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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Wednesday 29th 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
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Incorrect
75
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Complete Solutions

Q#ExplanationQuestionCorrectSolutionStatus
1Explain
Let the line x+y=1x+y=1 meet the circle x2+y2=4x^{2}+y^{2}=4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to
(A) 373 \sqrt{7}
(B) 2142 \sqrt{14}
(C) 575 \sqrt{7}
(D) 14\sqrt{14}
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2Explain
Let M and m respectively be the maximum and the minimum values of f(x)=∣1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x∣,x∈Rf(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 4 x\end{array}\right|, x \in R Then M4−m4\mathrm{M}^{4}-\mathrm{m}^{4} is equal to :
(A) 1280
(B) 1295
(C) 1040
(D) 1215
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3Explain
Two parabolas have the same focus (4,3)(4,3) and their directrices are the x -axis and the y -axis, respectively. If these parabolas intersects at the points AA and BB, then (AB)2(A B)^{2} is equal to
(A) 192
(B) 384
(C) 96
(D) 392
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4Explain
Let ABC be a triangle formed by the lines 7x−6y+3=0,x+2y−31=07 x-6 y+3=0, x+2 y-31=0 and 9x−2y−19=09 x-2 y-19=0, Let the point (h,k)(\mathrm{h}, \mathrm{k}) be the image of the centroid of ΔABC\Delta \mathrm{ABC} in the line 3x+6y−53=03 \mathrm{x}+6 \mathrm{y}-53=0. Then h2+k2+hk\mathrm{h}^{2}+\mathrm{k}^{2}+\mathrm{hk} is equal to
(A) 37
(B) 47
(C) 40
(D) 36
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5Explain
Let a⃗=2i^−j^+3k^,b⃗=3i^−5j^+k^\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}-5 \hat{j}+\hat{k} and c⃗\vec{c} be a vector such that a⃗×c⃗=c⃗×b⃗\vec{a} \times \vec{c}=\vec{c} \times \vec{b} and (a⃗+c⃗)⋅(b⃗+c⃗)=168(\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168. Then the maximum value of ∣c⃗∣2|\vec{c}|^{2} is :
(A) 77
(B) 462
(C) 308
(D) 154
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6Explain
Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set P is :
(A) 158
(B) 173
(C) 164
(D) 161
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7Explain
Let the area of the region {(x,y):2y≤x2+3\left\{(x, y): 2 y \leq x^{2}+3\right., y+∣x∣≤3,y≥∣x−1∣}\mathrm{y}+|\mathrm{x}| \leq 3, \mathrm{y} \geq|\mathrm{x}-1|\} be A. Then 6A is equal to:
(A) 16
(B) 12
(C) 18
(D) 14
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8Explain
The least value of nn for which the number of integral terms in the Binomial expansion of (73+1112)n(\sqrt[3]{7}+\sqrt[12]{11})^{\mathrm{n}} is 183 , is :
(A) 2184
(B) 2148
(C) 2172
(D) 2196
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9Explain
The number of solutions of the equation (9x−9x+2)(2x−7x+3)=0\left(\frac{9}{\mathrm{x}}-\frac{9}{\sqrt{\mathrm{x}}}+2\right)\left(\frac{2}{\mathrm{x}}-\frac{7}{\sqrt{\mathrm{x}}}+3\right)=0 is :
(A) 2
(B) 4
(C) 1
(D) 3
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10Explain
Let y=y(x)y=y(x) be the solution of the differential equation cos⁡x(log⁡e(cos⁡x))2dy+(sin⁡x−3ysin⁡xlog⁡e(cos⁡x))dx=0\cos x\left(\log _{e}(\cos x)\right)^{2} d y+\left(\sin x-3 y \sin x \log _{e}(\cos x)\right) d x=0, x∈(0,π2)x \in\left(0, \frac{\pi}{2}\right). If y(π4)=−1log⁡e2y\left(\frac{\pi}{4}\right)=\frac{-1}{\log _{e} 2}, then y(π6)y\left(\frac{\pi}{6}\right) is :
