Dataset Viewer
question
string | answer
string | options
list | correct_options
list | question_type
int64 |
|---|---|---|---|---|
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:
|
757
|
[
"760",
"755",
"750",
"757"
] |
[
3
] | 1 |
If $\lim_{t \to 0} \left(\int_0^1 (3x + 5)^t dx\right)^{\frac{1}{t}} = \frac{2}{56} \left(\frac{8}{5}\right)^{\frac{2}{\alpha}}$, then $\alpha$ is equal to:
|
64
|
[] |
[] | 0 |
If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
|
-1080
|
[
"-1080",
"-1020",
"-1200",
"-120"
] |
[
0
] | 1 |
Let $\mathbf{E}: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$ and $\mathbf{H}: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$. Let the distance between the foci of $\mathbf{E}$ and the foci of $\mathbf{H}$ be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $\mathbf{E}$ and $\mathbf{H}$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to:
|
8
|
[
"10",
"9",
"8",
"7"
] |
[
2
] | 1 |
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:
|
421
|
[
"423",
"420",
"421",
"422"
] |
[
2
] | 1 |
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:
|
97
|
[
"90",
"93",
"97",
"83"
] |
[
2
] | 1 |
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:
|
768
|
[] |
[] | 0 |
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:
|
7
|
[
"6",
"9",
"7",
"8"
] |
[
2
] | 1 |
If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C$, where $C$ is the constant of integration, then $\alpha + 2 \beta$ is equal to:
|
16
|
[] |
[] | 0 |
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:
|
5
|
[] |
[] | 0 |
In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to:
|
515
|
[
"525",
"510",
"515",
"505"
] |
[
2
] | 1 |
The number of integral terms in the expansion of \( \left( \frac{1}{5^2} + \frac{1}{7^8} \right)^{1016} \) is:
|
128
|
[
"127",
"130",
"129",
"128"
] |
[
3
] | 1 |
The product of the last two digits of \( (1919)^{1919} \) is _____
|
63
|
[] |
[] | 0 |
Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2 + 3a) x^2 + \left( \frac{a + 2}{a - 1} \right) x + b, a \neq 1$. If $f(x + y) = f(x) + f(y) + 1 - \frac{2}{7} xy$, then the value of $28 \sum_{i=1}^{5} |f(i)|$ is
|
675
|
[
"545",
"715",
"735",
"675"
] |
[
3
] | 1 |
Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$, and $(2,4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to:
|
22
|
[
"19",
"24",
"21",
"22"
] |
[
3
] | 1 |
Let $f$ be a differentiable function such that $2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt$, $x \geq 0$. Then $f(2)$ is equal to
|
19
|
[] |
[] | 0 |
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:
|
15
|
[
"15",
"25",
"30",
"20"
] |
[
0
] | 1 |
Let $A$, $B$, $C$ be three points in $xy$-plane, whose position vectors are given by $\sqrt{3} \hat{i}+\hat{j}$, $\hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\vec{OA}$ and $\vec{OB}$ is $\frac{\theta}{\sqrt{2}}$, then the sum of all the possible values of $a$ is:
|
1
|
[
"2",
"$\\frac{9}{2}$",
"1",
"0"
] |
[
2
] | 1 |
Let $E_1: \frac{x^2}{5} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ are constructed such that their centres and eccentricities are same as that of $E_1$, and the length of minor axis of $E_i$ is the length of major axis of $E_{i+1} (i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right)$, is equal to
|
54
|
[] |
[] | 0 |
If the system of equations $x + 2y - 3z = 2$, $2x + \lambda y + 5z = 5$, $14x + 3y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
|
12
|
[
"13",
"10",
"12",
"11"
] |
[
2
] | 1 |
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:
|
46
|
[
"42",
"46",
"48",
"40"
] |
[
1
] | 1 |
The roots of the quadratic equation $3x^2 - px + q = 0$ are $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2p$ is equal to:
|
474
|
[] |
[] | 0 |
If in the expansion of $(1 + x)^p (1 - x)^q$, the coefficients of $x$ and $x^2$ are 1 and -2, respectively, then $p^2 + q^2$ is equal to:
|
13
|
[
"18",
"13",
"8",
"20"
] |
[
1
] | 1 |
The number of singular matrices of order 2, whose elements are from the set \( \{2, 3, 6, 9\} \), is _____
|
36
|
[] |
[] | 0 |
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:
|
185
|
[
"184",
"186",
"185",
"187"
] |
[
2
] | 1 |
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:
|
38
|
[] |
[] | 0 |
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 _____
|
3
|
[] |
[] | 0 |
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:
|
50
|
[
"50",
"49",
"46",
"47"
] |
[
0
] | 1 |
Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to:
|
37
|
[
"47",
"37",
"36",
"40"
] |
[
1
] | 1 |
Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2n - 1}$. If $2A = 5B$, then $n$ is equal to:
|
21
|
[
"22",
"20",
"21",
"19"
] |
[
2
] | 1 |
Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:
|
6
|
[
"$-2\\sqrt{10}$",
"1",
"6",
"-6"
] |
[
2
] | 1 |
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 _____
|
22
|
[] |
[] | 0 |
Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + \left(a_5 + a_{10} + a_{15} + \ldots + a_{2020}\right) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to:
|
11132
|
[] |
[] | 0 |
Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1$. If $u$ and $v$ satisfy the equations $\begin{aligned} \alpha u+\beta v &= 18 \\ \gamma u+\delta v &= 20 \end{aligned}$ then $u+v$ equals:
|
5
|
[
"5",
"3",
"4",
"8"
] |
[
0
] | 1 |
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 _____
|
8
|
[] |
[] | 0 |
Let $\mathbf{S} = \{x: \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1)\}$. Then $\sum_{x \in \mathbf{S}} (2x - 1)^2$ is equal to
|
5
|
[] |
[] | 0 |
Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbf{N}$ for which $\lim_{x \to 0^+} \left( x \left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{p}{x}\right] \right) - x^2 \left( \left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{p^2}{x^2}\right] \right) \right) \geq 1$ is equal to
|
24
|
[] |
[] | 0 |
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:
|
4
|
[
"5",
"4",
"3",
"7"
] |
[
1
] | 1 |
If $y = y(x)$ is the solution of the differential equation $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right) \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{\pi^2 - 8}{4}$, then $y^2(0)$ is equal to:
|
4
|
[] |
[] | 0 |
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:
|
21
|
[] |
[] | 0 |
If the function \(f(x) = \begin{cases} \frac{2}{x} \left\{ \sin (k_1 + 1) x + \sin (k_2 - 1) x \right\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x > 0 \end{cases}\) is continuous at \(x = 0\), then \(k_1^2 + k_2^2\) is
|
10
|
[
"2",
"5",
"8",
"10"
] |
[
3
] | 1 |
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:
|
22
|
[
"9",
"13",
"22",
"31"
] |
[
2
] | 1 |
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:
|
14
|
[
"\\tfrac{23}{2}",
"14",
"\\tfrac{25}{2}",
"10"
] |
[
1
] | 1 |
If the system of linear equations: $\begin{aligned} x + y + 2z &= 6 \\ 2x + 3y + az &= a + 1 \\ -x - 3y + bz &= 2b \end{aligned}$ where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to:
|
16
|
[
"16",
"12",
"22",
"9"
] |
[
0
] | 1 |
Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{\frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1\right\}$. Then $n(B)$ is equal to:
|
31
|
[
"36",
"31",
"37",
"29"
] |
[
1
] | 1 |
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:
|
31
|
[
"49",
"31",
"43",
"37"
] |
[
1
] | 1 |
