r} \cdot (4\vec{i} + 5\vec{j} - \mu\vec{k}) = 0$ are parallel, then the values of $\lambda$ and $\mu$ are respectively :
- A. $\frac{1}{2}, -2$
- B. $\frac{1}{2}, -\frac{1}{2}$
- C. $\frac{1}{2}, 2$
- D. $\frac{1}{2}, -2$
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Q15
mcq
1 mark
The solution of $\frac{dy}{dx} + P(x)y = 0$ is :
- A. $x = ce^{-\int P dy}$
- B. $y = ce^{\int P dx}$
- C. $x = ce^{\int P dy}$
- D. $y = ce^{-\int P dx}$
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Q16
mcq
1 mark
$\int_{0}^{\pi/2} \sin^7 x dx$ =
- A. $\pi/2$
- B. $\int_{0}^{\pi/2} \cos^7 x dx$
- D. 1
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Q17
mcq
1 mark
The value of $\sin^{-1} \frac{1}{2} + \cos^{-1} \frac{1}{2}$ is :
- B. $\pi/2$
- C. $\pi/3$
- D. $\pi$
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Q18
mcq
1 mark
If A, B and C are invertible matrices of some order, then which one of the following is not true ?
- A. $\det(A^{-1}) = (\det A)^{-1}$
- B. $\text{adj } A = |A|A^{-1}$
- C. $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$
- D. $\text{adj }(AB) = (\text{adj } A)(\text{adj } B)$
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Q19
mcq
1 mark
The value of the complex number $(i^{25})^3$ is equal to :
- A. 1
- B. i
- C. -i
- D. -1
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Q20
mcq
1 mark
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in calculation of the volume is (in cubic cm) :
- A. 2
- B. 0.4
- C. 4.8
- D. 0.45
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Q21
long answer
Show that the three vectors $2\hat{i} + 3\hat{j} + 2\hat{k}$, $\hat{i} - \hat{j} + \hat{k}$ and $3\hat{i} + 3\hat{j} + \hat{k}$ are coplanar.
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Q22
long answer
Prove that $\lim_{x \to \infty} \frac{x^m}{e^x}$, where $m$ is a positive integer, is $\infty$.
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Q23
short answer
If $g(x) = x^2 + 1 \sin x$, then find $dg$.
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Q24
long answer
Show that the solution of $\frac{dy}{dx} = \frac{1-y^2}{1-x^2}$ is $\sin^{-1} y = \sin^{-1} x + C$ (or) $\sin^{-1} x = \sin^{-1} y + C$.
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Q25
long answer
If $X$ is the random variable with distribution function $F(x)$, given by:
$F(x) = 0$; $x < 0$
$F(x) = x^2$; $0 \le x < 1$
$F(x) = 1$; $1 \le x$
then prove that the p.d.f. is
$f(x) = 0$; $x \le 0$
$f(x) = 2x$; $0 < x < 1$
$f(x) = 0$; otherwise
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Q26
long answer
The probability density function of $X$ is given by $f(x) = k e^{-2x}$ for $x > 0$, $0$ for $x \le 0$. Prove that the value of $k$ is $2$.
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Q27
long answer
Show that the differential equation corresponding to $y = A \sin x$, where $A$ is an arbitrary constant, is $\frac{dy}{dx} = y \tan x$.
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Q28
long answer
Show that the rank of the matrix $\begin{bmatrix} 0 & 1 & 2 & 1 \\ 0 & 2 & 4 & 3 \\ 8 & 1 & 0 & 2 \end{bmatrix}$ is 3.
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Q29
long answer
If $A = \begin{bmatrix} 8 & -4 \\ 5 & -3 \end{bmatrix}$, verify that $A(\text{adj } A) = (\text{adj } A)A = |A|I$.
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Q30
long answer
Show that the points $-\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $-\frac{1}{2} - i\frac{\sqrt{3}}{2}$ are the vertices of an equilateral triangle of side length $\sqrt{3}$.
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Q31
long answer
If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Show that the volume of the cuboid is 60 cubic units.
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Q32
long answer
Prove that the equation of the parabola with focus (4, 0) and directrix $x = -4$ is $y^2 = 16x$.
