question
string | answer
string | options
list | correct_options
list | question_type
int64 |
|---|---|---|---|---|
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:
|
6
|
[] |
[] | 0 |
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:
|
164
|
[] |
[] | 0 |
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:
|
18
|
[
"15",
"18",
"24",
"13"
] |
[
1
] | 1 |
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:
|
379
|
[] |
[] | 0 |
The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is:
|
5
|
[
"6",
"5",
"7",
"4"
] |
[
1
] | 1 |
The square of the distance of the point $\left(\frac{10}{7}, \frac{22}{7}, 7\right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is:
|
66
|
[
"54",
"44",
"41",
"66"
] |
[
3
] | 1 |
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:
|
2
|
[
"0",
"2",
"1",
"3"
] |
[
1
] | 1 |
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:
|
5
|
[
"4",
"3",
"6",
"5"
] |
[
3
] | 1 |
If the set of all $a \in \mathbb{R}$, for which the equation $2x^2+(a-5)x+15=3a$ has no real root, is the interval $(\alpha, \beta)$, and $X=\{x \in \mathbb{Z}: \alpha < x < \beta\}$, then $\sum_{x \in X} x^2$ is equal to:
|
2139
|
[
"2109",
"2129",
"2119",
"2139"
] |
[
3
] | 1 |
If the area of the region $\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a > 0\}$ is $\frac{\pi^2 + 8e + 1}{e}$, then the value of $a$ is:
|
5
|
[
"8",
"7",
"5",
"6"
] |
[
2
] | 1 |
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:
|
25
|
[
"40",
"32",
"25",
"37"
] |
[
2
] | 1 |
Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
|
192
|
[
"392",
"384",
"192",
"96"
] |
[
2
] | 1 |
A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $(2, 5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of $r$ is the interval $(\alpha, \beta)$, then $3\beta - 2\alpha$ is equal to:
|
15
|
[
"10",
"15",
"12",
"14"
] |
[
1
] | 1 |
The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is:
|
18817
|
[
"16923",
"3763",
"33845",
"18817"
] |
[
3
] | 1 |
Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is:
|
957
|
[] |
[] | 0 |
Let $f:[0,3] \rightarrow A$ be defined by $f(x)=2x^3-15x^2+36x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $g(x)=\frac{x^{200}}{x^{2005}+1}$. If both the functions are onto and $S=\{x \in \mathbb{Z} : x \in A \text{ or } x \in B\}$, then $n(S)$ is equal to:
|
30
|
[
"29",
"30",
"31",
"36"
] |
[
1
] | 1 |
Let $L_1: \frac{x-1}{3} = \frac{y-1}{4} = \frac{z+1}{0}$ and $L_2: \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{0}$, $\alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26 \alpha (PB)^2$ is:
|
216
|
[] |
[] | 0 |
Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4 \hat{k}$, $\lambda > 0$, on the vector $\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is:
|
16
|
[] |
[] | 0 |
Let $A$ and $B$ be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is:
|
14
|
[] |
[] | 0 |
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:
|
11
|
[
"11",
"10",
"12",
"13"
] |
[
0
] | 1 |
The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is:
|
17280
|
[] |
[] | 0 |
Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_{0}^{x} t f(t) \, dt$. If $g(x^3)=x^6+x^7$, then value of $\sum_{r=1}^{15} f(r^3)$ is:
|
310
|
[
"270",
"340",
"320",
"310"
] |
[
3
] | 1 |
Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \operatorname{adj}(-6 \operatorname{adj}(3 A))) = 2^{m+n} \cdot 3^{mn}$, $m > n$. Then $4m + 2n$ is equal to:
|
34
|
[] |
[] | 0 |
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is $\frac{29}{45}$, then $n$ is equal to:
|
6
|
[
"6",
"3",
"5",
"4"
] |
[
0
] | 1 |
The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\frac{z}{2} + \frac{2}{z}\right| = 1$, is:
|
8
|
[
"4",
"8",
"10",
"6"
] |
[
1
] | 1 |
Number of functions $f : \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
|
392
|
[] |
[] | 0 |
The sum $1 + 3 + 11 + 25 + 45 + 71 + \dots$ up to 20 terms is equal to:
|
7240
|
[
"7240",
"7130",
"6982",
"8124"
] |
[
0
] | 1 |
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:
|
3
|
[
"3",
"6",
"5",
"4"
] |
[
0
] | 1 |
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:
|
449
|
[
"445",
"451",
"447",
"449"
] |
[
3
] | 1 |
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:
|
64
|
[
"56",
"65",
"64",
"70"
] |
[
2
] | 1 |
Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}{4}=\frac{y+1}{4}=\frac{z-3}{2}$ and $\frac{x-3}{4}=\frac{y-2}{3}=\frac{z+2}{4}$. If the line $L$ intersects the $yz$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is:
|
3
|
[
"$\\sqrt{10}$",
"$2\\sqrt{3}$",
"2",
"3"
] |
[
3
] | 1 |
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)
|
5760
|
[
"5880",
"960",
"840",
"5760"
] |
[
3
] | 1 |
The remainder, when $7^{103}$ is divided by 23, is equal to:
|
14
|
[
"6",
"17",
"9",
"14"
] |
[
3
] | 1 |
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:
|
81
|
[
"27",
"9",
"81",
"18"
] |
[
2
] | 1 |
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:
|
24
|
[
"24",
"27",
"18",
"21"
] |
[
0
] | 1 |
Let the area of a \(\triangle PQR\) with vertices \(P(5, 4)\), \(Q(-2, 4)\) and \(R(a, b)\) be 35 square units. If its orthocenter and centroid are \(O\left(2, \frac{14}{5}\right)\) and \(C(c, d)\) respectively, then \(c + 2d\) is equal to
|
3
|
[
"\\(\\frac{8}{3}\\)",
"\\(\\frac{7}{3}\\)",
"2",
"3"
] |
[
3
] | 1 |
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:
|
0
|
[
"0",
"-2",
"4",
"2"
] |
[
0
] | 1 |
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:
|
15
|
[
"18",
"12",
"9",
"15"
] |
[
3
] | 1 |
If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to:
|
2
|
[
"-2",
"6",
"-6",
"2"
] |
[
3
] | 1 |
If $\sum_{x=0}^5 \frac{{ }^{11} C_{2x}}{2x + 2} = \frac{m}{n}$, $\gcd(m, n) = 1$, then $m - n$ is equal to:
|
2035
|
[] |
[] | 0 |
Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right-angled triangle $AMN$ is inscribed in the triangle $OAB$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$, respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN:NB=\lambda:1$, then the sum of all possible value(s) of $\lambda$ is:
|
2
|
[
"2",
"$\\frac{5}{2}$",
"$\\frac{1}{2}$",
"$\\frac{15}{6}$"
] |
[
0
] | 1 |
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:
|
14
|
[] |
[] | 0 |
Let the range of the function $f(x) = 6 + 16 \cos x \cdot \cos \left(\frac{\pi}{3} - x\right) \cdot \cos \left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is:
|
11
|
[
"11",
"8",
"10",
"9"
] |
[
0
] | 1 |
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:
|
11
|
[
"1",
"0",
"10",
"11"
] |
[
3
] | 1 |
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:
|
-3
|
[
"2",
"-3",
"3",
"-2"
] |
[
1
] | 1 |
Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{87}{13}, \frac{-60}{13}\right)$, then $|\alpha \lambda|$ is equal to
|
91
|
[
"84",
"113",
"91",
"101"
] |
[
2
] | 1 |
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:
|
13
|
[
"26",
"13",
"12",
"\\frac{13}{2}"
] |
[
1
] | 1 |
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:
|
75
|
[
"80",
"72",
"96",
"75"
] |
[
3
] | 1 |
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:
|
34
|
[] |
[] | 0 |
Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat{i}+2\hat{j}+2\hat{k}$. If the distance of the point $Q$ from the point $R(1,3,3)$ is 5, then the square of the area of $\triangle PQR$ is equal to:
|
136
|
[
"148",
"136",
"144",
"140"
] |
[
1
] | 1 |
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 _____
|
30
|
[] |
[] | 0 |
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:
|
32
|
[
"18",
"32",
"22",
"26"
] |
[
1
] | 1 |
Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min \{1 + x + [x], x + 2[x]\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{cases}$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals
|
5
|
[] |
[] | 0 |
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:
|
24
|
[
"36",
"24",
"48",
"20"
] |
[
1
] | 1 |
Let $\mathbf{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geqslant 3\}$ and $\mathbf{B}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x|+|y| \leq 3\}$. If $\mathbf{C}=\{(x, y) \in \mathbf{A} \cap \mathbf{B} : x=0 \text{ or } y=0\}$, then $\sum_{(x, y) \in \mathbf{C}}|x+y|$ is:
|
12
|
[
"15",
"24",
"18",
"12"
] |
[
3
] | 1 |
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:
|
22
|
[] |
[] | 0 |
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:
|
441
|
[] |
[] | 0 |
Let the area of the region $\{(x, y): 2y \leq x^2 + 3, y + |x| \leq 3, y \geq |x-1|\}$ be $A$. Then $6A$ is equal to:
|
14
|
[
"16",
"12",
"14",
"18"
] |
[
2
] | 1 |
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:
|
183
|
[] |
[] | 0 |
Let $f(x) = \lim_{n \to \infty} \sum_{t=0}^{n} \left(\frac{\tan \left(\frac{x}{2^{n+1}}\right) + \tan^3 \left(\frac{x}{2^{n+1}}\right)}{1 - \tan^3 \left(\frac{x}{2^{n+1}}\right)}\right)$. Then $\lim_{x \to 0} \frac{x}{(x - f(x))}$ is equal to:
|
1
|
[] |
[] | 0 |
Let \(\left| \frac{z - \overline{z}}{2z + \overline{z}} \right| = \frac{1}{3}\), \(z \in \mathbb{C}\), be the equation of a circle with center at \(C\). If the area of the triangle, whose vertices are at the points \((0, 0)\), \(C\) and \((\alpha, 0)\) is 11 square units, then \(\alpha^2\) equals
|
100
|
[
"50",
"100",
"\\(\\frac{81}{25}\\)",
"\\(\\frac{121}{25}\\)"
] |
[
1
] | 1 |
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:
|
2477
|
[] |
[] | 0 |
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:
|
12
|
[
"12",
"11",
"13",
"14"
] |
[
0
] | 1 |
If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $\left(\sqrt{2}, \frac{4}{3}\right)$, and the length of the chord is $\frac{2\sqrt{6}}{3}$, then $\alpha$ is:
|
22
|
[
"20",
"",
"18",
"26"
] |
[
1
] | 1 |
Let the function, $f(x) = \begin{cases} -3 a x^2 - 2, & x < 1 \\ a^2 + b x, & x \geqslant 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is:
|
34
|
[] |
[] | 0 |
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:
|
36
|
[
"48",
"28",
"40",
"36"
] |
[
3
] | 1 |
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:
|
-1
|
[
"-1",
"28",
"27",
"1"
] |
[
0
] | 1 |
If the system of equations \((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5\), \(\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7\), \((\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9\) has infinitely many solutions, then \(\lambda^2 + \lambda\) is equal to
|
12
|
[
"6",
"10",
"20",
"12"
] |
[
3
] | 1 |
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:
|
26
|
[
"22",
"18",
"26",
"30"
] |
[
2
] | 1 |
Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $P=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $B=PAP^{T}$, $C=P^{T} B^{10} P$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m,n)=1$, then $m+n$ is:
|
65
|
[
"127",
"258",
"65",
"2049"
] |
[
2
] | 1 |
Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i-2)=30$, $\sum_{i=1}^{10}(x_i-\beta)^2=98$, $\beta>2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1-1)+4\beta$, $2(x_2-1)+4\beta$, $\ldots$, $2(x_{10}-1)+4\beta$, then $\frac{\beta \mu}{\sigma^2}$ is equal to:
|
100
|
[
"100",
"120",
"110",
"90"
] |
[
0
] | 1 |
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:
|
10
|
[
"10",
"8",
"11",
"9"
] |
[
0
] | 1 |
If the domain of the function $\log_5\left(18x-x^2-77\right)$ is $(\alpha, \beta)$ and the domain of the function $\log_{(\alpha-1)}\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to:
|
186
|
[
"195",
"179",
"186",
"174"
] |
[
2
] | 1 |
If $y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right)$, then $(x - y)^2 + 3y^2$ is equal to:
|
3
|
[] |
[] | 0 |
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:
|
338
|
[
"218",
"377",
"290",
"338"
] |
[
3
] | 1 |