Let a curve \(y = f(x)\) pass through the points \((0, 5)\) and \((\log_e 2, k)\). If the curve satisfies the differential equation \(2(3 + y) e^{2x} dx - (7 + e^{2x}) dy = 0\), then \(k\) is equal to
|
8
|
[
"4",
"32",
"8",
"16"
] |
[
2
] | 1 |
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:
|
2
|
[
"2",
"1",
"0",
"3"
] |
[
0
] | 1 |
Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[4]{4})^{12}$. Then $p+q$ is equal to:
|
283
|
[
"283",
"287",
"295",
"299"
] |
[
0
] | 1 |
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:
|
6
|
[
"6",
"8",
"10",
"7"
] |
[
0
] | 1 |
Let in a $\triangle ABC$, the length of the side $AC$ be 6, the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is:
|
21
|
[
"17",
"21",
"56",
"42"
] |
[
1
] | 1 |
The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is
|
125
|
[] |
[] | 0 |
The value of \(\int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left(\log_e x\right)^2 + 1}}{e^{\left(\log_e x\right)^2 + 1}} \right)^{-1} dx\) is
|
1
|
[
"2",
"\\(\\log_e 2\\)",
"1",
"\\(e^2\\)"
] |
[
2
] | 1 |
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:
|
732
|
[
"866",
"750",
"820",
"732"
] |
[
3
] | 1 |
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:
|
7
|
[
"7",
"4",
"6",
"-1"
] |
[
0
] | 1 |
For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is:
|
675
|
[
"540",
"675",
"1350",
"135"
] |
[
1
] | 1 |
Let $\int x^3 \sin x \, dx = g(x) + C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbb{Z}$, then $\alpha + \beta - \gamma$ equals:
|
55
|
[
"48",
"55",
"62",
"47"
] |
[
1
] | 1 |
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:
|
210
|
[
"210",
"150",
"240",
"120"
] |
[
0
] | 1 |
If $\sum_{r=1}^{30} \frac{r^3 (^{30}C_r)^2}{^{30}C_{r-1}} = \alpha \times 2^{29}$, then $\alpha$ is equal to:
|
465
|
[] |
[] | 0 |
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:
|
19
|
[] |
[] | 0 |
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:
|
4
|
[
"9",
"6",
"12",
"4"
] |
[
3
] | 1 |
The number of real roots of the equation \(x\lvert x - 2\rvert + 3\lvert x - 3\rvert + 1 = 0\) is:
|
1
|
[
"4",
"2",
"1",
"3"
] |
[
2
] | 1 |
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 _____
|
56
|
[] |
[] | 0 |
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:
|
147
|
[
"139",
"147",
"151",
"143"
] |
[
1
] | 1 |
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:
|
8
|
[] |
[] | 0 |
The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to:
|
15
|
[] |
[] | 0 |
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:
|
-3
|
[
"3/2",
"-3/2",
"3",
"-3"
] |
[
3
] | 1 |
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:
|
15
|
[
"15",
"13",
"16",
"14"
] |
[
0
] | 1 |
The remainder when \( (64^{64})^{64} \) is divided by 7 is equal to:
|
1
|
[
"4",
"1",
"3",
"6"
] |
[
1
] | 1 |
If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{18} + \beta^{18} + \alpha^{13} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{18} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to
|
441
|
[
"441",
"398",
"312",
"409"
] |
[
0
] | 1 |
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:
|
216
|
[
"72",
"18",
"36",
"216"
] |
[
3
] | 1 |
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:
|
11
|
[] |
[] | 0 |
If $\sin x + \sin^2 x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, then $\left(\cos^{12} x + \tan^{12} x\right) + 3\left(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x\right) + \left(\cos^6 x + \tan^6 x\right)$ is equal to:
|
2
|
[
"4",
"1",
"3",
"2"
] |
[
3
] | 1 |
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:
|
70
|
[
"74",
"68",
"70",
"78"
] |
[
2
] | 1 |
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is:
|
90
|
[
