Datasets: daman1209arora /jeebench

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1.61k
math
AC
43
MCQ(multiple)
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? (A) If $b>0$, then $f$ is an increasing function (B) If $b<0$, then $f$ is a decreasing function (C) $f(x) f(-x)=1$ for all $x \in \mathbb{R}$ (D) $f(x)-f(-x)=0$ for all $x \in \mathbb{R}$
math
44
MCQ(multiple)
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 intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE? (A) $1<e<\sqrt{2}$ (B) $\sqrt{2}<e<2$ (C) $\Delta=a^{4}$ (D) $\Delta=b^{4}$
math
ABD
45
MCQ(multiple)
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) $f$ is differentiable at every $x \in \mathbb{R}$ (B) If $g(0)=1$, then $g$ is differentiable at every $x \in \mathbb{R}$ (C) The derivative $f^{\prime}(1)$ is equal to 1 (D) The derivative $f^{\prime}(0)$ is equal to 1
math
ABC
46
MCQ(multiple)
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) $\alpha+\beta=2$ (B) $\delta-\gamma=3$ (C) $\delta+\beta=4$ (D) $\alpha+\beta+\gamma=\delta$
math
AC
47
MCQ(multiple)
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 $\overrightarrow{P S}$, respectively. If $|\vec{u}|+|\vec{v}|=|\vec{w}|$ and if the area of the parallelogram $P Q R S$ is 8 , then which of the following statements is/are TRUE? (A) $a+b=4$ (B) $a-b=2$ (C) The length of the diagonal $P R$ of the parallelogram $P Q R S$ is 4 (D) $\vec{w}$ is an angle bisector of the vectors $\overrightarrow{P Q}$ and $\overrightarrow{P S}$
math
ABD
48
MCQ(multiple)
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 $g(m, n)=\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\left(\begin{array}{c} n+p \\ p \end{array}\right)}$ where for any nonnegative integer $p$, $f(m, n, p)=\sum_{i=0}^{p}\left(\begin{array}{c} m \\ i \end{array}\right)\left(\begin{array}{c} n+i \\ p \end{array}\right)\left(\begin{array}{c} p+n \\ p-i \end{array}\right) .$ Then which of the following statements is/are TRUE? (A) $g(m, n)=g(n, m)$ for all positive integers $m, n$ (B) $g(m, n+1)=g(m+1, n)$ for all positive integers $m, n$ (C) $g(2 m, 2 n)=2 g(m, n)$ for all positive integers $m, n$ (D) $g(2 m, 2 n)=(g(m, n))^{2}$ for all positive integers $m, n$
math
495
49
Numeric
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?
math
1080
50
Numeric
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?
math
8
51
Numeric
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 probability that this perfect square is an odd number, then what is the value of $14 p$?
math
19
52
Numeric
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 $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots+f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$?
math
4
53
Numeric
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 value of $f(0)$?
math
0.5
54
Numeric
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 $0<$ $\lambda_{1}<\cdots<\lambda_{r}<1$. Then what is the value of $\lambda_{1}+\cdots+\lambda_{r}$?
phy
D
4
MCQ
A heavy nucleus $Q$ of half-life 20 minutes undergoes alpha-decay with probability of $60 \%$ and beta-decay with probability of $40 \%$. Initially, the number of $Q$ nuclei is 1000. The number of alpha-decays of $Q$ in the first one hour is (A) 50 (B) 75 (C) 350 (D) 525
phy
0.5
5
Numeric
A projectile is thrown from a point $\mathrm{O}$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, $0.5 \mathrm{~s}$ after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point $\mathrm{O}$. The acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{s}^2$. What is the value of $t$?
phy
7.5
6
Numeric
A projectile is thrown from a point $\mathrm{O}$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, $0.5 \mathrm{~s}$ after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point $\mathrm{O}$. The acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{s}^2$. What is the value of $x$?
phy
ABC
13
MCQ(multiple)
A particle of mass $M=0.2 \mathrm{~kg}$ is initially at rest in the $x y$-plane at a point $(x=-l$, $y=-h)$, where $l=10 \mathrm{~m}$ and $h=1 \mathrm{~m}$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \mathrm{~m} / \mathrm{s}^{2}$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{\tau}$, respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x, y$ and $z$-directions, respectively. If $\hat{k}=\hat{i} \times \hat{j}$ then which of the following statement(s) is(are) correct? (A) The particle arrives at the point $(x=l, y=-h)$ at time $t=2 \mathrm{~s}$ (B) $\vec{\tau}=2 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$ (C) $\vec{L}=4 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$ (D) $\vec{\tau}=\hat{k}$ when the particle passes through the point $(x=0, y=-h)$
phy
14
MCQ(multiple)
Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom? (A) The ratio of the longest wavelength to the shortest wavelength in Balmer series is $9 / 5$ (B) There is an overlap between the wavelength ranges of Balmer and Paschen series (C) The wavelengths of Lyman series are given by $\left(1+\frac{1}{m^{2}}\right) \lambda_{0}$, where $\lambda_{0}$ is the shortest wavelength of Lyman series and $m$ is an integer (D) The wavelength ranges of Lyman and Balmer series do not overlap
phy
4
17
Integer
An $\alpha$-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential $V$ and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $\alpha$-particle and the sulfur ion move in circular orbits of radii $r_{\alpha}$ and $r_{S}$, respectively. What is the ratio $\left(r_{S} / r_{\alpha}\right)$?
