question
string | answer
string | options
list | correct_options
list | question_type
int64 |
|---|---|---|---|---|
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:
|
48
|
[
"48",
"32",
"16",
"24"
] |
[
0
] | 1
|
Let $P$ be the foot of the perpendicular from the point $(1,2,2)$ on the line $L: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line $\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$, $\lambda \in \mathbb{R}$, intersect the line $L$ at $Q$. Then $2(PQ)^2$ is equal to:
|
27
|
[
"25",
"19",
"29",
"27"
] |
[
3
] | 1
|
Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is:
|
3
|
[] |
[] | 0
|
Let \(I(x) = \int \frac{dx}{(x - 11)^{\frac{a}{13}} (x + 15)^{\frac{c}{13}}}\). If \(I(37) - I(24) = \frac{1}{4} \left( \frac{1}{5^{\frac{1}{3}}} - \frac{1}{6^{\frac{1}{13}}} \right)\), \(b, c \in \mathbb{N}\), then \(3(b + c)\) is equal to
|
39
|
[
"22",
"39",
"40",
"26"
] |
[
1
] | 1
|
If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to
|
9
|
[
"9",
"1",
"7",
"8"
] |
[
0
] | 1
|
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:
|
80
|
[
"80",
"60",
"72",
"64"
] |
[
0
] | 1
|
Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$, and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:
|
33
|
[
"44",
"",
"",
"55"
] |
[
2
] | 1
|
Let the parabola $y = x^2 + p x - 3$ meet the coordinate axes at the points $P$, $Q$, and $R$. If the circle $C$ with centre at $(-1, -1)$ passes through the points $P$, $Q$, and $R$, then the area of $\triangle PQR$ is:
|
6
|
[
"7",
"4",
"6",
"5"
] |
[
2
] | 1
|
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:
|
24
|
[
"22",
"24",
"26",
"20"
] |
[
1
] | 1
|
Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a} + 2 \vec{b}$ and $3 \vec{a} - \lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is:
|
0
|
[
"2",
"1",
"0",
"3"
] |
[
2
] | 1
|
Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2 \log_e (x - 2) - x^2 + a x + 1$ is strictly increasing and $(b, c)$ be the largest open interval in which the function $g(x) = (x - 1)^3 (x + 2 - a)^2$ is strictly decreasing. Then $100(a + b - c)$ is equal to:
|
360
|
[
"420",
"360",
"160",
"280"
] |
[
1
] | 1
|
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:
|
24
|
[
"24",
"36",
"30",
"18"
] |
[
0
] | 1
|
If the area of the region \{(x,y): |x-5| \le y \le 4\sqrt{x}\} is A, then 3A is equal to _____
|
368
|
[] |
[] | 0
|
Let the area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ be $\alpha$. If $9\alpha=\beta\pi+\gamma$, $\beta, \gamma$ are integers, then the value of $|\beta-\gamma|$ equals:
|
33
|
[
"27",
"33",
"15",
"18"
] |
[
1
] | 1
|
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:
|
5
|
[
"3",
"4",
"5",
"7"
] |
[
2
] | 1
|
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:
|
155
|
[
"165",
"155",
"145",
"135"
] |
[
1
] | 1
|
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:
|
3
|
[
"5",
"4",
"2",
"3"
] |
[
3
] | 1
|
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:
|
20
|
[
"20",
"30",
"10",
"15"
] |
[
0
] | 1
|
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:
|
239
|
[] |
[] | 0
|
For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
|
7
|
[
"9",
"5",
"7",
"4"
] |
[
2
] | 1
|
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:
|
9
|
[
"4",
"9",
"3",
"8"
] |
[
1
] | 1
|
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:
|
8
|
[
"12",
"6",
"8",
"10"
] |
[
2
] | 1
|
The largest $n\in\mathbb{N}$ such that $3^n$ divides $50!$ is:
|
22
|
[
"21",
"22",
"20",
"23"
] |
[
1
] | 1
|
Let $z_1, z_2$, and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}$, $\arg(z_2) = 0$, and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is:
|
29
|
[
"24",
"29",
"41",
"31"
] |
[
1
] | 1
|
