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chem | JEE Adv 2022 Paper 1 | 90.39 | 37 | Numeric | 2 mol of $\mathrm{Hg}(g)$ is combusted in a fixed volume bomb calorimeter with excess of $\mathrm{O}_{2}$ at $298 \mathrm{~K}$ and 1 atm into $\mathrm{HgO}(s)$. During the reaction, temperature increases from $298.0 \mathrm{~K}$ to $312.8 \mathrm{~K}$. If heat capacity of the bomb calorimeter and enthalpy of formation of $\mathrm{Hg}(g)$ are $20.00 \mathrm{~kJ} \mathrm{~K}^{-1}$ and $61.32 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ at $298 \mathrm{~K}$, respectively, the calculated standard molar enthalpy of formation of $\mathrm{HgO}(s)$ at 298 $\mathrm{K}$ is $\mathrm{X} \mathrm{kJ} \mathrm{mol} \mathrm{m}^{-1}$. What is the value of $|\mathrm{X}|$?
[Given: Gas constant $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ] |
chem | JEE Adv 2022 Paper 1 | 0.77 | 38 | Numeric | What is the reduction potential $\left(E^{0}\right.$, in $\left.\mathrm{V}\right)$ of $\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})$?
[Given: $\left.E_{\left(\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{MnO}_{2}(\mathrm{~s})\right)}^{0}=1.68 \mathrm{~V} ; E_{\left(\mathrm{MnO}_{2}(\mathrm{~s}) / \mathrm{Mn}^{2+}(\mathrm{aq})\right)}^{0}=1.21 \mathrm{~V} ; E_{\left(\mathrm{Mn}^{2+}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})\right)}^{0}=-1.03 \mathrm{~V}\right]$ |
chem | JEE Adv 2022 Paper 1 | 10.02 | 39 | Numeric | A solution is prepared by mixing $0.01 \mathrm{~mol}$ each of $\mathrm{H}_{2} \mathrm{CO}_{3}, \mathrm{NaHCO}_{3}, \mathrm{Na}_{2} \mathrm{CO}_{3}$, and $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. What is the $p \mathrm{H}$ of the resulting solution?
[Given: $p \mathrm{~K}_{\mathrm{a} 1}$ and $p \mathrm{~K}_{\mathrm{a} 2}$ of $\mathrm{H}_{2} \mathrm{CO}_{3}$ are 6.37 and 10.32, respectively; $\log 2=0.30$ ] |
chem | JEE Adv 2022 Paper 1 | 0.32 | 40 | Numeric | The treatment of an aqueous solution of $3.74 \mathrm{~g}$ of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$ with excess KI results in a brown solution along with the formation of a precipitate. Passing $\mathrm{H}_{2} \mathrm{~S}$ through this brown solution gives another precipitate $\mathbf{X}$. What is the amount of $\mathbf{X}$ (in $\mathrm{g}$ )?
[Given: Atomic mass of $\mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~K}=39, \mathrm{Cu}=63, \mathrm{I}=127$ ] |
chem | JEE Adv 2022 Paper 1 | 2.38 | 41 | Numeric | Dissolving $1.24 \mathrm{~g}$ of white phosphorous in boiling NaOH solution in an inert atmosphere gives a gas $\mathbf{Q}$. What is the amount of $\mathrm{CuSO}_{4}$ (in g) required to completely consume the gas $\mathbf{Q}$?
[Given: Atomic mass of $\mathrm{H}=1, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{P}=31, \mathrm{~S}=32, \mathrm{Cu}=63$ ] |
chem | JEE Adv 2022 Paper 1 | AD | 45 | MCQ(multiple) | For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap of two $2 p_{z}$ orbitals is(are)
(A) $\sigma$ orbital has a total of two nodal planes.
(B) $\sigma^{*}$ orbital has one node in the $x z$-plane containing the molecular axis.
(C) $\pi$ orbital has one node in the plane which is perpendicular to the molecular axis and goes through the center of the molecule.
(D) $\pi^{*}$ orbital has one node in the $x y$-plane containing the molecular axis. |
chem | JEE Adv 2022 Paper 1 | AD | 46 | MCQ(multiple) | The correct option(s) related to adsorption processes is(are)
(A) Chemisorption results in a unimolecular layer.
(B) The enthalpy change during physisorption is in the range of 100 to $140 \mathrm{~kJ} \mathrm{~mol}^{-1}$.
(C) Chemisorption is an endothermic process.
(D) Lowering the temperature favors physisorption processes. |
chem | JEE Adv 2022 Paper 1 | BCD | 47 | MCQ(multiple) | The electrochemical extraction of aluminum from bauxite ore involves
(A) the reaction of $\mathrm{Al}_{2} \mathrm{O}_{3}$ with coke (C) at a temperature $>2500^{\circ} \mathrm{C}$.
(B) the neutralization of aluminate solution by passing $\mathrm{CO}_{2}$ gas to precipitate hydrated alumina $\left(\mathrm{Al}_{2} \mathrm{O}_{3} \cdot 3 \mathrm{H}_{2} \mathrm{O}\right)$.
(C) the dissolution of $\mathrm{Al}_{2} \mathrm{O}_{3}$ in hot aqueous $\mathrm{NaOH}$.
(D) the electrolysis of $\mathrm{Al}_{2} \mathrm{O}_{3}$ mixed with $\mathrm{Na}_{3} \mathrm{AlF}_{6}$ to give $\mathrm{Al}$ and $\mathrm{CO}_{2}$. |
chem | JEE Adv 2022 Paper 1 | AD | 48 | MCQ(multiple) | The treatment of galena with $\mathrm{HNO}_{3}$ produces a gas that is
(A) paramagnetic
(B) bent in geometry
(C) an acidic oxide
(D) colorless |
chem | JEE Adv 2022 Paper 1 | D | 52 | MCQ | LIST-I contains compounds and LIST-II contains reactions
LIST-I
(I) $\mathrm{H}_2 \mathrm{O}_2$
(II) $\mathrm{Mg}(\mathrm{OH})_2$
(III) $\mathrm{BaCl}_2$
(IV) $\mathrm{CaCO}_3$
LIST-II
(P) $\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$
(Q) $\mathrm{BaO}_{2}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow$
(R) $\mathrm{Ca}(\mathrm{OH})_{2}+\mathrm{MgCl}_{2} \rightarrow$
(S) $\mathrm{BaO}_{2}+\mathrm{HCl} \rightarrow$
(T) $\mathrm{Ca}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$
Match each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option\
(A) I $\rightarrow$ Q; II $\rightarrow$ P; III $\rightarrow \mathrm{S}$; IV $\rightarrow$ R
(B) I $\rightarrow$ T; II $\rightarrow$ P; III $\rightarrow$ Q; IV $\rightarrow \mathrm{R}$
(C) I $\rightarrow$ T; II $\rightarrow$ R; III $\rightarrow$ Q; IV $\rightarrow$ P
(D) I $\rightarrow$ Q; II $\rightarrow$ R; III $\rightarrow \mathrm{S}$; IV $\rightarrow \mathrm{P}$ |
chem | JEE Adv 2022 Paper 1 | A | 53 | MCQ | LIST-I contains metal species and LIST-II contains their properties.
