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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Thursday 23rd January 2025, Time: 3:00 PM to 6:00 PM

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JEE Main
Mathematics, Physics, Chemistry
Evening Session
3 hours

Paper Overview

75
Total Questions
0
Correct
0
Incorrect
75
N/A
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Complete Solutions

Q#ExplanationQuestionCorrectSolutionStatus
1Explain
If in the expansion of (1+x)p(1−x)q(1+\mathrm{x})^{\mathrm{p}}(1-\mathrm{x})^{\mathrm{q}}, the coefficients of xx and x2x^{2} are 1 and -2 , respectively, then p2+q2\mathrm{p}^{2}+\mathrm{q}^{2} is equal to :
(A) 8
(B) 18
(C) 13
(D) 20
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2Explain
Let A={(x,y)∈R×R:∣x+y∣≥3}\mathrm{A}=\{(\mathrm{x}, \mathrm{y}) \in \mathbf{R} \times \mathbf{R}:|\mathrm{x}+\mathrm{y}| \geq 3\} and B={(x,y)∈R×R:∣x∣+∣y∣≤3}\mathrm{B}=\{(\mathrm{x}, \mathrm{y}) \in \mathbf{R} \times \mathbf{R}:|\mathrm{x}|+|\mathrm{y}| \leq 3\}. If C={(x,y)∈A∩B:x=0\mathrm{C}=\{(\mathrm{x}, \mathrm{y}) \in \mathbf{A} \cap \mathbf{B}: \mathrm{x}=0 or y=0}\mathrm{y}=0\}, then ∑(x,y)∈C∣x+y∣\sum_{(x, y) \in C}|x+y| is :
(A) 15
(B) 18
(C) 24
(D) 12
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3Explain
The system of equations x+y+z=6x+y+z=6, x+2y+5z=9x+2 y+5 z=9, x+5y+λz=μ\mathrm{x}+5 \mathrm{y}+\lambda \mathrm{z}=\mu, has no solution if
(A) λ=17,μ≠18\lambda=17, \mu \neq 18
(B) λ≠17,μ≠18\lambda \neq 17, \mu \neq 18
(C) λ=15,μ≠17\lambda=15, \mu \neq 17
(D) λ=17,μ=18\lambda=17, \mu=18
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4Explain
Let ∫x3sin⁡xdx=g(x)+C\int x^{3} \sin x d x=g(x)+C, where CC is the constant of integration. If 8(g(π2)+g′(π2))=απ3+βπ2+γ,α,β,γ∈Z8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^{3}+\beta \pi^{2}+\gamma, \alpha, \beta, \gamma \in Z, Then α+β−γ\alpha+\beta-\gamma equals :
(A) 55
(B) 47
(C) 48
(D) 62
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5Explain
A rod of length eight units moves such that its ends AA and BB always lie on the lines x−y+2=0x-y+2=0 and y+2=0\mathrm{y}+2=0, respectively. If the locus of the point P , that divides the rod⁡AB\operatorname{rod} \mathrm{AB} internally in the ratio 2:12: 1 is 9(x2+αy2+βxy+γx+28y)−76=09\left(x^{2}+\alpha y^{2}+\beta x y+\gamma x+28 y\right)-76=0, then α−β−γ\alpha-\beta-\gamma is equal to :
(A) 24
(B) 23
(C) 21
(D) 22
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6Explain
The distance of the line x−22=y−63=z−34\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4} from the point (1,4,0)(1,4,0) along the line x1=y−22=z+33\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3} is :
(A) 17\sqrt{17}
(B) 14\sqrt{14}
(C) 15\sqrt{15}
(D) 13\sqrt{13}
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7Explain
Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1(r>0)r: 1(r>0). If OO is the origin and (OQ→⋅OA→)−15∣OP→×OA→∣2=10(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^{2}=10, then the value of r is :
(A) 14
(B) 3
(C) 7\sqrt{7}
(D) 7
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8Explain
If the area of the region {(x,y):−1≤x≤1,0≤y≤a+e∣x∣−e−x,a>0}\left\{(\mathrm{x}, \mathrm{y}):-1 \leq \mathrm{x} \leq 1,0 \leq \mathrm{y} \leq \mathrm{a}+\mathrm{e}^{|\mathrm{x}|}-\mathrm{e}^{-\mathrm{x}}, \mathrm{a}>0\right\} is e2+8e+1e\frac{\mathrm{e}^{2}+8 \mathrm{e}+1}{\mathrm{e}}, then the value of a is :
(A) 7
(B) 6
(C) 8
(D) 5
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9Explain
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min81 \mathrm{~cm}^{3} / \mathrm{min} and the thickness of the ice-cream layer decreases at the rate of 14π cm/min\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}. The surface area (in cm2\mathrm{cm}^{2} ) of the chocolate ball (without the ice-cream layer) is :
(A) 225π225 \pi
(B) 128π128 \pi
(C) 196π196 \pi
(D) 256π256 \pi
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10Explain
A board has 16 squares as shown in the figure : Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
(A) 45\frac{4}{5}
(B) 710\frac{7}{10}
(C) 35\frac{3}{5}
(D) 2330\frac{23}{30}
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11Explain
Let x=x(y)\mathrm{x}=\mathrm{x}(\mathrm{y}) be the solution of the differential equation y=(x−ydxdy)sin⁡(xy),y>0y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right), y>0 and x (1) = pi2\frac{pi}{2}. Then cos(x(2) is equal to :