If the equation \(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\) has equal roots, where \(a + c = 15\) and \(b = \frac{36}{5}\), then \(a^2 + c^2\) is equal to
|
117
|
[] |
[] | 0
|
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:
|
2520
|
[
"360",
"45",
"2520",
"1820"
] |
[
2
] | 1
|
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:
|
540
|
[
"600",
"450",
"630",
"540"
] |
[
3
] | 1
|
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:
|
54
|
[
"54",
"63",
"57",
"60"
] |
[
0
] | 1
|
If \(\displaystyle\lim_{x\to0}\bigl(\tfrac{\tan x}{x}\bigr)^{1/x^2}=p\), then \(96\log_e p\) is equal to:
|
32
|
[] |
[] | 0
|
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:
|
18
|
[
"19",
"20",
"17",
"18"
] |
[
3
] | 1
|
The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
|
36000
|
[
"36000",
"37000",
"34000",
"35000"
] |
[
0
] | 1
|
Let $f(x)=\int_0^x t\left(t^2-9t+20\right) dt$, $1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$ equals:
|
157
|
[
"253",
"154",
"125",
"157"
] |
[
3
] | 1
|
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:
|
64
|
[] |
[] | 0
|
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:
|
26
|
[
"26",
"29",
"25",
"30"
] |
[
0
] | 1
|
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:
|
3
|
[] |
[] | 0
|
Let $\mathbf{A}=\left[a_{ij}\right]=\left[\begin{array}{cc} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{array}\right]$. If $\mathbf{A}_{ij}$ is the cofactor of $\mathbf{a}_{ij}$, $\mathbf{C}_{ij}=\sum_{k=1}^2 \mathbf{a}_{ik} \mathbf{A}_{jk}, 1 \leq i, j \leq 2$, and $\mathbf{C}=\left[\mathbf{C}_{ij}\right]$, then $8|\mathbf{C}|$ is equal to:
|
242
|
[
"288",
"222",
"242",
"262"
] |
[
2
] | 1
|
If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose midpoint is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to:
|
58
|
[
"58",
"46",
"37",
"72"
] |
[
0
] | 1
|
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:
|
18
|
[
"18",
"20",
"19",
"17"
] |
[
0
] | 1
|
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:
|
18
|
[
"24",
"12",
"18",
"16"
] |
[
2
] | 1
|
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:
|
29
|
[
"31",
"27",
"29",
"24"
] |
[
2
] | 1
|
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:
|
2300
|
[
"4060",
"1040",
"2300",
"4960"
] |
[
2
] | 1
|
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:
|
53
|
[
"53",
"52",
"39",
"44"
] |
[
0
] | 1
|
The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[3]{11})^n$ is 183, is:
|
2184
|
[
"2184",
"2196",
"2148",
"2172"
] |
[
0
] | 1
|
If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36$, then $\alpha^2 + \beta$ is
|
14
|
[] |
[] | 0
|
Let $\mathbf{A} = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2 / \pi)} |\sin x| + \log_{(2 / \pi)} |\cos x| = 2\}$ and $\mathbf{B} = \{x \geqslant 0 : \sqrt{x} (\sqrt{x} - 4) - 3 |\sqrt{x} - 2| + 6 = 0\}$. Then $n(\mathbf{A} \cup \mathbf{B})$ is equal to:
|
8
|
[
"4",
"8",
"6",
"2"
] |
[
1
] | 1
|
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:
|
5
|
[
"0",
"9",
"5",
"20"
] |
[
2
] | 1
|
If $\alpha = 1 + \sum_{r=1}^6 (-3)^{r-1} {}^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is
|
5
|
[] |
[] | 0
|
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:
|
5
|
[
"6",
"7",
"5",
"8"
] |
[
2
] | 1
|
Let $\mathbf{S} = \{m \in \mathbf{Z}: \mathbf{A}^{m^2} + \mathbf{A}^m = 3 \mathbf{I} - \mathbf{A}^{-6}\}$, where $\mathbf{A} = \left[\begin{array}{cc} 2 & -1 \\ 1 & 0 \end{array}\right]$. Then $n(\mathbf{S})$ is equal to
|
2
|
[] |
[] | 0
|
Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:
|
8
|
[
"6",
"8",
"9",
"7"
] |
[
1
] | 1
|