LIST-I
(I) $\left[\mathrm{Cr}(\mathrm{CN})_{6}\right]^{4-}$
(II) $\left[\mathrm{RuCl}_{6}\right]^{2-}$
(III) $\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$
(IV) $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$
LIST-II
(P) $t_{2 \mathrm{~g}}$ orbitals contain 4 electrons
(Q) $\mu$ (spin-only $=4.9 \mathrm{BM}$
(R) low spin complex ion
(S) metal ion in $4+$ oxidation state
(T) $d^{4}$ species
[Given: Atomic number of $\mathrm{Cr}=24, \mathrm{Ru}=44, \mathrm{Fe}=26$ ]
Match each metal species in LIST-I with their properties in LIST-II, and choose the correct option
(A) I $\rightarrow$ R, T; II $\rightarrow$ P, S; III $\rightarrow$ Q, T; IV $\rightarrow$ P, Q
(B) I $\rightarrow$ R, S; II $\rightarrow$ P, T; III $\rightarrow$ P, Q; IV $\rightarrow$ Q, T
(C) I $\rightarrow \mathrm{P}, \mathrm{R} ; \mathrm{II} \rightarrow \mathrm{R}, \mathrm{S}$; III $\rightarrow \mathrm{R}, \mathrm{T}$; IV $\rightarrow \mathrm{P}, \mathrm{T}$
(D) I $\rightarrow$ Q, T; II $\rightarrow \mathrm{S}, \mathrm{T}$; III $\rightarrow \mathrm{P}, \mathrm{T}$; IV $\rightarrow \mathrm{Q}, \mathrm{R}$ |
chem | JEE Adv 2022 Paper 1 | D | 54 | MCQ | Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option.
LIST-I
(I) Aniline
(II) $o$-Cresol
(III) Cysteine
(IV) Caprolactam
LIST-II
(P) Sodium fusion extract of the compound on boiling with $\mathrm{FeSO}_{4}$, followed by acidification with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$, gives Prussian blue color.
(Q) Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
(R) Addition of the compound to a saturated solution of $\mathrm{NaHCO}_{3}$ results in effervescence.
(S) The compound reacts with bromine water to give a white precipitate.
(T) Treating the compound with neutral $\mathrm{FeCl}_{3}$ solution produces violet color.
(A) $\mathrm{I} \rightarrow \mathrm{P}, \mathrm{Q}$; II $\rightarrow \mathrm{S}$; III $\rightarrow \mathrm{Q}$, R; IV $\rightarrow \mathrm{P}$
(B) $\mathrm{I} \rightarrow \mathrm{P}$; II $\rightarrow \mathrm{R}, \mathrm{S}$; III $\rightarrow \mathrm{R}$; IV $\rightarrow \mathrm{Q}, \mathrm{S}$
(C) $\mathrm{I} \rightarrow \mathrm{Q}, \mathrm{S}$; II $\rightarrow \mathrm{P}, \mathrm{T}$; III $\rightarrow \mathrm{P}$; IV $\rightarrow \mathrm{S}$
(D) $\mathrm{I} \rightarrow \mathrm{P}, \mathrm{S}$; II $\rightarrow \mathrm{T}$; III $\rightarrow \mathrm{Q}, \mathrm{R}$; IV $\rightarrow \mathrm{P}$ |
math | JEE Adv 2022 Paper 2 | 1 | 1 | Integer | Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$. If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then what is the greatest integer less than or equal to
\[
\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^{2}
\]? |
math | JEE Adv 2022 Paper 2 | 8 | 2 | Integer | If $y(x)$ is the solution of the differential equation
\[
x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x>0, \quad y(1)=2,
\]
and the slope of the curve $y=y(x)$ is never zero, then what is the value of $10 y(\sqrt{2})$? |
math | JEE Adv 2022 Paper 2 | 5 | 3 | Numeric | What is the greatest integer less than or equal to
\[
\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x
\]? |
math | JEE Adv 2022 Paper 2 | 1 | 4 | Integer | What is the product of all positive real values of $x$ satisfying the equation
\[
x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}
\]? |
math | JEE Adv 2022 Paper 2 | 5 | 5 | Integer | If
\[
\beta=\lim _{x \rightarrow 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},
\]
then what is the value of $6 \beta$? |
math | JEE Adv 2022 Paper 2 | 3 | 6 | Integer | Let $\beta$ be a real number. Consider the matrix
\[
A=\left(\begin{array}{ccc}
\beta & 0 & 1 \\
2 & 1 & -2 \\
3 & 1 & -2
\end{array}\right)
\]
If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then what is the value of $9 \beta$? |
math | JEE Adv 2022 Paper 2 | 7 | 7 | Integer | Consider the hyperbola
\[
\frac{x^{2}}{100}-\frac{y^{2}}{64}=1
\]
with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1} P$ at $P_{1}$. Let $\delta$ be the distance of $P$ from the straight line $S P_{1}$, and $\beta=S_{1} P$. Then what is the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$? |
math | JEE Adv 2022 Paper 2 | 6 | 8 | Integer | Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by
\[
f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases}
\]
If $\alpha$ is the area of the region
\[
\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:|x| \leq \frac{3}{4}, 0 \leq y \leq \min \{f(x), g(x)\}\right\}
\]
then what is the value of $9 \alpha$? |
math | JEE Adv 2022 Paper 2 | AB | 9 | MCQ(multiple) | Let $P Q R S$ be a quadrilateral in a plane, where $Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$. If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
(A) $(0, \sqrt{2})$
(B) $(1,2)$
(C) $(\sqrt{2}, 3)$
(D) $(2 \sqrt{2}, 3 \sqrt{2})$ |
math | JEE Adv 2022 Paper 2 | ABC | 10 | MCQ(multiple) | Let
\[
\alpha=\sum_{k=1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)
\]
Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by
\[
g(x)=2^{\alpha x}+2^{\alpha(1-x)}
\]
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
(B) The maximum value of $g(x)$ is $1+2^{\frac{1}{3}}$
(C) The function $g(x)$ attains its maximum at more than one point
(D) The function $g(x)$ attains its minimum at more than one point |
math | JEE Adv 2022 Paper 2 | A | 11 | MCQ(multiple) | Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
\[
(\bar{z})^{2}+\frac{1}{z^{2}}
\]
are integers, then which of the following is/are possible value(s) of $|z|$ ?
(A) $\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$
(B) $\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$
(C) $\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$
(D) $\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$ |
math | JEE Adv 2022 Paper 2 | CD | 12 | MCQ(multiple) | Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?
(A) If $n=4$, then $(\sqrt{2}-1) r<R$
(B) If $n=5$, then $r<R$
(C) If $n=8$, then $(\sqrt{2}-1) r<R$
(D) If $n=12$, then $\sqrt{2}(\sqrt{3}+1) r>R$ |
math | JEE Adv 2022 Paper 2 | BCD | 13 | MCQ(multiple) | Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let
\[
\vec{a}=3 \hat{i}+\hat{j}-\hat{k},
\]
\[\vec{b}=\hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R},
\]
\[\vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb{R}
\]
be three vectors such that $b_{2} b_{3}>0, \vec{a} \cdot \vec{b}=0$ and
\[
\left(\begin{array}{ccc}
0 & -c_{3} & c_{2} \\
c_{3} & 0 & -c_{1} \\
-c_{2} & c_{1} & 0
\end{array}\right)\left(\begin{array}{l}
1 \\
b_{2} \\
b_{3}
\end{array}\right)=\left(\begin{array}{r}
3-c_{1} \\
1-c_{2} \\
-1-c_{3}
\end{array}\right) .
\]
Then, which of the following is/are TRUE ?
(A) $\vec{a} \cdot \vec{c}=0$
(B) $\vec{b} \cdot \vec{c}=0$
(C) $|\vec{b}|>\sqrt{10}$
(D) $|\vec{c}| \leq \sqrt{11}$ |
math | JEE Adv 2022 Paper 2 | C | 14 | MCQ(multiple) | For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
\[
\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0
\]
Then, which of the following statements is/are TRUE?