(A) 1−2(log⁡e2)21-2\left(\log _{\mathrm{e}} 2\right)^{2}
(B) 2(log⁡e2)2−12\left(\log _{\mathrm{e}} 2\right)^{2}-1
(C) 2(log⁡e2)−12\left(\log _{\mathrm{e}} 2\right)-1
(D) 1−2(log⁡e2)1-2\left(\log _{\mathrm{e}} 2\right)
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12Explain
Let the range of the function f(x)=6+16cos⁡x⋅cos⁡(π3−x)⋅cos⁡(π3+x)⋅sin⁡3x⋅cos⁡6x,x∈Rf(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in R be [α,β][\alpha, \beta]. Then the distance of the point (α,β)(\alpha, \beta) from the line 3x+4y+12=03 x+4 y+12=0 is :
(A) 11
(B) 8
(C) 10
(D) 9
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13Explain
Let the shortest distance from (a,0)(a, 0), a>0a>0, to the parabola y2=4xy^{2}=4 x be 4 . Then the equation of the circle passing through the point (a,0)(\mathrm{a}, 0) and the focus of the parabola, and having its centre on the axis of the parabola is:
(A) x2+y2−6x+5=0x^{2}+y^{2}-6 x+5=0
(B) x2+y2−4x+3=0x^{2}+y^{2}-4 x+3=0
(C) x2+y2−10x+9=0x^{2}+y^{2}-10 x+9=0
(D) x2+y2−8x+7=0x^{2}+y^{2}-8 x+7=0
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14Explain
Let X=R×RX=R \times R. Define a relation RR on XX as: (a1, b1)R(a2, b2)⇔b1=b2\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right) \mathrm{R}\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right) \Leftrightarrow \mathrm{b}_{1}=\mathrm{b}_{2}. Statement-I: R is an equivalence relation. Statement-II: For some (a,b)∈X(\mathrm{a}, \mathrm{b}) \in \mathrm{X}, the set S={(x,y)∈X:(x,y)R(a,b)}\mathrm{S}=\{(\mathrm{x}, \mathrm{y}) \in \mathrm{X}:(\mathrm{x}, \mathrm{y}) \mathrm{R}(\mathrm{a}, \mathrm{b})\} represents a line parallel to y=x\mathrm{y}=\mathrm{x}. In the light of the above statements, choose the correct answer from the options given below:
(A) Both Statement-I and Statement-II are false.
(B) Statement-I is true but Statement-II is false.
(C) Both Statement-I and Statement-II are true.
(D) Statement-I is false but Statement-II is true.
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15Explain
The length of the chord of the ellipse x24+y22=1\frac{x^{2}}{4}+\frac{y^{2}}{2}=1, whose mid-point is (1,12)\left(1, \frac{1}{2}\right), is:
(A) 2315\frac{2}{3} \sqrt{15}
(B) 5315\frac{5}{3} \sqrt{15}
(C) 1315\frac{1}{3} \sqrt{15}
(D) 15\sqrt{15}
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16Explain
Let A=[aij]\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] be a 3×33 \times 3 matrix such that A[010]=[001],A[413]=[010]\mathrm{A}\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], \mathrm{A}\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] and A[212]=[100]\mathrm{A}\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], then a23\mathrm{a}_{23} equals:
(A) -1
(B) 0
(C) 2
(D) 1
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17Explain
The number of complex numbers z , satisfying ∣z∣=1|\mathrm{z}|=1 and ∣zz‾+z‾z∣=1\left|\frac{\mathrm{z}}{\overline{\mathrm{z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1, is :
(A) 6
(B) 4
(C) 10
(D) 8
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18Explain
If the square of the shortest distance between the lines x−21=y−12=z+3−3\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3} and x+12=y+34=z+5−5\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5} is mn\frac{m}{n}, where m,nm, n are coprime numbers, then m+nm+n is equal to:
(A) 6
(B) 9
(C) 21
(D) 14
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19Explain
If I=∫0π2sin⁡32xsin⁡32x+cos⁡32xdx\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} \mathrm{x}}{\sin ^{\frac{3}{2}} \mathrm{x}+\cos ^{\frac{3}{2}} \mathrm{x}} \mathrm{dx}, then ∫02lxsin⁡xcos⁡xsin⁡4x+cos⁡4xdx\int_{0}^{2 l} \frac{x \sin x \cos x}{\sin ^{4} x+\cos ^{4} x} d x equals:
(A) π216\frac{\pi^{2}}{16}
(B) π24\frac{\pi^{2}}{4}
(C) π28\frac{\pi^{2}}{8}
(D) π212\frac{\pi^{2}}{12}
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20Explain
lim⁡x→∞(2x2−3x+5)(3x−1)x2(3x2+5x+4)(3x+2)x\lim _{x \rightarrow \infty} \frac{\left(2 x^{2}-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^{2}+5 x+4\right) \sqrt{(3 x+2)^{x}}} is equal to:
(A) 23e\frac{2}{\sqrt{3 e}}
(B) 2e3\frac{2 e}{\sqrt{3}}
(C) 2e3\frac{2 e}{3}
(D) 23e\frac{2}{3 \sqrt{\mathrm{e}}}
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21Explain
The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is ____\_\_\_\_ .