"90",
"84",
"122",
"108"
] |
[
0
] | 1 |
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:
|
194
|
[
"190",
"192",
"188",
"194"
] |
[
3
] | 1 |
Let $A(6,8)$, $B(10\cos\alpha, -10\sin\alpha)$ and $C(-10\sin\alpha, 10\cos\alpha)$, be the vertices of a triangle. If $L(a,9)$ and $G(h,k)$ be its orthocenter and centroid respectively, then $(5a - 3h + 6k + 100\sin 2\alpha)$ is equal to:
|
145
|
[] |
[] | 0 |
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
|
1944
|
[] |
[] | 0 |
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is
|
1405
|
[] |
[] | 0 |
Let the circle \(C\) touch the line \(x - y + 1 = 0\), have the centre on the positive \(x\)-axis, and cut off a chord of length \(\frac{4}{\sqrt{13}}\) along the line \(-3x + 2y = 1\). Let \(H\) be the hyperbola \(\frac{x^2}{\alpha^2} - \frac{y^2}{2\alpha} = 1\), whose one of the foci is the centre of \(C\) and the length of the transverse axis is the diameter of \(C\). Then \(2\alpha^2 + 3\beta^2\) is equal to
|
19
|
[] |
[] | 0 |
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:
|
130
|
[
"140",
"130",
"160",
"150"
] |
[
1
] | 1 |
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:
|
129
|
[
"131",
"130",
"129",
"128"
] |
[
2
] | 1 |
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:
|
56
|
[
"68",
"29",
"2",
"56"
] |
[
3
] | 1 |
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 _____
|
13
|
[] |
[] | 0 |
Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{xy}{x^2-1} = \frac{x^4 + 4x}{\sqrt{1-x^2}}, -1 < x < 1$ such that $f(0)=0$. If $6 \int_{-1/2}^{1/2} f(x) dx = 2\pi - \alpha$ then $\alpha^2$ is equal to:
|
27
|
[] |
[] | 0 |
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:
|
17160
|
[
"15120",
"19320",
"17160",
"18290"
] |
[
2
] | 1 |
$\cos \left( \sin^{-1} \frac{3}{8} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)$ is equal to:
|
0
|
[
"",
"0",
"$\\frac{32}{65}$",
"$\\frac{33}{65}$"
] |
[
1
] | 1 |
Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta) x - 1 = 0, \theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$, then $16(M + m)$ equals:
|
25
|
[
"24",
"25",
"17",
"27"
] |
[
1
] | 1 |
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:
|
4
|
[
"4",
"10",
"6",
"8"
] |
[
0
] | 1 |
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$, then the value of $64(\mu + \sigma^2)$ is:
|
48
|
[
"51",
"64",
"32",
"48"
] |
[
3
] | 1 |
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:
|
35
|
[
"32",
"33",
"34",
"35"
] |
[
3
] | 1 |
Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is
|
25
|
[
"20",
"90",
"45",
"25"
] |
[
3
] | 1 |
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:
|
36
|
[] |
[] | 0 |
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:
|
6
|
[
"4",
"10",
"8",
"6"
] |
[
3
] | 1 |
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:
|
14
|
[
"18",
"14",
"12",
"16"
] |
[
1
] | 1 |
If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{5}$ is $\frac{m}{n}$, where $m, n$ are coprime numbers, then $m+n$ is equal to:
|
14
|
[
"21",
"14",
"6",
"10"
] |
[
1
] | 1 |
Let $\mathbf{L}_1: \frac{x-1}{1} = \frac{y-2}{1} = \frac{z-1}{2}$ and $\mathbf{L}_2: \frac{x+1}{1} = \frac{y-2}{2} = \frac{z}{4}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals:
|
25
|
[
"20",
"18",
"25",
"16"
] |
[
2
] | 1 |
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:
|
17
|
[
"17",
"10",
"37",
"26"
] |
[
0
] | 1 |
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:
|
2
|
[] |
[] | 0 |
Let $H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2: -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 e_2^2$ is equal to:
|
55
|
[] |
[] | 0 |
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