(A) $y(x)$ is an increasing function
(B) $y(x)$ is a decreasing function
(C) There exists a real number $\beta$ such that the line $y=\beta$ intersects the curve $y=y(x)$ at infinitely many points
(D) $y(x)$ is a periodic function |
math | JEE Adv 2022 Paper 2 | A | 15 | MCQ | Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?
(A) 21816
(B) 85536
(C) 12096
(D) 156816 |
math | JEE Adv 2022 Paper 2 | A | 16 | MCQ | If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?
(A) $\left(\begin{array}{rr}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
(B) $\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)$
(C) $\left(\begin{array}{rr}3033 & 3032 \\ -3032 & -3031\end{array}\right)$
(D) $\left(\begin{array}{rr}3032 & 3031 \\ -3031 & -3030\end{array}\right)$ |
math | JEE Adv 2022 Paper 2 | C | 17 | MCQ | Suppose that
Box-I contains 8 red, 3 blue and 5 green balls,
Box-II contains 24 red, 9 blue and 15 green balls,
Box-III contains 1 blue, 12 green and 3 yellow balls,
Box-IV contains 10 green, 16 orange and 6 white balls.
A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to
(A) $\frac{15}{256}$
(B) $\frac{3}{16}$
(C) $\frac{5}{52}$
(D) $\frac{1}{8}$ |
math | JEE Adv 2022 Paper 2 | B | 18 | MCQ | For positive integer $n$, define
\[
f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}}
\]
Then, the value of $\lim _{n \rightarrow \infty} f(n)$ is equal to
(A) $3+\frac{4}{3} \log _{e} 7$
(B) $4-\frac{3}{4} \log _{e}\left(\frac{7}{3}\right)$
(C) $4-\frac{4}{3} \log _{e}\left(\frac{7}{3}\right)$
(D) $3+\frac{3}{4} \log _{e} 7$ |
phy | JEE Adv 2022 Paper 2 | 3 | 19 | Integer | A particle of mass $1 \mathrm{~kg}$ is subjected to a force which depends on the position as $\vec{F}=$ $-k(x \hat{i}+y \hat{j}) k g \mathrm{ks}^{-2}$ with $k=1 \mathrm{~kg} \mathrm{~s}^{-2}$. At time $t=0$, the particle's position $\vec{r}=$ $\left(\frac{1}{\sqrt{2}} \hat{i}+\sqrt{2} \hat{j}\right) m$ and its velocity $\vec{v}=\left(-\sqrt{2} \hat{i}+\sqrt{2} \hat{j}+\frac{2}{\pi} \hat{k}\right) m s^{-1}$. Let $v_{x}$ and $v_{y}$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z=0.5 \mathrm{~m}$, what is the value of $\left(x v_{y}-y v_{x}\right)$ in $m^{2} s^{-1}$? |
phy | JEE Adv 2022 Paper 2 | 2 | 20 | Integer | In a radioactive decay chain reaction, ${ }_{90}^{230} \mathrm{Th}$ nucleus decays into ${ }_{84}^{214} \mathrm{Po}$ nucleus. What is the ratio of the number of $\alpha$ to number of $\beta^{-}$particles emitted in this process? |
phy | JEE Adv 2022 Paper 2 | 4 | 22 | Integer | In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_{e}$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_{0}}$, where $\epsilon_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[e]^{\alpha}\left[m_{e}\right]^{\beta}[h]^{\gamma}[k]^{\delta}$. The value of $\alpha+\beta+\gamma+\delta$ is |
phy | JEE Adv 2022 Paper 2 | 6 | 25 | Integer | On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{~kg}$ is attached to a spring with natural length $l_{0}=0.1 \mathrm{~m}$. The spring constant is $k_{1}=0.009 \mathrm{Nm}^{-1}$ when the length of the spring $l>l_{0}$ and is $k_{2}=0.016 \mathrm{Nm}^{-1}$ when $l<l_{0}$. Initially the bob is released from $l=$ $0.15 \mathrm{~m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=(n \pi) s$, then what is the integer closest to $n$? |
phy | JEE Adv 2022 Paper 2 | CD | 29 | MCQ(multiple) | A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma=$ $\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a 1}$, the radius of the bubble is found to be $r_{1}$ and the temperature of the enclosed gas is $T_{1}$. When the atmospheric pressure is $P_{a 2}$, the radius of the bubble and the temperature of the enclosed gas are $r_{2}$ and $T_{2}$, respectively.
Which of the following statement(s) is(are) correct?
(A) If the surface of the bubble is a perfect heat insulator, then $\left(\frac{r_{1}}{r_{2}}\right)^{5}=\frac{P_{a 2}+\frac{2 S}{r_{2}}}{P_{a 1}+\frac{2 S}{r_{1}}}$.
(B) If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure.
(C) If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then $\left(\frac{r_{1}}{r_{2}}\right)^{3}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.
(D) If the surface of the bubble is a perfect heat insulator, then $\left(\frac{T_{2}}{T_{1}}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$. |
phy | JEE Adv 2022 Paper 2 | ACD | 30 | MCQ(multiple) | A disk of radius $\mathrm{R}$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
\[
V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right)
\]
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_{0}$ and $z_{0}>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \epsilon_{0}}{q \sigma}$. Which of the following statement(s) is(are) correct?
(A) For $\beta=\frac{1}{4}$ and $z_{0}=\frac{25}{7} R$, the particle reaches the origin.
(B) For $\beta=\frac{1}{4}$ and $z_{0}=\frac{3}{7} R$, the particle reaches the origin.
(C) For $\beta=\frac{1}{4}$ and $z_{0}=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_{0}$.
(D) For $\beta>1$ and $z_{0}>0$, the particle always reaches the origin. |
phy | JEE Adv 2022 Paper 2 | A | 34 | MCQ | When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0 \mathrm{~V}$. This potential drops to $0.6 \mathrm{~V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take $\frac{h c}{e}=1.24 \times$ $10^{-6} \mathrm{~J} \mathrm{mC}^{-1}$.]
(A) $1.72 \times 10^{-7} \mathrm{~m}, 1.20 \mathrm{eV}$
(B) $1.72 \times 10^{-7} \mathrm{~m}, 5.60 \mathrm{eV}$
(C) $3.78 \times 10^{-7} \mathrm{~m}, 5.60 \mathrm{eV}$
(D) $3.78 \times 10^{-7} \mathrm{~m}, 1.20 \mathrm{eV}$ |
phy | JEE Adv 2022 Paper 2 | C | 35 | MCQ | Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{~mm}$. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Measurement condition & Main scale reading & Circular scale reading \\
\hline
$\begin{array}{l}\text { Two arms of gauge touching } \\ \text { each other without wire }\end{array}$ & 0 division & 4 divisions \\
\hline
Attempt-1: With wire & 4 divisions & 20 divisions \\
\hline
Attempt-2: With wire & 4 divisions & 16 divisions \\
\hline
\end{tabular}
\end{center}
What are the diameter and cross-sectional area of the wire measured using the screw gauge?
(A) $2.22 \pm 0.02 \mathrm{~mm}, \pi(1.23 \pm 0.02) \mathrm{mm}^{2}$
(B) $2.22 \pm 0.01 \mathrm{~mm}, \pi(1.23 \pm 0.01) \mathrm{mm}^{2}$
(C) $2.14 \pm 0.02 \mathrm{~mm}, \pi(1.14 \pm 0.02) \mathrm{mm}^{2}$
(D) $2.14 \pm 0.01 \mathrm{~mm}, \pi(1.14 \pm 0.01) \mathrm{mm}^{2}$ |
chem | JEE Adv 2022 Paper 2 | 6 | 37 | Integer | Concentration of $\mathrm{H}_{2} \mathrm{SO}_{4}$ and $\mathrm{Na}_{2} \mathrm{SO}_{4}$ in a solution is $1 \mathrm{M}$ and $1.8 \times 10^{-2} \mathrm{M}$, respectively. Molar solubility of $\mathrm{PbSO}_{4}$ in the same solution is $\mathrm{X} \times 10^{-\mathrm{Y}} \mathrm{M}$ (expressed in scientific notation). What is the value of $\mathrm{Y}$?