chem
A
22
MCQ
The calculated spin only magnetic moments of $\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ and $\left[\mathrm{CuF}_{6}\right]^{3-}$ in $\mathrm{BM}$, respectively, are (Atomic numbers of $\mathrm{Cr}$ and $\mathrm{Cu}$ are 24 and 29, respectively) (A) 3.87 and 2.84 (B) 4.90 and 1.73 (C) 3.87 and 1.73 (D) 4.90 and 2.84
chem
100.1
27
Numeric
The boiling point of water in a 0.1 molal silver nitrate solution (solution $\mathbf{A}$ ) is $\mathbf{x}^{\circ} \mathrm{C}$. To this solution $\mathbf{A}$, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution $\mathbf{B}$. The difference in the boiling points of water in the two solutions $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}$. (Assume: Densities of the solutions $\mathbf{A}$ and $\mathbf{B}$ are the same as that of water and the soluble salts dissociate completely. Use: Molal elevation constant (Ebullioscopic Constant), $K_b=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$; Boiling point of pure water as $100^{\circ} \mathrm{C}$.) What is the value of $\mathbf{x}$?
chem
2.5
28
Numeric
The boiling point of water in a 0.1 molal silver nitrate solution (solution $\mathbf{A}$ ) is $\mathbf{x}^{\circ} \mathrm{C}$. To this solution $\mathbf{A}$, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution $\mathbf{B}$. The difference in the boiling points of water in the two solutions $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}$. (Assume: Densities of the solutions $\mathbf{A}$ and $\mathbf{B}$ are the same as that of water and the soluble salts dissociate completely. Use: Molal elevation constant (Ebullioscopic Constant), $K_b=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$; Boiling point of pure water as $100^{\circ} \mathrm{C}$.) What is the value of $|\mathbf{y}|$?
chem
BC
31
MCQ(multiple)
The correct statement(s) related to colloids is(are) (A) The process of precipitating colloidal sol by an electrolyte is called peptization. (B) Colloidal solution freezes at higher temperature than the true solution at the same concentration. (C) Surfactants form micelle above critical micelle concentration (CMC). CMC depends on temperature. (D) Micelles are macromolecular colloids.
chem
ACD
33
MCQ(multiple)
The correct statement(s) related to the metal extraction processes is(are) (A) A mixture of $\mathrm{PbS}$ and $\mathrm{PbO}$ undergoes self-reduction to produce $\mathrm{Pb}$ and $\mathrm{SO}_{2}$. (B) In the extraction process of copper from copper pyrites, silica is added to produce copper silicate. (C) Partial oxidation of sulphide ore of copper by roasting, followed by self-reduction produces blister copper. (D) In cyanide process, zinc powder is utilized to precipitate gold from $\mathrm{Na}\left[\mathrm{Au}(\mathrm{CN})_{2}\right]$.
chem
13
35
Integer
What is the maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of 1-methylcyclohex-1-ene using $\mathrm{Br}_{2}$ and UV light?
math
B
37
MCQ
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) $x^{2}+y^{2}+x+y=0$
math
A
38
MCQ
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) $\frac{13}{32}$
math
A
39
MCQ
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 replacement, from the set $F_{2}$ and let $S_{2}$ denote the set of these chosen elements. Let $G_{2}=G_{1} \cup S_{2}$. Finally, two elements are chosen at random, without replacement, from the set $G_{2}$ and let $S_{3}$ denote the set of these chosen elements. Let $E_{3}=E_{2} \cup S_{3}$. Given that $E_{1}=E_{3}$, let $p$ be the conditional probability of the event $S_{1}=\{1,2\}$. Then the value of $p$ is (A) $\frac{1}{5}$ (B) $\frac{3}{5}$ (C) $\frac{1}{2}$ (D) $\frac{2}{5}$
math
C
40
MCQ
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 below: \begin{aligned} & P:\left|z_{2}-z_{1}\right|+\left|z_{3}-z_{2}\right|+\cdots+\left|z_{10}-z_{9}\right|+\left|z_{1}-z_{10}\right| \leq 2 \pi \\ & Q:\left|z_{2}^{2}-z_{1}^{2}\right|+\left|z_{3}^{2}-z_{2}^{2}\right|+\cdots+\left|z_{10}^{2}-z_{9}^{2}\right|+\left|z_{1}^{2}-z_{10}^{2}\right| \leq 4 \pi \end{aligned} Then, (A) $P$ is TRUE and $Q$ is FALSE (B) $Q$ is TRUE and $P$ is FALSE (C) both $P$ and $Q$ are TRUE (D) both $P$ and $Q$ are FALSE
math
76.25
41
Numeric
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 $\frac{625}{4} p_{1}$?
math
24.5
42
Numeric
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 $\frac{125}{4} p_{2}$?
math
1.00
43
Numeric
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 $M=\left[\begin{array}{ccc} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$. What is the value of $|\mathrm{M}|$?
math
1.5
44
Numeric
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 $M=\left[\begin{array}{ccc} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$. What is the value of $D$?
math
9.00
45
Numeric
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 $\mathrm{L}_1$ and the distance of $\mathrm{P}$ from $\mathrm{L}_2$ is $\lambda^2$. The line $\mathrm{y}=2 \mathrm{x}+1$ meets $\mathrm{C}$ at two points $\mathrm{R}$ and $\mathrm{S}$, where the distance between $\mathrm{R}$ and $\mathrm{S}$ is $\sqrt{270}$. Let the perpendicular bisector of RS meet $\mathrm{C}$ at two distinct points $\mathrm{R}^{\prime}$ and $\mathrm{S}^{\prime}$. Let $\mathrm{D}$ be the square of the distance between $\mathrm{R}^{\prime}$ and S'. What is the value of $\lambda^{2}$?
