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JEE-MAIN EXAMINATION – APRIL 2024

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Thursday 04th April 2024, Time: 9:00 AM to 12:00 NOON

JEE Main

Mathematics, Physics, Chemistry

Morning Session

3 hours

Paper Overview

Total Questions

Correct

Incorrect

N/A

iExplain doesn't support the question format

Complete Solutions

Q#	Explanation	Question	Correct	Solution	Status
1	View ExplanationExplain	Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function given by $f(x)=\left\{\begin{array}{lll}\frac{1-\cos 2 x}{x^{2}} & , & x<0 \\ \alpha & , & x=0, \text { where } \alpha, \beta \in R . \text { If } \\ \frac{\beta \sqrt{1-\cos x}}{x} & , & x>0\end{array}\right.$ $f$ is continuous at $\mathrm{x}=0$ , then $\alpha^{2}+\beta^{2}$ is equal to: (A) 48 (B) 12 (C) 3 (D) 6	2	Available	Available
2	View ExplanationExplain	Three urns $\mathrm{A}, \mathrm{B}$ and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is : (A) $\frac{4}{17}$ (B) $\frac{5}{18}$ (C) $\frac{7}{18}$ (D) $\frac{5}{16}$	2	Available	Available
3	View ExplanationExplain	The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$ . A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is : (A) $x-y-(2+\sqrt{2})=0$ (B) $-x+y-(2-\sqrt{2})=0$ (C) $x+y-(2-\sqrt{2})=0$ (D) $x+y+(2-\sqrt{2})=0$	3	Available	Available
4	View ExplanationExplain	If the solution $y=y(x)$ of the differential equation $\left(x^{4}+2 x^{3}+3 x^{2}+2 x+2\right) d y-\left(2 x^{2}+2 x+3\right) d x=0$ satisfies $y(-1)=-\frac{\pi}{4}$ , then $y(0)$ is equal to: (A) $-\frac{\pi}{12}$ (B) 0 (C) $\frac{\pi}{4}$ (D) $\frac{\pi}{2}$	3	Available	Available
5	View ExplanationExplain	Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^{2}-3 x+8}{2 x^{2}+3 x+8}$ be $\frac{m}{n}$ , where $\operatorname{gcd}(m, n)=1$ . Then $m+n$ is equal to: (A) 182 (B) 217 (C) 195 (D) 201	4	Available	Available
6	View ExplanationExplain	One of the points of intersection of the curves $\mathrm{y}=1+3 \mathrm{x}-2 \mathrm{x}^{2}$ and $\mathrm{y}=\frac{1}{\mathrm{x}}$ is $\left(\frac{1}{2}, 2\right)$ . Let the area of the region enclosed by these curves be $\frac{1}{24}(\ell \sqrt{5}+m)-n \log _{e}(1+\sqrt{5})$ , where $\ell, m, n \in$ N . Then $\ell+\mathrm{m}+\mathrm{n}$ is equal to (A) 32 (B) 30 (C) 29 (D) 31	2	Available	Available
7	View ExplanationExplain	If the system of equations $x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0$ $x+(\cos \alpha) y+(\sin \alpha) z=0$ $x+(\sin \alpha) y-(\cos \alpha) z=0$ has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to : (A) $\frac{3 \pi}{4}$ (B) $\frac{7 \pi}{24}$ (C) $\frac{5 \pi}{24}$ (D) $\frac{11 \pi}{24}$	3	Available	Available
8	View ExplanationExplain	There are 5 points $\mathrm{P}_{1}, \mathrm{P}_{2}, \mathrm{P}_{3}, \mathrm{P}_{4}, \mathrm{P}_{5}$ on the side AB , excluding A and B , of a triangle ABC . Similarly there are 6 points $\mathrm{P}_{6}, \mathrm{P}_{7}, \ldots, \mathrm{P}_{11}$ on the side BC and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{18}$ as vertices, is : (A) 776 (B) 751 (C) 796 (D) 771	2	Available	Available
9	View ExplanationExplain	Let $f(\mathrm{x})=\left\{\begin{array}{cc}-2, & -2 \leq \mathrm{x} \leq 0 \\ \mathrm{x}-2, & 0<\mathrm{x} \leq 2\end{array}\right.$ and $\mathrm{h}(\mathrm{x})=\mathrm{f}(\|\mathrm{x}\|)+\|\mathrm{f}(\mathrm{x})\|$ . Then $\int_{-2}^{2} h(x) d x$ is equal to : (A) 2 (B) 4 (C) 1 (D) 6	1	Available	Available
10	View ExplanationExplain	The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to : (A) 3133 (B) 633 (C) 931 (D) 6131	1	Available	Available
