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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Wednesday 29th 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 the set of all a∈R\mathrm{a} \in \mathbf{R}, for which the equation 2x2+(a−5)x+15=3a2 \mathrm{x}^{2}+ (\mathrm{a}-5) \mathrm{x}+15=3 \mathrm{a} has no real root, is the interval (α,β)(\alpha, \beta), and X={x∈Z:α<x<β}X=\{x \in Z: \alpha<x<\beta\}, then ∑x∈Xx2\sum_{x \in X} x^{2} is equal to
(A) 2109
(B) 2129
(C) 2139
(D) 2119
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2Explain
If sin⁡x+sin⁡2x=1,x∈(0,π2)\sin x+\sin ^{2} x=1, x \in\left(0, \frac{\pi}{2}\right), then (cos⁡12x+tan⁡12x)+3(cos⁡10x+tan⁡10x+cos⁡8x+tan⁡8x)+(cos⁡6x+tan⁡6x)\left(\cos ^{12} \mathrm{x}+\tan ^{12} \mathrm{x}\right)+3\left(\cos ^{10} \mathrm{x}+\tan ^{10} \mathrm{x}+\cos ^{8} \mathrm{x}+\tan ^{8} \mathrm{x}\right)+\left(\cos ^{6} x+\tan ^{6} x\right) is equal to
(A) 4
(B) 3
(C) 2
(D) 1
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3Explain
Let the area enclosed between the curves ∣y∣=1−x2|\mathrm{y}|=1- \mathrm{x}^{2} and x2+y2=1\mathrm{x}^{2}+\mathrm{y}^{2}=1 be α\alpha. If 9α=βπ+γ;β,γ9 \alpha=\beta \pi+\gamma ; \beta, \gamma are integers, then the value of ∣β−γ∣|\beta-\gamma| equals
(A) 27
(B) 18
(C) 15
(D) 33
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4Explain
If the domain of the function log⁡5(18x−x2−77)\log _{5}\left(18 \mathrm{x}-\mathrm{x}^{2}-77\right) is (α,β)(\alpha, \beta) and the domain of the function log⁡(x−1)(2x2+3x−2x2−3x−4)\log _{(x-1)}\left(\frac{2 x^{2}+3 x-2}{x^{2}-3 x-4}\right) is (γ,δ)(\gamma, \delta), then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to :
(A) 195
(B) 174
(C) 186
(D) 179
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5Explain
Let the function f(x)=(x2−1)∣x2−ax+2∣+cos⁡∣x∣f(x)=\left(x^{2}-1\right)\left|x^{2}-a x+2\right|+\cos |x| be not differentiable at the two points x=α=2\mathrm{x}=\alpha=2 and x=β\mathrm{x}=\beta. Then the distance of the point ( α,β\alpha, \beta ) from the line 12x+5y+10=012 x+5 y+10=0 is equal to :
(A) 3
(B) 4
(C) 2
(D) 5
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6Explain
Let a straight line L pass through the point P(2,−1,3)\mathrm{P}(2,-1,3) and be perpendicular to the lines x−12=y+11=z−3−2\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{1}=\frac{\mathrm{z}-3}{-2} and x−31=y−23=z+24\frac{\mathrm{x}-3}{1}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}+2}{4}. If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is :
(A) 2
(B) 10\sqrt{10}
(C) 3
(D) 232 \sqrt{3}
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7Explain
Let S=N∪{0}\mathrm{S}=\mathbf{N} \cup\{0\}. Define a relation R\mathbf{R} from S to R\mathbf{R} by : R={(x,y):log⁡ey=xlog⁡e(25),x∈ S,y∈R}\mathbf{R}=\left\{(\mathrm{x}, \mathrm{y}): \log _{\mathrm{e}} \mathrm{y}=\mathrm{x} \log _{\mathrm{e}}\left(\frac{2}{5}\right), \mathrm{x} \in \mathrm{~S}, \mathrm{y} \in \mathbf{R}\right\}. Then, the sum of all the elements in the range of R\mathbf{R} is equal to
(A) 32\frac{3}{2}
(B) 53\frac{5}{3}
(C) 109\frac{10}{9}
(D) 52\frac{5}{2}
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8Explain
Let the line x+y=1x+y=1 meet the axes of xx and yy at AA and BB, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points MM and NN lie on the lines OBO B and AB , respectively. If the area of the triangle AMN is 49\frac{4}{9} of the area of the triangle OAB and AN:NB=λ:1\mathrm{AN}: \mathrm{NB}=\lambda: 1, then the sum of all possible value(s) of is λ\lambda :
(A) 12\frac{1}{2}
(B) 136\frac{13}{6}
(C) 52\frac{5}{2}
(D) 2
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9Explain
If αx+βy=109\alpha x+\beta y=109 is the equation of the chord of the ellipse x29+y24=1\frac{x^{2}}{9}+\frac{y^{2}}{4}=1, whose mid point is (52,12)\left(\frac{5}{2}, \frac{1}{2}\right), then α+β\alpha+\beta is equal to
(A) 37
(B) 46
(C) 58
(D) 72
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10Explain
If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440th 440^{\text {th }} position in this arrangement, is :
(A) PRNAKU
(B) PRKANU
