Reasoning-Benchmarks
Collection
A collection of mutiple benchmarks for large reasoning model evaluation • 25 items • Updated
problem stringlengths 107 1.36k | answer stringlengths 1 5 | datasource stringclasses 1
value |
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Let $-\frac{\pi}{6}<\theta<-\frac{\pi}{12}$. Suppose $\alpha_{1}$ and $\beta_{1}$ are the roots of the equation $x^{2}-2 x \sec \theta+1=0$ and $\alpha_{2}$ and $\beta_{2}$ are the roots of the equation $x^{2}+2 x \tan \theta-1=0$. If $\alpha_{1}>\beta_{1}$ and $\alpha_{2}>\beta_{2}$, then $\alpha_{1}+\beta_{2}$ equals... | C | jeebench-math |
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
(A) 380
(B) 320
(C) 260
(D) 95 | A | jeebench-math |
Let $S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set $S$ is equal to
(A) $-\frac{7 \pi}{9}$
(B) $-\frac{2 \pi}{9}$
(C) 0
(D) $\frac{5 \pi}{9}$ | C | jeebench-math |
A computer producing factory has only two plants $T_{1}$ and $T_{2}$. Plant $T_{1}$ produces $20 \%$ and plant $T_{2}$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that
$P$ (computer turns out to be defective given that it is produc... | C | jeebench-math |
The least value of $\alpha \in \mathbb{R}$ for which $4 \alpha x^{2}+\frac{1}{x} \geq 1$, for all $x>0$, is
(A) $\frac{1}{64}$
(B) $\frac{1}{32}$
(C) $\frac{1}{27}$
(D) $\frac{1}{25}$ | C | jeebench-math |
Consider a pyramid $O P Q R S$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $O P$ and $O R$ along the $x$-axis and the $y$-axis, respectively. The base $O P Q R$ of the pyramid is a square with $O P=3$. The point $S$ is directly above the mid-point $T$ of diagonal $O Q$ such that... | BCD | jeebench-math |
Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)=2-\frac{f(x)}{x}$ for all $x \in(0, \infty)$ and $f(1) \neq 1$. Then
(A) $\lim _{x \rightarrow 0+} f^{\prime}\left(\frac{1}{x}\right)=1$
(B) $\lim _{x \rightarrow 0+} x f\left(\frac{1}{x}\right)=2$
(C) $\lim _{x \rightar... | A | jeebench-math |
Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q=\left[q_{i j}\right]$ is a matrix such that $P Q=k I$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order 3 . If $q_{23}=-\frac{k}{8}$ and $\operatorname{det}(... | BC | jeebench-math |
In a triangle $X Y Z$, let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$, respectively, and $2 s=x+y+z$. If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle $X Y Z$ is $\frac{8 \pi}{3}$, then
(A) area of the triangle $X Y Z$ is $6 \sqrt{6}$
(B) the radius of circum... | ACD | jeebench-math |
A solution curve of the differential equation $\left(x^{2}+x y+4 x+2 y+4\right) \frac{d y}{d x}-y^{2}=0, x>0$, passes through the point $(1,3)$. Then the solution curve
(A) intersects $y=x+2$ exactly at one point
(B) intersects $y=x+2$ exactly at two points
(C) intersects $y=(x+2)^{2}$
(D) does NO'T intersect $y=(x... | AD | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}, \quad g: \mathbb{R} \rightarrow \mathbb{R}$ and $h: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x)=x^{3}+3 x+2, g(f(x))=x$ and $h(g(g(x)))=x$ for all $x \in \mathbb{R}$. Then
(A) $\quad g^{\prime}(2)=\frac{1}{15}$
(B) $h^{\prime}(1)=666$
(C) ... | BC | jeebench-math |
The circle $C_{1}: x^{2}+y^{2}=3$, with centre at $O$, intersects the parabola $x^{2}=2 y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_{1}$ at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$, respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $2 \sqrt{3}$ an... | ABC | jeebench-math |
Let $R S$ be the diameter of the circle $x^{2}+y^{2}=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$ ) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $R S$ at point $E$... | AC | jeebench-math |
