Question: If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to:

Answer: \frac{46}{3}

Choices:
- \frac{47}{3}
- \frac{46}{3}
- 18
- 13


Question: Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $\int_1^x f(t)\,dt = 5x f(x) - x^5 - 9$ for all $x \geq 1$, then the value of $f(3)$ is:

Answer: 32

Choices:
- 18
- 32
- 22
- 26


Question: The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is:

Choices:
- 4
- 10
- 6
- 8


Question: Let $A = \{1, 2, 3, \ldots, 10\}$ and $R$ be a relation on $A$ such that $R = \{(a, b) : a = 2b + 1\}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), \ldots, (a_k, a_{k+1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to:

Answer: 5

Choices:
- 6
- 7
- 5
- 8


Question: If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

Answer: \frac{4}{\sqrt{17}}

Choices:
- \frac{4}{\sqrt{17}}
- \frac{\sqrt{3}}{16}
- \frac{3}{\sqrt{19}}
- \frac{\sqrt{5}}{7}


Question: The line $L_1$ is parallel to the vector $\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through the point $(7, 6, 2)$ and the line $L_2$ is parallel to the vector $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ and passes through the point $(5, 3, 4)$. The shortest distance between the lines $L_1$ and $L_2$ is:

Answer: \frac{23}{\sqrt{38}}

Choices:
- \frac{23}{\sqrt{38}}
- \frac{21}{\sqrt{57}}
- \frac{23}{\sqrt{57}}
- \frac{21}{\sqrt{38}}


Question: Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_a^b \frac{9x^2}{1 + 5^x} \, dx$ is equal to:

Answer: 24

Choices:
- 24
- 27
- 18
- 21


Question:
If the system of equations \\
$2x + \lambda y + 3z = 5$ \\
$3x + 2y - z = 7$ \\
$4x + 5y + \mu z = 9$ \\
 has infinitely many solutions, then $(\lambda^2 + \mu^2)$ is equal to:

Answer: 26

Choices:
- 22
- 18
- 26
- 30


Question:
If $\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$, then the number of solutions of \\
$\sqrt{3}\csc^2\theta - 2(\sqrt{3} - 1)\csc\theta - 4 = 0$, is equal to:

Answer: 6

Choices:
- 6
- 8
- 10
- 7


Question:
Given three identical bags each containing 10 balls, whose colours are as follows: \\
\textbf{Bag I:} Red = 3, Blue = 2, Green = 5 \\
\textbf{Bag II:} Red = 4, Blue = 3, Green = 3 \\
\textbf{Bag III:} Red = 5, Blue = 1, Green = 4 \\
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is $p$ and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left( \frac{1}{p} + \frac{1}{q} \right)$ is:

Answer: 7

Choices:
- 6
- 9
- 7
- 8


Question: If the mean and the variance of $6,\ 4,\ a,\ 8,\ b,\ 12,\ 10,\ 13$ are $9$ and $9.25$ respectively, then $a + b + ab$ is equal to:

Answer: 103

Choices:
- 105
- 103
- 100
- 106


Question:
If the domain of the function \\
$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$ is $(a,\ b)$, then $(1 + a)^2 + b^2$ is equal to:

Answer: 26

Choices:
- 26
- 29
- 25
- 30


Question: $4\int_0^1 \left(\frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}}\right) dx - 3 \log_e(\sqrt{3})$ is equal to:

Answer: $2 - \sqrt{2} - \log_e(1 + \sqrt{2})$

Choices:
- $2 + \sqrt{2} + \log_e(1 + \sqrt{2})$
- $2 - \sqrt{2} - \log_e(1 + \sqrt{2})$
- $2 + \sqrt{2} - \log_e(1 + \sqrt{2})$
- $2 - \sqrt{2} + \log_e(1 + \sqrt{2})$


Question: If $\lim\limits_{x \to 0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$ is finite, then $(a + b)$ is equal to:

Answer: $\frac{1}{2}$

Choices:
- $\frac{1}{2}$
- 0
- $\frac{3}{4}$
- $-1$


Question: If $\sum_{r=0}^{10} \left(\frac{10^{r+1} - 1}{10^r}\right) \cdot {}^{11}C_{r+1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$, then $\alpha$ is equal to:

Answer: 20

Choices:
- 15
- 11
- 24
- 20


Question:
The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is: \\
(Figure: 3 rows of boxes, arranged in a T-shape, with a total of 8 boxes)

Answer: 5760

Choices:
- 5880
- 960
- 840
- 5760


Question: Let the point $P$ of the focal chord $PQ$ of the parabola $y^2 = 16x$ be $(1, -4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m : n$, $\gcd(m, n) = 1$, then $m^2 + n^2$ is equal to:

Answer: 17

Choices:
- 17
- 10
- 37
- 26


Question: Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{b} \times \vec{c} = \vec{d}$, then $|\vec{a} \cdot \vec{d}|$ is equal to:

Answer: 15

Choices:
- 18
- 12
- 9
- 15


Question: Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with coordinate axes be $48$ square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive x-axis, then the value of $b^2 + c^2$ is:

Answer: 97

Choices:
- 90
- 93
- 97
- 83


Question: Let $A$ be a $3 \times 3$ real matrix such that $A^2(A - 2I) - 4(A - I) = O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$, where $\alpha$, $\beta$ and $\gamma$ are real constants, then $\alpha + \beta + \gamma$ is equal to:

Answer: 12

Choices:
- 12
- 20
- 76
- 4


Question:
Let $y = y(x)$ be the solution of the differential equation
$\frac{dy}{dx} + 2y \sec^2 x = 2\sec^2 x + 3 \tan x \cdot \sec^2 x$, such that $y(0) = \frac{5}{4}$. Then $12 \left( y\left(\frac{\pi}{4}\right) - e^{-2} \right)$ is equal to:

Answer: 21


Question:
If the sum of the first 10 terms of the series
\( \frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots \) is \( \frac{m}{n} \), where gcd(m, n) = 1, then m + n is equal to:

Answer: 441


Question: If $y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right)$, then $(x - y)^2 + 3y^2$ is equal to:

Answer: 3


Question: Let $A(4, -2)$, $B(1, 1)$ and $C(9, -3)$ be the vertices of a triangle $ABC$. Then the maximum area of the parallelogram $AFDE$, formed with vertices $D$, $E$, and $F$ on the sides $BC$, $CA$ and $AB$ of the triangle $ABC$ respectively, is:

Answer: 3


Question: If the set of all $a \in \mathbb{R} \setminus \{1\}$, for which the roots of the equation $(1 - a)x^2 + 2(a - 3)x + 9 = 0$ are positive is $(-\infty, -\alpha] \cup [\beta, \gamma)$, then $2\alpha + \beta + \gamma$ is equal to:

Answer: 7


Question: The largest $n\in\mathbb{N}$ such that $3^n$ divides $50!$ is:

Answer: 22

Choices:
- 21
- 22
- 20
- 23


Question: The number of sequences of ten terms, whose terms are either $0$, $1$, or $2$, that contain exactly five $1$’s and exactly three $2$’s is equal to:

Answer: 2520

Choices:
- 360
- 45
- 2520
- 1820


Question: Let one focus of the hyperbola $H:\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\tfrac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9\,(e^2+l)$ is equal to:

Answer: 16

Choices:
- 14
- 15
- 16
- 12


Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function such that
$(\sin x\cos y)\bigl[f(2x+2y)-f(2x-2y)\bigr]=(\cos x\sin y)\bigl[f(2x+2y)+f(2x-2y)\bigr]$
for all $x,y\in\mathbb{R}$. If $f'(0)=\tfrac12$, then the value of $24\,f''\bigl(\tfrac{5\pi}{3}\bigr)$ is:

Answer: -3

Choices:
- 2
- -3
- 3
- -2


Question: Let $A=\begin{pmatrix}\alpha-1 & -1\\6 & \beta\end{pmatrix}$, $\alpha>0$, such that $\det(A)=0$ and $\alpha+\beta=1$. If $I$ denotes the $2\times2$ identity matrix, then the matrix $(I+A)^8$ is:

Answer: \begin{pmatrix}766 & -255\\1530 & -509\end{pmatrix}

Choices:
- \begin{pmatrix}4 & -1\\6 & -1\end{pmatrix}
- \begin{pmatrix}257 & -64\\514 & -127\end{pmatrix}
- \begin{pmatrix}1025 & -511\\2024 & -1024\end{pmatrix}
- \begin{pmatrix}766 & -255\\1530 & -509\end{pmatrix}


Question:
The term independent of $x$ in the expansion of
$\displaystyle\bigl(\tfrac{x+1}{x^{2/3}+1-x^{1/3}} - \tfrac{x+1}{x-x^{1/2}}\bigr)^{10}$,
for $x>1$, is:

Answer: 210

Choices:
- 210
- 150
- 240
- 120


Question: If $\theta \in [-2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta + (2-\sqrt{6})\cos\theta - \sqrt{3} = 0$ is equal to:

Answer: 8

Choices:
- 12
- 6
- 8
- 10


Question: Let $a_1,a_2,a_3,\dots$ be in an A.P. such that $\displaystyle\sum_{k=1}^{12}a_{2k-1}=-\tfrac{72}{5}a_1$, $a_1\neq0$. If $\displaystyle\sum_{k=1}^n a_k=0$, then $n$ is:

Answer: 11

Choices:
- 11
- 10
- 18
- 17


Question: If the function $f(x)=2x^3 - 9a x^2 + 12a^2 x + 1$, where $a>0$, attains its local maximum and local minimum at $p$ and $q$ respectively, such that $p^2 = q$, then $f(3)$ is equal to:

Answer: 37

Choices:
- 55
- 10
- 23
- 37


Question: Let $z$ be a complex number such that $|z|=1$. If $\displaystyle\frac{2 + k^2 z}{k + \overline{z}} = k z$, $k\in\mathbb{R}$, then the maximum distance of $k + i k^2$ from the circle $|z - (1 + 2i)| = 1$ is:

Answer: \sqrt{5} + 1

Choices:
- \sqrt{5} + 1
- 2
- 3
- \sqrt{3} + 1


Question: If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$, and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:

Answer: \frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} + 5\hat{k})

Choices:
- \frac{1}{\sqrt{155}}(-7\hat{i} + 9\hat{j} + 5\hat{k})
- \frac{1}{\sqrt{155}}(-7\hat{i} + 9\hat{j} - 5\hat{k})
- \frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} + 5\hat{k})
- \frac{1}{\sqrt{155}}(7\hat{i} + 9\hat{j} - 5\hat{k})


Question: Let $A$ be the set of all functions $f: \mathbb{Z} \to \mathbb{Z}$ and $R$ be a relation on $A$ such that $R = \{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\}$. Then $R$ is:

Answer: Symmetric but neither reflective nor transitive

Choices:
- Symmetric and transitive but not reflective
- Symmetric but neither reflective nor transitive
- Reflexive but neither symmetric nor transitive
- Transitive but neither reflexive nor symmetric


Question: For \(\alpha,\beta,\gamma\in\mathbb{R}\), if \(\displaystyle\lim_{x\to0}\frac{x^2\sin(\alpha x)+(\gamma-1)e^{x^2}}{\sin(2x)-\beta x}=3\), then \(\beta+\gamma-\alpha\) is equal to:

Answer: 7

Choices:
- 7
- 4
- 6
- -1


Question: Let \(P_n=\alpha^n+\beta^n,\ n\in\mathbb{N}\). If \(P_{10}=123,\ P_9=76,\ P_8=47\) and \(P_1=1\), then the quadratic equation having roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is:

Answer: x^2 + x - 1 = 0

Choices:
- x^2 - x + 1 = 0
- x^2 + x - 1 = 0
- x^2 - x - 1 = 0
- x^2 + x + 1 = 0


Question: If the system of linear equations $3x + y + \beta z = 3$, $2x + \alpha y - z = -3$, $x + 2y + z = 4$ has infinitely many solutions, then the value of $22\beta - 9\alpha$ is:

Answer: 31

Choices:
- 49
- 31
- 43
- 37


Question: If $S$ and $S'$ are the foci of the ellipse $\frac{x^2}{18} + \frac{y^2}{9} = 1$ and $P$ is a point on the ellipse, then $\min\bigl(SP\cdot S'P\bigr) + \max\bigl(SP\cdot S'P\bigr)$ is equal to:

Answer: 27

Choices:
- 3(1+\sqrt{2})
- 3(6+\sqrt{2})
- 9
- 27


Question: Let the vertices Q and R of the triangle PQR lie on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\), QR = 5, and the coordinates of the point P be \((0,2,3)\). If the area of the triangle PQR is \(\tfrac{m}{n}\), then:

Answer: 2m - 5\sqrt{21}\,n = 0

Choices:
- m - 5\sqrt{21}\,n = 0
- 2m - 5\sqrt{21}\,n = 0
- 5m - 2\sqrt{21}\,n = 0
- 5m - 21\sqrt{2}\,n = 0


Question: Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of ABCD is equal to:

Answer: \sqrt{110}

Choices:
- \sqrt{340}
- 12
- \sqrt{110}
- 7\sqrt{3}


Question:
Let $a\in\mathbb{R}$ and $A$ be a matrix of order $3\times3$ such that $\det(A)=-4$ and
$$A+I=\begin{pmatrix}1 & a & 1\\2 & 1 & 0\\a & 1 & 2\end{pmatrix},$$
where $I$ is the $3\times3$ identity. If
$$\det\bigl((a+1)\adj((a-1)A)\bigr)=2^m3^n,\quad m,n\in\{0,1,2,\dots,20\},$$
then $m+n$ is equal to:

Answer: 16

Choices:
- 14
- 17
- 15
- 16


Question: Let the focal chord $PQ$ of the parabola $y^2 = 4x$ make an angle of $60^\circ$ with the positive x-axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the y-axis at the point $(0,\alpha)$, then $5\alpha^2$ is equal to:

Answer: 15

Choices:
- 15
- 25
- 30
- 20


Question:
Let ⌊·⌋ denote the greatest integer function. If
\[
\displaystyle\int_{0}^{e^3} \Bigl\lfloor\frac{1}{e^{x-1}}\Bigr\rfloor\,dx = \alpha - \ln 2,
\]
then \(\alpha^3\) is equal to:

Answer: 8


Question: Let \(f:\mathbb{R}\to\mathbb{R}\) be a thrice‐differentiable odd function satisfying \(f''(x)=f(x)\), \(f(0)=0\), and \(f'(0)=3\). Then \(9f(\ln 3)\) is equal to:

Answer: 36


Question: If the area of the region \(\{(x,y):\;4 - x^2 \le y \le x^2,\;y \le 4,\;x \ge 0\}\) is \(\tfrac{80\sqrt{2}}{\alpha} - \beta\), where \(\alpha,\beta\in\mathbb{N}\), then \(\alpha + \beta\) is equal to:

Answer: 22


Question: Three distinct numbers are selected randomly from the set {1,2,3,…,40}. If the probability that the selected numbers are in an increasing G.P. is m/n, gcd(m,n)=1, then m+n is equal to:

Answer: 2477


Question: The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$ is equal to:

Answer: 768


Question: Let $f: \mathbb{R}\to\mathbb{R}$ be a function defined by $$f(x)=\lvert x+2\rvert - 2\lvert x\rvert.$$ If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is:

Choices:
- 5
- 3
- 2
- 4


Question: Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$- and $z$-axes, respectively, is half of the angle that this line makes with the positive $x$-axis. Then the sum of all possible values of the angle $\beta$ is:

Answer: $\tfrac{3\pi}{4}$

Choices:
- $\tfrac{3\pi}{4}$
- $\pi$
- $\tfrac{\pi}{2}$
- $\tfrac{3\pi}{2}$


Question: If the four distinct points $(4,6)$, $(-1,5)$, $(0,0)$ and $(k,3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to:

Answer: 35

Choices:
- 32
- 33
- 34
- 35


Question: Let the mean and variance of five observations $x_1=1$, $x_2=3$, $x_3=a$, $x_4=7$ and $x_5=b$, with $a>b$, be $5$ and $10$ respectively. Then the variance of the observations $n + x_n$ for $n=1,2,\dots,5$ is:

Answer: 16

Choices:
- 17
- 16.4
- 17.4
- 16


Question: Let $A=\{-2,-1,0,1,2,3\}$. Define a relation $R$ on $A$ by $xRy$ if and only if $y=\max(x,1)$. If $l$ is the number of elements in $R$, and $m$ and $n$ are the minimum numbers of ordered pairs that must be added to $R$ to make it reflexive and symmetric respectively, then $l+m+n$ is equal to:

Answer: 12

Choices:
- 12
- 11
- 13
- 14


Question: Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is:

Answer: 20

Choices:
- 20
- 30
- 10
- 15


Question: Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k,\tfrac{k}{2})$ from the line $3x+4y+5=0$ is:

Answer: 15

Choices:
- 15
- 5\sqrt{3}
- 15\sqrt{5}
- 12


Question: Line $L_1$ of slope $2$ and line $L_2$ of slope $\tfrac12$ intersect at the origin $O$. In the first quadrant, $P_1,P_2,\dots,P_{12}$ are 12 points on $L_1$ and $Q_1,Q_2,\dots,Q_9$ are 9 points on $L_2$. Then the total number of triangles that can be formed with vertices chosen from the 22 points $O,P_1,\dots,P_{12},Q_1,\dots,Q_9$ is:

Answer: 1134

Choices:
- 1080
- 1134
- 1026
- 1188


Question: Let $f$ be a function such that $f(x) + 3\,f\bigl(24/x\bigr) = 4x$, $x\neq0$. Then $f(3) + f(8)$ is equal to:

Answer: 11

Choices:
- 11
- 10
- 12
- 13


Question: The integral \(\displaystyle\int_{0}^{\pi}\frac{8x}{4\cos^2x + \sin^2x}\,dx\) is equal to:

Answer: $2\pi^2$

Choices:
- $2\pi^2$
- $4\pi^2$
- $\pi^2$
- $\tfrac{3\pi^2}{2}$


Question: The area of the region $\{(x,y):\lvert x-y\rvert \le y \le 4\sqrt{x}\}$ is:

Answer: $\tfrac{1024}{3}$

Choices:
- 512
- $\tfrac{1024}{3}$
- $\tfrac{512}{3}$
- $\tfrac{2048}{3}$


Question:
If the probability that the random variable $X$ takes the value $x$ is given by
$$P(X=x)=k\,(x+1)3^{-x},\quad x=0,1,2,\dots,$$
where $k$ is a constant, then $P(X\ge3)$ is equal to:

Answer: 1/9

Choices:
- 7/27
- 4/9
- 8/27
- 1/9


Question: Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 3(\tan^2x)\,y + 3y = \sec^2x$, with $y(0)=\tfrac{1}{3}+e^3$. Then $y\bigl(\tfrac{\pi}{4}\bigr)$ is equal to:

Answer: \tfrac{4}{3}

Choices:
- \tfrac{2}{3}
- \tfrac{4}{3}
- \tfrac{4}{3} + e^3
- \tfrac{2}{3} + e^3


Question: If $z_1, z_2, z_3\in\mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$, then $\displaystyle\sum_{k=1}^3 (z_k - z_0)^2$ is equal to:

Choices:
- 0
- 1
- i
- -i


Question: The number of solutions of the equation $$(4-\sqrt{3})\sin x - 2\sqrt{3}\cos^2x = -\frac{4}{1+\sqrt{3}},\quad x\in[-2\pi,\tfrac{5\pi}{2}]$$ is equal to:

Answer: 5

Choices:
- 4
- 3
- 6
- 5


Question: The shortest distance between the curves $$y^2 = 8x$$ and $$x^2 + y^2 + 12y + 35 = 0$$ is:

Answer: 2\sqrt{2} - 1

Choices:
- 2\sqrt{3} - 1
- \sqrt{2}
- 3\sqrt{2} - 1
- 2\sqrt{2} - 1


Question: Let $C$ be the circle of minimum area enclosing the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\tfrac12$ and foci $(\pm2,0)$. Let $PQR$ be a variable triangle whose vertex $P$ lies on $C$ and whose side $QR$ of length $29$ is parallel to the major axis of the ellipse and passes through the point where the ellipse intersects the negative $y$–axis. Then the maximum area of $\triangle PQR$ is:

Answer: $8(2+\sqrt{3})$

Choices:
- $6(3+\sqrt{2})$
- $8(3+\sqrt{2})$
- $6(2+\sqrt{3})$
- $8(2+\sqrt{3})$


Question: The distance of the point $(7,10,11)$ from the line $\displaystyle\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\displaystyle\frac{x-9}{2}=\frac{y-13}{-3}=\frac{z-17}{6}$ is:

Answer: 14

Choices:
- 18
- 14
- 12
- 16


Question: The sum \(1 + \tfrac{1+3}{2!} + \tfrac{1+3+5}{3!} + \tfrac{1+3+5+7}{4!} + \dots\) up to infinitely many terms is equal to:

Answer: 2e

Choices:
- 6e
- 4e
- 3e
- 2e


Question: If the domain of the function $f(x)=\log_7\bigl(1-\log_4(x^2-9x+18)\bigr)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to:

Answer: 18

Choices:
- 18
- 16
- 15
- 17


Question:
Let $I$ be the identity matrix of order $3\times3$ and let
$$A=\begin{pmatrix} \lambda & 2 & 3\\4 & 5 & 6\\7 & -1 & 2\end{pmatrix}$$
with $|A|=-1$. Let $B$ be the inverse of the matrix $\mathrm{adj}(A)\,\mathrm{adj}(A^2)$. Then $|\lambda B + I|$ is equal to:

Answer: 38


Question: Let \((1+x+x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \dots + a_{20} x^{20}\). If \((a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 = 121k\), then \(k\) is equal to:

Answer: 239


Question: If \(\displaystyle\lim_{x\to0}\bigl(\tfrac{\tan x}{x}\bigr)^{1/x^2}=p\), then \(96\log_e p\) is equal to:

Answer: 32


Question: Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}-3\hat{j}+3\hat{k}$, $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{d}$ be a vector such that $\vec{b}\times\vec{d}=\vec{c}\times\vec{d}$ and $\vec{a}\cdot\vec{d}=4$. Then $|\vec{a}\times\vec{d}|^2$ is equal to:

Answer: 128


Question: If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $$3x^2 - y^2 - ax + by + \gamma = 0,$$ then $a + b + \gamma$ is equal to:

Answer: 141


Question: Let $A$ be a matrix of order $3\times3$ with $\lvert A\rvert=5$. If \(\bigl\lvert 2\,\mathrm{adj}\bigl(3A\,\mathrm{adj}(2A)\bigr)\bigr\rvert=2^{\alpha}3^{\beta}5^{\gamma}\), where $\alpha,\beta,\gamma\in\mathbb{N}$, then $\alpha+\beta+\gamma$ is equal to:

Answer: 27

Choices:
- 25
- 26
- 27
- 28


Question: Let a line passing through the point $(4,1,0)$ intersect the line $L_1:\;\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ at the point $A(\alpha,\beta,\gamma)$ and the line $L_2:\;x-6 = y - z + 4$ at the point $B(a,b,c)$. Then $\det\begin{pmatrix}1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c\end{pmatrix}$ is equal to:

Answer: 8

Choices:
- 8
- 16
- 12
- 6


Question:
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}\,x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then
$$\frac{P_{25} + \sqrt{3}\,P_{24}}{2P_{23}} \;+\; \frac{Q_{25} - Q_{23}}{Q_{24}}$$
is equal to:

Answer: 5

Choices:
- 3
- 4
- 5
- 7


Question: The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is:

Answer: 18817

Choices:
- 16923
- 3763
- 33845
- 18817


Question: Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $0 \le x^2 + 2y \le 4$. Let $\ell$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added to $R$ to make it a reflexive relation. Then $\ell + m$ is equal to:

Answer: 18

Choices:
- 19
- 20
- 17
- 18


Question: A line passing through the point $P(\sqrt{5},\sqrt{5})$ intersects the ellipse $\dfrac{x^2}{36}+\dfrac{y^2}{25}=1$ at points $A$ and $B$ such that $(PA)\cdot(PB)$ is maximum. Then $5\bigl(PA^2+PB^2\bigr)$ is equal to:

Answer: 338

Choices:
- 218
- 377
- 290
- 338


Question: The sum $1 + 3 + 11 + 25 + 45 + 71 + \dots$ up to 20 terms is equal to:

Answer: 7240

Choices:
- 7240
- 7130
- 6982
- 8124


Question: If the domain of the function \(f(x)=\log_e\bigl(\tfrac{2x-3}{5+4x}\bigr)+\sin^{-1}\bigl(\tfrac{4+3x}{2-x}\bigr)\) is $[\alpha,\beta]$, then $\alpha^2+4\beta$ is equal to:

Choices:
- 5
- 4
- 3
- 7


Question: If \(\sum_{r=1}^9 \binom{r+3}{2} C_r = \alpha\bigl(\tfrac{3}{2}\bigr)^9 - \beta\), where \(\alpha,\beta\in\mathbb{N}\), then \((\alpha+\beta)^2\) is equal to:

Answer: 81

Choices:
- 27
- 9
- 81
- 18


Question: If \(y(x)=\begin{vmatrix} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix},\;x\in\mathbb{R},\) then \(\frac{d^2y}{dx^2}+y\) is equal to:

Answer: -1

Choices:
- -1
- 28
- 27
- 1


Question: The number of solutions of the equation $2x + 3\tan x = \pi$, $x\in[-2\pi,2\pi]\setminus\{\pm\tfrac{\pi}{2},\pm\tfrac{3\pi}{2}\}$ is:

Answer: 5

Choices:
- 6
- 5
- 4
- 3


Question: Let $g$ be a differentiable function such that $\displaystyle\int_0^x g(t)\,dt = x - \int_x^\pi g(t)\,dt$, $x\ge0$, and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x\,g(x)$ for $x\in[0,\tfrac{\pi}{2}]$. If $y(0)=0$, then $y\bigl(\tfrac{\pi}{3}\bigr)$ is equal to:

Answer: \frac{4\pi}{3}

Choices:
- \frac{2\pi}{3\sqrt{3}}
- \frac{4\pi}{3}
- \frac{2\pi}{3}
- \frac{4\pi}{\sqrt{3}}


Question: A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1:\;2x+y+6=0$ and $L_2:\;4x+2y-p=0,\;p>0$, at the points $A$ and $B$, respectively. If $AB=\tfrac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\tfrac{AM}{BM}$ is equal to:

Choices:
- 5
- 4
- 2
- 3


Question: Let $z\in\mathbb{C}$ be such that \(\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i\). Then the sum of all possible values of $z^2$ is:

Answer: -19 - 2i

Choices:
- 19 - 2i
- -19 - 2i
- 19 + 2i
- -19 + 2i


Question: Let f(x)=\int x^3\sqrt{3 - x^2}\,dx. If 5f(\sqrt{2}) = -4, then f(1) is equal to:

Answer: -6\sqrt{2}/5

Choices:
- -2\sqrt{2}/5
- -8\sqrt{2}/5
- -4\sqrt{2}/5
- -6\sqrt{2}/5


Question: Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \tfrac{111}{4}$, then $24(a_1 + a_2 + a_3)$ is equal to:

Answer: 129

Choices:
- 131
- 130
- 129
- 128


Question: Let the domain of the function \(f(x)=\log_2\bigl(\log_4\bigl(\log_6(3+4x-x^2)\bigr)\bigr)\) be \((a,b)\). If \(\displaystyle\int_0^a \lfloor x^2\rfloor\,dx = p - \sqrt{q} - \sqrt{r}\), with \(p,q,r\in\mathbb{N}\) and \(\gcd(p,q,r)=1\), where \(\lfloor\cdot\rfloor\) is the greatest integer function, then \(p+q+r\) is equal to:

Answer: 10

Choices:
- 10
- 8
- 11
- 9


Question: The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is:

Answer: \tfrac{7\sqrt{2}}{8}

Choices:
- \tfrac{7\sqrt{2}}{2}
- \tfrac{7\sqrt{2}}{16}
- \tfrac{7\sqrt{2}}{4}
- \tfrac{7\sqrt{2}}{8}


Question: Let \(f(x)=\begin{cases}(1+ax)^{1/x},&x<0,\\1+b,&x=0,\\\displaystyle\frac{\sqrt{x+4}-2}{\sqrt[3]{x+c}-2},&x>0\end{cases}\) be continuous at \(x=0\). Then \(e^a b c\) is equal to:

Answer: 48

Choices:
- 64
- 72
- 48
- 36


Question: Line L1 passes through the point (1, 2, 3) and is parallel to the z-axis. Line L2 passes through the point (λ, 5, 6) and is parallel to the y-axis. If for λ = λ₁, λ₂ with λ₂ < λ₁ the shortest distance between these two lines is 3, then the square of the distance of the point (λ₁, λ₂, 7) from the line L1 is:

Answer: 25

Choices:
- 40
- 32
- 25
- 37


Question: All five-letter words are made using all the letters A, B, C, D and E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by Wₙ. Let the probability P(Wₙ) of choosing the word Wₙ satisfy P(Wₙ) = 2·P(Wₙ₋₁) for n > 1. If P(CDBEA) = 2^α / (2^β − 1), with α, β ∈ ℕ, then α + β is equal to:

Answer: 183


Question: Let the product of the focal distances of the point P(4,2√3) on the hyperbola H: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then p² + q² is equal to:

Answer: 120


Question: Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}-\hat{k}$, $\vec{c}=\lambda\hat{j}+\mu\hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a}\times\vec{c}=\vec{b}\times\hat{d}$ and $\vec{c}\cdot\hat{d}=1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $\lvert3\lambda\hat{d}+\mu\vec{c}\rvert^2$ is equal to:

Answer: 5


Question: If the number of seven‐digit numbers such that the sum of their digits is even is $m\cdot n\cdot10^n$, where $m,n\in\{1,2,3,\dots,9\}$, then $m+n$ is equal to:

Answer: 14


Question: The area of the region bounded by the curve \(y = \max\{|x|,\,|x-2|\}\), the x-axis, and the lines \(x=-2\) and \(x=4\) is equal to:

Answer: 12


Question: Let $a>0$. If the function $f(x)=6x^{3}-45a x^{2}+108a^{2}x+1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1}x_{2}=54$, then $a+x_{1}+x_{2}$ is equal to:

Answer: 18

Choices:
- 15
- 18
- 24
- 13


Question: The sum of the infinite series $\cot^{-1}\left(\tfrac{7}{4}\right)+\cot^{-1}\left(\tfrac{19}{4}\right)+\cot^{-1}\left(\tfrac{39}{4}\right)+\cot^{-1}\left(\tfrac{67}{4}\right)+\ldots$ is:

Answer: \frac{\pi}{2}-\tan^{-1}\left(\frac{1}{2}\right)

Choices:
- \frac{\pi}{2}+\tan^{-1}\left(\frac{1}{2}\right)
- \frac{\pi}{2}-\cot^{-1}\left(\frac{1}{2}\right)
- \frac{\pi}{2}+\cot^{-1}\left(\frac{1}{2}\right)
- \frac{\pi}{2}-\tan^{-1}\left(\frac{1}{2}\right)


Question: Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1$ and $f'(2)=4$. If \(\displaystyle\lim_{x\to0}\bigl(f(2+x)\bigr)^{3/x}=e^{\alpha}\), then the number of times the curve \(y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha\) meets the $x$-axis is:

Choices:
- 2
- 1
- 0
- 3


Question: Let $A = \{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $xRy$ iff $2x - y \in \{0,1\}$. Let $\ell$ be the number of elements in $R$. Let $m$ and $n$ be the minimum numbers of elements that must be added to $R$ to make it reflexive and symmetric, respectively. Then $\ell + m + n$ is equal to:

Answer: 17

Choices:
- 18
- 17
- 15
- 16


Question: Let the product of \(\omega_1 = (8+i)\sin\theta + (7+4i)\cos\theta\) and \(\omega_2 = (1+8i)\sin\theta + (4+7i)\cos\theta\) be \(\alpha + i\beta\), where \(i=\sqrt{-1}\). Let \(p\) and \(q\) be the maximum and minimum values of \(\alpha+\beta\), respectively. Then \(p+q\) is equal to:

Answer: 130

Choices:
- 140
- 130
- 160
- 150


Question: Let the values of $p$, for which the shortest distance between the lines $\displaystyle\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\mathbf{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ is $\tfrac{1}{\sqrt6}$, be $a,b$ with $a<b$. Then the length of the latus rectum of the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is:

Answer: \tfrac{2}{3}

Choices:
- 9
- \tfrac{3}{2}
- \tfrac{2}{3}
- 18


Question: The axis of a parabola is the line \(y = x\) and its vertex and focus are in the first quadrant at distances \(\sqrt{2}\) and \(2\sqrt{2}\) from the origin, respectively. If the point \((1, k)\) lies on the parabola, then a possible value of \(k\) is:

Answer: 9

Choices:
- 4
- 9
- 3
- 8


Question: Let the domains of the functions $f(x)=\log_{10}\log_{10}\bigl(8-\log_{2}(x^2+4x+5)\bigr)$ and $g(x)=\sin^{-1}\!\bigl(\tfrac{7x+10}{x-2}\bigr)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to:

Answer: 15

Choices:
- 15
- 13
- 16
- 14


Question: A line passing through the point $A(-2,0)$ touches the parabola $P:\;y^2 = x - 2$ at the point $B$ in the first quadrant. The area of the region bounded by the line $AB$, the parabola $P$, and the $x$-axis is:

Answer: \tfrac{8}{3}

Choices:
- \tfrac{7}{3}
- 2
- \tfrac{8}{3}
- 3


Question: Let the sum of the focal distances of the point P(4,3) on the hyperbola \(H:\;\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\tfrac{8\sqrt{5}}{3}\). If for \(H\), the length of the latus rectum is \(l\) and the product of the focal distances of the point \(P\) is \(m\), then \(9l^2+6m\) is equal to:

Answer: 185

Choices:
- 184
- 186
- 185
- 187


Question: Let the matrix \(A=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}\) satisfy \(A^n = A^{n-2} + A^2 - I\) for \(n\ge3\). Then the sum of all the elements of \(A^{50}\) is:

Answer: 53

Choices:
- 53
- 52
- 39
- 44


Question: If the sum of the first 20 terms of the series \(\displaystyle\frac{4\cdot1}{4+3\cdot1^2+1^4}+\frac{4\cdot2}{4+3\cdot2^2+2^4}+\frac{4\cdot3}{4+3\cdot3^2+3^4}+\dots\) is \(\tfrac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to:

Answer: 421

Choices:
- 423
- 420
- 421
- 422


Question: If \(1^2\binom{15}{1}^2 + 2^2\binom{15}{2}^2 + 3^2\binom{15}{3}^2 + \dots + 15^2\binom{15}{15}^2 = 2^m\,3^n\,5^k\), where \(m,n,k\) are positive integers, then \(m + n + k\) is equal to:

Answer: 19

Choices:
- 19
- 21
- 18
- 20


Question: Let for two distinct values of $p$ the lines $y = x + p$ touch the ellipse $E:\;\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $A$ and $B$. Let the line $y = x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is equal to:

Answer: 24

Choices:
- 36
- 24
- 48
- 20


Question: Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of the A.P. in A and B respectively such that D = d + 3, d > 0. If \(\frac{p+q}{p-q} = \frac{19}{5}\), then p - q is equal to:

Answer: 540

Choices:
- 600
- 450
- 630
- 540


Question: If a curve $y=y(x)$ passes through the point $(1,\tfrac{\pi}{2})$ and satisfies the differential equation $\bigl(7x^4\cot y - e^x\csc y\bigr)\frac{dx}{dy} = x^5$, $x\ge1$, then at $x=2$, the value of $\cos y$ is:

Answer: \frac{2e^2 - e}{128}

Choices:
- \frac{2e^2 - e}{64}
- \frac{2e^2 + e}{64}
- \frac{2e^2 - e}{128}
- \frac{2e^2 + e}{128}