[Given: Solubility product of $\mathrm{PbSO}_{4}\left(K_{s p}\right)=1.6 \times 10^{-8}$. For $\mathrm{H}_{2} \mathrm{SO}_{4}, K_{a l}$ is very large and $\left.K_{a 2}=1.2 \times 10^{-2}\right]$ |
chem | JEE Adv 2022 Paper 2 | 5 | 38 | Integer | An aqueous solution is prepared by dissolving $0.1 \mathrm{~mol}$ of an ionic salt in $1.8 \mathrm{~kg}$ of water at $35^{\circ} \mathrm{C}$. The salt remains $90 \%$ dissociated in the solution. The vapour pressure of the solution is $59.724 \mathrm{~mm}$ of $\mathrm{Hg}$. Vapor pressure of water at $35{ }^{\circ} \mathrm{C}$ is $60.000 \mathrm{~mm}$ of $\mathrm{Hg}$. What is the number of ions present per formula unit of the ionic salt? |
chem | JEE Adv 2022 Paper 2 | 4 | 40 | Integer | The reaction of $\mathrm{Xe}$ and $\mathrm{O}_{2} \mathrm{~F}_{2}$ gives a Xe compound $\mathbf{P}$. What is the number of moles of $\mathrm{HF}$ produced by the complete hydrolysis of $1 \mathrm{~mol}$ of $\mathbf{P}$? |
chem | JEE Adv 2022 Paper 2 | 6 | 41 | Integer | Thermal decomposition of $\mathrm{AgNO}_{3}$ produces two paramagnetic gases. What is the total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons? |
chem | JEE Adv 2022 Paper 2 | BC | 45 | MCQ(multiple) | To check the principle of multiple proportions, a series of pure binary compounds $\left(\mathrm{P}_{\mathrm{m}} \mathrm{Q}_{\mathrm{n}}\right)$ were analyzed and their composition is tabulated below. The correct option(s) is(are)
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Compound & Weight \% of P & Weight \% of Q \\
\hline
$\mathbf{1}$ & 50 & 50 \\
\hline
$\mathbf{2}$ & 44.4 & 55.6 \\
\hline
$\mathbf{3}$ & 40 & 60 \\
\hline
\end{tabular}
\end{center}
(A) If empirical formula of compound 3 is $\mathrm{P}_{3} \mathrm{Q}_{4}$, then the empirical formula of compound 2 is $\mathrm{P}_{3} \mathrm{Q}_{5}$.
(B) If empirical formula of compound 3 is $\mathrm{P}_{3} \mathrm{Q}_{2}$ and atomic weight of element $\mathrm{P}$ is 20 , then the atomic weight of $\mathrm{Q}$ is 45 .
(C) If empirical formula of compound 2 is $\mathrm{PQ}$, then the empirical formula of the compound $\mathbf{1}$ is $\mathrm{P}_{5} \mathrm{Q}_{4}$.
(D) If atomic weight of $\mathrm{P}$ and $\mathrm{Q}$ are 70 and 35 , respectively, then the empirical formula of compound 1 is $\mathrm{P}_{2} \mathrm{Q}$. |
chem | JEE Adv 2022 Paper 2 | BCD | 46 | MCQ(multiple) | The correct option(s) about entropy (S) is(are)
$[\mathrm{R}=$ gas constant, $\mathrm{F}=$ Faraday constant, $\mathrm{T}=$ Temperature $]$
(A) For the reaction, $\mathrm{M}(s)+2 \mathrm{H}^{+}(a q) \rightarrow \mathrm{H}_{2}(g)+\mathrm{M}^{2+}(a q)$, if $\frac{d E_{c e l l}}{d T}=\frac{R}{F}$, then the entropy change of the reaction is $\mathrm{R}$ (assume that entropy and internal energy changes are temperature independent).
(B) The cell reaction, $\operatorname{Pt}(s)\left|\mathrm{H}_{2}(g, 1 \mathrm{bar})\right| \mathrm{H}^{+}(a q, 0.01 \mathrm{M}) \| \mathrm{H}^{+}(a q, 0.1 \mathrm{M})\left|\mathrm{H}_{2}(g, 1 \mathrm{bar})\right| \operatorname{Pt}(s)$, is an entropy driven process.
(C) For racemization of an optically active compound, $\Delta \mathrm{S}>0$.
(D) $\Delta \mathrm{S}>0$, for $\left[\mathrm{Ni}\left(\mathrm{H}_{2} \mathrm{O}_{6}\right]^{2+}+3\right.$ en $\rightarrow\left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+}+6 \mathrm{H}_{2} \mathrm{O}$ (where en $=$ ethylenediamine). |
chem | JEE Adv 2022 Paper 2 | ABC | 47 | MCQ(multiple) | The compound(s) which react(s) with $\mathrm{NH}_{3}$ to give boron nitride (BN) is(are)
(A) $\mathrm{B}$
(B) $\mathrm{B}_{2} \mathrm{H}_{6}$
(C) $\mathrm{B}_{2} \mathrm{O}_{3}$
(D) $\mathrm{HBF}_{4}$ |
chem | JEE Adv 2022 Paper 2 | ABC | 48 | MCQ(multiple) | The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 \mathrm{~K}$ is(are)
(A) Limestone is used to remove silicate impurity.
(B) Pig iron obtained from blast furnace contains about $4 \%$ carbon.
(C) Coke (C) converts $\mathrm{CO}_{2}$ to $\mathrm{CO}$.
(D) Exhaust gases consist of $\mathrm{NO}_{2}$ and $\mathrm{CO}$. |
chem | JEE Adv 2022 Paper 2 | BC | 50 | MCQ(multiple) | Among the following, the correct statement(s) about polymers is(are)
(A) The polymerization of chloroprene gives natural rubber.
(B) Teflon is prepared from tetrafluoroethene by heating it with persulphate catalyst at high pressures.
(C) PVC are thermoplastic polymers.