What is the total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{array}\right|=10$? | 2 | jeebench-math |
Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(1+x)^{2}+(1+x)^{3}+\cdots+(1+x)^{49}+(1+m x)^{50}$ is $(3 n+1){ }^{51} C_{3}$ for some positive integer $n$. Then what is the value of $n$? | 5 | jeebench-math |
What is the total number of distinct $x \in[0,1]$ for which $\int_{0}^{x} \frac{t^{2}}{1+t^{4}} d t=2 x-1$? | 1 | jeebench-math |
Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim _{x \rightarrow 0} \frac{x^{2} \sin (\beta x)}{\alpha x-\sin x}=1$.Then what is the value of $6(\alpha+\beta)$? | 7 | jeebench-math |
Let $z=\frac{-1+\sqrt{3} i}{2}$, where $i=\sqrt{-1}$, and $r, s \in\{1,2,3\}$. Let $P=\left[\begin{array}{cc}(-z)^{r} & z^{2 s} \\ z^{2 s} & z^{r}\end{array}\right]$ and $I$ be the identity matrix of order 2 . Then what is the total number of ordered pairs $(r, s)$ for which $P^{2}=-I$? | 1 | jeebench-math |
Let $P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right]$ and $I$ be the identity matrix of order 3. If $Q=\left[q_{i j}\right]$ is a matrix such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals
(A) 52
(B) 103
(C) 201
(D) 205 | B | jeebench-math |
Let $b_{i}>1$ for $i=1,2, \ldots, 101$. Suppose $\log _{e} b_{1}, \log _{e} b_{2}, \ldots, \log _{e} b_{101}$ are in Arithmetic Progression (A.P.) with the common difference $\log _{e} 2$. Suppose $a_{1}, a_{2}, \ldots, a_{101}$ are in A.P. such that $a_{1}=b_{1}$ and $a_{51}=b_{51}$. If $t=b_{1}+b_{2}+\cdots+b_{51}$ a... | B | jeebench-math |
The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{2} \cos x}{1+e^{x}} d x$ is equal to
(A) $\frac{\pi^{2}}{4}-2$
(B) $\frac{\pi^{2}}{4}+2$
(C) $\pi^{2}-e^{\frac{\pi}{2}}$
(D) $\pi^{2}+e^{\frac{\pi}{2}}$ | A | jeebench-math |
Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is
(A) $x+y-3 z=0$
(B) $3 x+z=0$
(C) $x-4 y+7 z=0$
(D) $2 x-y=0$ | C | jeebench-math |
Let $a, b \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=a \cos \left(\left|x^{3}-x\right|\right)+b|x| \sin \left(\left|x^{3}+x\right|\right)$. Then $f$ is
(A) differentiable at $x=0$ if $a=0$ and $b=1$
(B) differentiable at $x=1$ if $a=1$ and $b=0$
(C) NOT differentiable at $x=0$ if ... | AB | jeebench-math |
Let $f: \mathbb{R} \rightarrow(0, \infty)$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be twice differentiable functions such that $f^{\prime \prime}$ and $g^{\prime \prime}$ are continuous functions on $\mathbb{R}$. Suppose $f^{\prime}(2)=g(2)=0, f^{\prime \prime}(2) \neq 0$ and $g^{\prime}(2) \neq 0$. If $\lim _{x \ri... | AD | jeebench-math |
Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ be functions defined by $f(x)=\left[x^{2}-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb{R}$. Then
(A) $f$ is discon... | BC | jeebench-math |
Let $a, b \in \mathbb{R}$ and $a^{2}+b^{2} \neq 0$. Suppose $S=\left\{z \in \mathbb{C}: z=\frac{1}{a+i b t}, t \in \mathbb{R}, t \neq 0\right\}$, where $i=\sqrt{-1}$. If $z=x+i y$ and $z \in S$, then $(x, y)$ lies on
(A) the circle with radius $\frac{1}{2 a}$ and centre $\left(\frac{1}{2 a}, 0\right)$ for $a>0, b \neq... | ACD | jeebench-math |
Let $P$ be the point on the parabola $y^{2}=4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2}+y^{2}-4 x-16 y+64=0$. Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then
(A) $S P=2 \sqrt{5}$
(B) $S Q: Q P=(\sqrt{5}+1): 2$
(C) the $x$-intercept of the normal... | ACD | jeebench-math |
Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations
\[
\begin{aligned}
& a x+2 y=\lambda \\
& 3 x-2 y=\mu
\end{aligned}
\]
Which of the following statement(s) is(are) correct?