Let $A$ be a $3 \times 3$ matrix such that $X^T A X = 0$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}$, $A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}$, and $\det(\text{adj}(2(A + I))) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is
|
44
|
[] |
[] | 0
|
Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $\left|\frac{x+1}{x+b}\right| = 1$ and $\left|\begin{array}{ccc} x+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right| = 1$, $z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to:
|
10
|
[] |
[] | 0
|
Let the function $f(x)=\left(x^2+1\right)\left|x^2-ax+2\right|+\cos|x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12x+5y+10=0$ is equal to:
|
3
|
[
"5",
"4",
"3",
"2"
] |
[
2
] | 1
|
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:
|
16
|
[
"14",
"17",
"15",
"16"
] |
[
3
] | 1
|
In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:
|
144
|
[
"96",
"144",
"120",
"72"
] |
[
1
] | 1
|
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:
|
12
|
[] |
[] | 0
|
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:
|
98
|
[] |
[] | 0
|
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:
|
3
|
[
"5",
"3",
"2",
"4"
] |
[
1
] | 1
|
Let $A=\left[a_{ij}\right]$ be a matrix of order $3 \times 3$, with $a_{ij}=(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha+\beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then $\alpha+\beta$ is equal to:
|
224
|
[
"280",
"224",
"210",
"168"
] |
[
1
] | 1
|
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:
|
128
|
[] |
[] | 0
|
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
|
5148
|
[
"5148",
"6084",
"4356",
"14950"
] |
[
0
] | 1
|
The number of different 5-digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is
|
4607
|
[
"4608",
"5720",
"5719",
"4607"
] |
[
3
] | 1
|
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:
|
4
|
[
"4",
"6",
"5",
"8"
] |
[
0
] | 1
|
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:
|
7
|
[] |
[] | 0
|
The interior angles of a polygon with $n$ sides are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to:
|
20
|
[] |
[] | 0
|
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 _____
|
6
|
[] |
[] | 0
|
If $24 \int_0^{\frac{\pi}{4}} \left(\sin \left|4x - \frac{\pi}{12}\right| + |2 \sin x|\right) dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to:
|
12
|
[] |
[] | 0
|
Let $\mathrm{P}$ be the set of seven-digit numbers with sum of their digits equal to 11. If the numbers in $\mathrm{P}$ are formed by using the digits 1, 2, and 3 only, then the number of elements in the set $P$ is:
|
161
|
[
"173",
"164",
"158",
"161"
] |
[
3
] | 1
|
The number of solutions of the equation $\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0$ is:
|
4
|
[
"2",
"3",
"1",
"4"
] |
[
3
] | 1
|
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:
|
385
|
[
"540",
"385",
"420",
"215"
] |
[
1
] | 1
|
The number of real solution(s) of the equation $x^2 + 3x + 2 = \min \{|x - 3|, |x + 2|\}$ is:
|
2
|
[
"",
"0",
"2",
"3"
] |
[
2
] | 1
|
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:
|
20
|
[
"15",
"11",
"24",
"20"
] |
[
3
] | 1
|
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:
|
8
|
[
"8",
"16",
"12",
"6"
] |
[
0
] | 1
|
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:
|
36
|
[
"26",
"36",
"30",
"24"
] |
[
1
] | 1
|
Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to:
|
8925
|
[
"8750",
"9100",
"8925",
"8575"
] |
[
2
] | 1
|
Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbf{C}$. Then the minimum value of $|z_1 - z_2|$ is:
|
7
|
[
"13",
"10",
"3",
"7"
] |
[
3
] | 1
|
The value of \(\left( \sin 70^\circ \right) \left( \cot 10^\circ \cot 70^\circ - 1 \right)\) is
|
1
|
[
"\\(\\frac{2}{3}\\)",
"1",
"0",
"\\(\\frac{3}{2}\\)"
] |
[
1
] | 1
|
Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{d} = \vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - 2 \vec{a}|^2 = 8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $|10 - 3 \vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2$ is equal to
|
6
|
[] |
[] | 0
|
Let the point $A$ divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r:1$ ($r>0$). If $O$ is the origin and $(\overrightarrow{OQ} \cdot \overrightarrow{OA} - \frac{1}{5}|\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10)$, then the value of $r$ is:
|
7
|
[
"$\\sqrt{7}$",
"14",
"3",
"7"
] |
[
3
] | 1
|
The number of natural numbers between 212 and 999, such that the sum of their digits is 15, is:
|
64
|
[] |
[] | 0
|
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:
|
27
|
[
"25",
"26",
"27",
"28"
] |
[
2
] | 1
|
Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is
|
44
|
[
"52",
"48",
"44",
"40"
] |
[
2
] | 1
|
Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)$ is equal to
|
118
|
[
"92",
"118",
"102",
"108"
] |
[
1
] | 1
|
If $7 = 5 + \frac{1}{7} (5 + \alpha) + \frac{1}{77} (5 + 2\alpha) + \frac{1}{77} (5 + 3\alpha) + \infty$, then the value of $\alpha$ is:
|
6
|
[
"$\\frac{6}{7}$",
"6",
"$\\frac{1}{7}$",
"1"
] |
[
1
] | 1
|
For a $3 \times 3$ matrix $M$, let $\text{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and $\text{trace}(A)=3$. If $B=\text{adj}(\text{adj}(2 A))$, then the value of $|B|+\text{trace}(B)$ equals:
|
280
|
[
"56",
"132",
"174",
"280"
] |
[
3
] | 1
|
Let $\alpha, \beta$ be the roots of the equation $x^2 - ax - b = 0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n = \alpha^n - \beta^n$. If $\mathbf{P}_3 = -5\sqrt{7}i$, $\mathbf{P}_4 = -3\sqrt{7}i$, $\mathbf{P}_5 = 11\sqrt{7}i$ and $\mathbf{P}_6 = 45\sqrt{7}i$, then $|\alpha^4 + \beta^4|$ is equal to:
|
31
|
[] |
[] | 0
|
Let $M$ and $m$ respectively be the maximum and the minimum values of $\begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 4x \end{vmatrix}, x \in \mathbf{R}$. Then $M^4 - m^4$ is equal to:
|
1280
|
[
"1280",
"1295",
"1215",
"1040"
] |
[
0
] | 1
|
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:
|
16
|
[
"17",
"16.4",
"17.4",
"16"
] |
[
3
] | 1
|
Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to:
|
28
|
[] |
[] | 0
|
For some $a$, $b$, let $f(x) = \left| \begin{array}{ccc} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to:
|
16
|
[
"16",
"25",
"9",
"36"
] |
[
0
] | 1
|
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:
|
14
|
[
"10",
"14",
"13",
"12"
] |
[
1
] | 1
|
Let ${}^n C_{r-1} = 28$, ${}^n C_r = 56$ and ${}^n C_{r+1} = 70$. Let $A(4 \cos t, 4 \sin t)$, $B(2 \sin t, -2 \cos t)$ and $C(3r - n, r^2 - n - 1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$, is the locus of the centroid of triangle $ABC$, then $\alpha$ equals
|
20
|
[
"6",
"18",
"8",
"20"
] |
[
3
] | 1
|
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:
|
126
|
[
"125",
"126",
"120",
"115"
] |
[
1
] | 1
|
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:
|
3761
|
[
"3814",
"4027",
"3761",
"4003"
] |
[
2
] | 1
|
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:
|
1134
|
[
"1080",
"1134",
"1026",
"1188"
] |
[
1
] | 1
|
Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is
|
5120
|
[] |
[] | 0
|
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:
|
250
|
[
"250",
"210",
"240",
"220"
] |
[
0
] | 1
|
Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, then $a_{23}$ equals:
|
-1
|
[
"-1",
"2",
"1",
"0"
] |
[
0
] | 1
|
Let $f: \mathbb{R} - \{0\} \rightarrow (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is:
|
6
|
[
"7",
"6",
"1",
"9"
] |
[
1
] | 1
|
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 _____
|
46
|
[] |
[] | 0
|
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