Chemistry
1
The correct statement(s) related to processes involved in the extraction of metals is(are) :
2
In the following reactions, $\mathbf{P}, \mathbf{Q}, \mathbf{R}$, and $\mathbf{S}$ are the major products.
The correct statement(s) about $\mathbf{P}, \mathbf{Q}, \mathbf{R}$, and $\mathbf{S}$ is(are) :
The correct statement(s) about $\mathbf{P}, \mathbf{Q}, \mathbf{R}$, and $\mathbf{S}$ is(are) :
3
Consider the following reaction scheme and choose the correct option(s) for the major products $\mathbf{Q}$, $\mathbf{R}$ and $\mathbf{S}$.
4
In the scheme given below, $\mathbf{X}$ and $\mathbf{Y}$, respectively, are
5
Plotting $1 / \Lambda_{\mathrm{m}}$ against $\mathrm{c} \Lambda_{\mathrm{m}}$ for aqueous solutions of a monobasic weak acid $(\mathrm{HX})$ resulted in a straight line with $\mathrm{y}$-axis intercept of $\mathrm{P}$ and slope of $\mathrm{S}$. The ratio $\mathrm{P} / \mathrm{S}$ is
$$ \begin{aligned} & {\left[\Lambda_{\mathrm{m}}=\right.\text { molar conductivity }} \\\\ & \Lambda_{\mathrm{m}}^{\mathrm{o}}=\text { limiting molar conductivity } \\\\ & \mathrm{c}=\text { molar concentration } \\\\ & \left.\mathrm{K}_{\mathrm{a}}=\text { dissociation constant of } \mathrm{HX}\right] \end{aligned} $$
$$ \begin{aligned} & {\left[\Lambda_{\mathrm{m}}=\right.\text { molar conductivity }} \\\\ & \Lambda_{\mathrm{m}}^{\mathrm{o}}=\text { limiting molar conductivity } \\\\ & \mathrm{c}=\text { molar concentration } \\\\ & \left.\mathrm{K}_{\mathrm{a}}=\text { dissociation constant of } \mathrm{HX}\right] \end{aligned} $$
6
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
7
In the given reaction scheme, $\mathbf{P}$ is a phenyl alkyl ether, $\mathbf{Q}$ is an aromatic compound; $\mathbf{R}$ and $\mathbf{S}$ are the major products.
The correct statement about $\mathbf{S}$ is :
The correct statement about $\mathbf{S}$ is :
8
The stoichiometric reaction of $516 \mathrm{~g}$ of dimethyldichlorosilane with water results in a tetrameric cyclic product $\mathbf{X}$ in $75 \%$ yield. The weight (in g) of $\mathbf{X}$ obtained is _______.
[Use, molar mass $\left(\mathrm{g} ~\mathrm{mol}^{-1}\right): \mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Si}=28, \mathrm{Cl}=35.5$ ]
[Use, molar mass $\left(\mathrm{g} ~\mathrm{mol}^{-1}\right): \mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Si}=28, \mathrm{Cl}=35.5$ ]
9
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 $\mathbf{y} ~\mathrm{dm}^3 \mathrm{~mol}^{-1}$. The value of $\mathbf{x} / \mathbf{y}$ is __________.
[Use: Gas constant, $\mathrm{R}=8 \times 10^{-2} \mathrm{~L}$ atm $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ]
[Use: Gas constant, $\mathrm{R}=8 \times 10^{-2} \mathrm{~L}$ atm $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ]
10
The plot of $\log k_f$ versus $1 / T$ for a reversible reaction $\mathrm{A}(\mathrm{g}) \rightleftharpoons \mathrm{P}(\mathrm{g})$ is shown.