(A) If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
(B) If $a \n... | BCD | jeebench-math |
Let $\hat{u}=u_{1} \hat{i}+u_{2} \hat{j}+u_{3} \hat{k}$ be a unit vector in $\mathbb{R}^{3}$ and $\hat{w}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2 \hat{k})$. Given that there exists a vector $\vec{v}$ in $\mathbb{R}^{3}$ such that $|\hat{u} \times \vec{v}|=1$ and $\hat{w} \cdot(\hat{u} \times \vec{v})=1$. Which of the foll... | BC | jeebench-math |
If $2 x-y+1=0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$, then which of the following CANNOT be sides of a right angled triangle?
[A] $a, 4,1$
[B] $a, 4,2$
[C] $2 a, 8,1$
[D] $2 a, 4,1$ | ABC | jeebench-math |
If a chord, which is not a tangent, of the parabola $y^{2}=16 x$ has the equation $2 x+y=p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p, h$ and $k$ ?
[A] $p=-2, h=2, k=-4$
[B] $p=-1, h=1, k=-3$
[C] $p=2, h=3, k=-4$
[D] $p=5, h=4, k=-3$ | C | jeebench-math |
Let $f: \mathbb{R} \rightarrow(0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$ ?
[A] $x^{9}-f(x)$
[B] $x-\int_{0}^{\frac{\pi}{2}-x} f(t) \cos t d t$
[C] e^{x}-\int_{0}^{x} f(t) \sin t d t$
[D] f(x)+\int_{0}^{\frac{\pi}{2}} f(t) ... | AB | jeebench-math |
Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?
[A]$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]
[B]$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[C]$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 ... | BD | jeebench-math |
Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y \neq 0$. If the complex number $z=x+i y$ satisfies $\operatorname{Im}\left(\frac{a z+b}{z+1}\right)=\mathrm{y}$, then which of the following is(are) possible value(s) of $x ?$
[A]$-1+\sqrt{1-y^{2}}$
[B]$-1-\sqrt{1-y^{2}}$
[C]$1+\sqrt{1+y^{2}}$
[D]$1-\sq... | AB | jeebench-math |
Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3}, P(X \mid Y)=\frac{1}{2}$ and $P(Y \mid X)=\frac{2}{5}$. Then
[A] $P(Y)=\frac{4}{15}$
[B] $P\left(X^{\prime} \mid Y\right)=\frac{1}{2}$
[C] \quad P(X \cap Y)=\frac{1}{5}$
[D] $P(X \cup Y)=\frac{2}{5}$ | AB | jeebench-math |
For how many values of $p$, the circle $x^{2}+y^{2}+2 x+4 y-p=0$ and the coordinate axes have exactly three common points? | 2 | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0, f\left(\frac{\pi}{2}\right)=3$ and $f^{\prime}(0)=1$. If
\[
g(x)=\int_{x}^{\frac{\pi}{2}}\left[f^{\prime}(t) \operatorname{cosec} t-\cot t \operatorname{cosec} t f(t)\right] d t
\]
for $x \in\left(0, \frac{\pi}{2}\right]$, then... | 2 | jeebench-math |
For a real number $\alpha$, if the system
\[
\left[\begin{array}{ccc}
1 & \alpha & \alpha^{2} \\
\alpha & 1 & \alpha \\
\alpha^{2} & \alpha & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right]
\]
of linear equations, has infinit... | 1 | jeebench-math |
Words of length 10 are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, what is the value of $\frac{y}{9 x}$? | 5 | jeebench-math |
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side? | 6 | jeebench-math |
The equation of the plane passing through the point $(1,1,1)$ and perpendicular to the planes $2 x+y-2 z=5$ and $3 x-6 y-2 z=7$, is
[A] $14 x+2 y-15 z=1$
[B] $14 x-2 y+15 z=27$
[C] $\quad 14 x+2 y+15 z=31$
[D] $-14 x+2 y+15 z=3$ | C | jeebench-math |
Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that
\[
\overrightarrow{O P} \cdot \overrightarrow{O Q}+\overrightarrow{O R} \cdot \overrightarrow{O S}=\overrightarrow{O R} \cdot \overrightarrow{O P}+\overrightarrow{O Q} \cdot \overrightarrow{O S}=\overrightarrow{O Q} \cdot \overr... | D | jeebench-math |
If $y=y(x)$ satisfies the differential equation
\[
8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} d x, \quad x>0
\]
and $y(0)=\sqrt{7}$, then $y(256)=$
[A] 3
[B] 9
[C] 16
[D] 80 | A | jeebench-math |
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a twice differentiable function such that $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$, and $f\left(\frac{1}{2}\right)=\frac{1}{2}, f(1)=1$, then
[A] $f^{\prime}(1) \leq 0$
[B] $0<f^{\prime}(1) \leq \frac{1}{2}$
[C] $\frac{1}{2}<f^{\prime}(1) \leq 1$
[D] $f^{\prim... | D | jeebench-math |
How many $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5 ?$
[A] 126
[B] 198
[C] 162
[D] 135 | B | jeebench-math |
Let $S=\{1,2,3, \ldots, 9\}$. For $k=1,2, \ldots, 5$, let $N_{k}$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$
[A] 210
[B] 252
[C] 125
[D] 126 | D | jeebench-math |
Three randomly chosen nonnegative integers $x, y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is
[A] $\frac{36}{55}$
[B] $\frac{6}{11}$
[C] $\frac{1}{2}$
[D] $\frac{5}{11}$ | B | jeebench-math |
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f^{\prime}(x)>2 f(x)$ for all $x \in \mathbb{R}$, and $f(0)=1$, then
[A] $f(x)$ is increasing in $(0, \infty)$
[B] $f(x)$ is decreasing in $(0, \infty)$
[C] $\quad f(x)>e^{2 x}$ in $(0, \infty)$
[D] $f^{\prime}(x)<e^{2 x}$ in $(0, \in... | AC | jeebench-math |
Let $f(x)=\frac{1-x(1+|1-x|)}{|1-x|} \cos \left(\frac{1}{1-x}\right)$ for $x \neq 1$. Then
[A] $\lim _{x \rightarrow 1^{-}} f(x)=0$
[B] $\lim _{x \rightarrow 1^{-}} f(x)$ does not exist
[C] $\lim _{x \rightarrow 1^{+}} f(x)=0$
[D] $\lim _{x \rightarrow 1^{+}} f(x)$ does not exist | AD | jeebench-math |
If $f(x)=\left|\begin{array}{ccc}\cos (2 x) & \cos (2 x) & \sin (2 x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x\end{array}\right|$, then
[A] $f^{\prime}(x)=0$ at exactly three points in $(-\pi, \pi)$
[B] $f^{\prime}(x)=0$ at more than three points in $(-\pi, \pi)$
[C] $f(x)$ attains its maximum at $x... | BC | jeebench-math |
If the line $x=\alpha$ divides the area of region $R=\left\{(x, y) \in \mathbb{R}^{2}: x^{3} \leq y \leq x, 0 \leq x \leq 1\right\}$ into two equal parts, then
[A] $0<\alpha \leq \frac{1}{2}$
[B] $\frac{1}{2}<\alpha<1$
[C] $\quad 2 \alpha^{4}-4 \alpha^{2}+1=0$
[D] $\alpha^{4}+4 \alpha^{2}-1=0$ | BC | jeebench-math |
If $I=\sum_{k=1}^{98} \int_{k}^{k+1} \frac{k+1}{x(x+1)} d x$, then
[A] $I>\log _{e} 99$
[B] $I<\log _{e} 99$
[C] $I<\frac{49}{50}$
[D] $I>\frac{49}{50}$ | BD | jeebench-math |
For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument with $-\pi<\arg (z) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?
\end{itemize}
(A) $\arg (-1-i)=\frac{\pi}{4}$, where $i=\sqrt{-1}$
(B) The function $f: \mathbb{R} \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-... | ABD | jeebench-math |
In a triangle $P Q R$, let $\angle P Q R=30^{\circ}$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE?