(17280)
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22Explain
Let α,β\alpha, \beta be the roots of the equation x2−ax−b=0x^{2}-a x-b=0 with Im⁡(α)<Im⁡(β)\operatorname{Im}(\alpha)<\operatorname{Im}(\beta). Let Pn=αn−βnP_{n}=\alpha^{n}-\beta^{n}. If P3=−57i,P4=−37i,P5=117iP_{3}=-5 \sqrt{7} i, \quad P_{4}=-3 \sqrt{7} i, \quad P_{5}=11 \sqrt{7} i \quad and P6=457i\mathrm{P}_{6}=45 \sqrt{7} \mathrm{i}, then ∣α4+β4∣\left|\alpha^{4}+\beta^{4}\right| is equal to ____\_\_\_\_ .
(31)
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23Explain
The focus of the parabola y2=4x+16y^{2}=4 x+16 is the centre of the circle C of radius 5 . If the values of λ\lambda, for which C passes through the point of intersection of the lines 3x−y=03 x-y=0 and x+λy=4x+\lambda y=4, are λ1\lambda_{1} and λ2,λ1<λ2\lambda_{2}, \lambda_{1}<\lambda_{2}, then 12λ1+29λ212 \lambda_{1}+29 \lambda_{2} is equal to ____\_\_\_\_ .
(15)
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24Explain
The variance of the numbers 8,21,34,47,…,3208,21,34,47, \ldots, 320, is ____\_\_\_\_ .
(8788)
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25Explain
The roots of the quadratic equation 3x2−px+q=03 x^{2}-p x+q=0 are 10th 10^{\text {th }} and 11th 11^{\text {th }} terms of an arithmetic progression with common difference 32\frac{3}{2}. If the sum of the first 11 terms of this arithmetic progression is 88 , then q−2qq-2 q is equal to ____\_\_\_\_ .
(474)
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26Explain
A ball having kinetic energy KE, is projected at an angle of 60∘60^{\circ} from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?
(A) (KE)8\frac{(\mathrm{KE})}{8}
(B) (KE)4\frac{(\mathrm{KE})}{4}
(C) (KE)16\frac{(\mathrm{KE})}{16}
(D) (KE)2\frac{(\mathrm{KE})}{2}
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27Explain
Two charges 7μc7 \mu \mathrm{c} and −4μc-4 \mu \mathrm{c} are placed at ( -7 cm , 0,0)0,0) and ( 7 cm,0,07 \mathrm{~cm}, 0,0 ) respectively. Given, ϵ0=8.85×10−12C2 N−1 m−2\epsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}, the electrostatic potential energy of the charge configuration is :
(A) -1.5 J
(B) -2.0 J
(C) -1.2 J
(D) -1.8 J
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28Explain
The refractive index of the material of a glass prism is 3\sqrt{3}. The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?
