problem stringlengths 107 1.36k | answer stringlengths 1 5 | datasource stringclasses 1
value |
|---|---|---|
What is the value of
\[
\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)
\]
in the interval $\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$? | 0 | jeebench-math |
What is the value of the integral
\[
\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta
\]? | 0.5 | jeebench-math |
Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of
\[
a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)
\]
is
(A) 0
(B) 8000
(C) 8080
(D) 16000 | D | jeebench-math |
If the function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?
(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto | C | jeebench-math |
Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined by
\[
f(x)=e^{x-1}-e^{-|x-1|} \quad \text { and } \quad g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)
\]
Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is
... | A | jeebench-math |
Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4 \lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to ... | A | jeebench-math |
Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_{2}$... | B | jeebench-math |
Consider all rectangles lying in the region
\[
\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq x \leq \frac{\pi}{2} \text { and } 0 \leq y \leq 2 \sin (2 x)\right\}
\]
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
(A) $\frac{3 \pi}... | C | jeebench-math |
Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: \mathbb{R} \rightarrow \mathbb{R}$ be an arbitrary function. Let $f g: \mathbb{R} \rightarrow \mathbb{R}$ be the product function defined by $(f g)(x)=f(x) g(x)$. Then which of the following statements is/a... | AC | jeebench-math |
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\operatorname{adj}(\operatorname{adj} M)$, then which of the following statements is/are ALWAYS TRUE?
(A) $M=I$
(B) $\operatorname{det} M=1$
(C) $M^{2}=I$
(D) $(\operatorname{adj} M)^{2}=I$ | BCD | jeebench-math |
Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE?
(A) $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$
(B) $|z| \leq 2$ for all $z \in S$
(C) $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$
(D) The ... | BC | jeebench-math |
Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$, respectively. If
\[
\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2 y}{x+y+z}
\]
then which of the following statements is/are TRUE?
(A) $2 Y=X+Z$
(B) $Y=X+Z$
(C) $\tan \f... | BC | jeebench-math |
Let $L_{1}$ and $L_{2}$ be the following straight lines.
\[
L_{1}: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3} \text { and } L_{2}: \frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}
\]
Suppose the straight line
\[
L: \frac{x-\alpha}{l}=\frac{y-1}{m}=\frac{z-\gamma}{-2}
\]
lies in the plane containing $L_{1}$ and $L_{2}$, and ... | AB | jeebench-math |
Which of the following inequalities is/are TRUE?
(A) $\int_{0}^{1} x \cos x d x \geq \frac{3}{8}$
(B) $\int_{0}^{1} x \sin x d x \geq \frac{3}{10}$
(C) $\int_{0}^{1} x^{2} \cos x d x \geq \frac{1}{2}$
(D) $\int_{0}^{1} x^{2} \sin x d x \geq \frac{2}{9}$ | ABD | jeebench-math |
Let $m$ be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$, where $x_{1}, x_{2}, x_{3}$ are positive ... | 8 | jeebench-math |
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_{1}, b_{2}, b_{3}, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_{1}=b_{1}=c$, then what is the number of all possible values of $c$, for... | 1 | jeebench-math |
Let $f:[0,2] \rightarrow \mathbb{R}$ be the function defined by
\[
f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)
\]
If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then what is the value of $\beta-\alpha$? | 1 | jeebench-math |
In a triangle $P Q R$, let $\vec{a}=\overrightarrow{Q R}, \vec{b}=\overrightarrow{R P}$ and $\vec{c}=\overrightarrow{P Q}$. If
\[
|\vec{a}|=3, \quad|\vec{b}|=4 \quad \text { and } \quad \frac{\vec{a} \cdot(\vec{c}-\vec{b})}{\vec{c} \cdot(\vec{a}-\vec{b})}=\frac{|\vec{a}|}{|\vec{a}|+|\vec{b}|},
\]
then what is the val... | 108 | jeebench-math |
For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by
\[
S=\left\{\left(x^{2}-1\right)^{2}\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right): a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right... | 3 | jeebench-math |
Let $e$ denote the base of the natural logarithm. What is the value of the real number $a$ for which the right hand limit
\[
\lim _{x \rightarrow 0^{+}} \frac{(1-x)^{\frac{1}{x}}-e^{-1}}{x^{a}}
\]
is equal to a nonzero real number? | 1 | jeebench-math |
For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then what is the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}\left(z_... | 8 | jeebench-math |
The probability that a missile hits a target successfully is 0.75 . In order to destroy the target completely, at least three successful hits are required. Then what is the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95? | 6 | jeebench-math |
Let $O$ be the centre of the circle $x^{2}+y^{2}=r^{2}$, where $r>\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x+2 y=4$, then what is the value of $r$? | 2 | jeebench-math |
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is -18 , then what is the value of the determinant of $A$? | 5 | jeebench-math |
Let the functions $f:(-1,1) \rightarrow \mathbb{R}$ and $g:(-1,1) \rightarrow(-1,1)$ be defined by
\[
f(x)=|2 x-1|+|2 x+1| \text { and } g(x)=x-[x] .