(C) PRKAUN
(D) PRNAUK
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11Explain
Let α,β(α≠β)\alpha, \beta(\alpha \neq \beta) be the values of m , for which the equations x+y+z=1;x+2y+4z=m\mathrm{x}+\mathrm{y}+\mathrm{z}=1 ; \mathrm{x}+2 \mathrm{y}+4 \mathrm{z}=\mathrm{m} and x+4y+10z=m2x+4 y+10 z=m^{2} have infinitely many solutions. Then the value of ∑n=110(nα+nβ)\sum_{n=1}^{10}\left(n^{\alpha}+n^{\beta}\right) is equal to :
(A) 440
(B) 3080
(C) 3410
(D) 560
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12Explain
Let A=[aij]\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] be a matrix of order 3×33 \times 3, with aij=(2)i+j\mathrm{a}_{\mathrm{ij}}=(\sqrt{2})^{\mathrm{i}+\mathrm{j}}. If the sum of all the elements in the third row of A2\mathrm{A}^{2} is α+β2,α,β∈Z\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}, then α+β\alpha+\beta is equal to
(A) 280
(B) 168
(C) 210
(D) 224
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13Explain
Let PP be the foot of the perpendicular from the point (1,2,2)(1,2,2) on the line L:x−11=y+1−1=z−22\mathrm{L}: \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}+1}{-1}=\frac{\mathrm{z}-2}{2}. Let the line r→=(−i^+j^−2k^)+λ(i^−j^+k^),λ∈R\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}), \lambda \in \mathbf{R}, intersect the line L at Q . Then 2(PQ)22(\mathrm{PQ})^{2} is equal to:
(A) 27
(B) 25
(C) 29
(D) 19
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14Explain
Let a circle CC pass through the points (4,2)(4,2) and (0(0, 2 ), and its centre lie on 3x+2y+2=03 x+2 y+2=0. Then the length of the chord, of the circle C , whose midpoint is (1,2)(1,2), is:
(A) 3\sqrt{3}
(B) 232 \sqrt{3}
(C) 424 \sqrt{2}
(D) 222 \sqrt{2}
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15Explain
Let A=[aij]\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] be a 2×22 \times 2 matrix such that aij∈{0,1}\mathrm{a}_{\mathrm{ij}} \in\{0,1\} for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:
(A) 14\frac{1}{4}
(B) 38\frac{3}{8}
(C) 58\frac{5}{8}
(D) 34\frac{3}{4}
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16Explain
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains nn white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/4529 / 45, then n is equal to:
(A) 3
(B) 4
(C) 5
(D) 6
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17Explain
The remainder, when 71037^{103} is divided by 23 , is equal to:
(A) 14
(B) 9
(C) 17
(D) 6
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18Explain
Let f(x)=∫0xt(t2−9t+20)dt,1≤x≤5f(x)=\int_{0}^{x} t\left(t^{2}-9 t+20\right) d t, 1 \leq x \leq 5. If the range of ff is [α,β][\alpha, \beta], then 4(α+β)4(\alpha+\beta) equals:
(A) 157
(B) 253
(C) 125
(D) 154
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19Explain
Let â be a unit vector perpendicular to the vectors b→=i^−2j^+3k^\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} and c→=2i^+3j^−k^\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, and makes an angle of cos⁡−1(−13)\cos ^{-1}\left(-\frac{1}{3}\right) with the vector i^+j^+k^\hat{i}+\hat{j}+\hat{k}. If a^\hat{a} makes an angle of π3\frac{\pi}{3} with the vector i^+αj^+k^\hat{i}+\alpha \hat{j}+\hat{k}, then the value of α\alpha is :
(A) −3-\sqrt{3}
(B) 6\sqrt{6}
(C) −6-\sqrt{6}
(D) 3\sqrt{3}
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20Explain
If for the solution curve y=f(x)y=f(x) of the differential equation dydx+(tan⁡x)y=2+sec⁡x(1+2sec⁡x)2\frac{d y}{d x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^{2}}, x∈(−π2,π2),f(π3)=310x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}, then f(π4)f\left(\frac{\pi}{4}\right) is equal to:
(A) 93+310(4+3)\frac{9 \sqrt{3}+3}{10(4+\sqrt{3})}
(B) 3+110(4+3)\frac{\sqrt{3}+1}{10(4+\sqrt{3})}
(C) 5−322\frac{5-\sqrt{3}}{2 \sqrt{2}}
(D) 4−214\frac{4-\sqrt{2}}{14}
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21Explain
If 24∫0π4(sin⁡∣4x−π12∣+[2sin⁡x])dx=2π+α24 \int_{0}^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) d x=2 \pi+\alpha, where [⋅][\cdot] denotes the greatest integer function, then α\alpha is equal to ____\_\_\_\_ .