Question: The centre of a circle $C$ is at the centre of the ellipse $E:\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $\triangle PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is:

Answer: 13

Choices:
- 26
- 13
- 12
- \frac{13}{2}


Question: Let $f(x)+2f\bigl(\tfrac{1}{x}\bigr)=x^2+5$ and $2g(x)-3g\bigl(\tfrac{1}{x}\bigr)=x$, $x>0$. If $\alpha=\int_1^2 f(x)\,dx$ and $\beta=\int_1^2 g(x)\,dx$, then the value of $9\alpha+\beta$ is:

Answer: 11

Choices:
- 1
- 0
- 10
- 11


Question:
Let $A$ be the point of intersection of the lines
$$L_1:\;\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1},$$
and
$$L_2:\;\frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}.$$Let $B$ and $C$ be points on $L_1$ and $L_2$ respectively such that $AB=AC=\sqrt{15}$. Then the square of the area of triangle $ABC$ is:

Answer: 54

Choices:
- 54
- 63
- 57
- 60


Question: Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these observations is:

Choices:
- 1
- 3/4
- 2
- 1/2


Question: If \(\alpha\) is a root of the equation \(x^2 + x + 1 = 0\) and \(\displaystyle\sum_{k=1}^n\bigl(\alpha^k + \alpha^{-k}\bigr)^2 = 20\), then \(n\) is equal to:

Answer: 11


Question: If \(\displaystyle\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^{9}}\,dx = \frac{1}{m}\bigl((\sqrt{1+x^2}+x)^{n}(\sqrt{1+x^2}-x)\bigr)+C\), where \(C\) is the constant of integration and \(m,n\in\mathbb{N}\), then \(m+n\) is equal to:

Answer: 379


Question: A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card being a spade is \(\tfrac{11}{50}\), then n is equal to:

Answer: 2


Question: Let $m$ and $n$, $(m<n)$ be two 2-digit numbers. Then the total number of pairs $(m,n)$ such that $\gcd(m,n)=6$ is:

Answer: 64


Question: Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the centroid of the triangle ABC. Then $$6\bigl(|\overrightarrow{AG}|^2+|\overrightarrow{BG}|^2+|\overrightarrow{CG}|^2\bigr)$$ is equal to:

Answer: 164


Question: Let f, g: (1, \infty) \to \mathbb{R} be defined as f(x) = \frac{2x+3}{5x+2} and g(x) = \frac{2-3x}{1-x}. If the range of the function f\circ g: [2,4] \to \mathbb{R} is [\alpha, \beta], then \frac{1}{\beta - \alpha} is equal to:

Answer: 56

Choices:
- 68
- 29
- 2
- 56


Question: Consider the sets A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 \ge 25\}, B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}, C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : 2x^2 + y^2 \le 4\}, and D = A \cap B. The total number of one-one functions from the set D to the set C is:

Answer: 17160

Choices:
- 15120
- 19320
- 17160
- 18290


Question: Let A = \{1, 6, 11, 16, \ldots\} and B = \{9, 16, 23, 30, \ldots\} be the sets consisting of the first 2025 terms of two arithmetic progressions. Then n(A \cup B) is:

Answer: 3761

Choices:
- 3814
- 4027
- 3761
- 4003


Question: The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:

Answer: 129/182

Choices:
- 129/182
- 103/182
- 17/26
- 19/26


Question: For an integer n \ge 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)^{2n-3} is 16, then the distance of the point P(2n-1, n^2-4n) from the line x + y = 8 is:

Answer: 3\sqrt{2}

Choices:
- \sqrt{2}
- 2\sqrt{2}
- 5\sqrt{2}
- 3\sqrt{2}


Question: Let the shortest distance between the lines \(\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}\) and \(\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}\) be 3\sqrt{30}. Then the positive value of 5\alpha + \beta is:

Answer: 46

Choices:
- 42
- 46
- 48
- 40


Question: If \lim_{x\to1} \frac{(x-1)(6 + \lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1, where \lambda, \mu \in \mathbb{R}, then \lambda + \mu is equal to:

Answer: 18

Choices:
- 18
- 20
- 19
- 17


Question: Let f: [0, \infty) \to \mathbb{R} be a differentiable function such that f(x) = 1 - 2x + \int_0^x e^{t-x} f(t) \, dt for all x \in [0, \infty). Then the area of the region bounded by y = f(x) and the coordinate axes is:

Answer: 1/2

Choices:
- \sqrt{5}
- 1/2
- \sqrt{2}
- 2


Question: Let A and B be two distinct points on the line L: (x-6)/3 = (y-7)/2 = (z-7)/-2. Both A and B are at a distance 2\sqrt{17} from the foot of the perpendicular drawn from the point (1, 2, 3) to the line L. If O is the origin, then OA \cdot OB is equal to:

Answer: 47

Choices:
- 49
- 47
- 21
- 62


Question: Let f: \mathbb{R} \to \mathbb{R} be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x \in \mathbb{R}. If \lim_{n\to\infty} [f(x) - f(x/2^n)] = G(x), then \sum_{r=1}^{10} G(r^2) is equal to:

Answer: 385

Choices:
- 540
- 385
- 420
- 215


Question: 1 + 3 + 5^2 + 7 + 9^2 + \dots up to 40 terms is equal to:

Answer: 41880

Choices:
- 43890
- 41880
- 33980
- 40870


Question: In the expansion of \(\bigl(\frac{\sqrt{2}}{3} + \frac{1}{\sqrt{3}}\bigr)^n\), n \in \mathbb{N}, if the ratio of the 15th term from the beginning to the 15th term from the end is \tfrac{1}{6}, then the value of \binom{n}{3} is:

Answer: 2300

Choices:
- 4060
- 1040
- 2300
- 4960


Question: Considering the principal values of the inverse trigonometric functions, \sin^{-1}\bigl(\tfrac{\sqrt{3}}{2}x + \tfrac{1}{2}\sqrt{1-x^2}\bigr), -\tfrac{1}{2} < x < \tfrac{1}{\sqrt{2}}, is equal to:

Answer: \frac{\pi}{6} + \sin^{-1} x

Choices:
- \frac{\pi}{4} + \sin^{-1} x
- \frac{\pi}{6} + \sin^{-1} x
- -\frac{5\pi}{6} - \sin^{-1} x
- \frac{5\pi}{6} - \sin^{-1} x


Question: Consider two vectors \mathbf{u}=3\mathbf{i}-\mathbf{j} and \mathbf{v}=2\mathbf{i}+\mathbf{j}-\lambda\mathbf{k}, \lambda > 0. The angle between them is given by \cos^{-1}\bigl(\tfrac{\sqrt{5}}{2\sqrt{7}}\bigr). Let \mathbf{v}=\mathbf{v}_1+\mathbf{v}_2, where \mathbf{v}_1 is parallel to \mathbf{u} and \mathbf{v}_2 is perpendicular to \mathbf{u}. Then the value of |\mathbf{v}_1|^2 + |\mathbf{v}_2|^2 is:

Answer: 14

Choices:
- \tfrac{23}{2}
- 14
- \tfrac{25}{2}
- 10


Question: Let the three sides of a triangle be on the lines 4x - 7y + 10 = 0, x + y = 5, and 7x + 4y = 15. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines x = 0, y = 0, and x + y = 1 is:

Answer: \sqrt{5}

Choices:
- 5
- \sqrt{5}
- \sqrt{20}
- 20


Question: The value of \int_{-1}^{1} \frac{(1 + \sqrt{|x|} - x)e^x + (\sqrt{|x|} - x)e^{-x}}{e^x + e^{-x}} \, dx is equal to:

Answer: 1 + \frac{2\sqrt{2}}{3}

Choices:
- 3 - \frac{2\sqrt{2}}{3}
- 2 + \frac{2\sqrt{2}}{3}
- 1 - \frac{2\sqrt{2}}{3}
- 1 + \frac{2\sqrt{2}}{3}


Question: The length of the latus-rectum of the ellipse whose foci are (2,5) and (2,-3) and eccentricity is \tfrac{4}{5} is:

Answer: \tfrac{18}{5}

Choices:
- \tfrac{6}{5}
- \tfrac{50}{3}
- \tfrac{10}{3}
- \tfrac{18}{5}


Question: Consider the equation x^2 + 4x - n = 0, where n \in [20,100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to:

Answer: 6

Choices:
- 7
- 8
- 6
- 5


Question: A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is:

Answer: \tfrac{28}{75}

Choices:
- \tfrac{11}{15}
- \tfrac{28}{75}
- \tfrac{2}{15}
- \tfrac{3}{5}


Question: If 10\sin^4\theta + 15\cos^4\theta = 6, then the value of \displaystyle \frac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta} is:

Answer: \frac{2}{5}

Choices:
- \frac{2}{5}
- \frac{3}{4}
- \frac{3}{5}
- \frac{1}{5}


Question: If the area of the region \{(x,y): |x-5| \le y \le 4\sqrt{x}\} is A, then 3A is equal to _____

Answer: 368


Question: Let A = \begin{pmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{pmatrix}. If for some \theta \in (0,\pi), A^2 = A^T, then the sum of the diagonal elements of the matrix (A + I)^3 + (A - I)^3 - 6A is equal to:

Answer: 6


Question: Let A = \{z \in \mathbb{C} : |z - 2 - i| = 3\}, B = \{z \in \mathbb{C} : \Re(z - 2) = 2\} and S = A \cap B. Then \sum_{z \in S} |z|^2 is equal to _____

Answer: 22


Question: Let C be the circle x^2 + (y-1)^2 = 2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on the x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E_1 and E_2 at P(x_1,y_1), Q(x_2,y_2) and R(x_3,y_3) respectively. Given that P is the midpoint of the line segment QR and PQ = \frac{2\sqrt{2}}{3}, the value of 9(x_1y_1 + x_2y_2 + x_3y_3) is equal to _____

Answer: 46


Question: Let m and n be the number of points at which the function f(x)=\max\{x, x^3, x^5, \dots, x^{21}\}, x\in\mathbb{R}, is not differentiable and not continuous, respectively. Then m + n is equal to _____

Answer: 3


Question: If the orthocentre of the triangle formed by the lines \(y = x + 1\), \(y = 4x - 8\) and \(y = m x + c\) is at \((3, -1)\), then \(m - c\) is:

Choices:
- 0
- -2
- 4
- 2


Question: Let the vectors \(a\) and \(b\) be of the same magnitude such that \(\frac{\lvert a + b\rvert + \lvert a - b\rvert}{\lvert a + b\rvert - \lvert a - b\rvert} = \sqrt{2} + 1\). Then \(\frac{\lvert a + b\rvert^2}{\lvert a\rvert^2}\) is:

Answer: 2 + \sqrt{2}

Choices:
- 2 + 4\sqrt{2}
- 1 + \sqrt{2}
- 2 + \sqrt{2}
- 4 + 2\sqrt{2}


Question: Let \(A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}\) and \(B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}\). Then:

Answer: \(B \subset A\)

Choices:
- \(B \subset A\)
- \(A \cup B = \{(x, y) : -4 \le x \le 4,\; -1 \le y \le 11\}\)
- neither \(A \subset B\) nor \(B \subset A\)
- \(A \subset B\)


Question: If the range of the function \(f(x) = \frac{5 - x}{x^2 - 3x + 2}\), \(x \neq 1,2\), is \(( -\infty, \alpha) \cup [\beta, \infty)\), then \(\alpha^2 + \beta^2\) is equal to:

Answer: 194

Choices:
- 190
- 192
- 188
- 194


Question: A bag contains 19 unbiased coins and one coin with heads on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the drawn coin was unbiased is \(\frac{m}{n}\), with \(\gcd(m,n)=1\), then \(n^2 - m^2\) is equal to:

Answer: 80

Choices:
- 80
- 60
- 72
- 64


Question: Let a random variable X take values 0, 1, 2 and 3 with P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) = \(\tfrac{1 - 2p}{2}\). If \(E(X^2) = 2E(X)\), then the value of \(8p - 1\) is:

Choices:
- 0
- 2
- 1
- 3


Question: If the area of the region \(\{(x,y): 1 + x^2 \le y \le \min\{x + 7,\;11 - 3x\}\}\) is \(A\), then \(3A\) is equal to:

Answer: 50

Choices:
- 50
- 49
- 46
- 47


Question: Let \(f: \mathbb{R} \to \mathbb{R}\) be a polynomial function of degree four having extreme values at \(x = 4\) and \(x = 5\). If \(\displaystyle \lim_{x \to 0} \frac{f(x)}{x^2} = 5\), then \(f(2)\) is equal to:

Answer: 10

Choices:
- 12
- 10
- 8
- 14


Question: The number of solutions of the equation \(\cos 2\theta\cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}\) in \([-\tfrac{\pi}{2},\tfrac{\pi}{2}]\) is:

Answer: 7

Choices:
- 7
- 5
- 6
- 9


Question: Let \(a_n\) be the \(n\)th term of an A.P. If \(S_n = a_1 + a_2 + \dots + a_n = 700\) for some \(n\), \(a_6 = 7\) and \(S_7 = 7\), then \(a_n\) is equal to:

Answer: 64

Choices:
- 56
- 65
- 64
- 70


Question: If the locus of \(z \in \mathbb{C}\), such that \(\Re\bigl(\tfrac{z-1}{2z + i}\bigr) + \Re\bigl(\tfrac{\overline{z}-1}{2\overline{z} - i}\bigr) = 2\), is a circle of radius \(r\) and center \((a,b)\), then \(\tfrac{15ab}{r^2}\) is equal to:

Answer: 18

Choices:
- 24
- 12
- 18
- 16


Question: Let the length of a latus rectum of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) be 10. If its eccentricity is the minimum value of the function \(f(t) = t^2 + t + \tfrac{11}{12}\), \(t \in \mathbb{R}\), then \(a^2 + b^2\) is equal to:

Answer: 126

Choices:
- 125
- 126
- 120
- 115


Question: Let \(y = y(x)\) be the solution of the differential equation \((x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x\), with \(y(0) = 1\). Then \(\displaystyle \int_{-3}^{3} y(x)\,dx\) is:

Answer: 24

Choices:
- 24
- 36
- 30
- 18


Question: If the equation of the line passing through the point \((0, -\tfrac{1}{2}, 0)\) and perpendicular to the lines \(\mathbf{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})\) and \(\mathbf{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})\) is \(\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}\), then \(a + b + c + d\) is equal to:

Answer: 14

Choices:
- 10
- 14
- 13
- 12


Question: Let \(p\) be the number of all triangles that can be formed by joining the vertices of a regular polygon \(P\) of \(n\) sides and \(q\) be the number of all quadrilaterals that can be formed by joining the vertices of \(P\). If \(p + q = 126\), then the eccentricity of the ellipse \(\frac{x^2}{16} + \frac{y^2}{n} = 1\) is:

Answer: \frac{1}{\sqrt{2}}

Choices:
- \frac{3}{4}
- \frac{1}{2}
- \frac{\sqrt{7}}{4}
- \frac{1}{\sqrt{2}}


Question: Consider the lines \(L_1: x - 1 = y - 2 = z\) and \(L_2: x - 2 = y - 2 = z - 1\). Let the feet of the perpendiculars from the point \(P(5,1,-3)\) on the lines \(L_1\) and \(L_2\) be \(Q\) and \(R\) respectively. If the area of the triangle \(PQR\) is \(A\), then \(4A^2\) is equal to:

Answer: 147

Choices:
- 139
- 147
- 151
- 143


Question: The number of real roots of the equation \(x\lvert x - 2\rvert + 3\lvert x - 3\rvert + 1 = 0\) is:

Choices:
- 4
- 2
- 1
- 3


Question: Let \(e_1\) and \(e_2\) be the eccentricities of the ellipse \(\tfrac{x^2}{a^2}+\tfrac{y^2}{25}=1\) and the hyperbola \(\tfrac{x^2}{16}-\tfrac{y^2}{b^2}=1\), respectively. If \(b<5\) and \(e_1e_2=1\), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

Answer: \tfrac{3}{5}

Choices:
- \tfrac{4}{5}
- \tfrac{3}{5}
- \tfrac{\sqrt{7}}{4}
- \tfrac{\sqrt{3}}{2}


Question: Let the system of equations \(x + 5y - z = 1\), \(4x + 3y - 3z = 7\), and \(24x + y + \lambda z = \mu\), with \(\lambda,\mu \in \mathbb{R}\), have infinitely many solutions. Then the number of solutions of this system, if \(x,y,z\) are integers and satisfy \(7 \le x+y+z \le 77\), is:

Choices:
- 3
- 6
- 5
- 4


Question: If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:

Answer: 757

Choices:
- 760
- 755
- 750
- 757


Question: If the function \(f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}\) is continuous at \(x = 0\), then \(f(0)\) is equal to:

Answer: 2


Question:
If
\[ \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt{3x^{-24} + x^{-26}} \right) dx \]
is equal to
\[ \frac{\alpha}{3(\alpha + 1)} \left(3x^{\beta} + x^{\gamma}\right)^{\frac{\alpha + 1}{\alpha}} + C,\ x > 0, \]
where \(\alpha, \beta, \gamma \in \mathbb{Z}\), and \(C\) is the constant of integration, then \(\alpha + \beta + \gamma\) is equal to:

Answer: 19


Question:
For \( t > -1 \), let \( \alpha_t \) and \( \beta_t \) be the roots of the equation
\[ \left((t + 2)^{\frac{1}{7}} - 1\right)x^2 + \left((t + 2)^{\frac{1}{6}} - 1\right)x + \left((t + 2)^{\frac{1}{21}} - 1\right) = 0. \]
If \( \lim_{t \to -1^+} \alpha_t = a \) and \( \lim_{t \to -1^+} \beta_t = b \), then \( 72(a + b)^2 \) is equal to:

Answer: 98


Question: Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be (-5, 0) and 5x + 9 = 0, respectively. If the product of the focal distances of a point \( (\alpha, 2\sqrt{5}) \) on the hyperbola is p, then 4p is equal to:

Answer: 189


Question:
The sum of the series
\[ 2 \times 1 \times \binom{20}{4} - 3 \times 2 \times \binom{20}{5} + 4 \times 3 \times \binom{20}{6} - 5 \times 4 \times \binom{20}{7} + \ldots + 18 \times 17 \times \binom{20}{20} \]
is equal to:

Answer: 34


Question: Evaluate the limit: \( \lim_{x \to \infty} \frac{\tan\left(5x^{1/3}\right) \log_e(1 + 3x^2)}{\left(\tan^{-1}(3\sqrt{x})\right)^2 \left(e^{5x^{4/3}} - 1\right)} \) is equal to:

Answer: 1/3

Choices:
- 1/15
- 1
- 1/3
- 5/3


Question: If the shortest distance between the lines \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\) and \(\frac{x}{1} = \frac{y}{\alpha} = \frac{z - 5}{1}\) is \(\frac{5}{\sqrt{6}}\), then the sum of all possible values of \(\alpha\) is:

Answer: -3

Choices:
- 3/2
- -3/2
- 3
- -3


Question: Let \(x = -1\) and \(x = 2\) be the critical points of the function \(f(x) = x^3 + ax^2 + b \log_2|x| + 1,\ x \ne 0\). Let \(m\) and \(M\) respectively be the absolute minimum and the absolute maximum values of \(f\) in the interval \(\left[-2, -\frac{1}{2}\right]\). Then \(|M + m|\) is equal to: (Take \(\log_2 2 = 0.7\))

Answer: 21.1

Choices:
- 21.1
- 19.8
- 22.1
- 20.9


Question: The remainder when \( (64^{64})^{64} \) is divided by 7 is equal to:

Choices:
- 4
- 1
- 3
- 6


Question: Let P be the parabola, whose focus is \((-2, 1)\) and directrix is \(2x + y + 2 = 0\). Then the sum of the ordinates of the points on P, whose abscissa is \(-2\), is:

Answer: 3/2

Choices:
- 3/2
- 5/2
- 1/4
- 3/4


Question: Let \( y = y(x) \) be the solution curve of the differential equation \( x(x^2 + e^x)dy + (e^x(x - 2)y - x^3)dx = 0,\ x > 0 \), passing through the point \((1, 0)\). Then \( y(2) \) is equal to:

Answer: \( \frac{4}{4 + e^2} \)

Choices:
- \( \frac{4}{4 - e^2} \)
- \( \frac{2}{2 + e^2} \)
- \( \frac{2}{2 - e^2} \)
- \( \frac{4}{4 + e^2} \)


Question: From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:

Answer: 155

Choices:
- 165
- 155
- 145
- 135


Question: If for \( \theta \in \left[ -\frac{\pi}{3}, 0 \right] \), the points \( (x, y) = \left( 3 \tan\left(\theta + \frac{\pi}{3}\right), 2 \tan\left(\theta + \frac{\pi}{6}\right) \right) \) lie on \( xy + \alpha x + \beta y + \gamma = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Answer: 75

Choices:
- 80
- 72
- 96
- 75


Question: Let \( C_1 \) be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let \( C_2 \) be the circle with centre \((1, 3)\) that touches \( C_1 \) externally at the point \((\alpha, \beta)\). If \( (\beta - \alpha)^2 = \frac{m}{n} \), \( \gcd(m, n) = 1 \), then \( m + n \) is equal to:

Answer: 22

Choices:
- 9
- 13
- 22
- 31


Question: The integral \( \int_{0}^{\frac{\pi}{2}} \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx \) is equal to:

Answer: \( \frac{\pi}{3\sqrt{3}} (\pi + 6) \)

Choices:
- \( \frac{\pi}{\sqrt{3}} (\pi + 1) \)
- \( \frac{\pi}{\sqrt{3}} (\pi + 2) \)
- \( \frac{\pi}{3\sqrt{3}} (\pi + 6) \)
- \( \frac{\pi}{2\sqrt{3}} (\pi + 4) \)


Question:
Among the statements:
(S1): The set \( \{ z \in \mathbb{C} \setminus \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \} \) contains exactly two elements, and
(S2): The set \( \{ z \in \mathbb{C} \setminus \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \} \) contains infinitely many elements.
Which of the following is correct?

Answer: only (S2) is correct

Choices:
- both are incorrect
- only (S1) is correct
- only (S2) is correct
- both are correct


Question: The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake, one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are \( \mu \) and \( \sigma \) respectively, then \( 10(\mu + \sigma) \) is equal to:

Answer: 449

Choices:
- 445
- 451
- 447
- 449


Question: Let \( x_1, x_2, x_3, x_4 \) be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from \( x_1, x_2, x_3, x_4 \), then the resulting numbers are in an arithmetic progression. Then the value of \( \frac{1}{24} x_1 x_2 x_3 x_4 \) is:

Answer: 216

Choices:
- 72
- 18
- 36
- 216


Question: Let the set of all values of \( p \in \mathbb{R} \), for which both the roots of the equation \( x^2 - (p + 2)x + (2p + 9) = 0 \) are negative real numbers, be the interval \( (\alpha, \beta] \). Then \( \beta - 2\alpha \) is equal to:

Answer: 5

Choices:
- 0
- 9
- 5
- 20


Question: Let \( A \) be a \( 3 \times 3 \) matrix such that \( |\operatorname{adj} (\operatorname{adj}(\operatorname{adj} A))| = 81 \). If \[ S = \left\{ n \in \mathbb{Z} : |\operatorname{adj}(\operatorname{adj} A)|^{\frac{(n-1)^2}{2}} = |A|^{3n^2 - 5n - 4} \right\}, \] then \( \sum_{n \in S} \left| A^{n^2 + n} \right| \) is equal to:

Answer: 732

Choices:
- 866
- 750
- 820
- 732


Question: If the area of the region bounded by the curves \( y = 4 - \frac{x^2}{4} \) and \( y = \frac{x - 4}{2} \) is equal to \( \alpha \), then \( 6\alpha \) equals:

Answer: 250

Choices:
- 250
- 210
- 240
- 220


Question:
Let the system of equations:
\[\begin{aligned}
2x + 3y + 5z &= 9, \\
7x + 3y - 2z &= 8, \\
12x + 3y - (4 + \lambda)z &= 16 - \mu
\end{aligned}\]
have infinitely many solutions. Then the radius of the circle centred at \((\lambda, \mu)\) and touching the line \(4x = 3y\) is:

Answer: \( \frac{7}{5} \)

Choices:
- \( \frac{17}{5} \)
- \( \frac{7}{5} \)
- 7
- \( \frac{21}{5} \)


Question: Let the line \( L \) pass through \((1, 1, 1)\) and intersect the lines \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} \quad \text{and} \quad \frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1}. \] Then, which of the following points lies on the line \( L \)?

Answer: (7, 15, 13)

Choices:
- (4, 22, 7)
- (5, 4, 3)
- (10, -29, -50)
- (7, 15, 13)


Question: Let the angle \( \theta \), \( 0 < \theta < \frac{\pi}{2} \), be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), and \( \theta = \sin^{-1}\left(\frac{\sqrt{65}}{9}\right) \). If the vector \( \vec{c} = 3\hat{a} + 6\hat{b} + 9(\hat{a} \times \hat{b}) \), then the value of \( 9(\vec{c} \cdot \hat{a}) - 3(\vec{c} \cdot \hat{b}) \) is:

Answer: 29

Choices:
- 31
- 27
- 29
- 24


Question: Let ABC be the triangle such that the equations of lines AB and AC be \( 3y - x = 2 \) and \( x + y = 2 \), respectively, and the points B and C lie on the x-axis. If P is the orthocentre of the triangle ABC, then the area of the triangle PBC is equal to:

Answer: 6

Choices:
- 4
- 10
- 8
- 6


Question: The number of points of discontinuity of the function \( f(x) = \left\lfloor \frac{x^2}{2} \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor, x \in [0, 4] \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function, is _____

Answer: 8


Question: The number of relations on the set \( A = \{1, 2, 3\} \) containing at most 6 elements including \((1, 2)\), which are reflexive and transitive but not symmetric, is _____

Answer: 6


Question: Consider the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) having one of its foci at \( P = (-3, 0) \). If the latus rectum through its other focus subtends a right angle at P and \( a^2 b^2 = \alpha \sqrt{2} - \beta \), \( \alpha, \beta \in \mathbb{N} \), then

Answer: 1944


Question: The number of singular matrices of order 2, whose elements are from the set \( \{2, 3, 6, 9\} \), is _____

Answer: 36


Question: For \( n \geq 2 \), let \( S_n \) denote the set of all subsets of \( \{1, 2, \ldots, n\} \) with no two consecutive numbers. For example, \( \{1, 3, 5\} \in S_6 \), but \( \{1, 2, 4\} \notin S_6 \). Then \( n(S_5) \) is equal to _____

Answer: 13


Question: Let the values of \( \lambda \) for which the shortest distance between the lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x - \lambda}{3} = \frac{y - 4}{4} = \frac{z - 5}{5} \) is \( \frac{1}{\sqrt{6}} \) be \( \lambda_1 \) and \( \lambda_2 \). Then the radius of the circle passing through the points (0, 0), \( (\lambda_1, \lambda_2) \) and \( (\lambda_2, \lambda_1) \) is