\]
where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ g:(-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x... | 4 | jeebench-math |
What is the value of the limit
\[
\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}
\]? | 8 | jeebench-math |
Let $b$ be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation
\[
f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}
\]
for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE?
(... | AC | jeebench-math |
Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1,0)$, and suppose the normal to the hyperbola at $P$ cuts off equal... | AD | jeebench-math |
Let $\boldsymbol{f}: \mathbb{R} \rightarrow \mathbb{R}$ and $\boldsymbol{g}: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying
\[
f(x+y)=f(x)+f(y)+f(x) f(y) \text { and } f(x)=x g(x)
\]
for all $x, y \in \mathbb{R}$. If $\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?
(A... | ABD | jeebench-math |
Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2}+\beta^{2}+\gamma^{2} \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?
(A) $\... | ABC | jeebench-math |
Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{P Q}=a \hat{i}+b \hat{j}$ and $\overrightarrow{P S}=a \hat{i}-b \hat{j}$ are adjacent sides of a parallelogram $P Q R S$. Let $\vec{u}$ and $\vec{v}$ be the projection vectors of $\vec{w}=\hat{i}+\hat{j}$ along $\overrightarrow{P Q}$ and $\overrightarro... | AC | jeebench-math |
For nonnegative integers $s$ and $r$, let
\[
\left(\begin{array}{ll}
s \\
r
\end{array}\right)= \begin{cases}\frac{s !}{r !(s-r) !} & \text { if } r \leq s \\
0 & \text { if } r>s .\end{cases}
\]
For positive integers $m$ and $n$, let
\[
g(m, n)=\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\left(\begin{array}{c}
n+p \\
... | ABD | jeebench-math |
An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then what is the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021? | 495 | jeebench-math |
In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then what is the number of all possible ways in which this can be done? | 1080 | jeebench-math |
Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the p... | 8 | jeebench-math |
Let the function $f:[0,1] \rightarrow \mathbb{R}$ be defined by
\[
f(x)=\frac{4^{x}}{4^{x}+2}
\]
Then what is the value of
\[
f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots+f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)
\]? | 19 | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f^{\prime}$ is continuous and $f(\pi)=-6$. If $F:[0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x)=\int_{0}^{x} f(t) d t$, and if
\[
\int_{0}^{\pi}\left(f^{\prime}(x)+F(x)\right) \cos x d x=2,
\]
then what is the valu... | 4 | jeebench-math |
Let the function $f:(0, \pi) \rightarrow \mathbb{R}$ be defined by
\[
f(\theta)=(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{4} .