(12)
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22Explain
If lim⁡t→0(∫01(3x+5)tdx)1t=α5e(85)23\lim _{t \rightarrow 0}\left(\int_{0}^{1}(3 x+5)^{t} d x\right)^{\frac{1}{t}}=\frac{\alpha}{5 e}\left(\frac{8}{5}\right)^{\frac{2}{3}}, then α\alpha is equal to ____\_\_\_\_ .
(64)
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23Explain
Let a1,a2,…,a2024\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{2024} be an Arithmetic Progression such that a1+(a5+a10+a15+…+a2020)+a2024=a_{1}+\left(a_{5}+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}= 2233. Then a1+a2+a3+…+a2024a_{1}+a_{2}+a_{3}+\ldots+a_{2024} is equal to ____\_\_\_\_ .
(11132)
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24Explain
Let integers a,b∈[−3,3]\mathrm{a}, \mathrm{b} \in[-3,3] be such that a+b≠0\mathrm{a}+\mathrm{b} \neq 0. Then the number of all possible ordered pairs (a,b)(\mathrm{a}, \mathrm{b}), for which ∣z−az+b∣=1\left|\frac{\mathrm{z}-\mathrm{a}}{\mathrm{z}+\mathrm{b}}\right|=1 and ∣z+1ωω2ωz+ω21ω21z+ω∣=1,z∈C\left|\begin{array}{ccc}\mathrm{z}+1 & \omega & \omega^{2} \\ \omega & \mathrm{z}+\omega^{2} & 1 \\ \omega^{2} & 1 & \mathrm{z}+\omega\end{array}\right|=1, \mathrm{z} \in \mathrm{C}, where ω\omega and ω2\omega^{2} are the roots of x2+x+1=0\mathrm{x}^{2}+\mathrm{x}+ 1=0, is equal to ____\_\_\_\_ .
(10)
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25Explain
Let y2=12xy^{2}=12 x the parabola and SS be its focus. Let PQ be a focal chord of the parabola such that (SP) (SQ)=1474(\mathrm{SQ})=\frac{147}{4}. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2−αx−643y=β64 x^{2}+64 y^{2}-\alpha x-64 \sqrt{3} y=\beta, then β−α\beta-\alpha is equal to ____\_\_\_\_ .
(1328)
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26Explain
The difference of temperature in a material can convert heat energy into electrical energy. To harvest the heat energy, the material should have
(A) low thermal conductivity and low electrical conductivity
(B) high thermal conductivity and high electrical conductivity
(C) low thermal conductivity and high electrical conductivity
(D) high thermal conductivity and low electrical conductivity
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27Explain
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process. Reason (R) : In isothermal process, PV=\mathrm{PV}= constant, while in adiabatic process PVγ=\mathrm{PV}^{\gamma}= constant. Here γ\gamma is the ratio of specific heats, P is the pressure and V is the volume of the ideal gas. In the light of the above statements, choose the correct answer from the options given below :
(A) Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
(B) (A) is true but (R) is false
(C) Both (A) and (R) are true and (R) is the correct explanation of (A).
(D) (A) is false but (R) is true
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28Explain
An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density σ0\sigma_{0}. Choose the correct option from the following.
(A) Torque on dipole is zero and net force is directed away from the sheet.
(B) Torque on dipole is zero and net force acts towards the sheet.
