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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Thursday 23rd January 2025, Time: 9:00 AM to 12:00 NOON

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JEE Main
Mathematics, Physics, Chemistry
Morning Session
3 hours

Paper Overview

75
Total Questions
0
Correct
0
Incorrect
75
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Complete Solutions

Q#ExplanationQuestionCorrectSolutionStatus
1Explain
The value of ∫e2e41X(e((log⁡ex)2+1)−1e((log⁡ex)2+1)−1+e((6−log⁡ex)2+1)−1)dx\int_{e^{2}}^{e^{4}} \frac{1}{X}\left(\frac{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}}{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}+e^{\left(\left(6-\log _{e} x\right)^{2}+1\right)^{-1}}}\right) d x is
(A) log⁡e2\log _{\mathrm{e}} 2
(B) 2
(C) 1
(D) e2\mathrm{e}^{2}
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2Explain
Let I(x)=∫dx(x−11)1113(x+15)1513\mathrm{I}(\mathrm{x})=\int \frac{\mathrm{dx}}{(\mathrm{x}-11)^{\frac{11}{13}}(\mathrm{x}+15)^{\frac{15}{13}}}. If I(37)−I(24)=14(1 b113−1c113),b,c∈N\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathbb{N}, then 3(b+c)3(b+c) is equal to
(A) 40
(B) 39
(C) 22
(D) 26
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3Explain
If the function f(x)={2x{sin⁡(k1+1)x+sin⁡(k2−1)x},x<04,x=02xlog⁡e(2+k1x2+k2x),x>0f(x)=\left\{\begin{array}{ccc}\frac{2}{x}\left\{\sin \left(k_{1}+1\right) x+\sin \left(k_{2}-1\right) x\right\} & , & x<0 \\ 4 & , & x=0 \\ \frac{2}{x} \log _{e}\left(\frac{2+k_{1} x}{2+k_{2} x}\right) & , & x>0\end{array}\right. is continuous at x=0\mathrm{x}=0, then k12+k22\mathrm{k}_{1}{ }^{2}+\mathrm{k}_{2}{ }^{2} is equal to
(A) 8
(B) 20
(C) 5
(D) 10
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4Explain
If the line 3x−2y+12=03 x-2 y+12=0 intersects the parabola 4y=3x24 y=3 x^{2} at the points AA and BB, then at the vertex of the parabola, the line segment AB subtends an angle equal to
(A) tan⁡−1(119)\tan ^{-1}\left(\frac{11}{9}\right)
(B) π2−tan⁡−1(32)\frac{\pi}{2}-\tan ^{-1}\left(\frac{3}{2}\right)
(C) tan⁡−1(45)\tan ^{-1}\left(\frac{4}{5}\right)
(D) tan⁡−1(97)\tan ^{-1}\left(\frac{9}{7}\right)
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5Explain
Let a curve y=f(x)y=f(x) pass through the points (0,5)(0,5) and (log⁡e2,k)\left(\log _{\mathrm{e}} 2, \mathrm{k}\right). If the curve satisfies the differential equation 2(3+y)e2xdx−(7+e2x)dy=02(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0, then kk is equal to
(A) 16
(B) 8
(C) 32
(D) 4
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6Explain
Let f(x)=log⁡exf(x)=\log _{e} x and g(x)=x4−2x3+3x2−2x+22x2−2x+1g(x)=\frac{x^{4}-2 x^{3}+3 x^{2}-2 x+2}{2 x^{2}-2 x+1} . Then the domain of fog is
(A) R\mathbb{R}
(B) (0,∞)(0, \infty)
(C) [0,∞)[0, \infty)
(D) [1,∞)[1, \infty)
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7Explain
Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the arc⁡AC\operatorname{arc} \mathrm{AC}, divides the arc AC such that  length of arc⁡AB length of arc⁡BC=15\frac{\text { length of } \operatorname{arc} \mathrm{AB}}{\text { length of } \operatorname{arc} \mathrm{BC}}=\frac{1}{5}, and OC→=αOA→+βOB→\overrightarrow{\mathrm{OC}}=\alpha \overrightarrow{\mathrm{OA}}+\beta \overrightarrow{\mathrm{OB}}, then α=2(3−1)β\alpha=\sqrt{2}(\sqrt{3}-1) \beta is equal to
(A) 2−32-\sqrt{3}
(B) 232 \sqrt{3}
(C) 535 \sqrt{3}
(D) 2+32+\sqrt{3}
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8Explain
If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
(A) -1200
(B) -1080
(C) -1020
(D) -120
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9Explain
Let P be the foot of the perpendicular from the point Q(10,−3,−1)\mathrm{Q}(10,-3,-1) on the line x−37=y−2−1=z+1−2\frac{\mathrm{x}-3}{7}=\frac{\mathrm{y}-2}{-1}=\frac{\mathrm{z}+1}{-2}. Then the area of the right angled triangle PQRP Q R, where RR is the point (3,−2,1)(3,-2,1), is
(A) 9159 \sqrt{15}
(B) 30\sqrt{30}
(C) 8158 \sqrt{15}
(D) 3303 \sqrt{30}
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10Explain