(A) $\angle Q P R=45^{\circ}$
(B) The area of the triangle $P Q R$ is $25 \sqrt{3}$ and $\angle Q R P=120^{\circ}$
(C) The radi... | BCD | jeebench-math |
Let $P_{1}: 2 x+y-z=3$ and $P_{2}: x+2 y+z=2$ be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of $P_{1}$ and $P_{2}$ has direction ratios $1,2,-1$
(B) The line
\[
\frac{3 x-4}{9}=\frac{1-3 y}{9}=\frac{z}{3}
\]
is perpendicular to the line of intersection of $P_{1... | CD | jeebench-math |
For every twice differentiable function $f: \mathbb{R} \rightarrow[-2,2]$ with $(f(0))^{2}+\left(f^{\prime}(0)\right)^{2}=85$, which of the following statement(s) is (are) TRUE?
(A) There exist $r, s \in \mathbb{R}$, where $r<s$, such that $f$ is one-one on the open interval $(r, s)$
(B) There exists $x_{0} \in(-4,0)... | ABD | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two non-constant differentiable functions. If
\[
f^{\prime}(x)=\left(e^{(f(x)-g(x))}\right) g^{\prime}(x) \text { for all } x \in \mathbb{R}
\]
and $f(1)=g(2)=1$, then which of the following statement(s) is (are) TRUE?
(A) $f(2)... | BC | jeebench-math |
What is the value of
\[
\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}
\]? | 8 | jeebench-math |
What is the number of 5 digit numbers which are divisible by 4 , with digits from the set $\{1,2,3,4,5\}$ and the repetition of digits is allowed? | 625 | jeebench-math |
Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23, \ldots$. Then, what is the number of elements in the set $X \cup Y$? | 3748 | jeebench-math |
What is the number of real solutions of the equation
\[
\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^{i}\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^{i}-\sum_{i=1}^{\infty}(-x)^{i}\right)
\]
lying in the interval $\left(-\frac{1}{2}, \... | 2 | jeebench-math |
For each positive integer $n$, let
\[
y_{n}=\frac{1}{n}((n+1)(n+2) \cdots(n+n))^{\frac{1}{n}}
\]
For $x \in \mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\lim _{n \rightarrow \infty} y_{n}=L$, then what is the value of $[L]$? | 1 | jeebench-math |
Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b}=0$. For some $x, y \in \mathbb{R}$, let $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$. If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then what is the value of $8 \co... | 3 | jeebench-math |
Let $a, b, c$ be three non-zero real numbers such that the equation
\[
\sqrt{3} a \cos x+2 b \sin x=c, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\]
has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3}$. Then, what is the value of $\frac{b}{a}$? | 0.5 | jeebench-math |
A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P(0,0), Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_{2}$ takes away the region which lies between the side $P Q$ and a curve of the form $y=x^{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$ is exactly $30 \... | 4 | jeebench-math |
For any positive integer $n$, define $f_{n}:(0, \infty) \rightarrow \mathbb{R}$ as
\[
f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)
\]
(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. )... | D | jeebench-math |
Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+i y$ $(x, y \in \mathbb{R}, i=\sqrt{-1})$ of the equation $s z+t \bar{z}+r=0$, where $\bar{z}=x-i y$. Then, which of the following statement(s) is (are) TRUE?
(A) If $L$ has exactly one element, then $|s| \neq|t|$
(B) If $|s|=|t|$, then $L... | ACD | jeebench-math |
What is the value of the integral
\[
\int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x
\]? | 2 | jeebench-math |
Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1,0,1\}$. Then, what is the maximum possible value of the determinant of $P$? | 4 | jeebench-math |
Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then what is the value of $\frac{1}{5 !}(\beta-\alpha)$? | 119 | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation
\[
\frac{d y}{d x}=(2+5 y)(5 y-2)
\]
then what is the value of $\lim _{x \rightarrow-\infty} f(x)$? | 0.4 | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation
\[
f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y) \text { for all } x, y \in \mathbb{R} .