Answer: 5\sqrt{2}/3

Choices:
- 5\sqrt{2}/3
- 4
- \sqrt{2}/3
- 3


Question: Let \( \alpha \) be a solution of \( x^2 + x + 1 = 0 \), and for some a and b in \( \mathbb{R} \), \([4 \; a \; b] \begin{bmatrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{bmatrix} = [0 \; 0 \; 0]\). If \( \frac{4}{\alpha^4} + \frac{m}{\alpha} + \frac{n}{\alpha^2} = 3 \), then m + n is equal to

Answer: 11

Choices:
- 3
- 11
- 7
- 8


Question: Let the function \( f(x) = \frac{x}{3} + \frac{3}{x} + 3, \; x \ne 0 \) be strictly increasing in \( (-\infty, \alpha_1) \cup (\alpha_2, \infty) \) and strictly decreasing in \( (\alpha_1, \alpha_2) \cup (\alpha_4, \alpha_5) \). Then \( \sum_{i=1}^{5} \alpha_i^2 \) is equal to:

Answer: 36

Choices:
- 48
- 28
- 40
- 36


Question:
If \( \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots = \frac{\pi^4}{90} \),

\( \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots = \alpha \),

\( \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots = \beta \),

then \( \frac{\alpha}{\beta} \) is equal to

Answer: 15

Choices:
- 23
- 18
- 15
- 14


Question: If A and B are two events such that \( P(A) = 0.7 \), \( P(B) = 0.4 \) and \( P(A \cap \overline{B}) = 0.5 \), where \( \overline{B} \) denotes the complement of B, then \( P(B \mid (A \cup \overline{B})) \) is equal to:

Answer: 1/4

Choices:
- 1/4
- 1/2
- 1/6
- 1/3


Question: The sum of the squares of the roots of \( |x + 2|^2 + |x - 2| - 2 = 0 \) and the squares of the roots of \( x^2 - 2|x - 3| - 5 = 0 \), is:

Answer: 36

Choices:
- 26
- 36
- 30
- 24


Question: Let \( a \) be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle \( \alpha \) with the positive x-axis and the equations of its diagonals are \( (\sqrt{3} + 1)x + (\sqrt{3} - 1)y = 0 \) and \( (\sqrt{3} - 1)x - (\sqrt{3} + 1)y + 8\sqrt{3} = 0 \). Then \( a^2 \) is equal to:

Answer: 48

Choices:
- 48
- 32
- 16
- 24


Question: Let \( f(x) \) be a positive function and \[ I_1 = \int_{-\frac{1}{2}}^{1} 2x f(2x(1 - 2x)) \, dx \quad \text{and} \quad I_2 = \int_{-1}^{\frac{1}{2}} f(x(1 - x)) \, dx. \] Then the value of \( \frac{I_2}{I_1} \) is equal to:

Choices:
- 9
- 6
- 12
- 4


Question: Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). Let \( \vec{c} \) be a unit vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \) and be perpendicular to \( \vec{a} \). Then such a vector \( \vec{c} \) is:

Answer: \( \frac{1}{\sqrt{2}} (-\hat{i} + \hat{k}) \)

Choices:
- \( \frac{1}{\sqrt{5}} (\hat{j} - 2\hat{k}) \)
- \( \frac{1}{\sqrt{3}} (-\hat{i} + \hat{j} - \hat{k}) \)
- \( \frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k}) \)
- \( \frac{1}{\sqrt{2}} (-\hat{i} + \hat{k}) \)


Question: Let the ellipse \( 3x^2 + py^2 = 4 \) pass through the centre C of the circle \( x^2 + y^2 - 2x - 4y - 11 = 0 \) of radius \( r \). Let \( f_1, f_2 \) be the focal distances of the point C on the ellipse. Then \( 6f_1f_2 - r \) is equal to:

Answer: 70

Choices:
- 74
- 68
- 70
- 78


Question: The integral \( \int_{-1}^{\frac{3}{2}} \left| \pi^2 x \sin(\pi x) \right| \, dx \) is equal to:

Answer: 1 + 3\pi

Choices:
- 3 + 2\pi
- 4 + \pi
- 1 + 3\pi
- 2 + 3\pi


Question: A line passing through the point \( P(a, \theta) \) makes an acute angle \( \alpha \) with the positive x-axis. Let this line be rotated about the point P through an angle \( \frac{\alpha}{2} \) in the clockwise direction. If in the new position, the slope of the line is \( 2 - \sqrt{3} \) and its distance from the origin is \( \frac{1}{\sqrt{2}} \), then the value of \( 3a^2 \tan^2 \alpha - 2\sqrt{3} \) is:

Choices:
- 4
- 6
- 5
- 8


Question: There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

Answer: 210

Choices:
- 230
- 220
- 200
- 210


Question: Let \( A = \left\{ \theta \in [0, 2\pi] : 1 + 10\,\text{Re} \left( \frac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta} \right) = 0 \right\}. \) Then \( \sum_{\theta \in A} \theta^2 \) is equal to:

Answer: \( \frac{21}{4} \pi^2 \)

Choices:
- \( \frac{21}{4} \pi^2 \)
- \( 8\pi^2 \)
- \( \frac{27}{4} \pi^2 \)
- \( 6\pi^2 \)


Question: The number of integral terms in the expansion of \( \left( \frac{1}{5^2} + \frac{1}{7^8} \right)^{1016} \) is:

Answer: 128

Choices:
- 127
- 130
- 129
- 128


Question:
Let \( A = \{0, 1, 2, 3, 4, 5\} \). Let \( R \) be a relation on \( A \) defined by \( (x, y) \in R \) if and only if \( \max\{x, y\} \in \{3, 4\} \). Then among the statements
(S₁): The number of elements in R is 18, and
(S₂): The relation R is symmetric but neither reflexive nor transitive
Which of the following is correct?

Answer: only (S₂) is true

Choices:
- both are true
- both are false
- only (S₂) is true
- only (S₁) is true


Question: Let \( f(x) = x - 1 \) and \( g(x) = e^x \) for \( x \in \mathbb{R} \). If \[ \frac{dy}{dx} = \left( e^{-2\sqrt{x}} g(f(f(f(x)))) - \frac{y}{\sqrt{x}} \right), \quad y(0) = 0, \] then \( y(1) \) is:

Answer: \( \frac{e - 1}{e^4} \)

Choices:
- \( \frac{1 - e^2}{e^4} \)
- \( \frac{2e - 1}{e^3} \)
- \( \frac{e - 1}{e^4} \)
- \( \frac{1 - e^3}{e^4} \)


Question: The value of \[ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2\left(\frac{1}{2}\right)} + 1}{\tan\left(\frac{1}{2}\right)} \right) \] is equal to:

Answer: \( \pi - \frac{5}{4} \)

Choices:
- \( \pi - \frac{5}{4} \)
- \( \pi - \frac{3}{2} \)
- \( \pi + \frac{3}{2} \)
- \( \pi + \frac{5}{2} \)


Question: Let \( A = \begin{bmatrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2p & 8 + 3p + 2q \\ 6 & 12 + 3p & 20 + 6p + 3q \end{bmatrix} \). If \( \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n \), \( m, n \in \mathbb{N} \), then \( m + n \) is equal to:

Answer: 24

Choices:
- 22
- 24
- 26
- 20


Question:
Given below are two statements:

**Statement I:** \[ \lim_{x \to 0} \left( \frac{\tan^{-1}x + \log_e \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} \right) - 2x}{x^5} \right) = \frac{2}{5} \]

**Statement II:** \[ \lim_{x \to 1} \left( \frac{2}{x^{1 - x}} \right) = \frac{1}{e^2} \]

In the light of the above statements, choose the correct answer from the options given below:

Answer: Both Statement I and Statement II are true

Choices:
- Statement I is false but Statement II is true
- Statement I is true but Statement II is false
- Both Statement I and Statement II are false
- Both Statement I and Statement II are true


Question: Let the area of the bounded region \( \{(x, y) : 0 \leq 9x \leq y^2, \; y \geq 3x - 6 \} \) be \( A \). Then \( 6A \) is equal to _____

Answer: 15


Question: Let the domain of the function \( f(x) = \cos^{-1} \left( \frac{4x + 5}{3x - 7} \right) \) be \( [\alpha, \beta] \) and the domain of \( g(x) = \log_2(2 - 6\log_7(2x + 5)) \) be \( [\gamma, \delta] \). Then \( |7(\alpha + \beta) + 4(\gamma + \delta)| \) is equal to _____

Answer: 96


Question: Let the area of the triangle formed by the lines \( x + 2 = y - 1 = z \), \( \frac{x - 3}{5} = \frac{y}{-1} = \frac{z - 1}{1} \), and \( \frac{x}{-3} = \frac{y - 3}{3} = \frac{z - 2}{1} \) be \( A \). Then \( A^2 \) is equal to _____

Answer: 56


Question: The product of the last two digits of \( (1919)^{1919} \) is _____

Answer: 63


Question: Let \( r \) be the radius of the circle, which touches the x-axis at point \( (a, 0), \; a < 0 \) and the parabola \( y^2 = 9x \) at the point \( (4, 6) \). Then \( r \) is equal to _____

Answer: 30