(D) Ethene at 350-570 K temperature and 1000-2000 atm pressure in the presence of a peroxide initiator yields high density polythene. |
chem | JEE Adv 2022 Paper 2 | B | 51 | MCQ | Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in \%) of the resultant solid is closest to
(A) 25
(B) 35
(C) 55
(D) 75 |
chem | JEE Adv 2022 Paper 2 | C | 52 | MCQ | The reaction of $\mathrm{HClO}_{3}$ with $\mathrm{HCl}$ gives a paramagnetic gas, which upon reaction with $\mathrm{O}_{3}$ produces
(A) $\mathrm{Cl}_{2} \mathrm{O}$
(B) $\mathrm{ClO}_{2}$
(C) $\mathrm{Cl}_{2} \mathrm{O}_{6}$
(D) $\mathrm{Cl}_{2} \mathrm{O}_{7}$ |
chem | JEE Adv 2022 Paper 2 | C | 53 | MCQ | The reaction of $\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$ and $\mathrm{NaCl}$ in water produces a precipitate that dissolves upon the addition of $\mathrm{HCl}$ of appropriate concentration. The dissolution of the precipitate is due to the formation of
(A) $\mathrm{PbCl}_{2}$
(B) $\mathrm{PbCl}_{4}$
(C) $\left[\mathrm{PbCl}_{4}\right]^{2-}$
(D) $\left[\mathrm{PbCl}_{6}\right]^{2-}$ |
math | JEE Adv 2023 Paper 1 | ACD | 1 | MCQ(multiple) | Let $S=(0,1) \cup(1,2) \cup(3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 120
(D) Every continuous function from $S$ to $T$ is differentiable |
math | JEE Adv 2023 Paper 1 | AC | 2 | MCQ(multiple) | Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E: \frac{x^{2}}{6}+\frac{y^{2}}{3}=1$ and the parabola $P: y^{2}=12 x$. Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the points $A_{1}$ and $A_{2}$, respectively and the tangent $T_{2}$ touches $P$ and $E$ at the points $A_{4}$ and $A_{3}$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 35 square units
(B) The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 36 square units
(C) The tangents $T_{1}$ and $T_{2}$ meet the $x$-axis at the point $(-3,0)$
(D) The tangents $T_{1}$ and $T_{2}$ meet the $x$-axis at the point $(-6,0)$ |
math | JEE Adv 2023 Paper 1 | BCD | 3 | MCQ(multiple) | Let $f:[0,1] \rightarrow[0,1]$ be the function defined by $f(x)=\frac{x^{3}}{3}-x^{2}+\frac{5}{9} x+\frac{17}{36}$. Consider the square region $S=[0,1] \times[0,1]$. Let $G=\{(x, y) \in S: y>f(x)\}$ be called the green region and $R=\{(x, y) \in S: y<f(x)\}$ be called the red region. Let $L_{h}=\{(x, h) \in S: x \in[0,1]\}$ be the horizontal line drawn at a height $h \in[0,1]$. Then which of the following statements is(are) true?
(A) There exists an $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_{h}$ equals the area of the green region below the line $L_{h}$
(B) There exists an $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_{h}$ equals the area of the red region below the line $L_{h}$
(C) There exists an $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_{h}$ equals the area of the red region below the line $L_{h}$
(D) There exists an $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_{h}$ equals the area of the green region below the line $L_{h}$ |
math | JEE Adv 2023 Paper 1 | C | 4 | MCQ | Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int_{x^{2}}^{x} \sqrt{\frac{1-t}{t}} d t<g(x)<2 \sqrt{x}$ for all $x \in(0,1)$. Then $\lim _{x \rightarrow 0} f(x) g(x)$
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3 |
math | JEE Adv 2023 Paper 1 | A | 5 | MCQ | Let $Q$ be the cube with the set of vertices $\left\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}, x_{2}, x_{3} \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_{1}$ and $\ell_{2}$, let $d\left(\ell_{1}, \ell_{2}\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_{1}, \ell_{2}\right)$, as $\ell_{1}$ varies over $F$ and $\ell_{2}$ varies over $S$, is
(A) $\frac{1}{\sqrt{6}}$
(B) $\frac{1}{\sqrt{8}}$
(C) $\frac{1}{\sqrt{3}}$
(D) $\frac{1}{\sqrt{12}}$ |
math | JEE Adv 2023 Paper 1 | B | 6 | MCQ | Let $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^{2}}{8}+\frac{y^{2}}{20}<1\right.$ and $\left.y^{2}<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is
(A) $\frac{71}{220}$
(B) $\frac{73}{220}$
(C) $\frac{79}{220}$
(D) $\frac{83}{220}$ |
math | JEE Adv 2023 Paper 1 | A | 7 | MCQ | Let $P$ be a point on the parabola $y^{2}=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is
(A) $(2,3)$
(B) $(1,3)$
(C) $(2,4)$
(D) $(3,4)$ |
math | JEE Adv 2023 Paper 1 | 3 | 8 | Integer | Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for $x \in \mathbb{R}$. Then what is the number of real solutions of the equation
$\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1}(\tan x)$ in the set $\left(-\frac{3 \pi}{2},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$? |
math | JEE Adv 2023 Paper 1 | 8 | 9 | Integer | Let $n \geq 2$ be a natural number and $f:[0,1] \rightarrow \mathbb{R}$ be the function defined by
$$
f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \leq x \leq \frac{1}{2 n} \\ 2 n(2 n x-1) & \text { if } \frac{1}{2 n} \leq x \leq \frac{3}{4 n} \\ 4 n(1-n x) & \text { if } \frac{3}{4 n} \leq x \leq \frac{1}{n} \\ \frac{n}{n-1}(n x-1) & \text { if } \frac{1}{n} \leq x \leq 1\end{cases}
$$
If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f(x)$ is 4 , then what is the maximum value of the function $f$? |
math | JEE Adv 2023 Paper 1 | 1219 | 10 | Integer | Let $7 \overbrace{5 \cdots 5}^{r} 7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Consider the sum $S=77+757+7557+\cdots+7 \overbrace{5 \cdots 5}^{98}7$. If $S=\frac{7 \overbrace{5 \cdots 5}^{99}7+m}{n}$, where $m$ and $n$ are natural numbers less than 3000 , then what is the value of $m+n$? |
math | JEE Adv 2023 Paper 1 | 281 | 11 | Integer | Let $A=\left\{\frac{1967+1686 i \sin \theta}{7-3 i \cos \theta}: \theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then what is the value of $n$? |
math | JEE Adv 2023 Paper 1 | 45 | 12 | Integer | Let $P$ be the plane $\sqrt{3} x+2 y+3 z=16$ and let $S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: \alpha^{2}+\beta^{2}+\gamma^{2}=1\right.$ and the distance of $(\alpha, \beta, \gamma)$ from the plane $P$ is $\left.\frac{7}{2}\right\}$. Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u}-\vec{v}|=|\vec{v}-\vec{w}|=|\vec{w}-\vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then what is the value of $\frac{80}{\sqrt{3}} V$? |
math | JEE Adv 2023 Paper 1 | 3 | 13 | Integer | Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of $\left(a x^{2}+\frac{70}{27 b x}\right)^{4}$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left(a x-\frac{1}{b x^{2}}\right)^{7}$, then the value of $2 b$ is |
math | JEE Adv 2023 Paper 1 | A | 14 | MCQ | Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x+2 y+z=7$
$x+\alpha z=11$
$2 x-3 y+\beta z=\gamma$
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has
(Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has
(R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has
(S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma = 28$, then the system has
\textbf{List-II}
(1) a unique solution
(2) no solution
(3) infinitely many solutions
(4) $x=11, y=-2$ and $z=0$ as a solution
(5) $x=-15, y=4$ and $z=0$ as a solution
The correct option is:
(A) $(P) \rightarrow(3) \quad(Q) \rightarrow(2) \quad(R) \rightarrow(1) \quad(S) \rightarrow(4)$
(B) $(P) \rightarrow(3) \quad(Q) \rightarrow(2) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4)$
(C) $(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(4) \quad(S) \rightarrow(5)$
(D) $(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(1) \quad$ (S) $\rightarrow$ (3) |
math | JEE Adv 2023 Paper 1 | A | 15 | MCQ | Consider the given data with frequency distribution
$$
\begin{array}{ccccccc}
x_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\
f_{i} & 5 & 2 & 3 & 2 & 4 & 4
\end{array}
$$
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) The mean of the above data is
(Q) The median of the above data is
(R) The mean deviation about the mean of the above data is
(S) The mean deviation about the median of the above data is
\textbf{List-II}
(1) 2.5
(2) 5
(3) 6
(4) 2.7
(5) 2.4
The correct option is:
(A) $(P) \rightarrow(3) \quad$ (Q) $\rightarrow$ (2) $\quad$ (R) $\rightarrow$ (4) $\quad$ (S) $\rightarrow$ (5)
(B) $(P) \rightarrow(3) \quad(Q) \rightarrow(2) \quad(R) \rightarrow(1) \quad(S) \rightarrow(5)$
(C) $(P) \rightarrow(2) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(4) \quad(S) \rightarrow(1)$
(D) $(P) \rightarrow$ (3) $\quad$ (Q) $\rightarrow$ (3) $\quad$ (R) $\rightarrow$ (5) $\quad$ (S) $\rightarrow$ (5) |
math | JEE Adv 2023 Paper 1 | B | 16 | MCQ | Let $\ell_{1}$ and $\ell_{2}$ be the lines $\vec{r}_{1}=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_{2}=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_{1}$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $\ell_{2}$ and $H$. Let $H_{0}$ be a plane in $X$ for which $d\left(H_{0}\right)$ is the maximum value of $d(H)$ as $H$ varies over all planes in $X$.