\]
Then, the value of $\log _{e}(f(4))$ is | 2 | jeebench-math |
Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $P Q$ is perpendicular to the plane $x+y=3$ and the mid-point of $P Q$ lies in the plane $x+y=3$ ) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be 5 . If $R$ is the image of $P$ in the $x y$-plane,... | 8 | jeebench-math |
Consider the cube in the first octant with sides $O P, O Q$ and $O R$ of length 1 , along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0,0,0)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ su... | 0.5 | jeebench-math |
Let
\[
X=\left({ }^{10} C_{1}\right)^{2}+2\left({ }^{10} C_{2}\right)^{2}+3\left({ }^{10} C_{3}\right)^{2}+\cdots+10\left({ }^{10} C_{10}\right)^{2}
\]
where ${ }^{10} C_{r}, r \in\{1,2, \cdots, 10\}$ denote binomial coefficients. Then, what is the value of $\frac{1}{1430} X$? | 646 | jeebench-math |
Let $S$ be the set of all complex numbers $Z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $Z_{0}$ is such that $\frac{1}{\left|Z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$, then the principal argument of $\frac{4-z_{0}-\overline{z_{0}}}{Z_{0}-\overline{z_{0}}+2 i}$ ... | A | jeebench-math |
Let
\[
M=\left[\begin{array}{cc}
\sin ^{4} \theta & -1-\sin ^{2} \theta \\
1+\cos ^{2} \theta & \cos ^{4} \theta
\end{array}\right]=\alpha I+\beta M^{-1}
\]
where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If
$\alpha^{*}$ is the minimum of the ... | C | jeebench-math |
A line $y=m x+1$ intersects the circle $(x-3)^{2}+(y+2)^{2}=25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?
(A) $-3 \leq m<-1$
(B) $2 \leq m<4$
(C) $4 \leq m<6$
(D) $6 \leq m<8$ | B | jeebench-math |
The area of the region $\left\{(x, y): x y \leq 8,1 \leq y \leq x^{2}\right\}$ is
(A) $16 \log _{e} 2-\frac{14}{3}$
(B) $8 \log _{e} 2-\frac{14}{3}$
(C) $16 \log _{e} 2-6$
(D) $8 \log _{e} 2-\frac{7}{3}$ | A | jeebench-math |
Let
\[
M=\left[\begin{array}{lll}
0 & 1 & a \\
1 & 2 & 3 \\
3 & b & 1
\end{array}\right] \quad \text { and adj } M=\left[\begin{array}{rrr}
-1 & 1 & -1 \\
8 & -6 & 2 \\
-5 & 3 & -1
\end{array}\right]
\]
where $a$ and $b$ are real numbers. Which of the following options is/are correct?
(A) $a+b=3$
(B) $(\ope... | ABD | jeebench-math |
There are three bags $B_{1}, B_{2}$ and $B_{3}$. The bag $B_{1}$ contains 5 red and 5 green balls, $B_{2}$ contains 3 red and 5 green balls, and $B_{3}$ contains 5 red and 3 green balls. Bags $B_{1}, B_{2}$ and $B_{3}$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A ba... | AB | jeebench-math |
In a non-right-angled triangle $\triangle P Q R$, let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $P Q$ at $S$, the perpendicular from $P$ meets the side $Q R$ at $E$, and $R S$ and $P E$ intersect at $O$. If $p=\sqrt{3}, q=1$, and the r... | ACD | jeebench-math |
Define the collections $\left\{E_{1}, E_{2}, E_{3}, \ldots\right\}$ of ellipses and $\left\{R_{1}, R_{2}, R_{3}, \ldots\right\}$ of rectangles as follows:
$E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
$R_{1}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_{1}$;
$E_{n}:$ ellipse $\frac{x^{... | BC | jeebench-math |
Let $\Gamma$ denote a curve $y=y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_{P}$. If $P Y_{P}$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct?
(A) $y=\log _{e}\left(\fra... | AB | jeebench-math |
Let $L_{1}$ and $L_{2}$ denote the lines
\[
\vec{r}=\hat{i}+\lambda(-\hat{i}+2 \hat{j}+2 \hat{k}), \lambda \in \mathbb{R}
\]
and
\[ \vec{r}=\mu(2 \hat{i}-\hat{j}+2 \hat{k}), \mu \in \mathbb{R}
\]
respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which ... | ABC | jeebench-math |
Let $\omega \neq 1$ be a cube root of unity. Then what is the minimum of the set
\[
\left\{\left|a+b \omega+c \omega^{2}\right|^{2}: a, b, c \text { distinct non-zero integers }\right\}
\] equal? | 3 | jeebench-math |
Let $A P(a ; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0$. If
\[
A P(1 ; 3) \cap A P(2 ; 5) \cap A P(3 ; 7)=A P(a ; d)
\]
then what does $a+d$ equal? | 157 | jeebench-math |
Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0,1\}$. Let the events $E_{1}$ and $E_{2}$ be given by
\[
\begin{aligned}
& E_{1}=\{A \in S: \operatorname{det} A=0\} \text { and } \\
& E_{2}=\{A \in S: \text { sum of entries of } A \text { is } 7\} .