Let ∣zˉ−i2zˉ+i∣=13,z∈C\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in \mathbb{C}, be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points (0,0),C(0,0), \mathrm{C} and (α,0)(\alpha, 0) is 11 square units, then α2\alpha^{2} equals
(A) 100
(B) 50
(C) 12125\frac{121}{25}
(D) 8125\frac{81}{25}
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11Explain
Let R={(1,2),(2,3),(3,3)}R=\{(1,2),(2,3),(3,3)\} be a relation defined on the set {1,2,3,4}\{1,2,3,4\}. Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :
(A) 10
(B) 8
(C) 9
(D) 7
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12Explain
The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
(A) 34000
(B) 37000
(C) 36000
(D) 35000
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13Explain
Let the area of a △PQR\triangle \mathrm{PQR} with vertices P(5,4),Q(−2,4)\mathrm{P}(5,4), \mathrm{Q}(-2,4) and R(a,b)\mathrm{R}(\mathrm{a}, \mathrm{b}) be 35 square units. If its orthocenter and centroid are O(2,145)O\left(2, \frac{14}{5}\right) and C(c,d)C(c, d) respectively, then c+2 d\mathrm{c}+2 \mathrm{~d} is equal to
(A) 73\frac{7}{3}
(B) 3
(C) 2
(D) 83\frac{8}{3}
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14Explain
If π2≤x≤3π4\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}, then cos⁡−1(1213cos⁡x+513sin⁡x)\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right) is equal to
(A) x−tan⁡−143x-\tan ^{-1} \frac{4}{3}
(B) x−tan⁡−1512x-\tan ^{-1} \frac{5}{12}
(C) x+tan⁡−145x+\tan ^{-1} \frac{4}{5}
(D) x+tan⁡−1512x+\tan ^{-1} \frac{5}{12}
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15Explain
The value of (sin⁡70∘)(cot⁡10∘cot⁡70∘−1)\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right) is
(A) 1
(B) 0
(C) 3/23 / 2
(D) 2/32 / 3
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16Explain
Marks obtains by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is
(A) 48
(B) 44
(C) 40
(D) 52
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17Explain
Let the position vectors of the vertices A,B\mathrm{A}, \mathrm{B} and C of a tetrahedron ABCD be i^+2j^+k^,i^+3j^−2k^\hat{i}+2 \hat{j}+\hat{k}, \hat{i}+3 \hat{j}-2 \hat{k} and 2i^+j^−k^2 \hat{i}+\hat{j}-\hat{k} respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E . If the length of AD is 1103\frac{\sqrt{110}}{3} and the volume of the tetrahedron is 80562\frac{\sqrt{805}}{6 \sqrt{2}}, then the position vector of E is
(A) 12(i^+4j^+7k^)\frac{1}{2}(\hat{i}+4 \hat{j}+7 \hat{k})
(B) 112(7i^+4j^+3k^)\frac{1}{12}(7 \hat{i}+4 \hat{j}+3 \hat{k})
(C) 16(12i^+12j^+k^)\frac{1}{6}(12 \hat{i}+12 \hat{j}+\hat{k})
(D) 16(7i^+12j^+k^)\frac{1}{6}(7 \hat{i}+12 \hat{j}+\hat{k})
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18Explain
If A,B\mathrm{A}, \mathrm{B} and (adj⁡(A−1)+adj⁡(B−1))\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right) are non-singular matrices of same order, then the inverse of A(adj⁡(A−1)+adj⁡(B−1))−1 B\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}, is equal to
(A) AB−1+A−1BA B^{-1}+A^{-1} B
(B) adj⁡(B−1)+adj⁡(A−1)\operatorname{adj}\left(\mathrm{B}^{-1}\right)+\operatorname{adj}\left(\mathrm{A}^{-1}\right)
(C) 1∣AB∣(adj⁡(B)+adj⁡(A))\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))
(D) AB−1∣A∣+BA−1∣B∣\frac{A B^{-1}}{|A|}+\frac{B A^{-1}}{|B|}
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19Explain
If the system of equations (λ−1)x+(λ−4)y+λz=5(\lambda-1) x+(\lambda-4) y+\lambda z=5 λx+(λ−1)y+(λ−4)z=7\lambda x+(\lambda-1) y+(\lambda-4) z=7 (λ+1)x+(λ+2)y−(λ+2)z=9(\lambda+1) \mathrm{x}+(\lambda+2) \mathrm{y}-(\lambda+2) \mathrm{z}=9 has infinitely many solutions, then λ2+λ\lambda^{2}+\lambda is equal to
(A) 10
(B) 12
(C) 6
(D) 20
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20Explain
One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(A) 12\frac{1}{2}
(B) 35\frac{3}{5}
(C) 23\frac{2}{3}
(D) 49\frac{4}{9}
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21Explain
If the area of the larger portion bounded between the curves x2+y2=25x^{2}+y^{2}=25 and y=∣x−1∣y=|x-1| is 14(bπ+c)\frac{1}{4}(b \pi+c), b,c∈N\mathrm{b}, \mathrm{c} \in \mathbb{N}, then b+c\mathrm{b}+\mathrm{c} is equal to ____\_\_\_\_