11	View ExplanationExplain	Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\vec{C}$ . Then $\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)$ is : (A) $-\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{\sqrt{2}}{3} \hat{\mathrm{j}}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{\mathrm{k}}$ (B) $\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{1}{2} \hat{\mathrm{k}}$ (C) $\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{\mathrm{i}}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{\mathrm{j}}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{\mathrm{k}}$ (D) $\frac{\sqrt{2}}{3} \hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{k}}$	4	Available	Available
12	View ExplanationExplain	Let the first three terms $2, p$ and $q$ , with $q \neq 2$ , of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $n^{\text {th }}$ term of the A.P., then n is equal to (A) 151 (B) 169 (C) 177 (D) 163	4	Available	Available
13	View ExplanationExplain	Let $a, b \in R$ . Let the mean and the variance of 6 observations $-3,4,7,-6, \mathrm{a}, \mathrm{b}$ be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is : (A) $\frac{13}{3}$ (B) $\frac{16}{3}$ (C) $\frac{11}{3}$ (D) $\frac{14}{3}$	1	Available	Available
14	View ExplanationExplain	If 2 and 6 are the roots of the equation $\mathrm{ax}^{2}+\mathrm{bx}+1=0$ , then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$ , is : (A) $2 \mathrm{x}^{2}+11 \mathrm{x}+12=0$ (B) $4 x^{2}+14 x+12=0$ (C) $\mathrm{x}^{2}+10 \mathrm{x}+16=0$ (D) $x^{2}+8 x+12=0$	4	Available	Available
15	View ExplanationExplain	Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+\|z\|=0, z \in C$ . Then $4\left(\alpha^{2}+\beta^{2}\right)$ is equal to: (A) 6 (B) 4 (C) 8 (D) 2	2	Available	Available
16	View ExplanationExplain	Let the point, on the line passing through the points $\mathrm{P}(1,-2,3)$ and $\mathrm{Q}(5,-4,7)$ , farther from the origin and at a distance of 9 units from the point P , be $(\alpha, \beta, \gamma)$ . Then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to: (A) 155 (B) 150 (C) 160 (D) 165	1	Available	Available
17	View ExplanationExplain	A square is inscribed in the circle $x^{2}+y^{2}-10 x-6 y+30=0$ . One side of this square is parallel to $y=x+3$ . If $\left(x_{i}, y_{i}\right)$ are the vertices of the square, then $\sum\left(x_{i}^{2}+y_{i}^{2}\right)$ is equal to: (A) 148 (B) 156 (C) 160 (D) 152	4	Available	Available
18	View ExplanationExplain	If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{e}\left(\frac{3 x^{2}-8 x+5}{x^{2}-3 x-10}\right)$ is $\quad(\alpha, \beta]$ , then $3 \alpha+10 \beta$ is equal to : (A) 97 (B) 100 (C) 95 (D) 98	1	Available	Available
19	View ExplanationExplain	Let $f(\mathrm{x})=\mathrm{x}^{5}+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in \mathrm{R}$ . Consider a function $\mathrm{g}(\mathrm{x})$ such that (gof) $(\mathrm{x})=\mathrm{x}$ for all $\mathrm{x} \in \mathrm{R}$ . Then the value of $8 \mathrm{~g}^{\prime}(2)$ is : (A) 16 (B) 4 (C) 8 (D) 2	1	Available	Available
20	View ExplanationExplain	Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$ . If $\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^{8}$ , then $(\operatorname{det}(\mathrm{A}))^{2}$ is equal to : (A) 1 (B) 49 (C) 16 (D) 36	3	Available	Available
21	View ExplanationExplain	If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}$ , where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$ , then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to $\qquad$	(100)	Available	Available
22	View ExplanationExplain	In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $m+n$ is equal to $\qquad$	(45)	Available	Available
23	View ExplanationExplain	Let the solution $y=y(x)$ of the differential equation $\frac{d y}{d x}-y=1+4 \sin x$ satisfy $y(\pi)=1$ . Then $\mathrm{y}\left(\frac{\pi}{2}\right)+10$ is equal to $\qquad$	(7)	Available	Available