(A) 50∘50^{\circ}
(B) 60∘60^{\circ}
(C) 58∘58^{\circ}
(D) 48∘48^{\circ}
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29Explain
The equation of a transverse wave travelling along a string is y(x,t)=4.0sin⁡[20×10−3x+600t]mm\mathrm{y}(\mathrm{x}, \mathrm{t})=4.0 \sin \left[20 \times 10^{-3} \mathrm{x}+600 \mathrm{t}\right] \mathrm{mm}, where xx is in the mm and tt is in second. The velocity of the wave is :
(A) +30 m/s+30 \mathrm{~m} / \mathrm{s}
(B) −60 m/s-60 \mathrm{~m} / \mathrm{s}
(C) −30 m/s-30 \mathrm{~m} / \mathrm{s}
(D) +60 m/s+60 \mathrm{~m} / \mathrm{s}
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30Explain
The energy of a system is given as E(t)=α3e−βt\mathrm{E}(\mathrm{t})=\alpha^{3} \mathrm{e}^{-\beta \mathrm{t}}, where t is the time and β=0.3 s−1\beta=0.3 \mathrm{~s}^{-1}. The errors in the measurement of α\alpha and tt are 1.2%1.2 \% and 1.6%1.6 \%, respectively. At t=5 s\mathrm{t}=5 \mathrm{~s}, maximum percentage error in the energy is :
(A) 4%4 \%
(B) 11.6%11.6 \%
(C) 6%6 \%
(D) 8.4%8.4 \%
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31Explain
In photoelectric effect an em-wave is incident on a metal surface and electrons are ejected from the surface. If the work function of the metal is 2.14 eV and stopping potential is 2 V , what is the wavelength of the em-wave? (Given hc =1242eVnm=1242 \mathrm{eVnm} where h is the Planck's constant and c is the speed of light in vaccum.)
(A) 400 nm
(B) 600 nm
(C) 200 nm
(D) 300 nm
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32Explain
A circular disk of radius RR meter and mass MkgM k g is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that θ(t)=5t2−8t\theta(t)=5 t^{2}-8 t, where θ(t)\theta(t) is the angular position of the rotating disc as a function of time tt. How much power is delivered by the applied torque, when t=2 s\mathrm{t}=2 \mathrm{~s} ?
(A) 60MR260 \mathrm{MR}^{2}
(B) 72MR272 \mathrm{MR}^{2}
(C) 108MR2108 \mathrm{MR}^{2}
(D) 8MR28 \mathrm{MR}^{2}
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33Explain
Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is P1\mathrm{P}_{1}. The reading of the pressure gauge falls to P2P_{2} when the valve is opened. The speed of water flowing in the pipe is proportional to
(A) P1−P2\sqrt{\mathrm{P}_{1}-\mathrm{P}_{2}}
(B) (P1−P2)2\left(\mathrm{P}_{1}-\mathrm{P}_{2}\right)^{2}
(C) (P1−P2)4\left(P_{1}-P_{2}\right)^{4}
(D) P1−P2P_{1}-P_{2}
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34Explain
Match List-I with List-II. Choose the correct answer from the options given below:
(A) (A)-(I), (B)-(IV), (C)-(II), (D)-(III)
(B) (A)-(II), (B)-(I), (C)-(III), (D)-(IV)
(C) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
(D) (A)-(III), (B)-(II), (C)-(IV), (D)-(I)
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35Explain
If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon =27=27 days and gravitational attraction between the satellite and the moon is neglected.
(A) 1 day
(B) 81 days
(C) 27 days
(D) 3 days
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36Explain
Two point charges - 4μc4 \mu \mathrm{c} and 4μc4 \mu \mathrm{c}, constituting an electric dipole, are placed at (−9,0,0)cm(-9,0,0) \mathrm{cm} and ( 9,0,09,0,0 ) cm in a uniform electric field of strength 104NC−110^{4} \mathrm{NC}^{-1}. The work done on the dipole in rotating it from the equilibrium through 180∘180^{\circ} is :
(A) 14.4 mJ
(B) 18.4 mJ
(C) 12.4 mJ
(D) 16.4 mJ
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37Explain
A galvanometer having a coil of resistance 30Ω30 \Omega need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be 30XΩ\frac{30}{\mathrm{X}} \Omega, where X is
(A) 447
(B) 298
(C) 149
(D) 596
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38Explain
The width of one of the two slits in Young's double slit experiment is d while that of the other slit is xd . If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is 9:4 then what is the value of xx ? (Assume that the field strength varies according to the slit width.)
(A) 2
(B) 3
(C) 5
(D) 4
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39Explain
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The binding energy per nucleon is found to be practically independent of the atomic number A, for nuclei with mass numbers between 30 and 170. Reason (R) : Nuclear force is long range. In the light of the above statements, choose the correct answer from the options given below :
(A) (A) is false but (R) is true
(B) (A) is true but (R) is false
(C) Both (A) and (R) are true and (R) is the correct explanation of (A)
(D) Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
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40Explain
Water of mass m gram is slowly heated to increase the temperature from T1T_{1} to T2T_{2}. The change in entropy of the water, given specific heat of water is 1Jkg−1 K−11 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}, is :
(A) zero
(B) m(T2−T1)\mathrm{m}\left(\mathrm{T}_{2}-\mathrm{T}_{1}\right)
(C) mln(T1T2)m l n\left(\frac{T_{1}}{T_{2}}\right)
(D) mln(T2 T1)\mathrm{m} l \mathrm{n}\left(\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\right)
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41Explain
What is the current through the battery in the circuit shown below?