(77)
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22Explain
The sum of all rational terms in the expansion of (1+21/3+31/2)6\left(1+2^{1 / 3}+3^{1 / 2}\right)^{6} is equal to ____\_\_\_\_
(612)
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23Explain
Let the circle C touch the line x−y+1=0\mathrm{x}-\mathrm{y}+1=0, have the centre on the positive x -axis, and cut off a chord of length 413\frac{4}{\sqrt{13}} along the line −3x+2y=1-3 x+2 y=1. Let H be the hyperbola x2α2−y2β2=1\frac{\mathrm{x}^{2}}{\alpha^{2}}-\frac{\mathrm{y}^{2}}{\beta^{2}}=1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2α2+3β22 \alpha^{2}+3 \beta^{2} is equal to ____\_\_\_\_
(19)
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24Explain
If the set of all values of a , for which the equation 5x3−15x−a=05 \mathrm{x}^{3}-15 \mathrm{x}-\mathrm{a}=0 has three distinct real roots, is the interval (α,β)(\alpha, \beta), then β−2α\beta-2 \alpha is equal to ____\_\_\_\_
(30)
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25Explain
If the equation a(b−c)x2+b(c−a)x+c(a−b)=0a(b-c) x^{2}+b(c-a) x+c(a-b)=0 has equal roots, where a+c=15a+c=15 and b=365b=\frac{36}{5}, then a2+c2\mathrm{a}^{2}+\mathrm{c}^{2} is equal to ____\_\_\_\_
(117)
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26Explain
Regarding self-inductance : A : The self-inductance of the coil depends on its geometry. B : Self-inductance does not depend on the permeability of the medium. C : Self-induced e.m.f. opposes any change in the current in a circuit. D : Self-inductance is electromagnetic analogue of mass in mechanics. E : Work needs to be done against self-induced e.m.f. in establishing the current. Choose the correct answer from the options given below:
(A) A, B, C, D only
(B) A, C, D, E only
(C) A, B, C, E only
(D) B, C, D, E only
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27Explain
A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is yπ×10−2 s\mathrm{y} \pi \times 10^{-2} \mathrm{~s}, where the value of yy is (Acceleration due to gravity, g=10 m/s2g=10 \mathrm{~m} / \mathrm{s}^{2}, density of water =103 kg/m3=10^{3} \mathrm{~kg} / \mathrm{m}^{3} )
(A) 2
(B) 6
(C) 4
(D) 1
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28Explain
Given below are two statements: Statement-I : The hot water flows faster than cold water. Statement-II : Soap water has higher surface tension as compared to fresh water. In the light above statements, choose the correct answer from the options given below
(A) Statement-I is false but Statement II is true
(B) Statement-I is true but Statement II is false
(C) Both Statement-I and Statement-II are true
(D) Both Statement-I and Statement-II are false
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29Explain
A sub-atomic particle of mass 10−30 kg10^{-30} \mathrm{~kg} is moving with a velocity 2.21×106 m/s2.21 \times 10^{6} \mathrm{~m} / \mathrm{s}. Under the matter wave consideration, the particle will behave closely like ____\_\_\_\_ . ( h=6.63×10−34 J.s\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s} )
(A) Infra-red radiation
(B) X-rays
(C) Gamma rays
(D) Visible radiation
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30Explain
A spherical surface of radius of curvature RR, separates air from glass (refractive index =1.5=1.5 ). The centre of curvature is in the glass medium. A point object ' OO ' placed in air on the optic axis of the surface, so that its real image is formed at 'I' inside glass. The line OI intersects the spherical surface at P and PO=PI\mathrm{PO}=\mathrm{PI}. The distance PO equals to-
(A) 5 R
(B) 3R3 R
(C) 2 R
(D) 1.5 R
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31Explain
A radioactive nucleus n2n_{2} has 3 times the decay constant as compared to the decay constant of another radioactive nucleus n1\mathrm{n}_{1}. If initial number of both nuclei are the same, what is the ratio of number of nuclei of n2\mathrm{n}_{2} to the number of nuclei of n1\mathrm{n}_{1}, after one half-life of n1\mathrm{n}_{1} ?