24	View ExplanationExplain	If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k} \quad$ and $\quad \int_{0}^{\mathrm{k}}\left[\mathrm{x}^{2}\right] \mathrm{dx}=\alpha-\sqrt{\alpha}, \quad$ where denotes the greatest integer function, then $6 \alpha^{3}$ is equal to $\qquad$	(48)	Available	Available
25	View ExplanationExplain	Let A be a square matrix of order 2 such that $\|\mathrm{A}\|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $A^{2}+x A+y I=0$ lie on a hyperbola, whose transverse axis is parallel to the $x$ -axis, eccentricity is $e$ and the length of the latus rectum is $\ell$ , then $\mathrm{e}^{4}+\ell^{4}$ is equal to $\qquad$	(25)	Available	Available
26	View ExplanationExplain	Let $\mathrm{a}=1+\frac{{ }^{2} \mathrm{C}_{2}}{3!}+\frac{{ }^{3} \mathrm{C}_{2}}{4!}+\frac{{ }^{4} \mathrm{C}_{2}}{5!}+\ldots$ , $\mathrm{b}=1+\frac{{ }^{1} \mathrm{C}_{0}+{ }^{1} \mathrm{C}_{1}}{1!}+\frac{{ }^{2} \mathrm{C}_{0}+{ }^{2} \mathrm{C}_{1}+{ }^{2} \mathrm{C}_{2}}{2!}+\frac{{ }^{3} \mathrm{C}_{0}+{ }^{3} \mathrm{C}_{1}+{ }^{3} \mathrm{C}_{2}+{ }^{3} \mathrm{C}_{3}}{3!}+\ldots$ Then $\frac{2 b}{a^{2}}$ is equal to $\qquad$	(8)	Available	Available
27	View ExplanationExplain	Let A be a $3 \times 3$ matrix of non-negative real elements such that $\mathrm{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ . Then the maximum value of $\operatorname{det}(\mathrm{A})$ is $\qquad$	(27)	Available	Available
28	View ExplanationExplain	Let the length of the focal chord $P Q$ of the parabola $y^{2}=12 x$ be 15 units. If the distance of $P Q$ from the origin is $p$ , then $10 p^{2}$ is equal to $\qquad$	(72)	Available	Available
29	View ExplanationExplain	Let ABC be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-7 \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{\mathrm{i}}+\mathrm{b} \hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0$ . Then the square of the length of the largest side of the triangle ABC is	(54)	Available	Available
30	View ExplanationExplain	If $\int_{0}^{\frac{\pi}{4}} \frac{\sin ^{2} \mathrm{x}}{1+\sin \mathrm{x} \cos \mathrm{x}} \mathrm{dx}=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}$ , where a , $\mathrm{b} \in \mathrm{N}$ , then $\mathrm{a}+\mathrm{b}$ is equal to $\qquad$	(8)	Available	Available
31	View ExplanationExplain	An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then : (A) the electron will be accelerated along the axis. (B) the electron will continue to move with uniform velocity along the axis of the solenoid. (C) the electron path will be circular about the axis. (D) the electron will experience a force at $45^{\circ}$ to the axis and execute a helical path.	2	Available	Available
32	View ExplanationExplain	The electric field in an electromagnetic wave is given by $\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}} 40 \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right) \mathrm{NC}^{-1} . \quad$ The magnetic field induction of this wave is (in SI unit): (A) $\vec{B}=\hat{i} \frac{40}{c} \cos \omega\left(t-\frac{z}{c}\right)$ (B) $\vec{B}=\hat{j} 40 \cos \omega\left(t-\frac{z}{c}\right)$ (C) $\overrightarrow{\mathrm{B}}=\hat{\mathrm{k}} \frac{40}{\mathrm{c}} \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right)$ (D) $\vec{B}=\hat{j} \frac{40}{c} \cos \omega\left(t-\frac{z}{c}\right)$	4	Available	Available
33	View ExplanationExplain	Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left({ }_{0}^{1} \mathrm{n}\right)$ and uranium isotope $\left({ }_{92}^{235} \mathrm{U}\right)$ is correct: (A) ${ }_{56}^{144} \mathrm{Ba}+{ }_{36}^{89} \mathrm{Kr}+4{ }_{0}^{1} \mathrm{n}$ (B) ${ }_{56}^{140} \mathrm{Xe}+{ }_{38}^{94} \mathrm{Sr}+3{ }_{0}^{1} \mathrm{n}$ (C) ${ }_{51}^{153} \mathrm{Sb}+{ }_{41}^{99} \mathrm{Nb}+3{ }_{0}^{1} \mathrm{n}$ (D) ${ }_{56}^{144} \mathrm{Ba}+{ }_{36}^{89} \mathrm{Kr}+3{ }_{0}^{1} \mathrm{n}$	4	Available	Available