(A) 1.0 A
(B) 1.5 A
(C) 0.5 A
(D) 0.25 A
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42Explain
A plane electromagnetic wave of frequency 20 MHz travels in free space along the +x+x direction. At a particular point in space and time, the electric field vector of the wave is Ey=9.3Vm−1\mathrm{E}_{\mathrm{y}}=9.3 \mathrm{Vm}^{-} { }^{1}. Then, the magnetic field vector of the wave at that point is-
(A) Bz=9.3×10−8 T\mathrm{B}_{\mathrm{z}}=9.3 \times 10^{-8} \mathrm{~T}
(B) Bz=1.55×10−8 T\mathrm{B}_{\mathrm{z}}=1.55 \times 10^{-8} \mathrm{~T}
(C) Bz=6.2×10−8 T\mathrm{B}_{\mathrm{z}}=6.2 \times 10^{-8} \mathrm{~T}
(D) Bz=3.1×10−8 T\mathrm{B}_{\mathrm{z}}=3.1 \times 10^{-8} \mathrm{~T}
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43Explain
Using the given P-V diagram, the work done by an ideal gas along the path ABCD is-
(A) 4P0V04 P_{0} V_{0}
(B) 3P0V03 P_{0} V_{0}
(C) −4P0 V0-4 \mathrm{P}_{0} \mathrm{~V}_{0}
(D) −3P0 V0-3 \mathrm{P}_{0} \mathrm{~V}_{0}
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44Explain
A concave mirror of focal length ff in air is dipped in a liquid of refractive index μ\mu. Its focal length in the liquid will be :
(A) fμ\frac{f}{\mu}
(B) f(μ−1)\frac{f}{(\mu-1)}
(C) μf\mu f
(D) ff
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45Explain
A massless spring gets elongated by amount x1x_{1} under a tension of 5 N . Its elongation is x2\mathrm{x}_{2} under the tension of 7 N . For the elongation of (5x1−2x2)\left(5 \mathrm{x}_{1}-2 \mathrm{x}_{2}\right), the tension in the spring will be,
(A) 15 N
(B) 20 N
(C) 11 N
(D) 39 N
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46Explain
An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension 0.095 J/m20.095 \mathrm{~J} / \mathrm{m}^{2} and density 103 kg/m310^{3} \mathrm{~kg} / \mathrm{m}^{3}. The difference between pressure inside the bubble and atmospheric pressure ____\_\_\_\_ N/m2\mathrm{N} / \mathrm{m}^{2}. (Take g=10 m/s2\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2} )
(2190)
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47Explain
A satellite of mass M2\frac{\mathrm{M}}{2} is revolving around earth in a circular orbit at a height of R3\frac{\mathrm{R}}{3} from earth surface. The angular momentum of the satellite is MGMRxM \sqrt{\frac{G M R}{x}}. The value of xx is ____\_\_\_\_ , where M and RR are the mass and radius of earth, respectively. ( G is the gravitational constant)
(3)
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48Explain
At steady state the charge on the capacitor, as shown in the circuit below, is ____\_\_\_\_ μC\mu \mathrm{C}.
(16)
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49Explain
A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance 2.5μ F2.5 \mu \mathrm{~F}. The dielectric constant of the medium between the capacitor plates is 1 . It produces an instantaneous displacement current of 0.25 mA in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be ____\_\_\_\_ Vs−1\mathrm{Vs}^{-1}.
(100)
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50Explain
In a series LCR circuit, a resistor of 300Ω300 \Omega, a capacitor of 25 nF and an inductor of 100 mH are used. For maximum current in the circuit, the angular frequency of the ac source is ____\_\_\_\_ ×104\times 10^{4} radians s−1\mathrm{s}^{-1}.
(2)
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51Explain
The effect of temperature on spontaneity of reactions are represented as:
(A) (B) and (D) only
(B) (A) and (D) only
(C) (B) and (C) only
(D) (A) and (C) only
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52Explain
Standard electrode potentials for a few half cells are mentioned below: ECu2+/Cuo=0.34 V,EZn2+/Zno=−0.76 V\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^{\mathrm{o}}=0.34 \mathrm{~V}, \mathrm{E}_{\mathrm{Zn}^{2+} / \mathrm{Zn}}^{\mathrm{o}}=-0.76 \mathrm{~V} EAg+/Ago=0.80 V,EMg2+/Mgo=−2.37 V\mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{\mathrm{o}}=0.80 \mathrm{~V}, \mathrm{E}_{\mathrm{Mg}^{2+} / \mathrm{Mg}}^{\mathrm{o}}=-2.37 \mathrm{~V} Which one of the following cells gives the most negative value of ΔGo\Delta \mathrm{G}^{\mathrm{o}} ?
