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math | JEE Adv 2020 Paper 2 | 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?
(... |
math | JEE Adv 2020 Paper 2 | AD | 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... |
math | JEE Adv 2020 Paper 2 | 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... |
math | JEE Adv 2020 Paper 2 | 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) $\... |
math | JEE Adv 2020 Paper 2 | 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 $\overrightarro... |
math | JEE Adv 2020 Paper 2 | 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 \\
... |
math | JEE Adv 2020 Paper 2 | 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 | JEE Adv 2020 Paper 2 | 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 | JEE Adv 2020 Paper 2 | 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 p... |
math | JEE Adv 2020 Paper 2 | 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 | JEE Adv 2020 Paper 2 | 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 valu... |
math | JEE Adv 2020 Paper 2 | 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<\... |
phy | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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 s... |
phy | JEE Adv 2021 Paper 1 | 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 s... |
phy | JEE Adv 2021 Paper 1 | 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 a... |
phy | JEE Adv 2021 Paper 1 | AD | 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 $... |
phy | JEE Adv 2021 Paper 1 | 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 sulfu... |
chem | JEE Adv 2021 Paper 1 | 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... |
chem | JEE Adv 2021 Paper 1 | 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 t... |
chem | JEE Adv 2021 Paper 1 | 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 t... |
chem | JEE Adv 2021 Paper 1 | 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). ... |
chem | JEE Adv 2021 Paper 1 | 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 o... |
chem | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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}+... |
math | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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 repl... |
math | JEE Adv 2021 Paper 1 | 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 ... |
math | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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 \... |
math | JEE Adv 2021 Paper 1 | 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 \... |
math | JEE Adv 2021 Paper 1 | 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 $\mat... |
math | JEE Adv 2021 Paper 1 | 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 $\mat... |
math | JEE Adv 2021 Paper 1 | 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 ... |
math | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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... |
math | JEE Adv 2021 Paper 1 | 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... |
math | JEE Adv 2021 Paper 1 | 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 foll... |
math | JEE Adv 2021 Paper 1 | 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+... |
math | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 1 | 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 | JEE Adv 2021 Paper 2 | AD | 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... |
phy | JEE Adv 2021 Paper 2 | 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 ... |
phy | JEE Adv 2021 Paper 2 | 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 $P$ a... |
phy | JEE Adv 2021 Paper 2 | 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 ... |
phy | JEE Adv 2021 Paper 2 | 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 ... |
phy | JEE Adv 2021 Paper 2 | 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 circu... |
phy | JEE Adv 2021 Paper 2 | 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 circu... |
chem | JEE Adv 2021 Paper 2 | 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 reactio... |
chem | JEE Adv 2021 Paper 2 | 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... |
chem | JEE Adv 2021 Paper 2 | 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}(\ma... |
chem | JEE Adv 2021 Paper 2 | 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{P... |
chem | JEE Adv 2021 Paper 2 | 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}... |
chem | JEE Adv 2021 Paper 2 | 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}... |
chem | JEE Adv 2021 Paper 2 | 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 $\m... |
chem | JEE Adv 2021 Paper 2 | 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 $\m... |
chem | JEE Adv 2021 Paper 2 | 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... |
chem | JEE Adv 2021 Paper 2 | 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... |
chem | JEE Adv 2021 Paper 2 | 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 { kc... |
chem | JEE Adv 2021 Paper 2 | 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 whit... |
chem | JEE Adv 2021 Paper 2 | 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 whit... |
chem | JEE Adv 2021 Paper 2 | 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 $=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$... |
math | JEE Adv 2021 Paper 2 | 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 disti... |
math | JEE Adv 2021 Paper 2 | 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+... |
math | JEE Adv 2021 Paper 2 | 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}\... |
math | JEE Adv 2021 Paper 2 | 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... |
math | JEE Adv 2021 Paper 2 | 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
(... |
math | JEE Adv 2021 Paper 2 | 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 $(\al... |
math | JEE Adv 2021 Paper 2 | 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 $(\al... |
math | JEE Adv 2021 Paper 2 | 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}_{... |
math | JEE Adv 2021 Paper 2 | 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}_{... |
math | JEE Adv 2021 Paper 2 | 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 ... |
math | JEE Adv 2021 Paper 2 | 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 ... |
math | JEE Adv 2021 Paper 2 | 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 ... |
math | JEE Adv 2021 Paper 2 | 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 ... |
math | JEE Adv 2022 Paper 1 | 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 | JEE Adv 2022 Paper 1 | 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... |
math | JEE Adv 2022 Paper 1 | 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 ... |
math | JEE Adv 2022 Paper 1 | 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 | JEE Adv 2022 Paper 1 | 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 | JEE Adv 2022 Paper 1 | 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 rec... |
math | JEE Adv 2022 Paper 1 | 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 | JEE Adv 2022 Paper 1 | 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 | JEE Adv 2022 Paper 1 | 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 $a$ satisfie... |
math | JEE Adv 2022 Paper 1 | 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) $T_{30}=3454$
... |
math | JEE Adv 2022 Paper 1 | 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}... |
math | JEE Adv 2022 Paper 1 | 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 \ha... |
math | JEE Adv 2022 Paper 1 | 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 &... |
math | JEE Adv 2022 Paper 1 | 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\}... |
math | JEE Adv 2022 Paper 1 | 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 an... |
math | JEE Adv 2022 Paper 1 | 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... |
math | JEE Adv 2022 Paper 1 | 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 $E$ inters... |
phy | JEE Adv 2022 Paper 1 | 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 retain... |
phy | JEE Adv 2022 Paper 1 | 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. Th... |
phy | JEE Adv 2022 Paper 1 | 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 de... |
phy | JEE Adv 2022 Paper 1 | 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 strengt... |
phy | JEE Adv 2022 Paper 1 | 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}=\... |
phy | JEE Adv 2022 Paper 1 | 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) ... |
phy | JEE Adv 2022 Paper 1 | 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 s... |
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Chem Questions Involving Calculation
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