Pre-exponential factors for the forward and backward reactions are $10^{15} \mathrm{~s}^{-1}$ and $10^{11} \mathrm{~s}^{-1}$, respectively. If the value of $\log K$ for the reaction at $500 \mathrm{~K}$ is 6 , the value of $\left|\log k_b\right|$ at $250 \mathrm{~K}$ is ______.
$$ \begin{aligned} & {[K=\text { equilibrium constant of the reaction }} \\\\ & k_f=\text { rate constant of forward reaction } \\\\ & \left.k_b=\text { rate constant of backward reaction }\right] \end{aligned} $$
Pre-exponential factors for the forward and backward reactions are $10^{15} \mathrm{~s}^{-1}$ and $10^{11} \mathrm{~s}^{-1}$, respectively. If the value of $\log K$ for the reaction at $500 \mathrm{~K}$ is 6 , the value of $\left|\log k_b\right|$ at $250 \mathrm{~K}$ is ______.
$$ \begin{aligned} & {[K=\text { equilibrium constant of the reaction }} \\\\ & k_f=\text { rate constant of forward reaction } \\\\ & \left.k_b=\text { rate constant of backward reaction }\right] \end{aligned} $$
11
One mole of an ideal monoatomic gas undergoes two reversible processes $(\mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{B} \rightarrow \mathrm{C})$ as shown in the given figure:
$\mathrm{A} \rightarrow \mathrm{B}$ is an adiabatic process. If the total heat absorbed in the entire process $(\mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{B} \rightarrow \mathrm{C})$ is $\mathrm{R} T_2 \ln 10$, the value of $2 \log V_3$ is _______.
[Use, molar heat capacity of the gas at constant pressure, $C_{\mathrm{p}, \mathrm{m}}=\frac{5}{2} \mathrm{R}$ ]
$\mathrm{A} \rightarrow \mathrm{B}$ is an adiabatic process. If the total heat absorbed in the entire process $(\mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{B} \rightarrow \mathrm{C})$ is $\mathrm{R} T_2 \ln 10$, the value of $2 \log V_3$ is _______.
[Use, molar heat capacity of the gas at constant pressure, $C_{\mathrm{p}, \mathrm{m}}=\frac{5}{2} \mathrm{R}$ ]
12
In a one-litre flask, 6 moles of $A$ undergoes the reaction $A(\mathrm{~g}) \rightleftharpoons P(\mathrm{~g})$. The progress of product formation at two temperatures (in Kelvin), $\mathrm{T}_1$ and $\mathrm{T}_2$, is shown in the figure:
If $\mathrm{T}_1=2 \mathrm{~T}_2$ and $\left(\Delta \mathrm{G}_2^{\Theta}-\Delta \mathrm{G}_1^{\Theta}\right)=\mathrm{RT}_2 \ln \mathrm{x}$, then the value of $\mathrm{x}$ is _______.
$\left[\Delta \mathrm{G}_1^{\Theta}\right.$ and $\Delta \mathrm{G}_2^{\Theta}$ are standard Gibb's free energy change for the reaction at temperatures $\mathrm{T}_1$ and $\mathrm{T}_2$, respectively.]
If $\mathrm{T}_1=2 \mathrm{~T}_2$ and $\left(\Delta \mathrm{G}_2^{\Theta}-\Delta \mathrm{G}_1^{\Theta}\right)=\mathrm{RT}_2 \ln \mathrm{x}$, then the value of $\mathrm{x}$ is _______.
$\left[\Delta \mathrm{G}_1^{\Theta}\right.$ and $\Delta \mathrm{G}_2^{\Theta}$ are standard Gibb's free energy change for the reaction at temperatures $\mathrm{T}_1$ and $\mathrm{T}_2$, respectively.]
13
The total number of $s p^2$ hybridised carbon atoms in the major product $\mathbf{P}$ (a non-heterocyclic compound) of the following reaction is ______.
14
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.
| List - I | List - II |
|---|---|
| (P) $\mathrm{P}_2 \mathrm{O}_3+3 \mathrm{H}_2 \mathrm{O} \rightarrow$ | (1) $\mathrm{P}(\mathrm{O})\left(\mathrm{OCH}_3\right) \mathrm{Cl}_2$ |
| (Q) $\mathrm{P}_4+3 \mathrm{NaOH}+3 \mathrm{H}_2 \mathrm{O} \rightarrow$ | (2) $\mathrm{H}_3 \mathrm{PO}_3$ |
| (R) $\mathrm{PCl}_5+\mathrm{CH}_3 \mathrm{COOH} \rightarrow$ | (3) $\mathrm{PH}_3$ |
| (S) $\mathrm{H}_3 \mathrm{PO}_2+2 \mathrm{H}_2 \mathrm{O}+4 \mathrm{AgNO}_3 \rightarrow$ | (4) $\mathrm{POCl}_3$ |
| (5) $\mathrm{H}_3 \mathrm{PO}_4$ |
15
Match the electronic configurations in List-I with appropriate metal complex ions in List-II and choose the correct option.