(A) Zn∣Zn2+(1M)∥Ag+(1M)∣Ag\mathrm{Zn}\left|\mathrm{Zn}^{2+}(1 \mathrm{M}) \| \mathrm{Ag}^{+}(1 \mathrm{M})\right| \mathrm{Ag}
(B) Zn∣Zn2+(1M)∣∣Mg2+(1M)∣Mg\mathrm{Zn}\left|\mathrm{Zn}^{2+}(1 \mathrm{M})\right|\left|\mathrm{Mg}^{2+}(1 \mathrm{M})\right| \mathrm{Mg}
(C) Ag∣Ag+(1M)∣∣Mg2+(1M)∣Mg\mathrm{Ag}\left|\mathrm{Ag}^{+}(1 \mathrm{M})\right|\left|\mathrm{Mg}^{2+}(1 \mathrm{M})\right| \mathrm{Mg}
(D) Cu∣Cu2+(1M)∥Ag+(1M)∣Ag\mathrm{Cu}\left|\mathrm{Cu}^{2+}(1 \mathrm{M}) \| \mathrm{Ag}^{+}(1 \mathrm{M})\right| \mathrm{Ag}
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53Explain
The α\alpha - Helix and β\beta - Pleated sheet structures of protein are associated with its:
(A) quaternary structure
(B) primary structure
(C) secondary structure
(D) tertiary structure
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54Explain
Given below are two statements: In the light of the above statements, choose the correct answer from the options given below:
(A) Statement I true but Statement II is false
(B) Both Statement I and Statement II are true
(C) Statement I is false but Statement II is true
(D) Both Statement I and Statement II are false
2
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55Explain
Consider the reaction X2Y(g)⇌X2( g)+12Y2( g)\mathrm{X}_{2} \mathrm{Y}(\mathrm{g}) \rightleftharpoons \mathrm{X}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{Y}_{2}(\mathrm{~g}) The equation representing correct relationship between the degree of dissociation ( x ) of X2Y(g)\mathrm{X}_{2} \mathrm{Y}(\mathrm{g}) with its equilibrium constant Kp is ____\_\_\_\_ . Assume x to be very very small.
(A) x=2Kpp3x=\sqrt[3]{\frac{2 K p}{p}}
(B) x=2Kp2p3x=\sqrt[3]{\frac{2 K p^{2}}{p}}
(C) x=Kp2p3x=\sqrt[3]{\frac{K p}{2 p}}
(D) x=Kpp3x=\sqrt[3]{\frac{K p}{p}}
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56Explain
Identify A,B\mathrm{A}, \mathrm{B} and C in the given below reaction sequence: →HNO3 Pb(NO3)2→H2SO4 B\xrightarrow{\mathrm{HNO}_{3}} \mathrm{~Pb}\left(\mathrm{NO}_{3}\right)_{2} \xrightarrow{\mathrm{H}_{2} \mathrm{SO}_{4}} \mathrm{~B} acetate, Acetic acid, K2CrO4\mathrm{K}_{2} \mathrm{CrO}_{4}
(A) PbCl2,PbSO4,PbCrO4\mathrm{PbCl}_{2}, \mathrm{PbSO}_{4}, \mathrm{PbCrO}_{4}
(B) PbS,PbSO4,PbCrO4\mathrm{PbS}, \mathrm{PbSO}_{4}, \mathrm{PbCrO}_{4}
(C) PbS,PbSO4, Pb(CH3COO)2\mathrm{PbS}, \mathrm{PbSO}_{4}, \mathrm{~Pb}\left(\mathrm{CH}_{3} \mathrm{COO}\right)_{2}
(D) PbCl2, Pb(SO4)2,PbCrO4\mathrm{PbCl}_{2}, \mathrm{~Pb}\left(\mathrm{SO}_{4}\right)_{2}, \mathrm{PbCrO}_{4}
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57Explain
Given below are two statements: Statement (I): The boiling points of alcohols and phenols increase with increase in the number of C -atoms. Statement (II): The boiling points of alcohols and phenols are higher in comparison to other class of compounds such as ethers, haloalkanes. In the light of the above statements, choose the correct answer from the options given below:
(A) Both Statement I and Statement II are false
(B) Statement I is false but Statement II is true
(C) Statement I is true but Statement II is false
(D) Both Statement I and Statement II are true
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58Explain
When a non-volatile solute is added to the solvent, the vapour pressure of the solvent decreases by 10 mm of Hg . The mole fraction of the solute in the solution is 0.2 . What would be the mole fraction of the solvent if decrease in vapour pressure is 20 mm of Hg?