math
77.14
46
Numeric
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 $\mathrm{L}_1$ and the distance of $\mathrm{P}$ from $\mathrm{L}_2$ is $\lambda^2$. The line $\mathrm{y}=2 \mathrm{x}+1$ meets $\mathrm{C}$ at two points $\mathrm{R}$ and $\mathrm{S}$, where the distance between $\mathrm{R}$ and $\mathrm{S}$ is $\sqrt{270}$. Let the perpendicular bisector of RS meet $\mathrm{C}$ at two distinct points $\mathrm{R}^{\prime}$ and $\mathrm{S}^{\prime}$. Let $\mathrm{D}$ be the square of the distance between $\mathrm{R}^{\prime}$ and S'. What is the value of $D$?
math
ABD
47
MCQ(multiple)
For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{array}\right], P=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right] \text { and } F=\left[\begin{array}{ccc} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{array}\right]$ If $Q$ is a nonsingular matrix of order $3 \times 3$, then which of the following statements is (are) TRUE? (A) $F=P E P$ and $P^{2}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ (B) $\left|E Q+P F Q^{-1}\right|=|E Q|+\left|P F Q^{-1}\right|$ (C) $\left|(E F)^{3}\right|>|E F|^{2}$ (D) Sum of the diagonal entries of $P^{-1} E P+F$ is equal to the sum of diagonal entries of $E+P^{-1} F P$
math
AB
48
MCQ(multiple)
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 $\left[-\frac{3}{2}, 2\right]$
math
ABC
49
MCQ(multiple)
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) $P\left(E \cap F \cap G^{c}\right) \leq \frac{1}{40}$ (B) $P\left(E^{c} \cap F \cap G\right) \leq \frac{1}{15}$ (C) $P(E \cup F \cup G) \leq \frac{13}{24}$ (D) $P\left(E^{c} \cap F^{c} \cap G^{c}\right) \leq \frac{5}{12}$
math
ABC
50
MCQ(multiple)
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) $(I-F E)(I+F G E)=I$ (C) $E F G=G E F$ (D) $(I-F E)(I-F G E)=I$
math
AB
51
MCQ(multiple)
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 following statements is (are) TRUE ? (A) $S_{10}(x)=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x^{2}}{10 x}\right)$, for all $x>0$ (B) $\lim _{n \rightarrow \infty} \cot \left(S_{n}(x)\right)=x$, for all $x>0$ (C) The equation $S_{3}(x)=\frac{\pi}{4}$ has a root in $(0, \infty)$ (D) $\tan \left(S_{n}(x)\right) \leq \frac{1}{2}$, for all $n \geq 1$ and $x>0$
math
BD
52
MCQ(multiple)
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 $x^{2}+y^{2}+5 x-3 y+4=0$ Then which of the following statements is (are) TRUE ? (A) $\alpha=-1$ (B) $\alpha \beta=4$ (C) $\alpha \beta=-4$ (D) $\beta=4$
math
4
53
Integer
For $x \in \mathbb{R}$, what is the number of real roots of the equation $3 x^{2}-4\left|x^{2}-1\right|+x-1=0$?
math
2
54
Integer
In a triangle $A B C$, let $A B=\sqrt{23}, B C=3$ and $C A=4$. Then what is the value of $\frac{\cot A+\cot C}{\cot B}$?
phy
2
MCQ(multiple)
A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_{s}$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_{0}$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_{s}$, will the sound reaching the pipe lead to a resonance? (A) $u=0.8 v$ and $f_{s}=f_{0}$ (B) $u=0.8 v$ and $f_{s}=2 f_{0}$ (C) $u=0.8 v$ and $f_{s}=0.5 f_{0}$ (D) $u=0.5 v$ and $f_{s}=1.5 f_{0}$
phy
BD
4
MCQ(multiple)
A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_{0}$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_{0}$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity(ies)? (A) $\frac{\text { Energy }}{\text { Charge } \times \text { Current }}$ (B) $\frac{\text { Force }}{\text { Length } \times \text { Time }}$ (C) $\frac{\text { Energy }}{\text { Volume }}$ (D) $\frac{\text { Power }}{\text { Area }}$
phy
ACD
5
MCQ(multiple)
A heavy nucleus $N$, at rest, undergoes fission $N \rightarrow P+Q$, where $P$ and $Q$ are two lighter nuclei. Let $\delta=M_{N}-M_{P}-M_{Q}$, where $M_{P}, M_{Q}$ and $M_{N}$ are the masses of $P$, $Q$ and $N$, respectively. $E_{P}$ and $E_{Q}$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_{P}$ and $v_{Q}$, respectively. If $c$ is the speed of light, which of the following statement(s) is(are) correct? (A) $E_{P}+E_{Q}=c^{2} \delta$ (B) $E_{P}=\left(\frac{M_{P}}{M_{P}+M_{Q}}\right) c^{2} \delta$ (C) $\frac{v_{P}}{v_{Q}}=\frac{M_{Q}}{M_{P}}$ (D) The magnitude of momentum for $P$ as well as $Q$ is $c \sqrt{2 \mu \delta}$, where $\mu=\frac{M_{P} M_{Q}}{\left(M_{P}+M_{Q}\right)}$
phy
0.18
9
Numeric
A pendulum consists of a bob of mass $m=0.1 \mathrm{~kg}$ and a massless inextensible string of length $L=1.0 \mathrm{~m}$. It is suspended from a fixed point at height $H=0.9 \mathrm{~m}$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J \mathrm{~kg}-\mathrm{m}^2 / \mathrm{s}$. The kinetic energy of the pendulum just after the liftoff is $K$ Joules. What is the value of $J$?
phy
0.16
10
Numeric
A pendulum consists of a bob of mass $m=0.1 \mathrm{~kg}$ and a massless inextensible string of length $L=1.0 \mathrm{~m}$. It is suspended from a fixed point at height $H=0.9 \mathrm{~m}$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J \mathrm{~kg}-\mathrm{m}^2 / \mathrm{s}$. The kinetic energy of the pendulum just after the liftoff is $K$ Joules. What is the value of $K$?