34	View ExplanationExplain	In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image ( v ) are $\Delta \mathrm{u}$ and $\Delta \mathrm{v}$ , respectively. The error in the measurement of the focal length of the convex lens will be : (A) $\frac{\Delta u}{u}+\frac{\Delta v}{v}$ (B) $f^{2}\left[\frac{\Delta u}{u^{2}}+\frac{\Delta v}{v^{2}}\right]$ (C) $2 f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$ (D) $f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$	2	Available	Available
35	View ExplanationExplain	Given below are two statements : Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $P_{1}-P_{2}=\rho g\left(h_{2}-h_{1}\right)$ Statement II : In ventury tube shown $2 \mathrm{gh}=v_{1}^{2}-v_{2}^{2}$ ![] In the light of the above statements, choose the most appropriate answer from the options given below. (A) Both Statement I and Statement II are correct. (B) Statement I is incorrect but Statement II is correct. (C) Both Statement I and Statement II are incorrect. (D) Statement I is correct but Statement II is incorrect.	4	Diagram Question	Available
36	View ExplanationExplain	The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8 \Omega$ and $10 \Omega$ respectively. After inserting in a hot bath of temperature $400^{\circ} \mathrm{C}$ , the resistance of platinum wire is : (A) $2 \Omega$ (B) $16 \Omega$ (C) $8 \Omega$ (D) $10 \Omega$	2	Available	Available
37	View ExplanationExplain	A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is: (A) $\frac{\mathrm{GMm} \pi}{2 \mathrm{~L}^{2}}$ (B) 0 (C) $\frac{\mathrm{GmM} \pi^{2}}{\mathrm{~L}^{2}}$ (D) $\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^{2}}$	4	Available	Available
38	View ExplanationExplain	On celcius scale the temperature of body increases by $40^{\circ} \mathrm{C}$ . The increase in temperature on Fahrenheit scale is: (A) $70^{\circ} \mathrm{F}$ (B) $68^{\circ} \mathrm{F}$ (C) $72^{\circ} \mathrm{F}$ (D) $75^{\circ} \mathrm{F}$	3	Available	Available
39	View ExplanationExplain	An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D . Focal length of each of the convex lens is : (A) 20 cm (B) 50 cm (C) 500 cm (D) 25 cm	1	Available	Available
40	View ExplanationExplain	Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light ( $\mathrm{I}_{1}< \mathrm{I}_{2}$ ) of same wavelengths :	3	Diagram Question	Available
41	View ExplanationExplain	A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t . Which of the following curve best describes acceleration of the block with time :	2	Diagram Question	Available
42	View ExplanationExplain	If a rubber ball falls from a height $h$ and rebounds upto the height of $\mathrm{h} / 2$ . The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are : (A) $50 \%, \sqrt{\frac{\mathrm{gh}}{2}}$ (B) $50 \%, \sqrt{\mathrm{gh}}$ (C) $40 \%, \sqrt{2 \mathrm{gh}}$ (D) $50 \%, \sqrt{2 \mathrm{gh}}$	4	Available	Available
43	View ExplanationExplain	The equation of stationary wave is : $y=2 a \sin \left(\frac{2 \pi n t}{\lambda}\right) \cos \left(\frac{2 \pi x}{\lambda}\right)$ Which of the following is NOT correct (A) The dimensions of nt is [L] (B) The dimensions of $n$ is $\left[L^{-1}\right]$ (C) The dimensions of $n / \lambda$ is $[T]$ (D) The dimensions of $x$ is $[L]$	3	Available	Available
44	View ExplanationExplain	A body travels 102.5 m in $\mathrm{n}^{\text {th }}$ second and 115.0 m in $(n+2)^{\text {th }}$ second. The acceleration is : (A) $9 \mathrm{~m} / \mathrm{s}^{2}$ (B) $6.25 \mathrm{~m} / \mathrm{s}^{2}$ (C) $12.5 \mathrm{~m} / \mathrm{s}^{2}$ (D) $5 \mathrm{~m} / \mathrm{s}^{2}$	2	Available	Available