(C) Potential energy of dipole is minimum and torque is zero.
(D) Potential energy and torque both are maximum
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29Explain
In an experiment with photoelectric effect, the stopping potential.
(A) increases with increase in the wavelength of the incident light
(B) increases with increase in the intensity of the incident light
(C) is (1e)\left(\frac{1}{\mathrm{e}}\right) times the maximum kinetic energy of the emitted photoelectrons
(D) decreases with increase in the intensity of the incident light
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30Explain
A point charge causes an electric flux of −2×104Nm2C−1-2 \times 10^{4} \mathrm{Nm}^{2} \mathrm{C}^{-1} to pass through a spherical Gaussian surface of 8.0 cm radius, centred on the charge. The value of the point charge is : (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} )
(A) −17.7×10−8C-17.7 \times 10^{-8} \mathrm{C}
(B) −15.7×10−8C-15.7 \times 10^{-8} \mathrm{C}
(C) 17.7×10−8C17.7 \times 10^{-8} \mathrm{C}
(D) 15.7×10−8C15.7 \times 10^{-8} \mathrm{C}
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31Explain
A poly-atomic molecule ( Cv=3R,Cp=4R\mathrm{C}_{\mathrm{v}}=3 \mathrm{R}, \mathrm{C}_{\mathrm{p}}=4 \mathrm{R}, where RR is gas constant) goes from phase space point A(PA=105 Pa, VA=4×10−6 m3)\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^{5} \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^{3}\right) to point B(PB=5×104 Pa, VB=6×10−6 m3)\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5\right. \left.\times 10^{4} \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^{3}\right) to point C(PC=104 Pa\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^{4} \mathrm{~Pa}\right., VC=8×10−6 m3\mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^{3} ). A to B is an adiabatic path and B to C is an isothermal path. The net heat absorbed per unit mole by the system is :
(A) 500R(ln⁡3+ln⁡4)500 \mathrm{R}(\ln 3+\ln 4)
(B) 450R(ln⁡4−ln⁡3)450 R(\ln 4-\ln 3)
(C) 500Rln⁡2500 R \ln 2
(D) 400Rln⁡4400 R \ln 4
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32Explain
Two identical symmetric double convex lenses of focal length ff are cut into two equal parts L1, L2\mathrm{L}_{1}, \mathrm{~L}_{2} by AB plane and L3, L4\mathrm{L}_{3}, \mathrm{~L}_{4} by XY plane as shown in figure respectively. The ratio of focal lengths of lenses L1\mathrm{L}_{1} and L3\mathrm{L}_{3} is
(A) 1: 4
(B) 1: 1
(C) 2: 1
(D) 1: 2
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33Explain
A plane electromagnetic wave propagates along the +x+x direction in free space. The components of the electric field, E→\overrightarrow{\mathrm{E}} and magnetic field, B→\overrightarrow{\mathrm{B}} vectors associated with the wave in Cartesian frame are :
(A) Ey,Bx\mathrm{E}_{\mathrm{y}}, \mathrm{B}_{\mathrm{x}}
(B) Ey,Bz\mathrm{E}_{\mathrm{y}}, \mathrm{B}_{\mathrm{z}}
(C) Ex,ByE_{x}, B_{y}
(D) Ez, By\mathrm{E}_{z}, \mathrm{~B}_{\mathrm{y}}
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34Explain
Two concave refracting surfaces of equal radii of curvature and refractive index 1.5 face each other in air as shown in figure. A point object O is placed midway, between P and B. The separation between the images of O , formed by each refracting surface is :
(A) 0.214 R
(B) 0.114 R
(C) 0.411 R
(D) 0.124 R
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35Explain
Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1k_{1} and k2\mathrm{k}_{2}, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of BB is
(A) k1k2\sqrt{\frac{\mathrm{k}_{1}}{\mathrm{k}_{2}}}
(B) k1k2\frac{\mathrm{k}_{1}}{\mathrm{k}_{2}}
(C) k2k1\frac{\mathrm{k}_{2}}{\mathrm{k}_{1}}
(D) k2k1\sqrt{\frac{\mathrm{k}_{2}}{\mathrm{k}_{1}}}
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36Explain
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities vA=5 m/s,vB=2 m/s,vC=4 m/s\mathrm{v}_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s} If we wait sufficiently long for elastic collision to happen, then vA=4 m/s,vB=2 m/s,vC=5 m/s\mathrm{v}_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s} will be the final velocities. Reason (R) : In an elastic collision between identical masses, two objects exchange their velocities. In the light of the above statements, choose the correct answer from the options given below :
(A) Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
(B) (A) is true but (R) is false
(C) Both (A) and (R) are true and (R) is the correct explanation of (A).