(A) 1/41 / 4
(B) 1/81 / 8
(C) 4
(D) 8
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32Explain
Identify the valid statements relevant to the given circuit at the instant when the key is closed. A. There will be no current through resistor R. B. There will be maximum current in the connecting wires. C. Potential difference between the capacitor plates A and B is minimum. D. Charge on the capacitor plates is minimum. Choose the correct answer from the options given below :
(A) C, D only
(B) B, C, D only
(C) A, C only
(D) A, B, D only
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33Explain
The position of a particle moving on x -axis is given by x(t)=Asin⁡t+Bcos⁡2t+Ct2+D\mathrm{x}(\mathrm{t})=\mathrm{A} \sin \mathrm{t}+\mathrm{B} \cos ^{2} \mathrm{t}+\mathrm{Ct}^{2}+\mathrm{D}, where t is time. The dimension of ABCD\frac{A B C}{D} is-
(A) L
(B) L3 T−2\mathrm{L}^{3} \mathrm{~T}^{-2}
(C) L2 T−2\mathrm{L}^{2} \mathrm{~T}^{-2}
(D) L2\mathrm{L}^{2}
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34Explain
Match the 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-III, B-I, C-IV, D-II
(C) A-I, B-III, C-II, D-IV
(D) A-III, B-IV, C-I, D-II
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35Explain
Consider a moving coil galvanometer (MCG) : A : The torsional constant in moving coil galvanometer has dimensions [ML2 T−2]\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right] B : Increasing the current sensitivity may not necessarily increase the voltage sensitivity. C : If we increase number of turns (N) to its double (2N), then the voltage sensitivity doubles. D : MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer. E : Current sensitivity of MCG depends inversely on number of turns of coil. Choose the correct answer from the options given below :
(A) A, B only
(B) A, D, only
(C) B, D, E only
(D) A, B, E only
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36Explain
A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance rr as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance rr. The dipoles are made of opposite charge q separated by a distance 2 a . For the charge particle at P not to experience any net force, which of the following correctly describes the situation?
(A) ar−20\frac{a}{r}-20
(B) ar∼10\frac{a}{r} \sim 10
(C) ar∼0.5\frac{\mathrm{a}}{\mathrm{r}} \sim 0.5
(D) ar∼3\frac{a}{r} \sim 3
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37Explain
A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J , then the mass of the bullet is ____\_\_\_\_ grams. (Latent heat of fusion of lead =2.5×104JKg−1=2.5 \times 10^{4} \mathrm{JKg}^{-1} and specific heat capacity of lead =125JKg−1 K−1=125 \mathrm{JKg}^{-1} \mathrm{~K}^{-1} )
(A) 20
(B) 15
(C) 10
(D) 5
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38Explain
What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness ' hh ' in terms of the angle of incidence ' ii ' and angle of refraction ' rr ', if the glass slab is placed in air medium ?
(A) htan⁡(i−r)tan⁡r\frac{h \tan (i-r)}{\tan r}
(B) hcos⁡(i−r)sin⁡r\frac{h \cos (i-r)}{\sin r}
(C) h
(D) hsin⁡(i−r)cos⁡r\frac{h \sin (i-r)}{\cos r}
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39Explain
A solid sphere of mass ' mm ' and radius ' rr ' is allowed to roll without slipping from the highest point of an inclined plane of length ' LL ' and makes an angle 30∘30^{\circ} with the horizontal. The speed of the particle at the bottom of the plane is v1\mathrm{v}_{1}. If the angle of inclination is increased to 45∘45^{\circ} while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v2v_{2}. The ratio of v12:v22v_{1}{ }^{2}: v_{2}{ }^{2} is
(A) 1:21: \sqrt{2}
(B) 1:31: 3
(C) 1:21: 2
(D) 1:31: \sqrt{3}
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40Explain
Refer to the circuit diagram given in the figure, which of the following observation are correct? Total resistance of circuit is 6Ω6 \Omega. Current in Ammeter is 1 A. Potential across AB is 4 Volts. Potential across CD is 4 Volts. E. Total resistance of the circuit is 8Ω8 \Omega. Choose the correct answer from the options given below:
(A) A, B and D only
(B) A, C and D only
(C) B, C and E only
(D) A, B and C only
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41Explain
The electric flux is ϕ=ασ+βλ\phi=\alpha \sigma+\beta \lambda where λ\lambda and σ\sigma are linear and surface charge density, respectively, (αβ)\left(\frac{\alpha}{\beta}\right) represents
(A) charge
(B) electric field
(C) displacement
(D) area
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42Explain
Given a thin convex lens (refractive index μ2\mu_{2} ), kept in a liquid (refractive index μ1,μ1<μ2\mu_{1}, \mu_{1}<\mu_{2} ) having radii of curvature ∣R1∣\left|\mathrm{R}_{1}\right| and ∣R2∣\left|\mathrm{R}_{2}\right|. Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?