phy
100.00
11
Numeric
In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu F$ across a $200 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. The power consumed by the lamp is $500 \mathrm{~W}$ while the voltage drop across it is $100 \mathrm{~V}$. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$. Assume, $\pi \sqrt{3} \approx 5$. What is the value of $C$?
phy
60
12
Numeric
In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance $\mathrm{C} \mu F$ across a $200 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. The power consumed by the lamp is $500 \mathrm{~W}$ while the voltage drop across it is $100 \mathrm{~V}$. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$. Assume, $\pi \sqrt{3} \approx 5$. What is the value of $\varphi$?
chem
BCD
21
MCQ(multiple)
For the following reaction $2 \mathbf{X}+\mathbf{Y} \stackrel{k}{\rightarrow} \mathbf{P}$ the rate of reaction is $\frac{d[\mathbf{P}]}{d t}=k[\mathbf{X}]$. Two moles of $\mathbf{X}$ are mixed with one mole of $\mathbf{Y}$ to make 1.0 L of solution. At $50 \mathrm{~s}, 0.5$ mole of $\mathbf{Y}$ is left in the reaction mixture. The correct statement(s) about the reaction is(are) $($ Use $: \ln 2=0.693)$ (A) The rate constant, $k$, of the reaction is $13.86 \times 10^{-4} \mathrm{~s}^{-1}$. (B) Half-life of $\mathbf{X}$ is $50 \mathrm{~s}$. (C) At $50 \mathrm{~s},-\frac{d[\mathbf{X}]}{d t}=13.86 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$. (D) At $100 \mathrm{~s},-\frac{d[\mathrm{Y}]}{d t}=3.46 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$.
chem
ABC
22
MCQ(multiple)
Some standard electrode potentials at $298 \mathrm{~K}$ are given below: $\begin{array}{ll}\mathrm{Pb}^{2+} / \mathrm{Pb} & -0.13 \mathrm{~V} \\ \mathrm{Ni}^{2+} / \mathrm{Ni} & -0.24 \mathrm{~V} \\ \mathrm{Cd}^{2+} / \mathrm{Cd} & -0.40 \mathrm{~V} \\ \mathrm{Fe}^{2+} / \mathrm{Fe} & -0.44 \mathrm{~V}\end{array}$ To a solution containing $0.001 \mathrm{M}$ of $\mathbf{X}^{2+}$ and $0.1 \mathrm{M}$ of $\mathbf{Y}^{2+}$, the metal rods $\mathbf{X}$ and $\mathbf{Y}$ are inserted (at $298 \mathrm{~K}$ ) and connected by a conducting wire. This resulted in dissolution of $\mathbf{X}$. The correct combination(s) of $\mathbf{X}$ and $\mathbf{Y}$, respectively, is(are) (Given: Gas constant, $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, Faraday constant, $\mathrm{F}=96500 \mathrm{C} \mathrm{mol}^{-1}$ ) (A) $\mathrm{Cd}$ and $\mathrm{Ni}$ (B) $\mathrm{Cd}$ and $\mathrm{Fe}$ (C) $\mathrm{Ni}$ and $\mathrm{Pb}$ (D) $\mathrm{Ni}$ and $\mathrm{Fe}$
chem
ABD
23
MCQ(multiple)
The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are) (Note: $\mathrm{py}=$ pyridine Given: Atomic numbers of Fe, Co, Ni and $\mathrm{Cu}$ are 26, 27, 28 and 29, respectively) (A) $\left[\mathrm{FeCl}_{4}\right]^{-}$and $\left[\mathrm{Fe}(\mathrm{CO})_{4}\right]^{2-}$ (B) $\left[\mathrm{Co}(\mathrm{CO})_{4}\right]^{-}$and $\left[\mathrm{CoCl}_{4}\right]^{2-}$ (C) $\left[\mathrm{Ni}(\mathrm{CO})_{4}\right]$ and $\left[\mathrm{Ni}(\mathrm{CN})_{4}\right]^{2-}$ (D) $\left[\mathrm{Cu}(\mathrm{py})_{4}\right]^{+}$and $\left[\mathrm{Cu}(\mathrm{CN})_{4}\right]^{3-}$
chem
ABD
24
MCQ(multiple)
The correct statement(s) related to oxoacids of phosphorous is(are) (A) Upon heating, $\mathrm{H}_{3} \mathrm{PO}_{3}$ undergoes disproportionation reaction to produce $\mathrm{H}_{3} \mathrm{PO}_{4}$ and $\mathrm{PH}_{3}$. (B) While $\mathrm{H}_{3} \mathrm{PO}_{3}$ can act as reducing agent, $\mathrm{H}_{3} \mathrm{PO}_{4}$ cannot. (C) $\mathrm{H}_{3} \mathrm{PO}_{3}$ is a monobasic acid. (D) The $\mathrm{H}$ atom of $\mathrm{P}-\mathrm{H}$ bond in $\mathrm{H}_{3} \mathrm{PO}_{3}$ is not ionizable in water.
chem
0.22
25
Numeric
At $298 \mathrm{~K}$, the limiting molar conductivity of a weak monobasic acid is $4 \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. At $298 \mathrm{~K}$, for an aqueous solution of the acid the degree of dissociation is $\alpha$ and the molar conductivity is $\mathbf{y} \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. At $298 \mathrm{~K}$, upon 20 times dilution with water, the molar conductivity of the solution becomes $3 \mathbf{y} \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. What is the value of $\alpha$?