\]
Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_{1} \pi, \ldots, \lambda_{r} \pi\right\}$, where $0<$ $\lambda_{1}<\cdots<\... | 0.5 | jeebench-math |
Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is
(A) $x^{2}+y^{2}-3 x+y=0$
(B) $x^{2}+y^{2}+x+3 y=0$
(C) $x^{2}+y^{2}+2 y-1=0$
(D) $x^{2}+y^{2}+... | B | jeebench-math |
The area of the region
\[
\left\{(x, y): 0 \leq x \leq \frac{9}{4}, \quad 0 \leq y \leq 1, \quad x \geq 3 y, \quad x+y \geq 2\right\}
\]
is
(A) $\frac{11}{32}$
(B) $\frac{35}{96}$
(C) $\frac{37}{96}$
(D) $\frac{13}{32}$ | A | jeebench-math |
Consider three sets $E_{1}=\{1,2,3\}, F_{1}=\{1,3,4\}$ and $G_{1}=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_{1}$, and let $S_{1}$ denote the set of these chosen elements. Let $E_{2}=E_{1}-S_{1}$ and $F_{2}=F_{1} \cup S_{1}$. Now two elements are chosen at random, without repl... | A | jeebench-math |
Let $\theta_{1}, \theta_{2}, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_{1}+\theta_{2}+\cdots+\theta_{10}=2 \pi$. Define the complex numbers $z_{1}=e^{i \theta_{1}}, z_{k}=z_{k-1} e^{i \theta_{k}}$ for $k=2,3, \ldots, 10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given ... | C | jeebench-math |
Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be the probability that the maximum of chosen numbers is at least 81 and $p_2$ be the probability that the minimum of chosen numbers is at most 40 . What is the value of $\frac{625}{4} p_{1}$? | 76.25 | jeebench-math |
Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be the probability that the maximum of chosen numbers is at least 81 and $p_2$ be the probability that the minimum of chosen numbers is at most 40 . What is the value of $\frac{125}{4} p_{2}$? | 24.5 | jeebench-math |
Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
\[\begin{gathered}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{gathered}\]
is consistent. Let $|M|$ represent the determinant of the matrix
\[M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \... | 1.00 | jeebench-math |
Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
\[\begin{gathered}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{gathered}\]
is consistent. Let $|M|$ represent the determinant of the matrix
\[M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \... | 1.5 | jeebench-math |
Consider the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ defined by
$\mathrm{L}_1: \mathrm{x} \sqrt{2}+\mathrm{y}-1=0$ and $\mathrm{L}_2: \mathrm{x} \sqrt{2}-\mathrm{y}+1=0$
For a fixed constant $\lambda$, let $\mathrm{C}$ be the locus of a point $\mathrm{P}$ such that the product of the distance of $\mathrm{P}$ from $\mat... | 9.00 | jeebench-math |
Consider the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ defined by
$\mathrm{L}_1: \mathrm{x} \sqrt{2}+\mathrm{y}-1=0$ and $\mathrm{L}_2: \mathrm{x} \sqrt{2}-\mathrm{y}+1=0$
For a fixed constant $\lambda$, let $\mathrm{C}$ be the locus of a point $\mathrm{P}$ such that the product of the distance of $\mathrm{P}$ from $\mat... | 77.14 | jeebench-math |
For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let
\[
E=\left[\begin{array}{ccc}
1 & 2 & 3 \\
2 & 3 & 4 \\
8 & 13 & 18
\end{array}\right], P=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right] \text { and } F=\left[\begin{array}{ccc}
1 & 3 & 2 \\
8 & 18 ... | ABD | jeebench-math |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by
\[
f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}
\]
Then which of the following statements is (are) TRUE ?
(A) $f$ is decreasing in the interval $(-2,-1)$
(B) $f$ is increasing in the interval $(1,2)$
(C) $f$ is onto
(D) Range of $f$ is $\left[-\frac{3}{2}, 2\right]... | AB | jeebench-math |
Let $E, F$ and $G$ be three events having probabilities
\[
P(E)=\frac{1}{8}, P(F)=\frac{1}{6} \text { and } P(G)=\frac{1}{4} \text {, and let } P(E \cap F \cap G)=\frac{1}{10} \text {. }
\]
For any event $H$, if $H^{c}$ denotes its complement, then which of the following statements is (are) TRUE ?