(A) μ1∣R1∣⋅∣R2∣μ2(∣R1∣+∣R2∣)−μ1∣R1∣\frac{\mu_{1}\left|\mathrm{R}_{1}\right| \cdot\left|\mathrm{R}_{2}\right|}{\mu_{2}\left(\left|\mathrm{R}_{1}\right|+\left|\mathrm{R}_{2}\right|\right)-\mu_{1}\left|\mathrm{R}_{1}\right|}
(B) μ1∣R1∣⋅∣R2∣μ2(∣R1∣+∣R2∣)−μ1∣R2∣\frac{\mu_{1}\left|R_{1}\right| \cdot\left|R_{2}\right|}{\mu_{2}\left(\left|R_{1}\right|+\left|R_{2}\right|\right)-\mu_{1}\left|R_{2}\right|}
(C) μ1∣R1∣⋅∣R2∣μ2(2∣R1∣+∣R2∣)−μ1∣R1∣⋅∣R2∣\frac{\mu_{1}\left|\mathrm{R}_{1}\right| \cdot\left|\mathrm{R}_{2}\right|}{\mu_{2}\left(2\left|\mathrm{R}_{1}\right|+\left|\mathrm{R}_{2}\right|\right)-\mu_{1} \sqrt{\left|\mathrm{R}_{1}\right| \cdot\left|\mathrm{R}_{2}\right|}}
(D) (μ2+μ1)∣R1∣(μ2−μ1)\frac{\left(\mu_{2}+\mu_{1}\right)\left|R_{1}\right|}{\left(\mu_{2}-\mu_{1}\right)}
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43Explain
The electric field of an electromagnetic wave in free space is E→=57cos⁡[7.5×106t−5×10−3(3x+4y)](4i^−3j^)N/C\overrightarrow{\mathrm{E}}=57 \cos \left[7.5 \times 10^{6} \mathrm{t}-5 \times 10^{-3}(3 \mathrm{x}+4 \mathrm{y})\right] (4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}. The associated magnetic field in Tesla is-
(A) B→=573×108cos⁡[7.5×106t−5×10−3(3x+4y)](5k^)\overrightarrow{\mathrm{B}}=\frac{57}{3 \times 10^{8}} \cos \left[7.5 \times 10^{6} \mathrm{t}-5 \times 10^{-3}(3 \mathrm{x}+4 \mathrm{y})\right](5 \hat{\mathrm{k}})
(B) B→=573×108cos⁡[7.5×106t−5×10−3(3x+4y)](k^)\overrightarrow{\mathrm{B}}=\frac{57}{3 \times 10^{8}} \cos \left[7.5 \times 10^{6} \mathrm{t}-5 \times 10^{-3}(3 \mathrm{x}+4 \mathrm{y})\right](\hat{\mathrm{k}})
(C) B→=−573×108cos⁡[7.5×106t−5×10−3(3x+4y)](5k^)\overrightarrow{\mathrm{B}}=-\frac{57}{3 \times 10^{8}} \cos \left[7.5 \times 10^{6} \mathrm{t}-5 \times 10^{-3}(3 \mathrm{x}+4 \mathrm{y})\right](5 \hat{\mathrm{k}})
(D) B→=−573×108cos⁡[7.5×106t−5×10−3(3x+4y)]\overrightarrow{\mathrm{B}}=-\frac{57}{3 \times 10^{8}} \cos \left[7.5 \times 10^{6} \mathrm{t}-5 \times 10^{-3}(3 \mathrm{x}+4 \mathrm{y})\right] (k^)(\hat{\mathrm{k}})
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44Explain
The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by airplane in the first 30.5 second is ____\_\_\_\_ km.
(A) 9
(B) 6
(C) 3
(D) 12
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45Explain
Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be-
(A) 2.0 cm
(B) 0.5 cm
(C) 1.5 cm
(D) 1.0 cm
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46Explain
A positive ion A and a negative ion B has charges 6.67×10−19C6.67 \times 10^{-19} \mathrm{C} and 9.6×10−10C9.6 \times 10^{-10} \mathrm{C}, and masses 19.2×10−27 kg19.2 \times 10^{-27} \mathrm{~kg} and 9×10−27 kg9 \times 10^{-27} \mathrm{~kg} respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is P×10−13\mathrm{P} \times 10^{-13}, where the value of P is ____\_\_\_\_ . (Take 14πε0=9×109Nm2C−1\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} \mathrm{C}^{-1} and universal gravitational constant as 6.67×10−11Nm2 kg−26.67 \times 10^{-11} \mathrm{Nm}^{2} \mathrm{~kg}^{-2} )
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47Explain
Two particles are located at equal distance from origin. The position vectors of those are represented by A⃗=2i^+3nj^+2k^\vec{A}=2 \hat{i}+3 n \hat{j}+2 \hat{k} \quad and B⃗=2i^−2j^+4pk^\vec{B}=2 \hat{i}-2 \hat{j}+4 p \hat{k}, respectively. If both the vectors are at right angle to each other, the value of n−1\mathrm{n}^{-1} is ____\_\_\_\_ .
(3)
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48Explain
An ideal gas initially at 0∘C0^{\circ} \mathrm{C} temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is 3/23 / 2, the change in temperature due to the thermodynamics process is ____\_\_\_\_ K.