[Atomic Number: $\mathrm{Fe}=26, \mathrm{Mn}=25, \mathrm{Co}=27$ ]
[Atomic Number: $\mathrm{Fe}=26, \mathrm{Mn}=25, \mathrm{Co}=27$ ]
| List - I | List - II |
|---|---|
| (P) $t_{2 g}^6 e_g^0$ | (1) $\left[\mathrm{Fe}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+}$ |
| (Q) $t_{2 g}^3 e_g^2$ | (2) $\left[\mathrm{Mn}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+}$ |
| (R) $\mathrm{e}^2 \mathrm{t}_2^3$ | (3) $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}$ |
| (S) $t_{2 g}^4 e_g^2$ | (4) $\left[\mathrm{FeCl}_4\right]^{-}$ |
| (5) $\left[\mathrm{CoCl}_4\right]^{2-}$ |
16
Match the reactions in List-I with the features of their products in List-II and choose the correct option.
| List - I | List - II |
|---|---|
(P)  |
(1) Inversion of configuration |
(Q)  |
(2) Retention of configuration |
(R)  |
(3) Mixture of enantiomers |
(S)  |
(4) Mixture of structural isomers |
| (5) Mixture of diastereomers |
17
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.
| List - I | List - II |
|---|---|
| (P) Etard reaction | (1) Acetophenone $\stackrel{\mathrm{Zn}-\mathrm{Hg}, \mathrm{HCl}}{\longrightarrow}$ |
| (Q) Gattermann reaction | (2) $$ \text { Toluene } \underset{\text { (ii) } \mathrm{SOCl}_2}{\stackrel{\text { (i) } \mathrm{KMnO}_4, \mathrm{KOH}, \Delta}{\longrightarrow}} $$ |
| (R) Gattermann-Koch reaction | (3) $$ \text { Benzene } \underset{\text { anhyd. } \mathrm{AlCl}_3}{\stackrel{\mathrm{CH}_3 \mathrm{Cl}}{\longrightarrow}} $$ |
| (S) Rosenmund reduction | (4) $$ \text { Aniline } \underset{273-278 \mathrm{~K}}{\stackrel{\mathrm{NaNO}_2 / \mathrm{HCl}}{\longrightarrow}} $$ |
| (5) $$ \text { Phenol } \stackrel{\mathrm{Zn}, \Delta}{\longrightarrow} $$ |
Mathematics
1
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?
2
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?
3
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?
4
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 $\int\limits_{x^2}^x \sqrt{\frac{1-t}{t}} d t < g(x) < 2 \sqrt{x}$ for all $x \in(0,1)$.
Then $\lim\limits_{x \rightarrow 0} f(x) g(x)$
5
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 :
6
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 :
7
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
8
Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, for $x \in \mathbb{R}$. Then the number of real solutions of the equation $\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1}(\tan x)$ in the set $\left(-\frac{3 \pi}{2},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$ is equal to :
9
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 the maximum value of the function $f$ is :
$$ f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \leq x \leq \frac{1}{2 n} \\\\ 2 n(2 n x-1) & \text { if } \frac{1}{2 n} \leq x \leq \frac{3}{4 n} \\\\ 4 n(1-n x) & \text { if } \frac{3}{4 n} \leq x \leq \frac{1}{n} \\\\ \frac{n}{n-1}(n x-1) & \text { if } \frac{1}{n} \leq x \leq 1\end{cases} $$
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 the maximum value of the function $f$ is :
10
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 the value of $m+n$ is
11
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 the value of $n$ is
12
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 the value of $\frac{80}{\sqrt{3}} V$ is :
13
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 :
14
Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$$ \begin{aligned} & x+2 y+z=7 \\\\ & x+\alpha z=11 \\\\ & 2 x-3 y+\beta z=\gamma \end{aligned} $$
Match each entry in List-I to the correct entries in List-II.
The correct option is:
$$ \begin{aligned} & x+2 y+z=7 \\\\ & x+\alpha z=11 \\\\ & 2 x-3 y+\beta z=\gamma \end{aligned} $$
Match each entry in List-I to the correct entries in List-II.
| List - I | List - II |
|---|---|
| (P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has | (1) a unique solution |
| (Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has | (2) no solution |
| (R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has | (3) infinitely many solutions |
| (S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has | (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:
15
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.