(A) $P\left(E \cap... | ABC | jeebench-math |
For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3 \times 3$ identity matrix. Let $E$ and $F$ be two $3 \times 3$ matrices such that $(I-E F)$ is invertible. If $G=(I-E F)^{-1}$, then which of the following statements is (are) TRUE ?
(A) $|F E|=|I-F E||F G E|$
(B) $(I-F E)(I+F... | ABC | jeebench-math |
For any positive integer $n$, let $S_{n}:(0, \infty) \rightarrow \mathbb{R}$ be defined by
\[
S_{n}(x)=\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right)
\]
where for any $x \in \mathbb{R}, \cot ^{-1}(x) \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the foll... | AB | jeebench-math |
For any complex number $w=c+i d$, let $\arg (\mathrm{w}) \in(-\pi, \pi]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+i y$ satisfying $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, the ordered pair $(x, y)$ lies on the circle
\[
x^{2}+y^{2}+5 x-3 y+... | BD | jeebench-math |
For $x \in \mathbb{R}$, what is the number of real roots of the equation
\[
3 x^{2}-4\left|x^{2}-1\right|+x-1=0
\]? | 4 | jeebench-math |
In a triangle $A B C$, let $A B=\sqrt{23}, B C=3$ and $C A=4$. Then what is the value of
\[
\frac{\cot A+\cot C}{\cot B}
\]? | 2 | jeebench-math |
Let
\[
\begin{gathered}
S_{1}=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}, \\
S_{2}=\{(i, j): 1 \leq i<j+2 \leq 10, i, j \in\{1,2, \ldots, 10\}\} \\
S_{3}=\{(i, j, k, l): 1 \leq i<j<k<l, i, j, k, l \in\{1,2, \ldots, 10\}\}
\end{gathered}
\]
and
\[
S_{4}=\{(i, j, k, l): i, j, k \text { and } l \text { are disti... | ABD | jeebench-math |
Consider a triangle $P Q R$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$, respectively. Then which of the following statements is (are) TRUE ?
(A) $\cos P \geq 1-\frac{p^{2}}{2 q r}$
(B) $\cos R \geq\left(\frac{q-r}{p+q}\right) \cos P+\left(\frac{p-r}{p+q}\right) \cos Q$
(C) $\frac{q+... | AB | jeebench-math |
Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a continuous function such that
\[
f(0)=1 \text { and } \int_{0}^{\frac{\pi}{3}} f(t) d t=0
\]
Then which of the following statements is (are) TRUE ?
(A) The equation $f(x)-3 \cos 3 x=0$ has at least one solution in $\left(0, \frac{\pi}{3}\... | ABC | jeebench-math |
For any real numbers $\alpha$ and $\beta$, let $y_{\alpha, \beta}(x), x \in \mathbb{R}$, be the solution of the differential equation
\[
\frac{d y}{d x}+\alpha y=x e^{\beta x}, \quad y(1)=1
\]
Let $S=\left\{y_{\alpha, \beta}(x): \alpha, \beta \in \mathbb{R}\right\}$. Then which of the following functions belong(s) to... | AC | jeebench-math |
Let $E$ denote the parabola $y^{2}=8 x$. Let $P=(-2,4)$, and let $Q$ and $Q^{\prime}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{\prime}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE?