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Q33
long answer
A force $\vec{F} = 3\hat{i} + 10\hat{j} - 3\hat{k}$ acts on a particle which is displaced from the point with position vector $4\hat{i} - 3\hat{j} - 2\hat{k}$ to the point with position vector $6\hat{i} + \hat{j} - 3\hat{k}$. Show that the work done by the force is 69 units.
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Q34
long answer
Show that the point on the curve $y = x^2 - 5x + 4$ at which the tangent is parallel to the line $3x + y = 7$ is (1, 0).
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Q35
long answer
An egg of a particular bird is spherical in shape. If the radius to the inside of the shell is 4 mm and radius to the outside of the shell is 4.2 mm, prove that the approximate volume of the shell is $12.8\pi$ mm$^3$.
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Q36
long answer
Define an operation $*$ on $Q$, the set of all rational numbers, as follows: $a * b = \frac{a+b}{2}$. Examine the closure and commutative properties satisfied by $*$ on $Q$.
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Q37
long answer
Show that $\int_0^1 \frac{x^2}{1+x^2} dx = 1 - \frac{\pi}{4}$.
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Q38
long answer
Solve the system of equations $x - y + 2z = 2$, $2x + y + 4z = 7$, $4x - y + z = 4$ by Cramer’s rule.
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Q39
long answer
A camera is accidentally knocked off an edge of a cliff 400 ft. high. The camera falls a distance of $s = -16t^2$ in $t$ seconds. Show that the camera hits the ground when $t=5$ seconds and also prove that the velocity when it hits the ground is $-160$ ft./sec.
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Q40
long answer
If $z = x + iy$ is a complex number such that $\left| \frac{z-4}{z+4i} \right| = 1$, show that the locus of $z$ is the real axis or $y=0$.
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Q41
long answer
Show that $\int_0^1 \frac{\log(1+x)}{1+x^2} dx = \frac{\pi}{8} \log 2$.
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Q42
long answer
Prove that $\cos^{-1} \frac{1}{3} + \cos^{-1} \frac{1}{4} = \cos^{-1} \frac{17}{12}$.
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Q43
long answer
Find the eccentricity, centre, vertices and foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and also draw the rough diagram.
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Q44
long answer
Solve $(e^y + 1) \cos x dx + e^y \sin x dy = 0$.
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Q45
long answer
Show that $(p \to q) \equiv \neg p \vee q$.
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Q46
long answer
Using Vector method, prove that $\cos(A-B) = \cos A \cos B + \sin A \sin B$.
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Q47
long answer
Find two positive numbers whose product is 20 and their sum is minimum.
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Q48
long answer
On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6 m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Show that the angle of projection is $\tan^{-1} \frac{4}{3}$.
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Q49
long answer
A random variable $X$ has the following probability mass function: $X: 1, 2, 3, 4, 5$; $f(x): k, 2k, 3k, 2k, 3k$. Find: (i) the value of $k$. (ii) $P(2 \le X < 5)$. (iii) $P(3 < X)$.
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Q50
long answer
Show that the area of the region bounded by $3x - 2y = 0$, $x = -3$ and $x = 1$ is $\frac{15}{2}$.
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Q51
long answer
Show that the Cartesian equation of the plane passing through the points (1, 2, 3) and (2, 3, 1) and also perpendicular to the plane $3x - 2y + 4z - 5 = 0$ is $2y + z - 7 = 0$.
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Q52
short answer
X has the following probability mass function:
X 1 2 3 4 5
f(x) k 2k 3k 2k 3k
Find: (i) the value of k. (ii) P(2 ≤ X < 5). (iii) P(3 < X).
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Q53
mcq
1 mark
$\int_{0}^{\frac{\pi}{2}} \sin^7 x \, dx = $
- A. \frac{\pi}{2}
- B. $\int_{0}^{\frac{\pi}{2}} \cos^7 x \, dx$
- D. 1
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Q54
mcq
1 mark
The order and degree of the differential equation $(\frac{d^2y}{dx^2}) + (\frac{dy}{dx})^3 + x^4 = 0$ are :
- A. 2, 6
- B. 2, 3
- C. 2, 4
- D. 3, 3
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Q55
mcq
1 mark
If $|z|=1$, then the value of $|z+1| + |z-1|$ is :
- A. 1/z
- B. z
- C. 1
- D. z^2
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Q56
short answer
2 marks
If $z = (2+3i)(1-i)$ then prove that $\frac{1}{26} z = \frac{5}{26} + \frac{1}{26} i$.