45	View ExplanationExplain	To measure the internal resistance of a battery, potentiometer is used. For $\mathrm{R}=10 \Omega$ , the balance point is observed at $\ell=500 \mathrm{~cm}$ and for $\mathrm{R}=1 \Omega$ the balance point is observed at $\ell=400 \mathrm{~cm}$ . The internal resistance of the battery is approximately : (A) $0.2 \Omega$ (B) $0.4 \Omega$ (C) $0.1 \Omega$ (D) $0.3 \Omega$	4	Available	Available
46	View ExplanationExplain	An infinitely long positively charged straight thread has a linear charge density $\lambda \mathrm{Cm}^{-1}$ . An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :	2	Diagram Question	Available
47	View ExplanationExplain	The value of net resistance of the network as shown in the given figure is : (A) $\left(\frac{5}{2}\right) \Omega$ (B) $\left(\frac{15}{4}\right) \Omega$ (C) $6 \Omega$ (D) $\left(\frac{30}{11}\right) \Omega$	3	Diagram Question	Available
48	View ExplanationExplain	P-T diagram of an ideal gas having three different densities $\rho_{1}, \rho_{2}, \rho_{3}$ (in three different cases) is shown in the figure. Which of the following is correct : (A) $\rho_{2}<\rho_{3}$ (B) $\rho_{1}>\rho_{2}$ (C) $\rho_{1}<\rho_{2}$ (D) $\rho_{1}=\rho_{2}=\rho_{3}$	2	Diagram Question	Available
49	View ExplanationExplain	The co-ordinates of a particle moving in $x-y$ plane are given by: $x=2+4 t, y=3 t+8 t^{2}$ . The motion of the particle is : (A) non-uniformly accelerated. (B) uniformly accelerated having motion along a straight line. (C) uniform motion along a straight line. (D) uniformly accelerated having motion along a parabolic path.	4	Available	Available
50	View ExplanationExplain	In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to : (A) pure inductor. (B) pure capacitor. (C) pure resistor. (D) combination of an inductor and capacitor. Choose the correct answer from the options given below :	4	Available	Available
51	View ExplanationExplain	An infinite plane sheet of charge having uniform surface charge density $+\sigma_{s} \mathrm{C} / \mathrm{m}^{2}$ is placed on x-y plane. Another infinitely long line charge having uniform linear charge density $+\lambda_{\mathrm{e}} \mathrm{C} / \mathrm{m}$ is placed at $\mathrm{z}=4 \mathrm{~m}$ plane and parallel to y -axis. If the magnitude values $\left\|\sigma_{\mathrm{s}}\right\|=2\left\|\lambda_{\mathrm{e}}\right\|$ then at point ( $0,0,2$ ), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi \sqrt{n}: 1$ . The value of $n$ is $\qquad$ .	(16)	Available	Available
52	View ExplanationExplain	A hydrogen atom changes its state from $n=3$ to $\mathrm{n}=2$ . Due to recoil, the percentage change in the wave length of emitted light is approximately $1 \times 10^{-n}$ . The value of $n$ is $\qquad$ . [Given Rhc $=13.6 \mathrm{eV}, \mathrm{hc}=1242 \mathrm{eV} \mathrm{nm}$ , $\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J}$ s, mass of the hydrogen atom $\left.=1.6 \times 10^{-27} \mathrm{~kg}\right]$	(7)	Available	Available
53	View ExplanationExplain	The magnetic field existing in a region is given by $\overrightarrow{\mathrm{B}}=0.2(1+2 \mathrm{x}) \hat{\mathrm{k}} \mathrm{T}$ . A square loop of edge 50 cm carrying 0.5 A current is placed in $x-y$ plane with its edges parallel to the $x-y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is $\qquad$ mN.	(50)	Diagram Question	Available
54	View ExplanationExplain	A alternating current at any instant is given by $\mathrm{i}=\left[6+\sqrt{56} \sin \left(100 \pi \mathrm{t}+\frac{\pi}{3}\right)\right]$ A. The rms value of the current is $\qquad$ A.	(8)	Available	Available