\end{aligned}
\]
If a matr... | 0.5 | jeebench-math |
Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_{A}$ and $\Gamma_{B}$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_{A}$ and $\Gamma_{B}$ such that both the circles are on the same side... | 10 | jeebench-math |
If
\[
I=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{d x}{\left(1+e^{\sin x}\right)(2-\cos 2 x)}
\]
then what does $27 I^{2}$ equal? | 4 | jeebench-math |
Three lines are given by
\[
\vec{r} & =\lambda \hat{i}, \lambda \in \mathbb{R}
\]
\[\vec{r} & =\mu(\hat{i}+\hat{j}), \mu \in \mathbb{R}
\]
\[
\vec{r} =v(\hat{i}+\hat{j}+\hat{k}), v \in \mathbb{R}.
\]
Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ ... | 0.75 | jeebench-math |
Let
\[
\begin{aligned}
& P_{1}=I=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right], \quad P_{2}=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right], \quad P_{3}=\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right], \\
& ... | ABC | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has
PROPERTY 1 if $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite, and
PROPERTY 2 if $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h^{2}}$ exists and is finite.
Then which of the following options is/are correct?
(A) $... | AB | jeebench-math |
Let
\[
f(x)=\frac{\sin \pi x}{x^{2}}, \quad x>0
\]
Let $x_{1}<x_{2}<x_{3}<\cdots<x_{n}<\cdots$ be all the points of local maximum of $f$ and $y_{1}<y_{2}<y_{3}<\cdots<y_{n}<\cdots$ be all the points of local minimum of $f$.
Then which of the following options is/are correct?
(A) $x_{1}<y_{1}$
(B) $x_{n+1}-x_{n}>2$... | BCD | jeebench-math |
For $a \in \mathbb{R},|a|>1$, let
\[
\lim _{n \rightarrow \infty}\left(\frac{1+\sqrt[3]{2}+\cdots+\sqrt[3]{n}}{n^{7 / 3}\left(\frac{1}{(a n+1)^{2}}+\frac{1}{(a n+2)^{2}}+\cdots+\frac{1}{(a n+n)^{2}}\right)}\right)=54
\]
Then the possible value(s) of $a$ is/are
(A) -9
(B) -6
(C) 7
(D) 8 | AD | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define
\[
F(x)=\int_{0}^{x} f(t) d t, \quad x>0 .
\]
Then which of the following options is/are correct?
(A) $F$ has a local minimum at $x=1$
(B) $F$ has a local maximum at $x=2$
(C) $F$ has two local maxima and one local minimum in $(0... | ABD | jeebench-math |
Three lines
\[
\begin{aligned}
L_{1}: & \vec{r}=\lambda \hat{i}, \lambda \in \mathbb{R}, \\
L_{2}: & \vec{r}=\hat{k}+\mu \hat{j}, \mu \in \mathbb{R} \text { and } \\
L_{3}: & \vec{r}=\hat{i}+\hat{j}+v \hat{k}, \quad v \in \mathbb{R}
\end{aligned}
\]
are given. For which point(s) $Q$ on $L_{2}$ can we find a point... | AC | jeebench-math |
Suppose
\[
\operatorname{det}\left[\begin{array}{cc}
\sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \\
\sum_{k=0}^{n}{ }^{n} C_{k} k & \sum_{k=0}^{n}{ }^{n} C_{k} 3^{k}
\end{array}\right]=0
\]
holds for some positive integer $n$. Then what does $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}$? | 6.2 | jeebench-math |
Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then what is the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats? | 30 | jeebench-math |
Let $|X|$ denote the number of elements in a set $X$. Let $S=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then what is the number of ordered pairs $(A, B)$ such that $1 \leq|B|<|A|$? | 422 | jeebench-math |
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