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Q57
short answer
2 marks
If $\alpha$ and $\beta$ are the roots of $x^2-5x+6=0$ then prove that $\alpha^2-\beta^2 = \pm 5$.
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Q58
short answer
2 marks
For what value of x does $\sin x = \sin^{-1} x$?
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Q59
short answer
2 marks
Show that the three vectors $2\vec{i}+3\vec{j}+2\vec{k}, \vec{i}-\vec{j}+\vec{k}$ and $3\vec{i}+2\vec{j}+3\vec{k}$ are coplanar.
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Q60
mcq
then the value of $1/z$ is :
- A. 1/z
- B. z
- C. 1
- D. z
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Q61
long answer
If $z=(2+3i)(1-i)$ then prove that $z^{-1} = \frac{1}{26} + i \frac{5}{26}$.
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Q62
long answer
Show that the three vectors $2i - 3j + 2k$, $i - j + k$ and $3i + j + k$ are coplanar.
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Q63
long answer
If $X$ is the random variable with distribution function $F(x)$, given by: $F(x) = 0$ for $x < 0$; $F(x) = \frac{1}{2}(x^2+x)$ for $0 \le x < 1$; $F(x) = 1$ for $x \ge 1$, then prove that the p.d.f. is $f(x) = x + \frac{1}{2}$ for $0 < x < 1$; $0$ otherwise.
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Q64
long answer
Show that $\int_0^1 \frac{1}{1+x} dx = \log 2$.
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Q65
mcq
1 mark
The value of $\sin^{-1} \left(\frac{1}{2}\right) + \cos^{-1} \left(\frac{1}{2}\right)$ is :
- A. $0$
- B. $\frac{\pi}{2}$
- C. $\frac{\pi}{3}$
- D. $\pi$
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Q66
mcq
1 mark
The volume of the parallelepiped with its edges represented by the vectors $\vec{i} + \vec{j} + \vec{k}$, $\vec{i} + 2\vec{j} + \vec{k}$ and $\vec{i} + \vec{j} + \pi\vec{k}$ is :
- A. $\pi$
- B. $\frac{\pi}{2}$
- C. $\frac{\pi}{4}$
- D. $\frac{\pi}{3}$
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Q67
mcq
1 mark
$\int_0^{\pi/2} \sin^7 x \, dx = \dots$
- A. $\frac{\pi}{2}$
- B. $\int_0^{\pi/2} \cos^7 x \, dx$
- C. $0$
- D. $1$
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Q68
short answer
2 marks
If $z = (2 + i\sqrt{3}) / (1 - i)$ then prove that $|z| = \sqrt{7/2}$.
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Q69
short answer
2 marks
Show that the solution of $\frac{dy}{dx} = \frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$ is $\sin^{-1} y = \sin^{-1} x + C$.
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Q70
mcq
If X is a random variable, then which of the following cannot be the values of a and b?
- A. 7 and 19
- B. 0 and 12
- C. 16 and 24
- D. 5 and 17
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Q71
long answer
Show that $\int_{0}^{1} \frac{dx}{x^2 + x + 1} = \dots$ (incomplete text in source).
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Q72
long answer
Show that $\int_{0}^{1} \dots$ (incomplete text in source).
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Q73
long answer
Prove that $\cos^{-1}(1/3) + \cos^{-1}(4/5) = \cos^{-1}(17/15)$ (check source for correction).
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Q74
long answer
Prove that $\cos^{-1} \left( \frac{1}{3} \right) - \cos^{-1} \left( \frac{1}{4} \right) = \cos^{-1} \left( \frac{1}{12} \right)$.
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A random variable X has the following probability mass function : X 1 2 3 4 5 f(x) k 2k 3k 2k 3k
Q75
long answer
Find the value of $k$ for the probability mass function where $X: 1, 2, 3, 4, 5$ and $f(x): k, 2k, 3k, 2k, 3k$.
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Q76
long answer
Find $P(2 \le X < 5)$ given the probability mass function.
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