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) The value of $d\left(H_{0}\right)$ is
(Q) The distance of the point $(0,1,2)$ from $H_{0}$ is
(R) The distance of origin from $H_{0}$ is
(S) The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_{0}$ is
\textbf{List-II}
(1) $\sqrt{3}$
(2) $\frac{1}{\sqrt{3}}$
(3) 0
(4) $\sqrt{2}$
(5) $\frac{1}{\sqrt{2}}$
The correct option is:
(A) $(P) \rightarrow(2) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(5) \quad(S) \rightarrow(1)$
(B) $(P) \rightarrow(5) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(3) \quad(S) \rightarrow$ (1)
(C) $(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(3) \quad(S) \rightarrow$ (2)
(D) $(P) \rightarrow(5) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(4) \quad(S) \rightarrow(2)$ |
math | JEE Adv 2023 Paper 1 | B | 17 | MCQ | Let $z$ be a complex number satisfying $|z|^{3}+2 z^{2}+4 \bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.
\textbf{List-I}
(P) $|z|^{2}$ is equal to
(Q) $|Z-\bar{Z}|^{2}$ is equal to
(R) $|Z|^{2}+|Z+\bar{Z}|^{2}$ is equal to
(S) $|z+1|^{2}$ is equal to
\textbf{List-II}
(1) 12
(2) 4
(3) 8
(4) 10
(5) 7
The correct option is:
(A)$(P) \rightarrow(1) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow$ (4)
(B) $(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(3) \quad(S) \rightarrow(5)$
(C) $(P) \rightarrow(2) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(5) \quad(S) \rightarrow(1)$
(D) $(P) \rightarrow(2) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4)$ |
phy | JEE Adv 2023 Paper 1 | A | 21 | MCQ | A bar of mass $M=1.00 \mathrm{~kg}$ and length $L=0.20 \mathrm{~m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10 \mathrm{~kg}$ is moving on the same horizontal surface with $5.00 \mathrm{~m} \mathrm{~s}^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L / 2$ from the pivoted end and returns back on the same path with speed $\mathrm{v}$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?
(A) $\omega=6.98 \mathrm{rad} \mathrm{s}^{-1}$ and $\mathrm{v}=4.30 \mathrm{~m} \mathrm{~s}^{-1}$
(B) $\omega=3.75 \mathrm{rad} \mathrm{s}^{-1}$ and $\mathrm{v}=4.30 \mathrm{~m} \mathrm{~s}^{-1}$
(C) $\omega=3.75 \mathrm{rad} \mathrm{s}^{-1}$ and $\mathrm{v}=10.0 \mathrm{~m} \mathrm{~s}^{-1}$
(D) $\omega=6.80 \mathrm{rad} \mathrm{s}^{-1}$ and $\mathrm{v}=4.10 \mathrm{~m} \mathrm{~s}^{-1}$ |
phy | JEE Adv 2023 Paper 1 | A | 23 | MCQ | One mole of an ideal gas expands adiabatically from an initial state $\left(T_{\mathrm{A}}, V_{0}\right)$ to final state $\left(T_{\mathrm{f}}, 5 V_{0}\right)$. Another mole of the same gas expands isothermally from a different initial state $\left(T_{\mathrm{B}}, V_{0}\right)$ to the same final state $\left(T_{\mathrm{f}}, 5 V_{0}\right)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$ ?
(A) $5^{\gamma-1}$
(B) $5^{1-\gamma}$
(C) $5^{\gamma}$
(D) $5^{1+\gamma}$ |
phy | JEE Adv 2023 Paper 1 | A | 24 | MCQ | Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{P}}=R / 3$. The accelerations of $\mathrm{P}$ and $\mathrm{Q}$ due to Earth's gravity are $g_{\mathrm{P}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{P}} / g_{\mathrm{Q}}=36 / 25$, what is the value of $h_{\mathrm{Q}}$ ?
(A) $3 R / 5$
(B) $R / 6$
(C) $6 R / 5$
(D) $5 R / 6$ |
phy | JEE Adv 2023 Paper 1 | 3 | 25 | Integer | A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n=4$ to level $n=3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is $1.95 \mathrm{eV}$. If the photoelectric threshold wavelength for the target metal is $310 \mathrm{~nm}$, what is the value of $Z$?
[Given: $h c=1240 \mathrm{eV}-\mathrm{nm}$ and $R h c=13.6 \mathrm{eV}$, where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum] |
phy | JEE Adv 2023 Paper 1 | 1 | 27 | Integer | In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 \mathrm{~cm}$ and the distance of its real image from the lens is $20 \pm 0.2 \mathrm{~cm}$. The error in the determination of focal length of the lens is $n \%$. What is the value of $n$? |
phy | JEE Adv 2023 Paper 1 | 121 | 28 | Integer | A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $(\gamma=5 / 3)$ and one mole of an ideal diatomic gas $(\gamma=7 / 5)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. What is the change in its internal energy in Joule. |
phy | JEE Adv 2023 Paper 1 | 40 | 29 | Integer | A person of height $1.6 \mathrm{~m}$ is walking away from a lamp post of height $4 \mathrm{~m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{~cm} \mathrm{~s}^{-1}$, then what is the speed of the tip of the person's shadow on the ground with respect to the person in $\mathrm{cm} \mathrm{s}^{-1}$? |
phy | JEE Adv 2023 Paper 1 | A | 31 | MCQ | List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.
\textbf{List-I}
(P) ${ }_{92}^{238} U \rightarrow{ }_{91}^{234} \mathrm{~Pa}$
(Q) ${ }_{82}^{214} \mathrm{~Pb} \rightarrow{ }_{82}^{210} \mathrm{~Pb}$
(R) ${ }_{81}^{210} \mathrm{Tl} \rightarrow{ }_{82}^{206} \mathrm{~Pb}$
(S) ${ }_{91}^{228} \mathrm{~Pa} \rightarrow{ }_{88}^{224} \mathrm{Ra}$
\textbf{List-II}
(1) one $\alpha$ particle and one $\beta^{+}$particle
(2) three $\beta^{-}$particles and one $\alpha$ particle
(3) two $\beta^{-}$particles and one $\alpha$ particle
(4) one $\alpha$ particle and one $\beta^{-}$particle
(5) one $\alpha$ particle and two $\beta^{+}$particles
(A) $P \rightarrow 4, Q \rightarrow 3, R \rightarrow 2, S \rightarrow 1$
(B) $P \rightarrow 4, Q \rightarrow 1, R \rightarrow 2, S \rightarrow 5$
(C) $P \rightarrow 5, Q \rightarrow 3, R \rightarrow 1, S \rightarrow 4$
(D) $P \rightarrow 5, Q \rightarrow 1, R \rightarrow 3, S \rightarrow 2$ |
phy | JEE Adv 2023 Paper 1 | C | 32 | MCQ | Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.