(A) 2log⁡e(3)−log⁡e(4)\frac{2}{\log _{e}(3)-\log _{e}(4)}
(B) 1log⁡e(4)−log⁡e(3)\frac{1}{\log _{e}(4)-\log _{e}(3)}
(C) −1log⁡e(4)-\frac{1}{\log _{e}(4)}
(D) 1log⁡e(3)−log⁡e(4)\frac{1}{\log _{e}(3)-\log _{e}(4)}
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11Explain
Define a relation RR on the interval [0,π2)\left[0, \frac{\pi}{2}\right) by xRyx R y if and only if sec⁡2x−tan⁡2y=1\sec ^{2} x-\tan ^{2} y=1. Then RR is :
(A) an equivalence relation
(B) both reflexive and transitive but not symmetric
(C) both reflexive and symmetric but not transitive
(D) reflexive but neither symmetric not transitive
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12Explain
Let the ellipse, E1:x2a2+y2b2=1,a>bE_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b and E2:x2 A2+y2 B2=1, A<B\mathrm{E}_{2}: \frac{\mathrm{x}^{2}}{\mathrm{~A}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~B}^{2}}=1, \mathrm{~A}<\mathrm{B} have same eccentricity 13\frac{1}{\sqrt{3}}. Let the product of their lengths of latus rectums be 323\frac{32}{\sqrt{3}}, and the distance between the foci of E1\mathrm{E}_{1} be 4. If E1\mathrm{E}_{1} and E2\mathrm{E}_{2} meet at A,B,C\mathrm{A}, \mathrm{B}, \mathrm{C} and D , then the area of the quadrilateral ABCD equals:
(A) 666 \sqrt{6}
(B) 1865\frac{18 \sqrt{6}}{5}
(C) 1265\frac{12 \sqrt{6}}{5}
(D) 2465\frac{24 \sqrt{6}}{5}
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13Explain
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th 11^{\text {th }} term is :
(A) 84
(B) 122
(C) 90
(D) 108
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14Explain
Let a⃗=i^+2j^+k^\vec{a}=\hat{i}+2 \hat{j}+\hat{k} and b⃗=2i^+7j^+3k^\vec{b}=2 \hat{i}+7 \hat{j}+3 \hat{k}. Let L1:r→=(−i^+2j^+k^)+λa→,λ∈R\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda \overrightarrow{\mathrm{a}}, \lambda \in \mathrm{R} and L2:r⃗=(j^+k^)+μb⃗,μ∈RL_{2}: \vec{r}=(\hat{j}+\hat{k})+\mu \vec{b}, \mu \in R be two lines. If the line L3L_{3} passes through the point of intersection of L1L_{1} and L2L_{2}, and is parallel to a⃗+b⃗\vec{a}+\vec{b}, then L3L_{3} passes through the point:
(A) (8,26,12)(8,26,12)
(B) (2,8,5)(2,8,5)
(C) (−1,−1,1)(-1,-1,1)
(D) (5,17,4)(5,17,4)
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15Explain
The value of lim⁡n→∞(∑K=1nk3+6k2+11k+5(k+3)!)\lim _{n \rightarrow \infty}\left(\sum_{K=1}^{n} \frac{k^{3}+6 k^{2}+11 k+5}{(k+3)!}\right) is :
(A) 43\frac{4}{3}
(B) 2
(C) 73\frac{7}{3}
(D) 53\frac{5}{3}
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16Explain
The integral 80∫0π4(sin⁡θ+cos⁡θ9+16sin⁡2θ)dθ80 \int_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) \mathrm{d} \theta is equal to :
(A) 3log⁡e43 \log _{\mathrm{e}} 4
(B) 6log⁡e46 \log _{e} 4
(C) 4log⁡e34 \log _{e} 3
(D) 2log⁡e32 \log _{e} 3
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17Explain
Let L1:x−11=y−2−1=z−12\mathrm{L}_{1}: \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-2}{-1}=\frac{\mathrm{z}-1}{2} and L2:x+1−1=y−22=z1\mathrm{L}_{2}: \frac{\mathrm{x}+1}{-1}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}}{1} be two lines. Let L3L_{3} be a line passing through the point ( α,β,γ\alpha, \beta, \gamma ) and be perpendicular to both L1\mathrm{L}_{1} and L2\mathrm{L}_{2}. If L3\mathrm{L}_{3} intersects L1\mathrm{L}_{1}, then ∣5α−11β−8γ∣|5 \alpha-11 \beta-8 \gamma| equals :