chem
0.86
26
Numeric
At $298 \mathrm{~K}$, the limiting molar conductivity of a weak monobasic acid is $4 \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. At $298 \mathrm{~K}$, for an aqueous solution of the acid the degree of dissociation is $\alpha$ and the molar conductivity is $\mathbf{y} \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. At $298 \mathrm{~K}$, upon 20 times dilution with water, the molar conductivity of the solution becomes $3 \mathbf{y} \times 10^2 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. What is the value of $\mathbf{y}$?
chem
3.57
27
Numeric
Reaction of $\mathbf{x} \mathrm{g}$ of $\mathrm{Sn}$ with $\mathrm{HCl}$ quantitatively produced a salt. Entire amount of the salt reacted with $\mathbf{y} g$ of nitrobenzene in the presence of required amount of $\mathrm{HCl}$ to produce $1.29 \mathrm{~g}$ of an organic salt (quantitatively). (Use Molar masses (in $\mathrm{g} \mathrm{mol}^{-1}$ ) of $\mathrm{H}, \mathrm{C}, \mathrm{N}, \mathrm{O}, \mathrm{Cl}$ and Sn as 1, 12, 14, 16, 35 and 119 , respectively). What is the value of $\mathbf{x}$?
chem
1.23
28
Numeric
Reaction of $\mathbf{x} \mathrm{g}$ of $\mathrm{Sn}$ with $\mathrm{HCl}$ quantitatively produced a salt. Entire amount of the salt reacted with $\mathbf{y} g$ of nitrobenzene in the presence of required amount of $\mathrm{HCl}$ to produce $1.29 \mathrm{~g}$ of an organic salt (quantitatively). (Use Molar masses (in $\mathrm{g} \mathrm{mol}^{-1}$ ) of $\mathrm{H}, \mathrm{C}, \mathrm{N}, \mathrm{O}, \mathrm{Cl}$ and Sn as 1, 12, 14, 16, 35 and 119 , respectively). What is the value of $\mathbf{y}$?
chem
1.87
29
Numeric
A sample $(5.6 \mathrm{~g})$ containing iron is completely dissolved in cold dilute $\mathrm{HCl}$ to prepare a $250 \mathrm{~mL}$ of solution. Titration of $25.0 \mathrm{~mL}$ of this solution requires $12.5 \mathrm{~mL}$ of $0.03 \mathrm{M} \mathrm{KMnO}_4$ solution to reach the end point. Number of moles of $\mathrm{Fe}^{2+}$ present in $250 \mathrm{~mL}$ solution is $\mathbf{x} \times 10^{-2}$ (consider complete dissolution of $\mathrm{FeCl}_2$ ). The amount of iron present in the sample is $\mathbf{y} \%$ by weight. (Assume: $\mathrm{KMnO}_4$ reacts only with $\mathrm{Fe}^{2+}$ in the solution Use: Molar mass of iron as $56 \mathrm{~g} \mathrm{~mol}^{-1}$ ). What is the value of $\mathbf{x}$?
chem
18.75
30
Numeric
A sample $(5.6 \mathrm{~g})$ containing iron is completely dissolved in cold dilute $\mathrm{HCl}$ to prepare a $250 \mathrm{~mL}$ of solution. Titration of $25.0 \mathrm{~mL}$ of this solution requires $12.5 \mathrm{~mL}$ of $0.03 \mathrm{M} \mathrm{KMnO}_4$ solution to reach the end point. Number of moles of $\mathrm{Fe}^{2+}$ present in $250 \mathrm{~mL}$ solution is $\mathbf{x} \times 10^{-2}$ (consider complete dissolution of $\mathrm{FeCl}_2$ ). The amount of iron present in the sample is $\mathbf{y} \%$ by weight. (Assume: $\mathrm{KMnO}_4$ reacts only with $\mathrm{Fe}^{2+}$ in the solution Use: Molar mass of iron as $56 \mathrm{~g} \mathrm{~mol}^{-1}$ ). What is the value of $\mathbf{y}$?
chem
A
31
MCQ
Correct match of the $\mathbf{C}-\mathbf{H}$ bonds (shown in bold) in Column $\mathbf{J}$ with their BDE in Column $\mathbf{K}$ is \begin{center} \begin{tabular}{|c|c|} \hline $\begin{array}{l}\text { Column J } \\ \text { Molecule }\end{array}$ & $\begin{array}{c}\text { Column K } \\ \text { BDE }\left(\text { kcal mol }^{-1}\right)\end{array}$ \\ \hline (P) $\mathbf{H}-\mathrm{CH}\left(\mathrm{CH}_{3}\right)_{2}$ & (i) 132 \\ \hline (Q) $\mathbf{H}-\mathrm{CH}_{2} \mathrm{Ph}$ & (ii) 110 \\ \hline (R) $\mathbf{H}-\mathrm{CH}=\mathrm{CH}_{2}$ & (iii) 95 \\ \hline (S) $\mathrm{H}-\mathrm{C} \equiv \mathrm{CH}$ & (iv) 88 \\ \hline \end{tabular} \end{center} (A) $P$ - iii, $Q$ - iv, R - ii, S - i (B) $P-i, Q-i i, R-$ iii, S - iv (C) P - iii, Q - ii, R - i, S - iv (D) $P-i i, Q-i, R-i v, S-i i i$
chem
C
33
MCQ
The reaction of $\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]$ with freshly prepared $\mathrm{FeSO}_4$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$ with the $\mathrm{FeSO}_4$ solution in complete absence of air produces a white precipitate $\mathbf{X}$, which turns blue in air. Mixing the $\mathrm{FeSO}_4$ solution with $\mathrm{NaNO}_3$, followed by a slow addition of concentrated $\mathrm{H}_2 \mathrm{SO}_4$ through the side of the test tube produces a brown ring. Precipitate $\mathbf{X}$ is (A) $\mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}$ (B) $\mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ (C) $\mathrm{K}_{2} \mathrm{Fe}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ (D) $\mathrm{KFe}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$