[Given: Wien's constant as $2.9 \times 10^{-3} \mathrm{~m}-\mathrm{K}$ and $\frac{h c}{e}=1.24 \times 10^{-6} \mathrm{~V}-\mathrm{m}$ ]
\textbf{List-I}
(P) $2000 \mathrm{~K}$
(Q) $3000 \mathrm{~K}$
(R) $5000 \mathrm{~K}$
(S) $10000 \mathrm{~K}$
\textbf{List-II}
(1) The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function $4 \mathrm{eV}$.
(2) The radiation at peak wavelength is visible to human eye.
(3) The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction.
(4) The power emitted per unit area is $1 / 16$ of that emitted by a blackbody at temperature $6000 \mathrm{~K}$.
(5) The radiation at peak emission wavelength can be used to image human bones.
(A) $P \rightarrow 3, Q \rightarrow 5, R \rightarrow 2, S \rightarrow 3$
(B) $P \rightarrow 3, Q \rightarrow 2, R \rightarrow 4, S \rightarrow 1$
(C) $P \rightarrow 3, Q \rightarrow 4, R \rightarrow 2, S \rightarrow 1$
(D) $P \rightarrow 1, Q \rightarrow 2, R \rightarrow 5, S \rightarrow 3$ |
phy | JEE Adv 2023 Paper 1 | B | 33 | MCQ | A series LCR circuit is connected to a $45 \sin (\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^{5} \mathrm{rad} \mathrm{s}^{-1}$ and current amplitude at resonance is $I_{0}$. When the angular frequency of the source is $\omega=8 \times 10^{4} \mathrm{rad} \mathrm{s}^{-1}$, the current amplitude in the circuit is $0.05 I_{0}$. If $L=50 \mathrm{mH}$, match each entry in List-I with an appropriate value from List-II and choose the correct option.
\textbf{List-I}
(P) $I_{0}$ in $\mathrm{mA}$
(Q) The quality factor of the circuit
(R) The bandwidth of the circuit in $\mathrm{rad} \mathrm{s}^{-1}$
(S) The peak power dissipated at resonance in Watt
\textbf{List-II}
(1) 44.4
(2) 18
(3) 400
(4) 2250
(5) 500
(A) $P \rightarrow 2, Q \rightarrow 3, R \rightarrow 5, S \rightarrow 1$
(B) $P \rightarrow 3, Q \rightarrow 1, R \rightarrow 4, S \rightarrow 2$
(C) $P \rightarrow 4, Q \rightarrow 5, R \rightarrow 3, S \rightarrow 1$
(D) $P \rightarrow 4, Q \rightarrow 2, R \rightarrow 1, S \rightarrow 5$ |
chem | JEE Adv 2023 Paper 1 | BCD | 35 | MCQ(multiple) | The correct statement(s) related to processes involved in the extraction of metals is(are)
(A) Roasting of Malachite produces Cuprite.
(B) Calcination of Calamine produces Zincite.
(C) Copper pyrites is heated with silica in a reverberatory furnace to remove iron.
(D) Impure silver is treated with aqueous KCN in the presence of oxygen followed by reduction with zinc metal. |
chem | JEE Adv 2023 Paper 1 | A | 39 | MCQ | Plotting $1 / \Lambda_{\mathrm{m}}$ against $\mathrm{c} \Lambda_{\mathrm{m}}$ for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y-axis intercept of $\mathrm{P}$ and slope of $\mathrm{S}$. The ratio $\mathrm{P} / \mathrm{S}$ is
$\left[\Lambda_{\mathrm{m}}=\right.$ molar conductivity
$\Lambda_{\mathrm{m}}^{\mathrm{o}}=$ limiting molar conductivity
$\mathrm{c}=$ molar concentration
$\mathrm{K}_{\mathrm{a}}=$ dissociation constant of $\mathrm{HX}$ ]
(A) $\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\mathrm{o}}$
(B) $\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\mathrm{o}} / 2$
(C) $2 \mathrm{~K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\mathrm{o}}$
(D) $1 /\left(\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\mathrm{o}}\right)$ |
chem | JEE Adv 2023 Paper 1 | B | 40 | MCQ | On decreasing the $p \mathrm{H}$ from 7 to 2, the solubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from $10^{-4} \mathrm{~mol} \mathrm{~L}^{-1}$ to $10^{-3} \mathrm{~mol} \mathrm{~L}^{-1}$. The $p\mathrm{~K}_{\mathrm{a}}$ of $\mathrm{HX}$ is
(A) 3
(B) 4
(C) 5
(D) 2 |
chem | JEE Adv 2023 Paper 1 | 222 | 42 | Integer | The stoichiometric reaction of $516 \mathrm{~g}$ of dimethyldichlorosilane with water results in a tetrameric cyclic product $\mathbf{X}$ in $75 \%$ yield. What is the weight (in g) obtained of $\mathbf{X}$?
[Use, molar mass $\left(\mathrm{g} \mathrm{mol}^{-1}\right): \mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Si}=28, \mathrm{Cl}=35.5$ ] |
chem | JEE Adv 2023 Paper 1 | 100 | 43 | Integer | A gas has a compressibility factor of 0.5 and a molar volume of $0.4 \mathrm{dm}^{3} \mathrm{~mol}^{-1}$ at a temperature of $800 \mathrm{~K}$ and pressure $\mathbf{x}$ atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be $\mathbf{y} \mathrm{dm}^{3} \mathrm{~mol}^{-1}$. What is the value of $\mathbf{x} / \mathbf{y}$?
[Use: Gas constant, $\mathrm{R}=8 \times 10^{-2} \mathrm{~L} \mathrm{~atm} \mathrm{~K} \mathrm{Kol}^{-1}$ ] |
chem | JEE Adv 2023 Paper 1 | D | 48 | MCQ | Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.
\textbf{List-I}
(P) $\mathrm{P}_{2} \mathrm{O}_{3}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow$
(Q) $\mathrm{P}_{4}+3 \mathrm{NaOH}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow$
(R) $\mathrm{PCl}_{5}+\mathrm{CH}_{3} \mathrm{COOH} \rightarrow$
(S) $\mathrm{H}_{3} \mathrm{PO}_{2}+2 \mathrm{H}_{2} \mathrm{O}+4 \mathrm{AgNO}_{3} \rightarrow$
\textbf{List-II}
(1) $\mathrm{P}(\mathrm{O})\left(\mathrm{OCH}_{3}\right) \mathrm{Cl}_{2}$
(2) $\mathrm{H}_{3} \mathrm{PO}_{3}$
(3) $\mathrm{PH}_{3}$
(4) $\mathrm{POCl}_{3}$
(5) $\mathrm{H}_{3} \mathrm{PO}_{4}$
(A) $\mathrm{P} \rightarrow 2$; $\mathrm{Q} \rightarrow 3$; $\mathrm{R} \rightarrow 1$; $\mathrm{S} \rightarrow 5$
(B) $\mathrm{P} \rightarrow 3$; Q $\rightarrow 5$; R $\rightarrow 4$; $\mathrm{S} \rightarrow 2$
(C) $\mathrm{P} \rightarrow 5 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 1$; S $\rightarrow 3$
(D) P $\rightarrow 2$; Q $\rightarrow 3$; R $\rightarrow 4$; $\mathrm{S} \rightarrow 5$ |
chem | JEE Adv 2023 Paper 1 | D | 49 | MCQ | Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option.