(A) The triangle $P F Q$ is a right-angled triangle
(... | ABD | jeebench-math |
Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^2 \leq 4-x\right\}$. Let $\mathcal{F}$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal{F}$. Let $(\al... | 1.5 | jeebench-math |
Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^2 \leq 4-x\right\}$. Let $\mathcal{F}$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal{F}$. Let $(\al... | 2.00 | jeebench-math |
Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ be defined by
\[f_1(x)=\int_0^x \prod_{j=1}^{21}(t-j)^j d t, x>0\]
and
\[f_2(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0\]
where, for any positive integer $\mathrm{n}$ and real numbers $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{... | 57 | jeebench-math |
Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ be defined by
\[f_1(x)=\int_0^x \prod_{j=1}^{21}(t-j)^j d t, x>0\]
and
\[f_2(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0\]
where, for any positive integer $\mathrm{n}$ and real numbers $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{... | 6 | jeebench-math |
Let $\mathrm{g}_i:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, \mathrm{i}=1,2$, and $f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $\mathrm{g}_1(\mathrm{x})=1, \mathrm{~g}_2(\mathrm{x})=|4 \mathrm{x}-\pi|$ and $f(\mathrm{x})=\sin ^2 \mathrm{x}$, for ... | 2 | jeebench-math |
Let
\[\mathrm{M}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2 \leq \mathrm{r}^2\right\}\]
where $\mathrm{r}>0$. Consider the geometric progression $a_n=\frac{1}{2^{n-1}}, n=1,2,3, \ldots$. Let $S_0=0$ and, for $n \geq 1$, let $S_n$ denote the sum of the first $n$ terms of ... | D | jeebench-math |
Let
\[\mathrm{M}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2 \leq \mathrm{r}^2\right\}\]
where $\mathrm{r}>0$. Consider the geometric progression $a_n=\frac{1}{2^{n-1}}, n=1,2,3, \ldots$. Let $S_0=0$ and, for $n \geq 1$, let $S_n$ denote the sum of the first $n$ terms of ... | B | jeebench-math |
Let $\psi_1:[0, \infty) \rightarrow \mathbb{R}, \psi_2:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}$ and $g:[0, \infty) \rightarrow \mathbb{R}$ be functions such that
\[\begin{aligned}
& f(0)=\mathrm{g}(0)=0, \\
& \psi_1(\mathrm{x})=\mathrm{e}^{-\mathrm{x}}+\mathrm{x}, \quad \mathrm{x} \geq ... | C | jeebench-math |
Considering only the principal values of the inverse trigonometric functions, what is the value of
\[
\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^{2}}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^{2}}+\tan ^{-1} \frac{\sqrt{2}}{\pi}
\]? | 2.35 | jeebench-math |
Let $\alpha$ be a positive real number. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g:(\alpha, \infty) \rightarrow \mathbb{R}$ be the functions defined by
\[
f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x... | 0.5 | jeebench-math |
In a study about a pandemic, data of 900 persons was collected. It was found that
190 persons had symptom of fever,
220 persons had symptom of cough,
220 persons had symptom of breathing problem,
330 persons had symptom of fever or cough or both,
350 persons had symptom of cough or breathing problem or both,
340 ... | 0.8 | jeebench-math |
Let $z$ be a complex number with non-zero imaginary part. If
\[
\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}
\]
is a real number, then the value of $|z|^{2}$ is | 0.5 | jeebench-math |
Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers,what is the number of distinct roots of the equation
\[
\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)
\]? | 4 | jeebench-math |
Let $l_{1}, l_{2}, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_{1}$, and let $w_{1}, w_{2}, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_{2}$, where $d_{1} d_{2}=10$. For each $i=1,2, \ldots, 100$, let $R_{i}$ be a rec... | 18900 | jeebench-math |
What is the number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$? | 569 | jeebench-math |
Let $A B C$ be the triangle with $A B=1, A C=3$ and $\angle B A C=\frac{\pi}{2}$. If a circle of radius $r>0$ touches the sides $A B, A C$ and also touches internally the circumcircle of the triangle $A B C$, then what is the value of $r$? | 0.83 | jeebench-math |
Consider the equation
\[
\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) .
\]
Which of the following statements is/are TRUE?
(A) No $a$ satisfies the above equation
(B) An integer $a$ satisfie... | CD | jeebench-math |
Let $a_{1}, a_{2}, a_{3}, \ldots$ be an arithmetic progression with $a_{1}=7$ and common difference 8 . Let $T_{1}, T_{2}, T_{3}, \ldots$ be such that $T_{1}=3$ and $T_{n+1}-T_{n}=a_{n}$ for $n \geq 1$. Then, which of the following is/are TRUE ?