(D) (A) is false but (R) is true
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37Explain
A sand dropper drops sand of mass m(t)m(t) on a conveyer belt at a rate proportional to the square root of speed (v)(\mathrm{v}) of the belt, i.e. dmdt∝v\frac{\mathrm{dm}}{\mathrm{dt}} \propto \sqrt{\mathrm{v}}. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?
(A) P2∝v3P^{2} \propto v^{3}
(B) P∝V\mathrm{P} \propto \sqrt{\mathrm{V}}
(C) P∝vP \propto v
(D) P2∝v5P^{2} \propto v^{5}
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38Explain
A convex lens mode of glass (refractive index == 1.5) has focal length 24 cm in air. When it is totally immersed in water (refractive index =1.33=1.33 ), its focal length changes to
(A) 72 cm
(B) 96 cm
(C) 24 cm
(D) 48 cm
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39Explain
A capacitor, C1=6μ F\mathrm{C}_{1}=6 \mu \mathrm{~F} is charged to a potential difference of V0=5 V\mathrm{V}_{0}=5 \mathrm{~V} using a 5 V battery. The battery is removed and another capacitor, C2=12μF\mathrm{C}_{2}=12 \mu \mathrm{F} is inserted in place of the battery. When the switch ' S ' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the charges ( q1\mathrm{q}_{1} and q2\mathrm{q}_{2} ) on the capacitors C1\mathrm{C}_{1} and C2\mathrm{C}_{2} when equilibrium condition is reached.
(A) q1=15μC,q2=30μC\mathrm{q}_{1}=15 \mu \mathrm{C}, \mathrm{q}_{2}=30 \mu \mathrm{C}
(B) q1=30μC,q2=15μC\mathrm{q}_{1}=30 \mu \mathrm{C}, \mathrm{q}_{2}=15 \mu \mathrm{C}
(C) q1=10μC,q2=20μC\mathrm{q}_{1}=10 \mu \mathrm{C}, \mathrm{q}_{2}=20 \mu \mathrm{C}
(D) q1=20μC,q2=10μC\mathrm{q}_{1}=20 \mu \mathrm{C}, \mathrm{q}_{2}=10 \mu \mathrm{C}
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40Explain
Three equal masses m are kept at vertices ( A , B, C) of an equilateral triangle of side aa in free space. At t=0\mathrm{t}=0, they are given an initial velocity V⃗A=V0AC→,V⃗B=V0BA→\vec{V}_{A}=V_{0} \overrightarrow{A C}, \quad \vec{V}_{B}=V_{0} \overrightarrow{B A} and V⃗C=V0CB→\vec{V}_{C}=V_{0} \overrightarrow{C B}. Here, AC→,CB→\overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{CB}} and BA→\overrightarrow{\mathrm{BA}} are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is :
(A) 12amV0\frac{1}{2} \mathrm{amV}_{0}
(B) 3amV03 \mathrm{a} \mathrm{mV}_{0}
(C) 32amV0\frac{\sqrt{3}}{2} \mathrm{amV}_{0}
(D) 32amV0\frac{3}{2} \mathrm{amV}_{0}
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41Explain
Match List-I with List-II. Choose the correct answer from the options given below:
(A) (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
(B) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(C) (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(D) (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
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42Explain
Match List-I with List-II. Choose the correct answer from the options given below:
(A) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(B) (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
(C) (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
(D) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
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43Explain
The truth table for the circuit given below is :
(A) A=0,B=0,Y=0; A=0,B=1,Y=1; A=1,B=0,Y=1; A=1,B=1,Y=0
(B) A=0,B=0,Y=0; A=1,B=0,Y=0; A=1,B=1,Y=0; A=0,B=1,Y=1
(C) A=0,B=0,Y=0; A=1,B=0,Y=1; A=0,B=1,Y=0; A=1,B=1,Y=0
(D) A=0,B=0,Y=0; A=1,B=1,Y=1; A=1,B=0,Y=1; A=0,B=1,Y=1
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44Explain
A cup of coffee cools from 90∘C90^{\circ} \mathrm{C} to 80∘C80^{\circ} \mathrm{C} in t minutes when the room temperature is 20∘C20^{\circ} \mathrm{C}. The time taken by the similar cup of coffee to cool from 80∘C80^{\circ} \mathrm{C} to 60∘C60^{\circ} \mathrm{C} at the same room temperature is :
(A) 135t\frac{13}{5} \mathrm{t}
(B) 1013t\frac{10}{13} \mathrm{t}
(C) 1310t\frac{13}{10} \mathrm{t}
(D) 513t\frac{5}{13} \mathrm{t}
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45Explain
The number of spectral lines emitted by atomic hydrogen that is in the 4th 4^{\text {th }} energy level, is
(A) 6
(B) 0
(C) 3
(D) 1
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46Explain
The magnetic field inside a 200 turns solenoid of radius 10 cm is 2.9×10−42.9 \times 10^{-4} Tesla. If the solenoid carries a current of 0.29 A , then the length of the solenoid is ____\_\_\_\_ πcm\pi \mathrm{cm}.