(A) 18
(B) 16
(C) 25
(D) 20
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18Explain
Let x1,x2,……x10\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \ldots \mathrm{x}_{10} be ten observations such that ∑i=110(xi−2)=30,∑i=110(xi−β)2=98,β>2\sum_{i=1}^{10}\left(x_{i}-2\right)=30, \sum_{i=1}^{10}\left(x_{i}-\beta\right)^{2}=98, \beta>2 and their variance is 45\frac{4}{5}. If μ\mu and σ2\sigma^{2} are respectively the mean and the variance of 2(x1−1)+4β,2(x2−1)+4β,….,2(x10−1)+4β2\left(x_{1}-1\right)+4 \beta, 2\left(x_{2}-1\right)+ 4 \beta, \ldots ., 2\left(\mathrm{x}_{10}-1\right)+4 \beta, then βμσ2\frac{\beta \mu}{\sigma^{2}} is equal to :
(A) 100
(B) 110
(C) 120
(D) 90
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19Explain
Let ∣z1−8−2i∣≤1\left|z_{1}-8-2 i\right| \leq 1 and ∣z2−2+6i∣≤2\left|z_{2}-2+6 i\right| \leq 2, z1,z2∈C\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{C}. Then the minimum value of ∣z1−z2∣\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| is :
(A) 3
(B) 7
(C) 13
(D) 10
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20Explain
Let A=[aij]=[log⁡5128log⁡45log⁡58log⁡425]\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]=\left[\begin{array}{cc}\log _{5} 128 & \log _{4} 5 \\ \log _{5} 8 & \log _{4} 25\end{array}\right]. If Aij\mathrm{A}_{\mathrm{ij}} is the cofactor of aij,Cij=∑k=12aikAjk,1≤i\mathrm{a}_{\mathrm{ij}}, \mathrm{C}_{\mathrm{ij}}=\sum_{\mathrm{k}=1}^{2} \mathrm{a}_{\mathrm{ik}} \mathrm{A}_{\mathrm{jk}}, 1 \leq \mathrm{i}, j≤2\mathrm{j} \leq 2, and C=[Cij]\mathrm{C}=\left[\mathrm{C}_{\mathrm{ij}}\right], then 8∣C∣8|\mathrm{C}| is equal to :
(A) 262
(B) 288
(C) 242
(D) 222
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21Explain
Let f:(0,∞)→R\mathrm{f}:(0, \infty) \rightarrow \mathrm{R} be a twice differentiable function. If for some a≠0,∫01f(λx)dλ=af(x)a \neq 0, \int_{0}^{1} f(\lambda x) d \lambda=a f(x), f(1)=1f(1)=1 and f(16)=18f(16)=\frac{1}{8}, then 16−f′(116)16-f^{\prime}\left(\frac{1}{16}\right) is equal to ____\_\_\_\_ ,
(112)
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22Explain
Let S={m∈Z:Am2+Am=3I−A−6}S=\left\{m \in Z: A^{m^{2}}+A^{m}=3 I-A^{-6}\right\}, where A=[2−110]\mathrm{A}=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]. Then n(S)\mathrm{n}(\mathrm{S}) is equal to ____\_\_\_\_ .
(2)
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23Explain
Let [t][t] be the greatest integer less than or equal to tt. Then the least value of p∈N\mathrm{p} \in \mathrm{N} for which lim⁡x→0+(x([1x]+[2x]+…..+[px])−x2([1x2]+[22x2]+….+[92x2]))≥1\lim _{x \rightarrow 0^{+}}\left(x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\ldots . .+\left[\frac{p}{x}\right]\right)-x^{2}\left(\left[\frac{1}{x^{2}}\right]+\left[\frac{2^{2}}{x^{2}}\right]+\ldots .+\left[\frac{9^{2}}{x^{2}}\right]\right)\right) \geq 1 is equal to ____\_\_\_\_.
(24)
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24Explain
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is 4 ____\_\_\_\_ .
(1405)
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25Explain
Let S={x:cos⁡−1x=π+sin⁡−1x+sin⁡−1(2x+1)}S=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}. Then ∑x∈S(2x−1)2\sum_{\mathrm{x} \in \mathrm{S}}(2 \mathrm{x}-1)^{2} is equal to ____\_\_\_\_ .