55	View ExplanationExplain	Twelve wires each having resistance $2 \Omega$ are joined to form a cube. A battery of 6 V emf is joined across point $a$ and $c$ . The voltage difference between e and f is $\qquad$ V.	(1)	Diagram Question	Available
56	View ExplanationExplain	A soap bubble is blown to a diameter of 7 cm . 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne $/ \mathrm{cm}$ then the new radius is $\qquad$ cm. Take $:\left(\pi=\frac{22}{7}\right)$ .	(7)	Available	Available
57	View ExplanationExplain	Two wavelengths $\lambda_{1}$ and $\lambda_{2}$ are used in Young's double slit experiment $\lambda_{1}=450 \mathrm{~nm}$ and $\lambda_{2}=650 \mathrm{~nm}$ . The minimum order of fringe produced by $\lambda_{2}$ which overlaps with the fringe produced by $\lambda_{1}$ is $n$ . The value of $n$ is $\qquad$ .	(9)	Available	Available
58	View ExplanationExplain	An elastic spring under tension of 3 N has a length Its length is b under tension 2 N . For its length $(3 a-2 b)$ , the value of tension will be $\qquad$ N.	(5)	Available	Available
59	View ExplanationExplain	Two forces $\overrightarrow{\mathrm{F}}_{1}$ and $\overrightarrow{\mathrm{F}}_{2}$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\overrightarrow{\mathrm{F}}_{1}$ and $\overrightarrow{\mathrm{F}}_{2}$ is $\cos ^{-1}\left(\frac{1}{\mathrm{n}}\right)$ . The value of $\|\mathrm{n}\|$ is $\qquad$ .	(6)	Available	Available
60	View ExplanationExplain	A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights $h_{1}$ and $h_{2}$ , respectively, above the initial level. The ratio $h_{1}: h_{2}$ is $\frac{n}{10}$ . The value of n is $\qquad$ .	(7)	Available	Available
61	View ExplanationExplain	What pressure (bar) of $\mathrm{H}_{2}$ would be required to make emf of hydrogen electrode zero in pure water at $25^{\circ} \mathrm{C}$ ? (A) $10^{-14}$ (B) $10^{-7}$ (C) 1 (D) 0.5 \section*{NTA Ans. (3)}	1	Available	Available
62	View ExplanationExplain	The correct sequence of ligands in the order of decreasing field strength is : (A) $\mathrm{CO}>\mathrm{H}_{2} \mathrm{O}>\mathrm{F}^{-}>\mathrm{S}^{2-}$ (B) ${ }^{-} \mathrm{OH}>\mathrm{F}^{-}>\mathrm{NH}_{3}>\mathrm{CN}^{-}$ (C) $\mathrm{NCS}^{-}>\mathrm{EDTA}^{4-}>\mathrm{CN}^{-}>\mathrm{CO}$ (D) $\mathrm{S}^{2-}>^{-} \mathrm{OH}>\mathrm{EDTA}^{4-}>\mathrm{CO}$	1	Available	Available
63	View ExplanationExplain	Match List -I with List II: Choose the correct answer from the options given below:	1	Diagram Question	Available
64	View ExplanationExplain	What will be the decreasing order of basic strength of the following conjugate bases? ${ }^{-} \mathrm{OH}, \mathrm{R} \overline{\mathrm{O}}, \mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}, \mathrm{C} \overline{\mathrm{l}}$ (A) $\mathrm{C} \overline{1}>\mathrm{OH}>\mathrm{R} \overline{\mathrm{O}}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}$ (B) $\mathrm{R} \overline{\mathrm{O}}>{ }^{-} \mathrm{OH}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}>\mathrm{C} \overline{1}$ (C) ${ }^{-} \mathrm{OH}>\mathrm{R} \overline{\mathrm{O}}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}>\mathrm{C} \overline{1}$ (D) $\mathrm{C} \overline{\mathrm{l}}>\mathrm{R} \overline{\mathrm{O}}>{ }^{-} \mathrm{OH}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}$	2	Available	Available
65	View ExplanationExplain	In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to: (A) prevent interference by phosphate ions (B) decrease concentration of ${ }^{-} \mathrm{OH}$ ions (C) increase concentration of $\mathrm{Cl}^{-}$ ions (D) increase concentration of $\mathrm{NH}_{4}{ }^{+}$ ions	2	Available	Available
66	View ExplanationExplain	Identify(B) and (C) and how are(A) and (C) related?	3	Diagram Question	Available