(273)
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49Explain
A force f=x2yi^+y2j^f=x^{2} y \hat{i}+y^{2} \hat{j} acts on a particle in a plane x+y=10\mathrm{x}+\mathrm{y}=10. The work done by this force during a displacement from (0,0)(0,0) to (4 m,2 m)(4 \mathrm{~m}, 2 \mathrm{~m}) is ____\_\_\_\_ Joule (round off to the nearest integer)
(152)
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50Explain
In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of 8 A/s8 \mathrm{~A} / \mathrm{s}. At an instant when RR is 12Ω12 \Omega, the value of the current in the circuit will be ____\_\_\_\_ A.
(3)
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51Explain
The element that does not belong to the same period of the remaining elements (modern periodic table) is:
(A) Palladium
(B) Iridium
(C) Osmium
(D) Platinum
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52Explain
Heat treatment of muscular pain involves radiation of wavelength of about 900 nm . Which spectral line of H atom is suitable for this? Given: Rydberg constant RH=105 cm−1, h=6.6×10−34 J s,c=3×108 m/sR_{H}=10^{5} \mathrm{~cm}^{-1}, \mathrm{~h}=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} )
(A) Paschen series, ∞→3\infty \rightarrow 3
(B) Lyman series, ∞→1\infty \rightarrow 1
(C) Balmer series, ∞→2\infty \rightarrow 2
(D) Paschen series, 5→35 \rightarrow 3
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53Explain
The incorrect statements among the following is
(A) PH3\mathrm{PH}_{3} shows lower proton affinity than NH3\mathrm{NH}_{3}.
(B) PF3\mathrm{PF}_{3} exists but NF5\mathrm{NF}_{5} does not.
(C) NO2\mathrm{NO}_{2} can dimerise easily.
(D) SO2\mathrm{SO}_{2} can act as an oxidizing agent, but not as a reducing agent.
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54Explain
CrCl3.xNH3\mathrm{CrCl}_{3} . \mathrm{xNH}_{3} can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558∘C0.558^{\circ} \mathrm{C}. Assuming 100%100 \% ionisation of this complex and coordination number of Cr is 6 , the complex will be (Given Kf=1.86 K kg mol−1\mathrm{K}_{\mathrm{f}}=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1} )
(A) [Cr(NH3)6]Cl3\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right] \mathrm{Cl}_{3}
(B) [Cr(NH3)4Cl2]Cl\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right] \mathrm{Cl}
(C) [Cr(NH3)5Cl]Cl2\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right] \mathrm{Cl}_{2}
(D) [Cr(NH3)3Cl3]\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{3} \mathrm{Cl}_{3}\right]
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55Explain
FeO42−→+2.0 VFe3+→0.8 VFe2+→−0.5 VFe0\mathrm{FeO}_{4}^{2-} \xrightarrow{+2.0 \mathrm{~V}} \mathrm{Fe}^{3+} \xrightarrow{0.8 \mathrm{~V}} \mathrm{Fe}^{2+} \xrightarrow{-0.5 \mathrm{~V}} \mathrm{Fe}^{0} In the above diagram, the standard electrode potentials are given in volts (over the arrow). The value of EFeO42−/Fe2+Θ\mathrm{E}_{\mathrm{FeO}_{4}^{2-} / \mathrm{Fe}^{2+}}^{\Theta} is
(A) 1.7 V
(B) 1.2 V
(C) 2.1 V
(D) 1.4 V
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56Explain
Match the 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-IV, B-I, C-III, D-II
(C) A-IV, B-III, C-I, D-II
(D) A-II, B-I, C-III, D-IV
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57Explain
Given below are two statements: Statement I: Fructose does not contain an aldehydic group but still reduces Tollen's reagent Statement II : In the presence of base, fructose undergoes rearrangement to give glucose. 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