(5)
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26Explain
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged. Reason (R) : By using the choke coil, the voltage across the tube is reduced by a factor (R/R2+ω2L2)\left(R / \sqrt{R^{2}+\omega^{2} L^{2}}\right), where ω\omega is frequency of the supply across resistor R and inductor L . If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage. In the light of the above statements, choose the most appropriate 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 false but (R) is true.
(C) Both (A) and (R) are true and (R) is the correct explanation of (A).
(D) (A) is true but (R) is false.
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27Explain
Two projectiles are fired with same initial speed from same point on ground at angles of (45∘−α)\left(45^{\circ}-\alpha\right) and ( 45∘+α45^{\circ}+\alpha ), respectively, with the horizontal direction. The ratio of their maximum heights attained is :
(A) 1−tan⁡α1+tan⁡α\frac{1-\tan \alpha}{1+\tan \alpha}
(B) 1+sin⁡α1−sin⁡α\frac{1+\sin \alpha}{1-\sin \alpha}
(C) 1−sin⁡2α1+sin⁡2α\frac{1-\sin 2 \alpha}{1+\sin 2 \alpha}
(D) 1+sin⁡2α1−sin⁡2α\frac{1+\sin 2 \alpha}{1-\sin 2 \alpha}
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28Explain
An electric dipole of mass m , charge q , and length ll is placed in a uniform electric field E→=E0i^\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \hat{\mathrm{i}}. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :
(A) 12π2 mlqE0\frac{1}{2 \pi} \sqrt{\frac{2 \mathrm{~m} l}{\mathrm{qE}_{0}}}
(B) 2π mlqE02 \pi \sqrt{\frac{\mathrm{~m} l}{\mathrm{qE}_{0}}}
(C) 12π ml2qE0\frac{1}{2 \pi} \sqrt{\frac{\mathrm{~m} l}{2 \mathrm{qE}_{0}}}
(D) 2πml2qE02 \pi \sqrt{\frac{\mathrm{ml}}{2 \mathrm{qE}_{0}}}
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29Explain
The pair of physical quantities not having same dimensions is :
(A) Torque and energy
(B) Surface tension and impulse
(C) Angular momentum and Planck's constant
(D) Pressure and Young's modulus
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30Explain
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. Reason (R) : Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below:
(A) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(B) Both (A) and (R) are true and (R) is the correct explanation of (A).
(C) (A) is true but (R) is false.
(D) (A) is false but (R) is true.
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31Explain
The expression given below shows the variation of velocity (v) with time ( t ), v=At2+BtC+tv=\mathrm{At}^{2}+\frac{\mathrm{Bt}}{\mathrm{C}+\mathrm{t}}. The dimension of ABC is :
(A) [M0 L2 T−3]\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-3}\right]
(B) [M0 L1 T−3]\left[\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-3}\right]
(C) [M0 L1 T−2]\left[\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-2}\right]
(D) [M0 L2 T−2]\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]
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32Explain
Consider I1\mathrm{I}_{1} and I2\mathrm{I}_{2} are the currents flowing simultaneously in two nearby coils 1&21 \& 2, respectively. If L1=L_{1}= self inductance of coil 1 , M12=\mathrm{M}_{12}= mutual inductance of coil 1 with respect to coil 2 , then the value of induced emf in coil 1 will be
(A) ε1=−L1dI1dt+M12dI2dt\varepsilon_{1}=-L_{1} \frac{d I_{1}}{d t}+M_{12} \frac{d I_{2}}{d t}