(8)
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47Explain
A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A . If the rate of change of potential difference between the plates is 7×108V/s7 \times 10^{8} \mathrm{V} / \mathrm{s} then the integer value of the distance between the parallel plates is - (Take, ϵ0=9×10−12 F m,π=227\epsilon_{0}=9 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7} ) …μm\ldots \mu \mathrm{m} .
(1320)
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48Explain
A physical quantity Q is related to four observables a,b,c,da, b, c, d as follows: Q=ab4cdQ=\frac{a b^{4}}{c d} where, a=(60±3)Pa;b=(20±0.1)m\mathrm{a}=(60 \pm 3) \mathrm{Pa} ; \mathrm{b}=(20 \pm 0.1) \mathrm{m}; c=(40±0.2)Nsm−2\mathrm{c}=(40 \pm 0.2) \mathrm{Nsm}^{-2} and d=(50±0.1)m\mathrm{d}=(50 \pm 0.1) \mathrm{m}, then the percentage error in Q is X1000\frac{\mathrm{X}}{1000}, where x=\mathrm{x}= ____\_\_\_\_ .
(77)
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49Explain
Two planets, A and B are orbiting a common star in circular orbits of radii RAR_{A} and RBR_{B}, respectively, with RB=2RAR_{B}=2 R_{A}. The planet BB is 424 \sqrt{2} times more massive than planet AA. The ratio (LBLA)\left(\frac{L_{B}}{L_{A}}\right) of angular momentum (LB)\left(\mathrm{L}_{\mathrm{B}}\right) of planet B to that of planet A(LA)\mathrm{A}\left(\mathrm{L}_{\mathrm{A}}\right) is closest to integer ____\_\_\_\_ .
(8)
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50Explain
Two cars P and Q are moving on a road in the same direction. Acceleration of car PP increases linearly with time whereas car Q moves with a constant acceleration. Both cars cross each other at time t=0\mathrm{t}=0, for the first time. The maximum possible number of crossing(s) (including the crossing at t=0\mathrm{t}=0 ) is ____\_\_\_\_ -
(3)
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51Explain
The calculated spin-only magnetic moments of K3[Fe(OH)6]\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{OH})_{6}\right] and K4[Fe(OH)6]\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{OH})_{6}\right] respectively are :
(A) 4.90 and 4.90 B.M.
(B) 5.92 and 4.90 B.M.
(C) 3.87 and 4.90 B.M.
(D) 4.90 and 5.92 B.M.
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52Explain
For hydrogen like species, which of the following graphs provides the most appropriate representation of E vs Z plot for a constant n ? [E: Energy of the stationary state, Z : atomic number, n=\mathrm{n}= principal quantum number]
2
Diagram Question
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53Explain
Given below are two statements : Statement (I) : In partition chromatography, stationary phase is thin film of liquid present in the inert support. Statement (II) : In paper chromatography, the material of paper acts as a stationary phase. 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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54Explain
Identify the essential amino acids from below : (A) Valine (B) Proline (C) Lysine (D) Threonine (E) Tyrosine Choose the correct answer from the options given below :
(A) (A),(C) and (D) only
(B) (A), (C) and (E) only
(C) (B), (C) and (E) only
(D) (C), (D) and (E) only
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55Explain
Which among the following halides will generate the most stable carbocation in Nucleophillic substitution reaction?