67	View ExplanationExplain	One of the commonly used electrode is calomel electrode. Under which of the following categories calomel electrode comes ? (A) Metal - Insoluble Salt - Anion electrodes (B) Oxidation - Reduction electrodes (C) Gas - Ion electrodes (D) Metal ion - Metal electrodes	1	Available	Available
68	View ExplanationExplain	Number of complexes from the following with even number of unpaired "d" electrons is $\qquad$ . $\left[\mathrm{V}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}, \quad\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}, \quad\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$ , $\left[\mathrm{Ni}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+},\left[\mathrm{Cu}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$ [Given atomic numbers: $\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Fe}=26$ , $\mathrm{Ni}=28, \mathrm{Cu}=29$ ] (A) 2 (B) 4 (C) 5 (D) 1	1	Available	Available
69	View ExplanationExplain	Which one of the following molecules has maximum dipole moment? (A) $\mathrm{NF}_{3}$ (B) $\mathrm{CH}_{4}$ (C) $\mathrm{NH}_{3}$ (D) $\mathrm{PF}_{5}$	3	Available	Available
70	View ExplanationExplain	Number of molecules/ions from the following in which the central atom is involved in $\mathrm{sp}^{3}$ hybridization is $\qquad$ . $\mathrm{NO}_{3}^{-}, \mathrm{BCl}_{3}, \mathrm{ClO}_{2}^{-}, \mathrm{ClO}_{3}$ (A) 2 (B) 4 (C) 3 (D) 1	1	Available	Available
71	View ExplanationExplain	Which among the following is incorrect statement? (A) Electromeric effect dominates over inductive effect (B) The electromeric effect is, temporary effect (C) The organic compound shows electromeric effect in the presence of the reagent only (D) Hydrogen ion $\left(\mathrm{H}^{+}\right)$ shows negative electromeric effect	4	Available	Available
72	View ExplanationExplain	Given below are two statements : Statement I: Acidity of $\alpha$ -hydrogens of aldehydes and ketones is responsible for Aldol reaction. Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product. In the light of above statements, choose the most appropriate answer from the options given below. (A) Both Statement I and Statement II are correct. (B) Both Statement I and Statement II are incorrect. (C) Statement I is incorrect but Statement II is correct. (D) Statement I is correct but Statement II is incorrect.	4	Available	Available
73	View ExplanationExplain	Which of the following nitrogen containing compound does not give Lassaigne's test ? (A) Phenyl hydrazine (B) Glycene (C) Urea (D) Hydrazine	4	Available	Available
74	View ExplanationExplain	Which of the following is the correct structure of L-Glucose?	1	Diagram Question	Available
75	View ExplanationExplain	The element which shows only one oxidation state other than its elemental form is : (A) Cobalt (B) Scandium (C) Titanium (D) Nickel	2	Available	Available
76	View ExplanationExplain	Identify the product in the following reaction :	4	Diagram Question	Available
77	View ExplanationExplain	Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is $\qquad$ . B, C, N, S, O, F, P, Al, Si (A) 7 (B) 5 (C) 6 (D) 3	4	Available	Available
78	View ExplanationExplain	The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is : (Given : Molar Mass $\mathrm{Na}: 23$ and $\mathrm{Cl}: 35.5 \mathrm{gmol}^{-1}$ ) (A) 20 (B) 0.2 (C) 2 (D) 4	2	Available	Available
79	View ExplanationExplain	Identify the correct set of reagents or reaction conditions ' $X$ ' and ' $Y$ ' in the following set of transformation. (A) $\mathrm{X}=$ conc.alc. $\mathrm{NaOH}, 80^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{Br}_{2} / \mathrm{CHCl}_{3}$ (B) $\mathrm{X}=$ dil.aq. $\mathrm{NaOH}, 20^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{HBr}$ /acetic acid (C) $\mathrm{X}=$ conc.alc. $\mathrm{NaOH}, 80^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{HBr}$ /acetic acid (D) $\mathrm{X}=$ dil.aq. $\mathrm{NaOH}, 20^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{Br}_{2} / \mathrm{CHCl}_{3}$	3	Diagram Question	Available