The correct option is:
$$ \begin{array}{ccccccc} x_i & 3 & 8 & 11 & 10 & 5 & 4 \\ f_i & 5 & 2 & 3 & 2 & 4 & 4 \end{array} $$
Match each entry in List-I to the correct entries in List-II.
| List - I | List - II |
|---|---|
| (P) The mean of the above data is | (1) 2.5 |
| (Q) The median of the above data is | (2) 5 |
| (R) The mean deviation about the mean of the above data is | (3) 6 |
| (S) The mean deviation about the median of the above data is | (4) 2.7 |
| (5) 2.4 |
The correct option is:
16
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.
The correct option is:
Match each entry in List-I to the correct entries in List-II.
| List - I | List - II |
|---|---|
| (P) The value of $d\left(H_0\right)$ is | (1) $\sqrt{3}$ |
| (Q) The distance of the point $(0,1,2)$ from $H_0$ is | (2) $\frac{1}{\sqrt{3}}$ |
| (R) The distance of origin from $H_0$ is | (3) 0 |
| (S) The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_0$ is | (4) $\sqrt{2}$ |
| (5) $\frac{1}{\sqrt{2}}$ |
The correct option is:
17
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.
The correct option is:
Match each entry in List-I to the correct entries in List-II.
| List - I | List - II |
|---|---|
| (P) $|z|^2$ is equal to | (1) 12 |
| (Q) $|z-\bar{z}|^2$ is equal to | (2) 4 |
| (R) $|z|^2+|z+\bar{z}|^2$ is equal to | (3) 8 |
| (S) $|z+1|^2$ is equal to | (4) 10 |
| (5) 7 |
The correct option is:
Physics
1
A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3 h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0=u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1 / \sqrt{3}$. Which of the following statement(s) is(are) correct?
2
A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is $\delta=60^{\circ}$ (see Figure-1). The angle of minimum deviation for red light from the same prism is $\delta_{\min }=30^{\circ}$ (see Figure-2). The refractive index of the prism material for blue light is $\sqrt{3}$. Which of the following statement(s) is(are) correct?
3
In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement(s) is(are) correct?
[Given: $e^{-1}=0.36$ ]
[Given: $e^{-1}=0.36$ ]
4
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 v. After this elastic collision, the bar rotates with an angular velocity $\omega$.
Which of the following statement is correct?
Which of the following statement is correct?
5
A container has a base of $50 \mathrm{~cm} \times 5 \mathrm{~cm}$ and height $50 \mathrm{~cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50 \mathrm{~cm} \times 50 \mathrm{~cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of $250 \mathrm{~cm}^3 \mathrm{~s}^{-1}$. What is the value of the capacitance of the container after 10 seconds?
[Given: Permittivity of free space $\epsilon_0=9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]
[Given: Permittivity of free space $\epsilon_0=9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]
6
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 $T_{\mathrm{A}} / T_{\mathrm{B}}$ ?
7
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 $h_{\mathrm{Q}}$ ?
8
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 $310 \mathrm{~nm}$, the value of $Z$ is _________.
[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]
[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]
9
An optical arrangement consists of two concave mirrors $M_1$ and $M_2$, and a convex lens $L$ with a common principal axis, as shown in the figure. The focal length of $\mathrm{L}$ is $10 \mathrm{~cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \mathrm{~cm}$ and $24 \mathrm{~cm}$, respectively. The distance between $L$ and $M_2$ is $20 \mathrm{~cm}$. A point object $S$ is placed at the mid-point between $\mathrm{L}$ and $M_2$ on the axis. When the distance between $\mathrm{L}$ and $\mathrm{M}_1$ is $n / 7 \mathrm{~cm}$, one of the images coincides with $\mathrm{S}$. The value of $n$ is _______.
10
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 $n \%$. The value of $n$ is ______.
11
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. The change in its internal energy is _______ Joule.
12
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 $60 \mathrm{~cm} \mathrm{~s}^{-1}$, the speed of the tip of the person's shadow on the ground with respect to the person is ____________ $\mathrm{cm}~ \mathrm{s}^{-1}$.