4
Diagram Question
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56Explain
Consider the equilibrium\nCO( g)+3H2( g)⇌CH4( g)+H2O( g)\mathrm{CO}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{CH}_{4}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{~g}) If the pressure applied over the system increases by two fold at constant temperature then (A) Concentration of reactants and products increases. (B) Equilibrium will shift in forward direction. (C) Equilibrium constant increases since concentration of products increases. (D) Equilibrium constant remains unchanged as concentration of reactants and products remain same. Choose the correct answer from the options given below :
(A) (A) and (B) only
(B) (A), (B) and (D) only
(C) (B) and (C) only
(D) (A), (B) and (C) only
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57Explain
Given below are two statements : Statement (I): NaCl is added to the ice at 0∘C0^{\circ} \mathrm{C}, present in the ice cream box to prevent the melting of ice cream. Statement (II) : On addition of NaCl to ice at 0∘C0^{\circ} \mathrm{C}, there is a depression in freezing point. In the light of the above statements, choose the correct answer from the options given below :
(A) Statement I is false but Statement II is true
(B) Both Statement I and Statement II are true
(C) Both Statement I and Statement II are false
(D) Statement I is true but Statement II is false
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58Explain
Given below are two statements : Statement (I) : On nitration of m-xylene with HNO3,H2SO4\mathrm{HNO}_{3}, \mathrm{H}_{2} \mathrm{SO}_{4} followed by oxidation, 4-nitrobenzene-1, 3-dicarboxylic acid is obtained as the major product. Statement (II): CH3\mathrm{CH}_{3} group is o/p-directing whileNO2\mathrm{NO}_{2} group is m-directing group. 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) Both Statement I and Statement II are true
(D) Statement I is true but Statement II is false
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59Explain
0.1 M solution of KI reacts with excess of H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} and KIO3\mathrm{KIO}_{3} solution. According to equation\n5I−+IO3−+6H+→3I2+3H2O5 \mathrm{I}^{-}+\mathrm{IO}_{3}^{-}+6 \mathrm{H}^{+} \rightarrow 3 \mathrm{I}_{2}+3 \mathrm{H}_{2} \mathrm{O}\nIdentify the correct statements : (A) 200 mL of KI solution reacts with 0.004 mol of KIO3\mathrm{KIO}_{3} (B) 200 mL of KI solution reacts with 0.006 mol of H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} (C) 0.5 L of KI solution produced 0.005 mol of I2\mathrm{I}_{2} (D) Equivalent weight of KIO3\mathrm{KIO}_{3} is equal to ( Molecular weight 5)\left(\frac{\text { Molecular weight }}{5}\right) Choose the correct answer from the options given below:
(A) (A) and (D) only
(B) (B) and (C) only
(C) (A) and (B) only
(D) (C) and (D) only
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60Explain
Match List-I with List-II: Choose the correct answer from the options given below :
(A) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
(B) (A)-(III), (B)-(II), (C)-(IV), (D)-(I)
(C) (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
(D) (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
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Diagram Question
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61Explain
O2\mathrm{O}_{2} gas will be evolved as a product of electrolysis of : (A) an aqueous solution of AgNO3\mathrm{AgNO}_{3} using silver electrodes. (B) an aqueous solution of AgNO3\mathrm{AgNO}_{3} using platinum electrodes. (C) a dilute solution of H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} using platinum electrodes. (D) a high concentration solution of H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} using platinum electrodes. Choose the correct answer from the options given below :
(A) (B) and (C) only
(B) (A) and (D) only
(C) (B) and (D) only
(D) (A) and (C) only
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62Explain
Identify the homoleptic complexes with odd number of d electrons in the central metal. (A) [FeO4]2−\left[\mathrm{FeO}_{4}\right]^{2-} (B) [Fe(CN)6]3−\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-} (C) [Fe(CN)5NO]2−\left[\mathrm{Fe}(\mathrm{CN})_{5} \mathrm{NO}\right]^{2-} (D) [CoCl4]2−\left[\mathrm{CoCl}_{4}\right]^{2-} (E) [Co(H2O)3 F3]\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{3} \mathrm{~F}_{3}\right] Choose the correct answer from the options given below :
(A) (B) and (D) only
(B) (C) and (E) only
(C) (A), (B) and (D) only
(D) (A), (C) and (E) only
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63Explain
Total number of sigma (σ)(\sigma) ____\_\_\_\_ and pi⁡(π)\operatorname{pi}(\pi) ____\_\_\_\_ bonds respectively present in hex-1-en-4-yne are :
(A) 13 and 3
(B) 11 and 3
(C) 3 and 13
(D) 14 and 3
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64Explain
If C(\quad \mathrm{C}( diamond )→C() \rightarrow \mathrm{C}( graphite )+XkJmol−1)+\mathrm{X} \mathrm{kJ} \mathrm{mol}^{-1} C( diamond )+O2( g)→CO2( g)+Y kJ mol−1\mathrm{C}(\text { diamond })+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{Y} \mathrm{~kJ} \mathrm{~mol}^{-1} C( graphite )+O2( g)→CO2( g)+Z kJ mol−1\mathrm{C}(\text { graphite })+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{Z} \mathrm{~kJ} \mathrm{~mol}^{-1} At constant temperature. Then
(A) X=Y+ZX=Y+Z
(B) −X=Y+Z-X=Y+Z
(C) X=−Y+ZX=-Y+Z
(D) X=Y−ZX=Y-Z
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65Explain
Given below are two statements : Statement (I): It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle. Statement (II) : If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then the uncertainty in the measurement of velocity is ≥hπ×12 m\geq \sqrt{\frac{\mathrm{h}}{\pi}} \times \frac{1}{2 \mathrm{~m}}. 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) Statement I is false but Statement II is true.