80	View ExplanationExplain	The correct order of first ionization enthalpy values of the following elements is : (A) O (B) N (C) Be (D) F (E) B Choose the correct answer from the options given below:	2	Available	Available
81	View ExplanationExplain	The enthalpy of formation of ethane $\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)$ from ethylene by addition of hydrogen where the bondenergies of $\mathrm{C}-\mathrm{H}, \mathrm{C}-\mathrm{C}, \mathrm{H}-\mathrm{H}$ are $414 \mathrm{~kJ}, 347 \mathrm{~kJ}$ , 615 kJ and 435 kJ respectively is - $\qquad$ kJ .	(125)	Available	Available
82	View ExplanationExplain	The number of correct reaction(s) among the following is $\qquad$ .	1	Diagram Question	Available
83	View ExplanationExplain	Xg of ethylamine is subjected to reaction with $\mathrm{NaNO}_{2} / \mathrm{HCl}$ followed by water; evolved dinitrogen gas which occupied 2.24 L volume at STP. X is $\qquad$ $\times 10^{-1} \mathrm{~g}$ .	(45)	Available	Available
84	View ExplanationExplain	The de-Broglie's wavelength of an electron in the $4^{\text {th }}$ orbit is $\qquad$ $\pi \mathrm{a}_{0} .\left(\mathrm{a}_{0}=\right.$ Bohr's radius $)$	(8)	Available	Available
85	View ExplanationExplain	Only 2 mL of $\mathrm{KMnO}_{4}$ solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid ( 2 M ) in acidic medium. The molarity of $\mathrm{KMnO}_{4}$ solution should be $\qquad$ M.	(50)	Available	Available
86	View ExplanationExplain	Consider the following reaction $\mathrm{MnO}_{2}+\mathrm{KOH}+\mathrm{O}_{2} \rightarrow \mathrm{~A}+\mathrm{H}_{2} \mathrm{O}$ . Product 'A' in neutral or acidic medium disproportionate to give products ' B ' and ' C ' along with water. The sum of spin-only magnetic moment values of B and C is $\qquad$ BM. (nearest integer) (Given atomic number of Mn is 25 )	(4)	Available	Available
87	View ExplanationExplain	Consider the following transformation involving first order elementary reaction in each step at constant temperature as shown below. Some details of the above reaction are listed below. If the overall rate constant of the above transformation ( k ) is given as $\mathrm{k}=\frac{\mathrm{k}_{1} \mathrm{k}_{2}}{\mathrm{k}_{3}}$ and the overall activation energy ( $\mathrm{E}_{\mathrm{a}}$ ) is $400 \mathrm{~kJ} \mathrm{~mol}^{-1}$ , then the value of $E a_{3}$ is $\qquad$ $\mathrm{kJ} \mathrm{mol}^{-1}$ (nearest integer)	(100)	Diagram Question	Available
88	View ExplanationExplain	2.5 g of a non-volatile, non-electrolyte is dissolved in 100 g of water at $25^{\circ} \mathrm{C}$ . The solution showed a boiling point elevation by $2^{\circ} \mathrm{C}$ . Assuming the solute concentration in negligible with respect to the solvent concentration, the vapour pressure of the resulting aqueous solution is $\qquad$ mm of Hg (nearest integer) [Given : Molal boiling point elevation constant of water $\left(\mathrm{K}_{\mathrm{b}}\right)=0.52 \mathrm{K} .\mathrm{~kg} \mathrm{~mol}^{-1}$ , 1 atm pressure $=760 \mathrm{~mm}$ of Hg , molar mass of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$ ]	(707)	Available	Available
89	View ExplanationExplain	The number of different chain isomers for $\mathrm{C}_{7} \mathrm{H}_{16}$ is $\qquad$ .	(9)	Available	Available
90	View ExplanationExplain	Number of molecules/species from the following having one unpaired electron is $\qquad$ . $\mathrm{O}_{2}, \mathrm{O}_{2}^{-1}, \mathrm{NO}, \mathrm{CN}^{-1}, \mathrm{O}_{2}^{2-}$	(2)	Available	Available