chem
D
34
MCQ
The reaction of $\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]$ with freshly prepared $\mathrm{FeSO}_4$ solution produces a dark blue precipitate called Turnbull's blue. Reaction of $\mathrm{K}_4\left[\mathrm{Fe}(\mathrm{CN})_6\right]$ with the $\mathrm{FeSO}_4$ solution in complete absence of air produces a white precipitate $\mathbf{X}$, which turns blue in air. Mixing the $\mathrm{FeSO}_4$ solution with $\mathrm{NaNO}_3$, followed by a slow addition of concentrated $\mathrm{H}_2 \mathrm{SO}_4$ through the side of the test tube produces a brown ring. Among the following, the brown ring is due to the formation of (A) $\left[\mathrm{Fe}(\mathrm{NO})_{2}\left(\mathrm{SO}_{4}\right)_{2}\right]^{2-}$ (B) $\left[\mathrm{Fe}(\mathrm{NO})_{2}\left(\mathrm{H}_{2} \mathrm{O}\right)_{4}\right]^{3+}$ (C) $\left[\mathrm{Fe}(\mathrm{NO})_{4}\left(\mathrm{SO}_{4}\right)_{2}\right]$ (D) $\left[\mathrm{Fe}(\mathrm{NO})\left(\mathrm{H}_{2} \mathrm{O}\right)_{5}\right]^{2+}$
chem
30
36
Integer
Consider a helium (He) atom that absorbs a photon of wavelength $330 \mathrm{~nm}$. What is the change in the velocity (in $\mathrm{cm} \mathrm{s}^{-1}$ ) of He atom after the photon absorption? (Assume: Momentum is conserved when photon is absorbed. Use: Planck constant $=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$, Avogadro number $=6 \times 10^{23} \mathrm{~mol}^{-1}$, Molar mass of $\mathrm{He}=4 \mathrm{~g} \mathrm{~mol}^{-1}$ )
math
ABD
37
MCQ(multiple)
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 $S_{4}=\{(i, j, k, l): i, j, k \text { and } l \text { are distinct elements in }\{1,2, \ldots, 10\}\}$ If the total number of elements in the set $S_{r}$ is $n_{r}, r=1,2,3,4$, then which of the following statements is (are) TRUE ? (A) $n_{1}=1000$ (B) $n_{2}=44$ (C) $n_{3}=220$ (D) $\frac{n_{4}}{12}=420$
math
AB
38
MCQ(multiple)
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) $\frac{q+r}{p}<2 \frac{\sqrt{\sin Q \sin R}}{\sin P}$ (D) If $p<q$ and $p<r$, then $\cos Q>\frac{p}{r}$ and $\cos R>\frac{p}{q}$
math
ABC
39
MCQ(multiple)
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 $\left(0, \frac{\pi}{3}\right)$ (B) The equation $f(x)-3 \sin 3 x=-\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$ (C) $\lim _{x \rightarrow 0} \frac{x \int_{0}^{x} f(t) d t}{1-e^{x^{2}}}=-1$ (D) $\lim _{x \rightarrow 0} \frac{\sin x \int_{0}^{x} f(t) d t}{x^{2}}=-1$
math
AC
40
MCQ(multiple)
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 the set $S$ ? (A) $f(x)=\frac{x^{2}}{2} e^{-x}+\left(e-\frac{1}{2}\right) e^{-x}$ (B) $f(x)=-\frac{x^{2}}{2} e^{-x}+\left(e+\frac{1}{2}\right) e^{-x}$ (C) $f(x)=\frac{e^{x}}{2}\left(x-\frac{1}{2}\right)+\left(e-\frac{e^{2}}{4}\right) e^{-x}$ (D) $f(x)=\frac{e^{x}}{2}\left(\frac{1}{2}-x\right)+\left(e+\frac{e^{2}}{4}\right) e^{-x}$
math
ABD
42
MCQ(multiple)
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 (B) The triangle $Q P Q^{\prime}$ is a right-angled triangle (C) The distance between $P$ and $F$ is $5 \sqrt{2}$ (D) $F$ lies on the line joining $Q$ and $Q^{\prime}$
math
1.5
43
Numeric
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 $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^2=4-x$. What is the radius of the circle $C$?
math
2.00
44
Numeric
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 $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^2=4-x$. What is the value of $\alpha$?
math
57
45
Numeric
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 $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}, \prod_{i=1}^{\mathrm{n}} \mathrm{a}_i$ denotes the product of $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}$. Let $\mathrm{m}_i$ and $\mathrm{n}_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i, i=1,2$, in the interval $(0, \infty)$. What is the value of $2 m_{1}+3 n_{1}+m_{1} n_{1}$?
math
6
46
Numeric
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 $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}, \prod_{i=1}^{\mathrm{n}} \mathrm{a}_i$ denotes the product of $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}$. Let $\mathrm{m}_i$ and $\mathrm{n}_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i, i=1,2$, in the interval $(0, \infty)$. What is the value of $6 m_{2}+4 n_{2}+8 m_{2} n_{2}$?
math
2
47
Numeric
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 all $\mathrm{x} \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$ Define $\mathrm{S}_i=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(\mathrm{x}) \cdot \mathrm{g}_i(\mathrm{x}) \mathrm{dx}, i=1,2$. What is the value of $\frac{16 S_{1}}{\pi}$?