(D) Both Statement I and Statement II are false.
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66Explain
Which one of the following reaction sequences will give an azo dye ?
1
Diagram Question
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67Explain
Drug X becomes ineffective after 50% decomposition. The original concentration of drug in a bottle was 16mg/mL16 \mathrm{mg} / \mathrm{mL} which becomes 4 mg/mL\mathrm{mg} / \mathrm{mL} in 12 months. The expiry time of the drug in months is ____\_\_\_\_ . Assume that the decomposition of the drug follows first order kinetics.
(A) 12
(B) 2
(C) 3
(D) 6
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68Explain
The type of oxide formed by the element among Li,Na,Be,Mg,B\mathrm{Li}, \mathrm{Na}, \mathrm{Be}, \mathrm{Mg}, \mathrm{B} and Al that has the least atomic radius is :
(A) A2O3\mathrm{A}_{2} \mathrm{O}_{3}
(B) AO2\mathrm{AO}_{2}
(C) AO
(D) A2O\mathrm{A}_{2} \mathrm{O}
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69Explain
First ionisation enthalpy values of first four group 15 elements are given below. Choose the correct value for the element that is a main component of apatite family :
(A) 1012 kJ mol−11012 \mathrm{~kJ} \mathrm{~mol}^{-1}
(B) 1402 kJ mol−11402 \mathrm{~kJ} \mathrm{~mol}^{-1}
(C) 834 kJ mol−1834 \mathrm{~kJ} \mathrm{~mol}^{-1}
(D) 947 kJ mol−1947 \mathrm{~kJ} \mathrm{~mol}^{-1}
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70Explain
Which one of the following, with HBr will give a phenol?
2
Diagram Question
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71Explain
Consider the following low-spin complexes K3[Co(NO2)6],K4[Fe(CN)6],K3[Fe(CN)6]\mathrm{K}_{3}\left[\mathrm{Co}\left(\mathrm{NO}_{2}\right)_{6}\right], \mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right], \mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right], Cu2[Fe(CN)6]\mathrm{Cu}_{2}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] and Zn2[Fe(CN)6]\mathrm{Zn}_{2}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]. The sum of the spin-only magnetic moment values of complexes having yellow colour is ____\_\_\_\_ B.M. (answer is nearest integer)
(0)
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72Explain
Isomeric hydrocarbons →\rightarrow negative Baeyer's test (Molecular formula C9H12\mathrm{C}_{9} \mathrm{H}_{12} ). The total number of isomers from above with four different non-aliphatic substitution sites is -
(2)
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73Explain
In the Claisen-Schmidt reaction to prepare, dibenzalacetone from 5.3 g benzaldehyde, a total of 3.51 g of product was obtained. The percentage yield in this reaction was ____\_\_\_\_ %\%.
(60)
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74Explain
In the sulphur estimation, 0.20 g of a pure organic compound gave 0.40 g of barium sulphate. The percentage of sulphur in the compound is ____\_\_\_\_ ×10−1\times 10^{-1} %. (Molar mass : O=16, S=32,Ba=137\mathrm{O}=16, \mathrm{~S}=32, \mathrm{Ba}=137 in gmol−1\mathrm{g} \mathrm{mol}^{-1} )
(275)
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75Explain
Total number of non bonded electrons present in NO2−\mathrm{NO}_{2}^{-}ion based on Lewis theory is ____\_\_\_\_ .
(12)
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