(B) ε1=−L1dI1dt−M12dI1dt\varepsilon_{1}=-L_{1} \frac{d I_{1}}{d t}-M_{12} \frac{d I_{1}}{d t}
(C) ε1=−L1dI1dt−M12dI2dt\varepsilon_{1}=-L_{1} \frac{d I_{1}}{d t}-M_{12} \frac{d I_{2}}{d t}
(D) ε1=−L1dI2dt−M12dI1dt\varepsilon_{1}=-\mathrm{L}_{1} \frac{\mathrm{dI}_{2}}{\mathrm{dt}}-\mathrm{M}_{12} \frac{\mathrm{dI}_{1}}{\mathrm{dt}}
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33Explain
At the interface between two materials having refractive indices n1\mathrm{n}_{1} and n2\mathrm{n}_{2}, the critical angle for reflection of an em wave is θ1c\theta_{1 c}. The n2n_{2} material is replaced by another material having refractive index n3\mathrm{n}_{3}, such that the critical angle at the interface between n1\mathrm{n}_{1} and n3\mathrm{n}_{3} materials is θ2C\theta_{2 \mathrm{C}}. If n3>n2>n1\mathrm{n}_{3}>\mathrm{n}_{2}>\mathrm{n}_{1}; n2n3=25\frac{\mathrm{n}_{2}}{\mathrm{n}_{3}}=\frac{2}{5} and sin⁡θ2C−sin⁡θ1C=12\sin \theta_{2 \mathrm{C}}-\sin \theta_{1 \mathrm{C}}=\frac{1}{2}, then θ1C\theta_{1 \mathrm{C}} is
(A) sin⁡−1(16n1)\sin ^{-1}\left(\frac{1}{6 n_{1}}\right)
(B) sin⁡−1(23n1)\sin ^{-1}\left(\frac{2}{3 n_{1}}\right)
(C) sin⁡−1(56n1)\sin ^{-1}\left(\frac{5}{6 n_{1}}\right)
(D) sin⁡−1(13n1)\sin ^{-1}\left(\frac{1}{3 n_{1}}\right)
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34Explain
Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be
(A) [a/4,3a/2][\mathrm{a} / 4,3 \mathrm{a} / 2]
(B) [a/2,2a][\mathrm{a} / 2,2 \mathrm{a}]
(C) [a/2,3a][a / 2,3 a]
(D) [a/4,2a][\mathrm{a} / 4,2 \mathrm{a}]
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35Explain
As shown below, bob A of a pendulum having massless string of length ' R ' is released from 60∘60^{\circ} to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the centre. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take gg as acceleration due to gravity)
(A) 13Rg\frac{1}{3} \sqrt{\mathrm{Rg}}
(B) Rg⁡\sqrt{\operatorname{Rg}}
(C) 43Rg\frac{4}{3} \sqrt{R g}
(D) 23Rg\frac{2}{3} \sqrt{\mathrm{Rg}}
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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) : Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance. Reason (R) : A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with frequency of incident radiation. In the light of the above statements, choose the most appropriate 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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37Explain
A coil of area A and N turns is rotating with angular velocity ω\omega in a uniform magnetic field B⃗\vec{B} about an axis perpendicular to B→\overrightarrow{\mathrm{B}}. Magnetic flux φ\varphi and induced emf ε\varepsilon across it, at an instant when B→\overrightarrow{\mathrm{B}} is parallel to the plane of coil, are :
(A) φ=AB,ε=0\varphi=\mathrm{AB}, \varepsilon=0
(B) φ=0,ε=NABω\varphi=0, \varepsilon=\mathrm{NAB} \omega
(C) φ=0,ε=0\varphi=0, \varepsilon=0
(D) φ=AB,ε=NABω\varphi=\mathrm{AB}, \varepsilon=\mathrm{NAB} \omega
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38Explain
The fractional compression (ΔVV)\left(\frac{\Delta \mathrm{V}}{\mathrm{V}}\right) of water at the depth of 2.5 km below the sea level is ____\_\_\_\_ \%. Given, the Bulk modulus of water =2×109Nm−2=2 \times 10^{9} \mathrm{Nm}^{-2}, density of water =103 kg m−3=10^{3} \mathrm{~kg} \mathrm{~m}^{-3}, acceleration due to gravity =g=10 ms−2=\mathrm{g}=10 \mathrm{~ms}^{-2}.
