question
string | answer
string | options
list | correct_options
list | question_type
int64 |
|---|---|---|---|---|
Suppose that the number of terms in an A.P. is $2k, k \in \mathbb{N}$. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then $k$ is equal to:
|
5
|
[
"6",
"5",
"8",
"4"
] |
[
1
] | 1
|
1 + 3 + 5^2 + 7 + 9^2 + \dots up to 40 terms is equal to:
|
41880
|
[
"43890",
"41880",
"33980",
"40870"
] |
[
1
] | 1
|
The sum of all rational terms in the expansion of \(\left(1 + 2^{1/2} + 3^{1/2}\right)^6\) is equal to
|
612
|
[] |
[] | 0
|
Let for some function $y = f(x)$, $\int_0^x t f(t) \, dt = x^2 f(x)$, $x > 0$ and $f(2) = 3$. Then $f(6)$ is equal to
|
1
|
[
"",
"3",
"6",
"2"
] |
[
0
] | 1
|
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to:
|
14
|
[
"4",
"14",
"13",
"11"
] |
[
1
] | 1
|
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:
|
16
|
[
"14",
"15",
"16",
"12"
] |
[
2
] | 1
|
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:
|
7
|
[
"7",
"5",
"6",
"9"
] |
[
0
] | 1
|
If the area of the larger portion bounded between the curves \(x^2 + y^2 = 25\) and \(y = |x - 1|\) is \(\frac{1}{4}(b \pi + c)\), \(b, c \in \mathbb{N}\), then \(b + c\) is equal to
|
77
|
[] |
[] | 0
|
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:
|
189
|
[] |
[] | 0
|
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:
|
120
|
[] |
[] | 0
|
If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{25 x^2 \cos^2 x}{2(x + e)^2} \, dx = \pi (\alpha \pi^2 + \beta)$, $\alpha, \beta \in \mathbb{Z}$, then $(\alpha + \beta)^2$ equals
|
100
|
[
"64",
"196",
"144",
"100"
] |
[
3
] | 1
|
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:
|
37
|
[
"55",
"10",
"23",
"37"
] |
[
3
] | 1
|
If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to
|
57
|
[
"57",
"59",
"55",
"56"
] |
[
0
] | 1
|
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:
|
10
|
[
"12",
"10",
"8",
"14"
] |
[
1
] | 1
|
For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
|
35
|
[
"20",
"10",
"35",
"70"
] |
[
2
] | 1
|
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:
|
12
|
[
"12",
"20",
"76",
"4"
] |
[
0
] | 1
|
If the components of $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\vec{b}=\hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ and $\frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})$, then $\alpha^2+\beta^2+\gamma^2$ is equal to:
|
26
|
[
"26",
"18",
"2",
"16"
] |
[
0
] | 1
|
Let \(\mathbf{R} = \{(1, 2), (2, 3), (3, 3)\}\) be a relation defined on the set \(\{1, 2, 3, 4\}\). Then the minimum number of elements, needed to be added in \(\mathbf{R}\) so that \(\mathbf{R}\) becomes an equivalence relation, is
|
7
|
[
"10",
"7",
"8",
"9"
] |
[
1
] | 1
|
Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ) = \frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to:
|
1328
|
[] |
[] | 0
|
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:
|
103
|
[
"105",
"103",
"100",
"106"
] |
[
1
] | 1
|
Let $f(x) = 7 \tan^8 x + 7 \tan^8 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_0^{\pi/4} f(x) \mathrm{d}x$ and $I_2 = \int_0^{\pi/4} x f(x) \mathrm{d}x$. Then $7 I_1 + 12 I_2$ is equal to:
|
1
|
[
"2",
"1",
"$2\\pi$",
"$\\pi$"
] |
[
1
] | 1
|
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
|
15
|
[
"23",
"18",
"15",
"14"
] |
[
2
] | 1
|
Let $f: \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6 f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If the $\lim_{x \rightarrow 0} \left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to
|
4
|
[
"5",
"3",
"4",
"6"
] |
[
2
] | 1
|
Let $\mathbf{A}=\{1,2,3,4\}$ and $\mathbf{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: \mathbf{A} \to \mathbf{B}$ such that $1 \in f(\mathbf{A})$ is equal to:
|
151
|
[
"151",
"139",
"163",
"127"
] |
[
0
] | 1
|
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:
|
6
|
[
"7",
"8",
"6",
"5"
] |
[
2
] | 1
|
The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is:
|
8788
|
[] |
[] | 0
|
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
|
11
|
[
"3",
"11",
"7",
"8"
] |
[
1
] | 1
|
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 \(\log_2 2 = 0.7\))
|
21.1
|
[
"21.1",
"19.8",
"22.1",
"20.9"
] |
[
0
] | 1
|
Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S = \{-3, -2, -1, 1, 2\}$. Let $S_1 = \{A = [a_{ij}] \in M: A = A^T \text{ and } a_{ij} \in S, \forall i, j\}$, $S_2 = \{A = [a_{ij}] \in M: A = -A^T \text{ and } a_{ij} \in S, \forall i, j\}$, $S_3 = \{A = [a_{ij}] \in M: a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j\}$. If $n(S_1 \cup S_2 \cup S_3) = 125 \alpha$, then $\alpha$ equals
|
1613
|
[] |
[] | 0
|
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:
|
48
|
[
"64",
"72",
"48",
"36"
] |
[
2
] | 1
|
If the set of all values of \(a\), for which the equation \(5x^3 - 15x - a = 0\) has three distinct real roots, is the interval \((\alpha, \beta)\), then \(\beta - 2\alpha\) is equal to
|
30
|
[] |
[] | 0
|
Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x) f'(y) + f'(x) f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to:
|
2525
|
[
"2525",
"5220",
"2384",
"2406"
] |
[
0
] | 1
|
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:
|
11
|
[
"11",
"10",
"18",
"17"
] |
[
0
] | 1
|
The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to
|
14
|
[
"14",
"21",
"28",
"7"
] |
[
0
] | 1
|
Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1$, $x+2y+4z=m$, and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to:
|
440
|
[
"3080",
"560",
"3410",
"440"
] |
[
3
] | 1
|
Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to:
|
784
|
[
"628",
"812",
"526",
"784"
] |
[
3
] | 1
|
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