13
Two point-like objects of masses $20 ~\mathrm{gm}$ and $30 ~\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10 \mathrm{~cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8} \mathrm{~N} \mathrm{~m} ~\mathrm{rad}^{-1}$. The angular frequency of the oscillations in $n \times 10^{-3} ~\mathrm{rad} ~\mathrm{s}^{-1}$. The value of $n$ is _________ .
14
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.
| List - I | List - II |
|---|---|
| (P) ${ }_{92}^{238} U \rightarrow{ }_{91}^{234} \mathrm{~Pa}$ | (1) one $\alpha$ particle and one $\beta^{+}$particle |
| (Q) ${ }_{82}^{214} \mathrm{~Pb} \rightarrow{ }_{82}^{210} \mathrm{~Pb}$ | (2) three $\beta^{-}$particles and one $\alpha$ particle |
| (R) ${ }_{81}^{210} \mathrm{Tl} \rightarrow{ }_{82}^{206} \mathrm{~Pb}$ | (3) two $\beta^{-}$particles and one $\alpha$ particle |
| (S) ${ }_{91}^{228} \mathrm{~Pa} \rightarrow{ }_{88}^{224} \mathrm{Ra}$ | (4) one $\alpha$ particle and one $\beta^{-}$particle |
| (5) one $\alpha$ particle and two $\beta^{+}$particles |
15
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 $\frac{h c}{e}=1.24 \times 10^{-6} \mathrm{~V}-\mathrm{m}$ ]
[Given: Wien's constant as $2.9 \times 10^{-3} \mathrm{~m}-\mathrm{K}$ and $\frac{h c}{e}=1.24 \times 10^{-6} \mathrm{~V}-\mathrm{m}$ ]
| List - I | List - II |
|---|---|
| (P) $2000 \mathrm{~K}$ | (1) The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function $4 \mathrm{eV}$. |
| (Q) $3000 \mathrm{~K}$ | (2) The radiation at peak wavelength is visible to human eye. |
| (R) $5000 \mathrm{~K}$ | (3) The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction. |
| (S) $10000 \mathrm{~K}$ | (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. |
16
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.
| List - I | List - II |
|---|---|
| (P) $I_0$ in $\mathrm{mA}$ | (1) 44.4 |
| (Q) The quality factor of the circuit | (2) 18 |
| (R) The bandwidth of the circuit in $\mathrm{rad}~ \mathrm{s}^{-1}$ | (3) 400 |
| (S) The peak power dissipated at resonance in Watt | (4) 2250 |
| (5) 500 |
17
A thin conducting rod $M N$ of mass $20 ~\mathrm{gm}$, length $25 \mathrm{~cm}$ and resistance $10 ~\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0=4 \mathrm{~T}$ directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t=0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-I with an appropriate value from List-II, and choose the correct option.
[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $e^{-1}=0.4$ ]
[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $e^{-1}=0.4$ ]
| List - I | List - II |
|---|---|
| (P) At $t=0.2 \mathrm{~s}$, the magnitude of the induced emf in Volt | (1) 0.07 |
| (Q) At $t=0.2 \mathrm{~s}$, the magnitude of the magnetic force in Newton | (2) 0.14 |
| (R) At $t=0.2 \mathrm{~s}$, the power dissipated as heat in Watt | (3) 1.20 |
| (S) The magnitude of terminal velocity of the rod in $\mathrm{m} \mathrm{s}^{-1}$ | (4) 0.12 |
| (5) 2.00 |
1
JEE Advanced 2023 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3 h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0=u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1 / \sqrt{3}$. Which of the following statement(s) is(are) correct?
2
JEE Advanced 2023 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is $\delta=60^{\circ}$ (see Figure-1). The angle of minimum deviation for red light from the same prism is $\delta_{\min }=30^{\circ}$ (see Figure-2). The refractive index of the prism material for blue light is $\sqrt{3}$. Which of the following statement(s) is(are) correct?
3
JEE Advanced 2023 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement(s) is(are) correct?
[Given: $e^{-1}=0.36$ ]
[Given: $e^{-1}=0.36$ ]
4
JEE Advanced 2023 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
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 v. After this elastic collision, the bar rotates with an angular velocity $\omega$.
Which of the following statement is correct?
Which of the following statement is correct?
Subject
Chemistry
17
Mathematics
17
Physics
17
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