(A) 1.75
(B) 1.0
(C) 1.5
(D) 1.25
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39Explain
If λ\lambda and K are de Broglie Wavelength and kinetic energy, respectively, of a particle with constant mass. The correct graphical representation for the particle will be :-
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40Explain
For the circuit shown above, equivalent GATE is :
(A) OR gate
(B) NOT gate
(C) AND gate
(D) NAND gate
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41Explain
A body of mass ' mm ' connected to a massless and unstretchable string goes in verticle circle of radius ' R ' under gravity g . The other end of the string is fixed at the center of circle. If velocity at top of circular path is ngRn \sqrt{g R}, where, n≥1n \geq 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is
(A) nn+4\frac{n}{n+4}
(B) n+4n\frac{n+4}{n}
(C) n2n2+4\frac{n^{2}}{n^{2}+4}
(D) n2+4n2\frac{n^{2}+4}{n^{2}}
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42Explain
Let uu and vv be the distances of the object and the image from a lens of focal length ff. The correct graphical representation of u and v for a convex lens when ∣u∣>f|\mathrm{u}|>f, is
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43Explain
Match List-I with List-II. Choose the correct answer from the options given below :
(A) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
(B) (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(C) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(D) (A)-(III), (B)-(II), (C)-(IV), (D)-(I)
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44Explain
The workdone in an adiabatic change in an ideal gas depends upon only :
(A) change in its pressure
(B) change in its specific heat
(C) change in its volume
(D) change in its temperature
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45Explain
Given below are two statements: one is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (A) : Electromagnetic waves carry energy but not momentum. Reason (R): Mass of a photon is zero. In the light of the above statements, choose the most appropriate answer from the options given below :
(A) (A) is true but (R) is false.
(B) (A) is false but (R) is true.
(C) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(D) Both (A) and (R) are true and (R) is the correct explanation of (A).
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46Explain
The coordinates of a particle with respect to origin in a given reference frame is (1,1,1)(1,1,1) meters. If a force of F→=i^−j^+k^\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}} acts on the particle, then the magnitude of torque (with respect to origin) in zz-direction is ____\_\_\_\_ .
(2)
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47Explain
A container of fixed volume contains a gas at 27∘C27^{\circ} \mathrm{C}. To double the pressure of the gas, the temperature of gas should be raised to ____\_\_\_\_ ∘C{ }^{\circ} \mathrm{C}.
(327)
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48Explain
Two light beams fall on a transparent material block at point 1 and 2 with angle θ1\theta_{1} and θ2\theta_{2}, respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2,d=43 cm2, d=4 \sqrt{3} \mathrm{~cm} and θ1=θ2=cos⁡−1(n22n1)\theta_{1}=\theta_{2}=\cos ^{-1}\left(\frac{\mathrm{n}_{2}}{2 \mathrm{n}_{1}}\right), where refractive index of the block n2>\mathrm{n}_{2}> refractive index of the outside medium n1n_{1}, then the thickness of the block is ____\_\_\_\_ cm.
(6)
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49Explain
In a hydraulic lift, the surface area of the input piston is 6 cm26 \mathrm{~cm}^{2} and that of the output piston is 1500 cm21500 \mathrm{~cm}^{2}. If 100 N force is applied to the input piston to raise the output piston by 20 cm , then the work done is ____\_\_\_\_ kJ.
(5)
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50Explain
The maximum speed of a boat in still water is 27 km/h27 \mathrm{~km} / \mathrm{h}. Now this boat is moving downstream in a river flowing at 9 km/h9 \mathrm{~km} / \mathrm{h}. A man in the boat throws a ball vertically upwards with speed of 10 m/s10 \mathrm{~m} / \mathrm{s}. Range of the ball as observed by an observer at rest on the river bank, is ____\_\_\_\_ cm . (Take g=10 m/s2\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2} )
(2000)
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51Explain
Total number of nucleophiles from the following is :NH3,PhSH,(H3C)2 S,H2C=CH2,O⊖H,H3O⊕\mathrm{NH}_{3}, \quad \mathrm{PhSH}, \quad\left(\mathrm{H}_{3} \mathrm{C}\right)_{2} \mathrm{~S}, \quad \mathrm{H}_{2} \mathrm{C}=\mathrm{CH}_{2}, \quad \stackrel{\ominus}{\mathrm{O}} \mathrm{H}, \quad \mathrm{H}_{3} \mathrm{O}^{\oplus}, (CH3)2CO,>=NCH3\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO},>=\mathrm{NCH}_{3}
(A) 5
(B) 4
(C) 7
(D) 6
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52Explain
The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.