(A) $T_{20}=1604$
(B) $\sum_{k=1}^{20} T_{k}=10510$
(C) $T_{30}=3454$
... | BC | jeebench-math |
Let $P_{1}$ and $P_{2}$ be two planes given by
\[
\begin{aligned}
& P_{1}: 10 x+15 y+12 z-60=0, \\
& P_{2}: \quad-2 x+5 y+4 z-20=0 .
\end{aligned}
\]
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?
(A) $\frac{x-1}{0}=\frac{y-1}{0}=\frac{z-1}{5}... | ABD | jeebench-math |
Let $S$ be the reflection of a point $Q$ with respect to the plane given by
\[
\vec{r}=-(t+p) \hat{i}+t \hat{j}+(1+p) \hat{k}
\]
where $t, p$ are real parameters and $\hat{i}, \hat{j}, \hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat{i}+15 \ha... | ABC | jeebench-math |
Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be the function defined by
where
\[
g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}
\]
$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 &... | AC | jeebench-math |
Consider the following lists:
List-I
(I) $\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$
(II) $\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$
(III) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}... | B | jeebench-math |
Two players, $P_{1}$ and $P_{2}$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$, respectively. If $x>y$, then $P_{1}$ scores 5 points an... | A | jeebench-math |
Let $p, q, r$ be nonzero real numbers that are, respectively, the $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ terms of a harmonic progression. Consider the system of linear equations
\[
\begin{gathered}
x+y+z=1 \\
10 x+100 y+1000 z=0 \\
q r x+p r y+p q z=0
\end{gathered}
\]
List-I
(I) If $\frac... | B | jeebench-math |
Consider the ellipse
\[
\frac{x^{2}}{4}+\frac{y^{2}}{3}=1
\]
Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ inters... | C | jeebench-math |
Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$. If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then what is the greatest integer less than or equal to
\[
\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}... | 1 | jeebench-math |
If $y(x)$ is the solution of the differential equation
\[
x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x>0, \quad y(1)=2,
\]
and the slope of the curve $y=y(x)$ is never zero, then what is the value of $10 y(\sqrt{2})$? | 8 | jeebench-math |
What is the greatest integer less than or equal to
\[
\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x
\]? | 5 | jeebench-math |
What is the product of all positive real values of $x$ satisfying the equation
\[
x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}
\]? | 1 | jeebench-math |
If
\[
\beta=\lim _{x \rightarrow 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},
\]
then what is the value of $6 \beta$? | 5 | jeebench-math |
Let $\beta$ be a real number. Consider the matrix
\[
A=\left(\begin{array}{ccc}
\beta & 0 & 1 \\
2 & 1 & -2 \\
3 & 1 & -2
\end{array}\right)
\]
If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then what is the value of $9 \beta$? | 3 | jeebench-math |
Consider the hyperbola
\[
\frac{x^{2}}{100}-\frac{y^{2}}{64}=1
\]
with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having th... | 7 | jeebench-math |
Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by
\[
f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases}
\]
If $\alpha$ is the area of the region
\[
\left\{(x, y) \in \mathbb{R} \times... | 6 | jeebench-math |
Let $P Q R S$ be a quadrilateral in a plane, where $Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$. If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
(A) $(0, \... | AB | jeebench-math |
Let
\[
\alpha=\sum_{k=1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)
\]
Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by
\[
g(x)=2^{\alpha x}+2^{\alpha(1-x)}
\]
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
(B) The maximum value of $g(... | ABC | jeebench-math |
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
\[
(\bar{z})^{2}+\frac{1}{z^{2}}
\]
are integers, then which of the following is/are possible value(s) of $|z|$ ?
(A) $\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}... | A | jeebench-math |
Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ exter... | CD | jeebench-math |
Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let
\[
\vec{a}=3 \hat{i}+\hat{j}-\hat{k},
\]
\[\vec{b}=\hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R},
\]
\[\vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb... | BCD | jeebench-math |
For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
\[
\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0
\]
Then, which of the following statements is/are TRUE?
(A) $y(x)$ is an increasing function
(B) $y(x)$ is a decreasing function
(C) There exists a r... | C | jeebench-math |
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