2.8×10−3 mol2.8 \times 10^{-3} \mathrm{~mol} of CO2\mathrm{CO}_{2} is left after removing 102110^{21} molecules from its ' xx ' mgm g sample. The mass of CO2\mathrm{CO}_{2} taken initially is Given : NA=6.02×1023 mol−1\mathrm{N}_{\mathrm{A}}=6.02 \times 10^{23} \mathrm{~mol}^{-1}
(A) 196.2 mg
(B) 98.3 mg
(C) 150.4 mg
(D) 48.2 mg
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59Explain
Ice at −5∘C-5^{\circ} \mathrm{C} is heated to become vapor with temperature of 110∘C110^{\circ} \mathrm{C} at atmospheric pressure. The entropy change associated with this process can be obtained from :
(A) ∫268 K383 KCpdT+ΔHmelting 273+ΔHboiling 373\int_{268 \mathrm{~K}}^{383 \mathrm{~K}} \mathrm{C}_{\mathrm{p}} \mathrm{dT}+\frac{\Delta \mathrm{H}_{\text {melting }}}{273}+\frac{\Delta \mathrm{H}_{\text {boiling }}}{373}
(B) ∫268 K273 KCp,mTdT+ΔHm, fusion Tf+ΔHm, vaporisation Tb\int_{268 \mathrm{~K}}^{273 \mathrm{~K}} \frac{\mathrm{C}_{\mathrm{p}, \mathrm{m}}}{\mathrm{T}} \mathrm{dT}+\frac{\Delta \mathrm{H}_{\mathrm{m}}, \text { fusion }}{\mathrm{T}_{\mathrm{f}}}+\frac{\Delta \mathrm{H}_{\mathrm{m}, \text { vaporisation }}}{\mathrm{T}_{\mathrm{b}}} +∫273 K373 KCp,mdTT+∫373 K383 KCp,mdTT+\int_{273 \mathrm{~K}}^{373 \mathrm{~K}} \frac{\mathrm{C}_{\mathrm{p}, \mathrm{m}} \mathrm{dT}}{\mathrm{T}}+\int_{373 \mathrm{~K}}^{383 \mathrm{~K}} \frac{\mathrm{C}_{\mathrm{p}, \mathrm{m}} \mathrm{dT}}{\mathrm{T}}
(C) ∫268K383KCpdT+qrev T\int_{268 K}^{383 K} \mathrm{C}_{\mathrm{p}} \mathrm{dT}+\frac{\mathrm{q}_{\text {rev }}}{\mathrm{T}}
(D) ∫268 K273 KCp,mdT+ΔHm, fusion Tf+ΔHm, vaporisation Tb\int_{268 \mathrm{~K}}^{273 \mathrm{~K}} \mathrm{C}_{\mathrm{p}, \mathrm{m}} \mathrm{dT}+\frac{\Delta \mathrm{H}_{\mathrm{m}}, \text { fusion }}{\mathrm{T}_{\mathrm{f}}}+\frac{\Delta \mathrm{H}_{\mathrm{m}, \text { vaporisation }}}{\mathrm{T}_{\mathrm{b}}} +∫273K373KCp,mdT+∫373K383KCp,mdT+\int_{273 K}^{373 K} C_{p, m} d T+\int_{373 K}^{383 K} C_{p, m} d T
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60Explain
The d-electronic configuration of an octahedral Co (II) complex having magnetic moment of 3.95 BM is :
(A) t2g6eg1t_{2 g}^{6} e_{g}^{1}
(B) t2g3eg0t_{2 g}^{3} e_{g}^{0}
(C) t2g5eg2t_{2 g}^{5} e_{g}^{2}
(D) e4t23e^{4} t_{2}^{3}
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61Explain
The complex that shows Facial - Meridional isomerism is
(A) [Co(NH3)3Cl3]\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3} \mathrm{Cl}_{3}\right]
(B) [Co(NH3)4Cl2]+\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right]^{+}
(C) [Co(en)3]3+\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}
(D) [Co(en)2Cl2]+\left[\mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]^{+}
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62Explain
The major product of the following reaction is : CH3CH2CH=O→ reflux  excess HCHO alkali \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}=\mathrm{O} \xrightarrow[\text { reflux }]{\substack{\text { excess } \mathrm{HCHO} \\ \text { alkali }}} ?
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63Explain
The correct stability order of the following species/molecules is :
(A) q>r>pq>r>p
(B) r>q>pr>q>p
(C) q>p>rq>p>r
(D) p>q>rp>q>r
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64Explain
Propane molecule on chlorination under photochemical condition gives two di-chloro products, " x " and " y ". Amongst " x " and " y ", " x " is an optically active molecule. How many tri-chloro products (consider only structural isomers) will be obtained from "x" when it is further treated with chlorine under the photochemical condition?