(A) 0.6
(B) 0.4
(C) 0.2
(D) 0.8
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59Explain
Given below are two statements: Statement (I): For a given shell, the total number of allowed orbitals is given by n2n^{2}. Statement (II) : For any subshell, the spatial orientation of the orbitals is given by −l-l to +l+l values including zero. In the light of the above statements, choose the correct answer from the options given below:
(A) Statement I is true but Statement II is false
(B) Statement I is false but Statement II is true
(C) Both Statement I and Statement II are true
(D) Both Statement I and Statement II are false
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60Explain
The ascending order of relative rate of solvolysis of following compounds is
(A) (D) << (A) << (B) << (C)
(B) (2) (C) << (B) << (A) << (D)
(C) (3) (D) << (B) << (A) << (C)
(D) (4) (C) << (D) << (B) << (A)
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61Explain
Match List - I with List - II. Choose the correct answer from the options given below:
(A) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
(B) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(C) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
(D) (A)-(I), (B)-(IV), (C)-(III), (D)-(II)
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62Explain
Which of the following graphs most appropriately represents a zero order reaction ?
2
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63Explain
Match List - I with List - II. Choose the correct answer from the options given below:
(A) (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(B) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
(C) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
(D) (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
2
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64Explain
Identify the coordination complexes in which the central metal ion has d4\mathrm{d}^{4} configuration. Choose the correct answer from the options given below:
(A) (C) and (E) only
(B) (B), (C) and (D) only
(C) (B) and (D) only
(D) (A), (B) and (E) only
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65Explain
Given below are the atomic numbers of some group 14 elements. The atomic number of the element with lowest melting point is :
(A) 14
(B) 6
(C) 82
(D) 50
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66Explain
pH of water is 7 at 25∘C25^{\circ} \mathrm{C}. If water is heated to 80∘C80^{\circ} \mathrm{C}, it's pH will :
(A) Decrease
(B) Remains the same
(C) H+\mathrm{H}^{+}concentration increases, OH−\mathrm{OH}^{-}concentration decreases
(D) Increase
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67Explain
Identify the products [A][\mathrm{A}] and [B][\mathrm{B}], respectively in the following reaction :
3
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68Explain
Consider a binary solution of two volatile liquid components 1 and 2x12 \mathrm{x}_{1} and y1\mathrm{y}_{1} are the mole fractions of component 1 in liquid and vapour phase, respectively. The slope and intercept of the linear plot of 1x1vs1y1\frac{1}{\mathrm{x}_{1}} \mathrm{vs} \frac{1}{\mathrm{y}_{1}} are given respectively as :
(A) P10P20,P20−P10P20\frac{\mathrm{P}_{1}^{0}}{\mathrm{P}_{2}^{0}}, \frac{\mathrm{P}_{2}^{0}-\mathrm{P}_{1}^{0}}{\mathrm{P}_{2}^{0}}
(B) P20P10,P10−P20P20\frac{\mathrm{P}_{2}^{0}}{\mathrm{P}_{1}^{0}}, \frac{\mathrm{P}_{1}^{0}-\mathrm{P}_{2}^{0}}{\mathrm{P}_{2}^{0}}
(C) P10P20,P10−P20P20\frac{\mathrm{P}_{1}^{0}}{\mathrm{P}_{2}^{0}}, \frac{\mathrm{P}_{1}^{0}-\mathrm{P}_{2}^{0}}{\mathrm{P}_{2}^{0}}
(D) P20P10,P20−P10P20\frac{\mathrm{P}_{2}^{0}}{\mathrm{P}_{1}^{0}}, \frac{\mathrm{P}_{2}^{0}-\mathrm{P}_{1}^{0}}{\mathrm{P}_{2}^{0}}
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69Explain
Given below are two statements about X-ray spectra of elements : Statement (I): A plot of v(v=\sqrt{v}(v= frequency of X-rays emitted) vs atomic mass is a straight line. Statement (II) : A plot of ν(ν=\nu(\nu= frequency of X-rays emitted) vs atomic number is a straight line. In the light of the above statements choose the correct answer from the options given below:
(A) Statement I is true but Statement II is false
(B) Both Statement I and Statement II are true
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
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70Explain
Consider the following reactions K2Cr2O7→−H2OKOH[A]→−H2OH2SO4[ B]+K2SO4\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \xrightarrow[-\mathrm{H}_{2} \mathrm{O}]{\mathrm{KOH}}[\mathrm{A}] \xrightarrow[-\mathrm{H}_{2} \mathrm{O}]{\mathrm{H}_{2} \mathrm{SO}_{4}}[\mathrm{~B}]+\mathrm{K}_{2} \mathrm{SO}_{4} The products [A][\mathrm{A}] and [B][\mathrm{B}], respectively are :
(A) K2Cr(OH)6\mathrm{K}_{2} \mathrm{Cr}(\mathrm{OH})_{6} and Cr2O3\mathrm{Cr}_{2} \mathrm{O}_{3}
(B) K2CrO4\mathrm{K}_{2} \mathrm{CrO}_{4} and Cr2O3\mathrm{Cr}_{2} \mathrm{O}_{3}
(C) K2CrO4\mathrm{K}_{2} \mathrm{CrO}_{4} and K2Cr2O7\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}
(D) K2CrO4\mathrm{K}_{2} \mathrm{CrO}_{4} and CrO
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71Explain
0.01 mole of an organic compound ( X ) containing 10%10 \% hydrogen, on complete combustion produced 0.9 gH2O0.9 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}. Molar mass of ( X ) is ____\_\_\_\_ gmol−1\mathrm{g} \mathrm{mol}^{-1}.