math
D
49
MCQ
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 this progression. For $n \geq 1$, let $C_n$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a_n$, and $D_n$ denote the circle with center $\left(S_{n-1}, S_{n-1}\right)$ and radius $a_n$. Consider $M$ with $r=\frac{1025}{513}$. Let $k$ be the number of all those circles $C_{n}$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then (A) $k+2 l=22$ (B) $2 k+l=26$ (C) $2 k+3 l=34$ (D) $3 k+2 l=40$
math
B
50
MCQ
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 this progression. For $n \geq 1$, let $C_n$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a_n$, and $D_n$ denote the circle with center $\left(S_{n-1}, S_{n-1}\right)$ and radius $a_n$. Consider $M$ with $r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}}$. The number of all those circles $D_{n}$ that are inside $M$ is (A) 198 (B) 199 (C) 200 (D) 201
math
C
51
MCQ
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 0, \\ & \psi_2(\mathrm{x})=\mathrm{x}^2-2 \mathrm{x}-2 \mathrm{e}^{-\mathrm{x}}+2, \mathrm{x} \geq 0, \\ & f(\mathrm{x})=\int_{-\mathrm{x}}^{\mathrm{x}}\left(|\mathrm{t}|-\mathrm{t}^2\right) \mathrm{e}^{-\mathrm{t}^2} \mathrm{dt}, \mathrm{x}>0 \end{aligned} and $g(x)=\int_0^{x^2} \sqrt{t} e^{-t} d t, x>0$. Which of the following statements is TRUE? (A) $f(\sqrt{\ln 3})+g(\sqrt{\ln 3})=\frac{1}{3}$ (B) For every $x>1$, there exists an $\alpha \in(1, x)$ such that $\psi_{1}(x)=1+\alpha x$ (C) For every $x>0$, there exists a $\beta \in(0, x)$ such that $\psi_{2}(x)=2 x\left(\psi_{1}(\beta)-1\right)$ (D) $f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$
math
2.35
1
Numeric
Considering only the principal values of the inverse trigonometric functions, what is the value of $\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^{2}}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^{2}}+\tan ^{-1} \frac{\sqrt{2}}{\pi}$?
math
0.5
2
Numeric
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 $f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)}$ Then what is the value of $\lim _{x \rightarrow \alpha^{+}} f(g(x))$?
math
0.8
3
Numeric
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 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then what the probability that the person has at most one symptom?
math
0.5
4
Numeric
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
math
4
5
Numeric
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 $\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)$?
math
18900
6
Numeric
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 rectangle with length $l_{i}$, width $w_{i}$ and area $A_{i}$. If $A_{51}-A_{50}=1000$, then what is the value of $A_{100}-A_{90}$?
math
569
7
Numeric
What is the number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$?
math
0.83
8
Numeric
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 $r$?
math
CD
9
MCQ(multiple)
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 $a$ satisfies the above equation (C) An irrational number $a$ satisfies the above equation (D) More than one $a$ satisfy the above equation
math
BC
10
MCQ(multiple)
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) $T_{30}=3454$ (D) $\sum_{k=1}^{30} T_{k}=35610$
math
ABD
11
MCQ(multiple)
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) $\frac{x-1}{0}=\frac{y-1}{0}=\frac{z-1}{5}$ (B) $\frac{x-6}{-5}=\frac{y}{2}=\frac{z}{3}$ (C) $\frac{x}{-2}=\frac{y-4}{5}=\frac{z}{4}$ (D) $\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}$
math
ABC
12
MCQ(multiple)
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 $10 \hat{i}+15 \hat{j}+20 \hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE ? (A) $3(\alpha+\beta)=-101$ (B) $3(\beta+\gamma)=-71$ (C) $3(\gamma+\alpha)=-86$ (D) $3(\alpha+\beta+\gamma)=-121$
math
AC
14
MCQ(multiple)
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 $g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$ $f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$ Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE? (A) $p\left(\frac{3+\sqrt{2}}{4}\right)<0$ (B) $p\left(\frac{1+3 \sqrt{2}}{4}\right)>0$ (C) $p\left(\frac{5 \sqrt{2}-1}{4}\right)>0$ (D) $p\left(\frac{5-\sqrt{2}}{4}\right)<0$
math
B
15
MCQ
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) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$ (IV) $\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$ List-II (P) has two elements (Q) has three elements (R) has four elements (S) has five elements (T) has six elements The correct option is: (A) (I) $\rightarrow$ (P); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (S) (B) (I) $\rightarrow$ (P); (II) $\rightarrow$ (P); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (R) (C) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (P); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (S) (D) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (R)
math
A
16
MCQ
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 and $P_{2}$ scores 0 point. If $x=y$, then each player scores 2 points. If $x<y$, then $P_{1}$ scores 0 point and $P_{2}$ scores 5 points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$, respectively, after playing the $i^{\text {th }}$ round. List-I (I) Probability of $\left(X_2 \geq Y_2\right)$ is (II) Probability of $\left(X_2>Y_2\right)$ is (III) Probability of $\left(X_3=Y_3\right)$ is (IV) Probability of $\left(X_3>Y_3\right)$ is List-II (P) $\frac{3}{8}$ (Q) $\frac{11}{16}$ (R) $\frac{5}{16}$ (S) $\frac{355}{864}$ (T) $\frac{77}{432}$ The correct option is: (A) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (S) (B) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (T) (C) (I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (S) (D) (I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (T)
math
B
17
MCQ
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 $\frac{q}{r}=10$, then the system of linear equations has (II) If $\frac{p}{r} \neq 100$, then the system of linear equations has (III) If $\frac{p}{q} \neq 10$, then the system of linear equations has (IV) If $\frac{p}{q}=10$, then the system of linear equations has List-II (P) $x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution (Q) $x=\frac{10}{9}, y=-\frac{1}{9}, z=0$ as a solution (III) If $\frac{p}{q} \neq 10$, then the system of linear (R) infinitely many solutions (IV) If $\frac{p}{q}=10$, then the system of linear (S) no solution (T) at least one solution The correct option is: (A) (I) $\rightarrow$ (T); (II) $\rightarrow$ (R); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (T) (B) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (R) (C) (I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (R) (D) (I) $\rightarrow$ (T); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (T)
math
C
18
MCQ
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 $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis. List-I (I) If $\phi=\frac{\pi}{4}$, then the area of the triangle $F G H$ is (II) If $\phi=\frac{\pi}{3}$, then the area of the triangle $F G H$ is (III) If $\phi=\frac{\pi}{6}$, then the area of the triangle $F G H$ is (IV) If $\phi=\frac{\pi}{12}$, then the area of the triangle $F G H$ is List-II (P) $\frac{(\sqrt{3}-1)^4}{8}$ (Q) 1 (R) $\frac{3}{4}$ (S) $\frac{1}{2 \sqrt{3}}$ (T) $\frac{3 \sqrt{3}}{2}$ The correct option is: (A) (I) $\rightarrow$ (R); (II) $\rightarrow$ (S); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (P) (B) $\quad$ (I) $\rightarrow$ (R); (II) $\rightarrow$ (T); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (P) (C) $\quad$ (I) $\rightarrow$ (Q); (II) $\rightarrow$ (T); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (P) (D) $\quad$ (I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (P)
phy
2.3
19
Numeric
Two spherical stars $A$ and $B$ have densities $\rho_{A}$ and $\rho_{B}$, respectively. $A$ and $B$ have the same radius, and their masses $M_{A}$ and $M_{B}$ are related by $M_{B}=2 M_{A}$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_{A}$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_{A}$. If $v_{A}$ and $v_{B}$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$. What is the value of $n$?