(A) 4
(B) 2
(C) 5
(D) 3
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65Explain
What amount of bromine will be required to convert 2 g of phenol into 2, 4, 6-tribromophenol ? (Given molar mass in gmol−1\mathrm{g} \mathrm{mol}^{-1} of C,H,O,Br\mathrm{C}, \mathrm{H}, \mathrm{O}, \mathrm{Br} are 12, 1, 16, 80 respectively)
(A) 10.22 g
(B) 6.0 g
(C) 4.0 g
(D) 20.44 g
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66Explain
The correct set of ions (aqueous solution) with same colour from the following is :
(A) V2+,Cr3+,Mn3+\mathrm{V}^{2+}, \mathrm{Cr}^{3+}, \mathrm{Mn}^{3+}
(B) Zn2+,V3+,Fe3+\mathrm{Zn}^{2+}, \mathrm{V}^{3+}, \mathrm{Fe}^{3+}
(C) Ti4+,V4+,Mn2+\mathrm{Ti}^{4+}, \mathrm{V}^{4+}, \mathrm{Mn}^{2+}
(D) Sc3+,Ti3+,Cr2+\mathrm{Sc}^{3+}, \mathrm{Ti}^{3+}, \mathrm{Cr}^{2+}
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67Explain
Given below are two statements : Statement I : In Lassaigne's test, the covalent organic molecules are transformed into ionic compounds. Statement II : The sodium fusion extract of an organic compound having N and S gives prussian blue colour with FeSO4\mathrm{FeSO}_{4} and Na4[Fe(CN)6]\mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] In the light of the above statements, choose the correct answer from the options given below
(A) Both Statement I and Statement II are true
(B) Both Statement I and Statement II are false
(C) Statement I is false but Statement II is true
(D) Statement I is true but Statement II is false
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68Explain
Which of the following happens when NH4OH\mathrm{NH}_{4} \mathrm{OH} is added gradually to the solution containing 1MA2+1 \mathrm{M} \mathrm{A}^{2+} and 1MB3+1 \mathrm{M} \mathrm{B}^{3+} ions ? Given : Ksp[A(OH)2]=9×10−10\mathrm{K}_{\mathrm{sp}}\left[\mathrm{A}(\mathrm{OH})_{2}\right]=9 \times 10^{-10} and
Ksp[ B(OH)3]=27×10−18 at 298 K.\mathrm{K}_{\mathrm{sp}}\left[\mathrm{~B}(\mathrm{OH})_{3}\right]=27 \times 10^{-18} \text { at } 298 \mathrm{~K} .
(A) B(OH)3\mathrm{B}(\mathrm{OH})_{3} will precipitate before A(OH)2\mathrm{A}(\mathrm{OH})_{2}
(B) A(OH)2\mathrm{A}(\mathrm{OH})_{2} and B(OH)3\mathrm{B}(\mathrm{OH})_{3} will precipitate together
(C) A(OH)2\mathrm{A}(\mathrm{OH})_{2} will precipitate before B(OH)3\mathrm{B}(\mathrm{OH})_{3}
(D) Both A(OH)2\mathrm{A}(\mathrm{OH})_{2} and B(OH)3\mathrm{B}(\mathrm{OH})_{3} do not show precipitation with NH4OH\mathrm{NH}_{4} \mathrm{OH}
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69Explain
Match the LIST-I with LIST-II Choose the correct answer from the options given below :
(A) A-IV, B-II, C-I, D-III
(B) A-III, B-II, C-I, D-IV
(C) A-IV, B-I, C-III, D-II
(D) A-II, B-IV, C-III, D-I
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70Explain
Which among the following react with Hinsberg's reagent? Choose the correct answer from the options given below:
(A) B and D only
(B) C and D only
(C) A, B and E only
(D) A, C and E only
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71Explain
If 1 mM solution of ethylamine produces pH=9\mathrm{pH}=9, then the ionization constant ( Kb\mathrm{K}_{\mathrm{b}} ) of ethylamine is 10−x10^{-x}. The value of xx is ____\_\_\_\_ (nearest integer). [The degree of ionization of ethylamine can be neglected with respect to unity.]
(7)
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72Explain
During "S" estimation, 160 mg of an organic compound gives 466 mg of barium sulphate. The percentage of Sulphur in the given compound is ____\_\_\_\_ %\%. (Given molar mass in gmol−1\mathrm{g} \mathrm{mol}^{-1} of Ba:137, S:32\mathrm{Ba}: 137, \mathrm{~S}: 32, O : 16)
(40)
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73Explain
Consider the following sequence of reactions to produce major product (A) Molar mass of product (A) is ____\_\_\_\_ gmol−1\mathrm{g} \mathrm{mol}^{-1}. (Given molar mass in gmol−1\mathrm{g} \mathrm{mol}^{-1} of C:12,H:1\mathrm{C}: 12, \mathrm{H}: 1, O:16,Br:80, N:14,P:31\mathrm{O}: 16, \mathrm{Br}: 80, \mathrm{~N}: 14, \mathrm{P}: 31 )
(171)
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74Explain
For the thermal decomposition of N2O5( g)\mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{~g}) at constant volume, the following table can be formed, for the reaction mentioned below : 2 N2O5( g)→2 N2O4( g)+O2( g)2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) x = ____\_\_\_\_ ×10−3 atm\times 10^{-3} \mathrm{~atm} [nearest integer] Given : Rate constant for the reaction is 4.606×10−2 s−14.606 \times 10^{-2} \mathrm{~s}^{-1}.
(900)
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75Explain
The standard enthalpy and standard entropy of decomposition of N2O4\mathrm{N}_{2} \mathrm{O}_{4} to NO2\mathrm{NO}_{2} are 55.0 kJ mol−155.0 \mathrm{~kJ} \mathrm{~mol}^{-1} and 175.0 J/K/mol175.0 \mathrm{~J} / \mathrm{K} / \mathrm{mol} respectively. The standard free energy change for this reaction at 25∘C25^{\circ} \mathrm{C} in Jmol−1\mathrm{J} \mathrm{mol}^{-1} is ____\_\_\_\_ (Nearest integer)
(2850)
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