[Atomic Number: $\mathrm{Fe}=26, \mathrm{Mn}=25$, $\mathrm{Co}=27$ ]
\textbf{List-I}
(P) $t_{2 g}^{6} e_{g}^{0}$
(Q) $t_{2 g}^{3} e_{g}^{2}$
(R) $\mathrm{e}^{2} \mathrm{t}_{2}^{3}$
(S) $t_{2 g}^{4} e_{g}^{2}$
\textbf{List-II}
(1) $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$
(2) $\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$
(3) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$
(4) $\left[\mathrm{FeCl}_{4}\right]^{-}$
(5) $\left[\mathrm{CoCl}_{4}\right]^{2-}$
(A) $\mathrm{P} \rightarrow 1$; Q $\rightarrow 4$; $\mathrm{R} \rightarrow 2$; $\mathrm{S} \rightarrow 3$
(B) $\mathrm{P} \rightarrow 1 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 5$
(C) $\mathrm{P} \rightarrow 3 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 5 ; \mathrm{S} \rightarrow 1$
(D) $\mathrm{P} \rightarrow 3 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 4$; $\mathrm{S} \rightarrow 1$ |
chem | JEE Adv 2023 Paper 1 | D | 51 | MCQ | The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.
\textbf{List-I}
(P) Etard reaction
(Q) Gattermann reaction
(R) Gattermann-Koch reaction
(S) Rosenmund reduction
\textbf{List-II}
(1) Acetophenone $\stackrel{\mathrm{Zn}-\mathrm{Hg}, \mathrm{HCl}}{\longrightarrow}$
(2) Toluene $\underset{\text{(ii)}\mathrm{SOCl}_{2}}{\stackrel{\text { (i) } \mathrm{KMnO}_{4}, \mathrm{KOH}, \Delta}{\longrightarrow}}$
(3) Benzene $\underset{\text { anhyd. } \mathrm{AlCl}_{3}}{\stackrel{\mathrm{CH}_{3} \mathrm{Cl}}{\longrightarrow}}$
(4) Aniline $\underset{273-278 \mathrm{~K}}{\stackrel{\mathrm{NaNO}_{2} / \mathrm{HCl}}{\longrightarrow}}$
(5) Phenol $\stackrel{\mathrm{Zn}, \Delta}{\longrightarrow}$
(A) $\mathrm{P} \rightarrow 2$; $\mathrm{Q} \rightarrow 4 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 3$
(B) $\mathrm{P} \rightarrow 1$; $\mathrm{Q} \rightarrow 3$; $\mathrm{R} \rightarrow 5$; $\mathrm{S} \rightarrow 2$
(C) $\mathrm{P} \rightarrow 3$; $\mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 1$; $\mathrm{S} \rightarrow 4$
(D) $\mathrm{P} \rightarrow 3$; $\mathrm{Q} \rightarrow 4$; R $\rightarrow 5$; $\mathrm{S} \rightarrow 2$ |
math | JEE Adv 2023 Paper 2 | C | 1 | MCQ | Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is
(A) $\frac{e^{2}+4}{3}$
(B) $\frac{\log _{e} 4+e}{3}$
(C) $\frac{4 e^{2}}{3}$
(D) $\frac{e^{2}-4}{3}$ |
math | JEE Adv 2023 Paper 2 | B | 2 | MCQ | Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is
(A) $\frac{1}{3}$
(B) $\frac{5}{21}$
(C) $\frac{4}{21}$
(D) $\frac{2}{7}$ |
math | JEE Adv 2023 Paper 2 | C | 3 | MCQ | For any $y \in \mathbb{R}$, let $\cot ^{-1}(y) \in(0, \pi)$ and $\tan ^{-1}(y) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the sum of all the solutions of the equation $\tan ^{-1}\left(\frac{6 y}{9-y^{2}}\right)+\cot ^{-1}\left(\frac{9-y^{2}}{6 y}\right)=\frac{2 \pi}{3}$ for $0<|y|<3$, is equal to
(A) $2 \sqrt{3}-3$
(B) $3-2 \sqrt{3}$
(C) $4 \sqrt{3}-6$
(D) $6-4 \sqrt{3}$ |
math | JEE Adv 2023 Paper 2 | B | 4 | MCQ | Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?
(A) The points $P, Q, R$ and $S$ are NOT coplanar
(B) $\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ internally in the ratio $5: 4$
(C) $\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ externally in the ratio $5: 4$
(D) The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is 95 |
math | JEE Adv 2023 Paper 2 | BC | 5 | MCQ(multiple) | Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$, be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$, otherwise $a_{i j}=0$. Then which of the following statements is(are) true?
(A) $M$ is invertible
(B) There exists a nonzero column matrix $\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)$ such that $M\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)=\left(\begin{array}{c}-a_{1} \\ -a_{2} \\ -a_{3}\end{array}\right)$
(C) The set $\left\{X \in \mathbb{R}^{3}: M X=\mathbf{0}\right\} \neq\{\boldsymbol{0}\}$, where $\mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$
(D) The matrix $(M-2 I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix |
math | JEE Adv 2023 Paper 2 | AB | 6 | MCQ(multiple) | Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^{2}\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in $(0,1)$
(B) There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
(C) The function $f$ is NOT differentiable at more than three points in $(0,1)$
(D) The minimum value of the function $f$ is $-\frac{1}{512}$ |
math | JEE Adv 2023 Paper 2 | ABC | 7 | MCQ(multiple) | Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^{2} f}{d x^{2}}(x)>0$ for all $x \in(-1,1)$. For $f \in S$, let $X_{f}$ be the number of points $x \in(-1,1)$ for which $f(x)=x$. Then which of the following statements is(are) true?
(A) There exists a function $f \in S$ such that $X_{f}=0$
(B) For every function $f \in S$, we have $X_{f} \leq 2$
(C) There exists a function $f \in S$ such that $X_{f}=2$
(D) There does NOT exist any function $f$ in $S$ such that $X_{f}=1$ |
math | JEE Adv 2023 Paper 2 | 0 | 8 | Integer | For $x \in \mathbb{R}$, let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then what is the minimum value of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\int_{0}^{x \tan ^{-1} x} \frac{e^{(t-\cos t)}}{1+t^{2023}} d t$? |
math | JEE Adv 2023 Paper 2 | 16 | 9 | Integer | For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation $\left(x^{2}-5\right) \frac{d y}{d x}-2 x y=-2 x\left(x^{2}-5\right)^{2}$ such that $y(2)=7$.
Then what is the maximum value of the function $y(x)$? |
math | JEE Adv 2023 Paper 2 | 31 | 10 | Integer | Let $X$ be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5 . Then what is the value of $38 p$ equal to? |
math | JEE Adv 2023 Paper 2 | 512 | 11 | Integer | Let $A_{1}, A_{2}, A_{3}, \ldots, A_{8}$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A_{i}$ denote the distance between the points $P$ and $A_{i}$ for $i=1,2, \ldots, 8$. If $P$ varies over the circle, then what is the maximum value of the product $P A_{1} \cdot P A_{2} \cdots P A_{8}? |
math | JEE Adv 2023 Paper 2 | 3780 | 12 | Integer | $R=\left\{\left(\begin{array}{lll}a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0\end{array}\right): a, b, c, d \in\{0,3,5,7,11,13,17,19\}\right\}$. Then what is the number of invertible matrices in $R$? |
math | JEE Adv 2023 Paper 2 | 2 | 13 | Integer | Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The tangent $P Q$ touches $C_{1}$ at $P$ and $C_{2}$ at $Q$. The tangent $S T$ touches $C_{1}$ at $S$ and $C_{2}$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B=\sqrt{5}$, then what is the value of $r^{2}$? |
math | JEE Adv 2023 Paper 2 | 1008 | 14 | Numeric | Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. Let $a$ be the area of the triangle $A B C$. Then what is the value of $(64 a)^{2}$? |
math | JEE Adv 2023 Paper 2 | 0.25 | 15 | Numeric | Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. Then what is the inradius of the triangle ABC? |