Let the position vectors of three vertices of a triangle be $4 \vec{p} + \vec{q} - 3 \vec{r}$, $-5 \vec{p} + \vec{q} + 2 \vec{r}$, and $2 \vec{p} - \vec{q} + 2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{2}$ and $\alpha \vec{p} + \beta \vec{q} + \gamma \vec{r}$ respectively, then $\alpha + 2 \beta + 5 \gamma$ is equal to:
|
3
|
[
"3",
"4",
"1",
"6"
] |
[
0
] | 1 |
Let $\overrightarrow{\mathrm{a}}=2\hat{\imath}-\hat{\jmath}+3\hat{k}$, $\overrightarrow{\mathrm{b}}=3\hat{\imath}-5\hat{\jmath}+\hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$ and $(\vec{a}+\vec{c}) \cdot (\vec{b}+\vec{c})=168$. Then the maximum value of $|\vec{c}|^2$ is:
|
308
|
[
"462",
"77",
"154",
"308"
] |
[
3
] | 1 |
Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right) = 0$, then $y'\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right)$ is equal to:
|
1
|
[] |
[] | 0 |
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 _____
|
15
|
[] |
[] | 0 |
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:
|
1
|
[
"1",
"3/4",
"2",
"1/2"
] |
[
0
] | 1 |
Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3} \alpha$ is equal to
|
4
|
[
"$3 + \\sqrt{3}$",
"$4$",
"$4 - \\sqrt{3}$",
"$3$"
] |
[
1
] | 1 |
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:
|
141
|
[] |
[] | 0 |
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:
|
0
|
[
"0",
"1",
"i",
"-i"
] |
[
0
] | 1 |
A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2:1$ is $9\left(x^2 + \alpha y^2 + \beta xy + \gamma x + 28 y\right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to:
|
23
|
[
"22",
"21",
"23",
"24"
] |
[
2
] | 1 |
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:
|
47
|
[
"49",
"47",
"21",
"62"
] |
[
1
] | 1 |
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:
|
18
|
[
"18",
"16",
"15",
"17"
] |
[
0
] | 1 |
Let $f : (0, \infty) \to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2 f(2)$ is equal to:
|
39
|
[
"39",
"19",
"29",
"23"
] |
[
0
] | 1 |
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:
|
210
|
[
"230",
"220",
"200",
"210"
] |
[
3
] | 1 |
If the equation of the parabola with vertex $\mathbf{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:
|
9
|
[
"7",
"9",
"8",
"6"
] |
[
1
] | 1 |
Let $T_r$ be the $r^{\text{th}}$ term of an A.P. If for some $m$, $T_m = \frac{1}{25}$, $T_{25} = \frac{1}{20}$, and $20 \sum_{r=1}^{25} T_r = 13$, then $5m \sum_{r=m}^{2m} T_r$ is equal to
|
126
|
[
"98",
"126",
"142",
"112"
] |
[
1
] | 1 |
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:
|
2
|
[] |
[] | 0 |
Let $f: (0, \infty) \to \mathbf{R}$ be a twice differentiable function. If for some $a \neq 0$, $\int_0^1 f(\lambda x) d\lambda = a f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to
|
112
|
[] |
[] | 0 |
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:
|
27
|
[
"3(1+\\sqrt{2})",
"3(6+\\sqrt{2})",
"9",
"27"
] |
[
3
] | 1 |
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:
|
19
|
[
"19",
"21",
"18",
"20"
] |
[
0
] | 1 |
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:
|
5
|
[
"6",
"5",
"4",
"3"
] |
[
1
] | 1 |
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:
|
17
|
[
"18",
"17",
"15",
"16"
] |
[
1
] | 1 |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x \in \mathbb{R}$, $\int_{0}^{2} x F'(x) \, dx = 6$ and $\int_{0}^{2} x^2 F''(x) \, dx = 40$, then $F'(2) + \int_{0}^{2} F(x) \, dx$ is equal to:
|
11
|
[
"",
"13",
"15",
"9"
] |
[
0
] | 1 |
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:
|
15
|
[
"15",
"5\\sqrt{3}",
"15\\sqrt{5}",
"12"
] |
[
0
] | 1 |
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 _____
|
96
|
[] |
[] | 0 |
If $\sum_{r=1}^{13} \left\{\frac{1}{\sin \left(\frac{1}{4}+(r-1) \frac{2}{3}\right) \sin \left(\frac{7}{4}+\frac{2}{7}\right)}\right\}=a \sqrt{3}+b$, $a, b \in \mathbb{Z}$, then $a^2+b^2$ is equal to:
|
8
|
[
"10",
"4",
"2",
"8"
] |
[
3
] | 1 |
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