(100)
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72Explain
Consider the following sequence of reactions. Total number of sp3\mathrm{sp}^{3} hybridised carbon atoms in the major product C formed is ____\_\_\_\_ .
(4)
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73Explain
When 81.0 g of aluminium is allowed to react with 128.0 g of oxygen gas, the mass of aluminium oxide produced in grams is ____\_\_\_\_ . (Nearest integer) Given : Molar mass of Al is 27.0 g mol−127.0 \mathrm{~g} \mathrm{~mol}^{-1} Molar mass of O is 16.0 g mol−116.0 \mathrm{~g} \mathrm{~mol}^{-1}
(153)
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74Explain
The bond dissociation enthalpy of X2ΔHbond o\mathrm{X}_{2} \Delta \mathrm{H}_{\text {bond }}^{\mathrm{o}} calculated from the given data is ____\_\_\_\_ kJmol−1\mathrm{kJ} \mathrm{mol}^{-1}. (Nearest integer) M+X−(s)→M+(g)+X−(g)ΔH∘lattice =800 kJ mol−1\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \rightarrow \mathrm{M}^{+}(\mathrm{g})+\mathrm{X}^{-}(\mathrm{g}) \Delta \mathrm{H}^{\circ}{ }_{\text {lattice }}=800 \mathrm{~kJ} \mathrm{~mol}^{-1} M(s)→M(g)ΔH∘sub =100 kJ mol−1\mathrm{M}(\mathrm{s}) \rightarrow \mathrm{M}(\mathrm{g}) \Delta \mathrm{H}^{\circ}{ }_{\text {sub }}=100 \mathrm{~kJ} \mathrm{~mol}^{-1} M(g)→M+(g)−+e−(g)ΔH∘i=500 kJ mol−1\mathrm{M}(\mathrm{g}) \rightarrow \mathrm{M}^{+}(\mathrm{g})^{-}+\mathrm{e}^{-}(\mathrm{g}) \Delta \mathrm{H}^{\circ}{ }_{\mathrm{i}}=500 \mathrm{~kJ} \mathrm{~mol}^{-1} X(g)+e−(g)→X−(g)ΔH∘eg=−300 kJ mol−1\mathrm{X}(\mathrm{g})+\mathrm{e}^{-}(\mathrm{g}) \rightarrow \mathrm{X}^{-}(\mathrm{g}) \Delta \mathrm{H}^{\circ}{ }_{\mathrm{eg}}=-300 \mathrm{~kJ} \mathrm{~mol}^{-1} M(s)+12X2( g)→M+X−(s)ΔH∘f=−400 kJ mol−1\mathrm{M}(\mathrm{s})+\frac{1}{2} \mathrm{X}_{2}(\mathrm{~g}) \rightarrow \mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \Delta \mathrm{H}^{\circ}{ }_{\mathrm{f}}=-400 \mathrm{~kJ} \mathrm{~mol}^{-1} [Given : M+X−\mathrm{M}^{+} \mathrm{X}^{-}is a pure ionic compound and X forms a diatomic molecule X2\mathrm{X}_{2} is gaseous state]
(200)
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75Explain
A compound ' X ' absorbs 2 moles of hydrogen and ' X ' upon oxidation with KMnO4∣H+\mathrm{KMnO}_{4} \mid \mathrm{H}^{+}gives . The total number of σ\sigma bonds present in the compound ' X ' is ____\_\_\_\_ .
(27)
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