phy
2.32
20
Numeric
The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${ }_{7}{ }_{7} \mathrm{~N}+$ ${ }_{2}^{4} \mathrm{He} \rightarrow{ }_{1}^{1} \mathrm{H}+{ }_{8}^{19} \mathrm{O}$ in a laboratory frame is $n$ (in $M e V$ ). Assume that ${ }_{7}^{16} \mathrm{~N}$ is at rest in the laboratory frame. The masses of ${ }_{7}^{16} \mathrm{~N},{ }_{2}^{4} \mathrm{He},{ }_{1}^{1} \mathrm{H}$ and ${ }_{8}^{19} \mathrm{O}$ can be taken to be $16.006 u, 4.003 u, 1.008 u$ and 19.003 $u$, respectively, where $1 u=930 \mathrm{MeV}^{-2}$. What is the value of $n$?
phy
0.52
23
Numeric
At time $t=0$, a disk of radius $1 \mathrm{~m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha=\frac{2}{3} \mathrm{rads}^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi} s$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $m$ ) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. What is the value of $x$? $\left[\right.$ Take $\left.g=10 m s^{-2}.\right]$
phy
4
25
Numeric
Consider an LC circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C=10^{-3} \mathrm{~F}$, kept on a plane. The area of the circuit is $1 \mathrm{~m}^{2}$. It is placed in a constant magnetic field of strength $B_{0}$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_{0}+\beta t$ with $\beta=0.04 \mathrm{Ts}^{-1}$. What is the maximum magnitude of the current in the circuit in $m A$?
phy
0.95
26
Numeric
A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$. When the acceleration due to gravity is $g$, the range of the projectile is $d$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{0.81}$, then the new range is $d^{\prime}=n d$. What is the value of $n$?
phy
ABD
32
MCQ(multiple)
The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_{b}^{p}$ and the binding energy of a neutron be $E_{b}^{n}$ in the nucleus. Which of the following statement(s) is(are) correct? (A) $E_{b}^{p}-E_{b}^{n}$ is proportional to $Z(Z-1)$ where $Z$ is the atomic number of the nucleus. (B) $E_{b}^{p}-E_{b}^{n}$ is proportional to $A^{-\frac{1}{3}}$ where $A$ is the mass number of the nucleus. (C) $E_{b}^{p}-E_{b}^{n}$ is positive. (D) $E_{b}^{p}$ increases if the nucleus undergoes a beta decay emitting a positron.
phy
C
35
MCQ
List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process. List-I (I) $10^{-3} \mathrm{~kg}$ of water at $100^{\circ} \mathrm{C}$ is converted to steam at the same temperature, at a pressure of $10^5 \mathrm{~Pa}$. The volume of the system changes from $10^{-6} \mathrm{~m}^3$ to $10^{-3} \mathrm{~m}^3$ in the process. Latent heat of water $=2250 \mathrm{~kJ} / \mathrm{kg}$. (II) 0.2 moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{~K}$ undergoes an isobaric expansion to volume $3 \mathrm{~V}$. Assume $R=8.0 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. (III) One mole of a monatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3} m^3$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$. (IV) Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{~kJ}$ of heat and undergoes isobaric expansion. List-II (P) $2 \mathrm{~kJ}$ (Q) $7 \mathrm{~kJ}$ (R) $4 \mathrm{~kJ}$ (S) $5 \mathrm{~kJ}$ (T) $3 \mathrm{~kJ}$ Which one of the following options is correct?\ (A) $\mathrm{I} \rightarrow \mathrm{T}$, II $\rightarrow$ R, III $\rightarrow \mathrm{S}$, IV $\rightarrow$ Q, (B) I $\rightarrow \mathrm{S}$, II $\rightarrow$ P, III $\rightarrow \mathrm{T}$, IV $\rightarrow$ P, (C) I $\rightarrow$ P, II $\rightarrow$ R, III $\rightarrow$ T, IV $\rightarrow$ Q. (D) I $\rightarrow$ Q, II $\rightarrow$ R, III $\rightarrow \mathrm{S}$, IV $\rightarrow \mathrm{T}$,