(A) ESn4+/Sn2+⊖=+1.15 V\mathrm{E}_{\mathrm{Sn}^{4+} / \mathrm{Sn}^{2+}}^{\ominus}=+1.15 \mathrm{~V}
(B) ETl3+/Tl⊖=+1.26 V\mathrm{E}_{\mathrm{T} l^{3+} / \mathrm{T} l}^{\ominus}=+1.26 \mathrm{~V}
(C) EAl3+/Al⊖=−1.66 V\mathrm{E}_{\mathrm{Al}^{3+} / \mathrm{Al}}^{\ominus}=-1.66 \mathrm{~V}
(D) EPb4+/Pb2+⊖=+1.67 V\mathrm{E}_{\mathrm{Pb}^{4+} / \mathrm{Pb}^{2+}}^{\ominus}=+1.67 \mathrm{~V}
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53Explain
The molar conductivity of a weak electrolyte when plotted against the square root of its concentration, which of the following is expected to be observed?
(A) A small decrease in molar conductivity is observed at infinite dilution.
(B) A small increase in molar conductivity is observed at infinite dilution.
(C) Molar conductivity increases sharply with increase in concentration.
(D) Molar conductivity decreases sharply with increase in concentration.
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54Explain
At temperature T , compound AB2( g)\mathrm{AB}_{2(\mathrm{~g})} dissociates as AB2( g)⇌AB(g)+12 B2( g)\mathrm{AB}_{2(\mathrm{~g})} \rightleftharpoons \mathrm{AB}_{(\mathrm{g})}+\frac{1}{2} \mathrm{~B}_{2(\mathrm{~g})} having degree of dissociation x (small compared to unity). The correct expression for x in terms of Kp\mathrm{K}_{\mathrm{p}} and p is
(A) 2Kpp3\sqrt[3]{\frac{2 K_{p}}{p}}
(B) 2Kpp4\sqrt[4]{\frac{2 K_{p}}{p}}
(C) 2Kp2p3\sqrt[3]{\frac{2 K_{p}^{2}}{p}}
(D) Kp\sqrt{\mathrm{K}_{\mathrm{p}}}
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55Explain
Match List-I with List-II. Choose the correct answer from the options given below:
(A) (A)-(III), (B)-(II), (C)-(IV), (D)-(I)
(B) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
(C) (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
(D) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
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56Explain
Choose the correct statements. (A) Weight of a substance is the amount of matter present in it. (B) Mass is the force exerted by gravity on an object. (C) Volume is the amount of space occupied by a substance. (D) Temperatures below 0∘C0^{\circ} \mathrm{C} are possible in Celsius scale, but in Kelvin scale negative temperature is not possible. (E) Precision refers to the closeness of various measurements for the same quantity.
(A) (B), (C) and (D) Only
(B) (A), (B) and (C) Only
(C) (A), (D) and (E) Only
(D) (C), (D) and (E) Only
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57Explain
The correct increasing order of stability of the complexes based on Δo\Delta_{\mathrm{o}} value is : (I) [Mn(CN)6]3−\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{3-} (II) [Co(CN)6]4−\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{4-} (III) [Fe(CN)6]4−\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-} (IV) [Fe(CN)6]3−\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}
(A) II < III < I < IV
(B) IV < III < II < I
(C) I < II < IV < III
(D) III < II < IV < I
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58Explain
Match List-I with List-II. Choose the correct answer from the options given below :
(A) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
(B) (A)-(III), (B)-(I), (C)-(II), (D)-(IV)
(C) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)
(D) (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
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59Explain
In the following substitution reaction : Product ' P ' formed is :
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60Explain
For a Mg∣Mg2+(aq)∥Ag+(aq)∣Ag\mathrm{Mg}\left|\mathrm{Mg}^{2+}(\mathrm{aq}) \| \mathrm{Ag}^{+}(\mathrm{aq})\right| \mathrm{Ag} the correct Nernst Equation is :
(A) Ecell =Ecell o−RT2 Fln⁡[Ag+][Mg2+]\mathrm{E}_{\text {cell }}=\mathrm{E}_{\text {cell }}^{\mathrm{o}}-\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Ag}^{+}\right]}{\left[\mathrm{Mg}^{2+}\right]}
(B) Ecell =Ecell o+RT2 Fln⁡[Ag+]2[Mg2+]\mathrm{E}_{\text {cell }}=\mathrm{E}_{\text {cell }}^{\mathrm{o}}+\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Ag}^{+}\right]^{2}}{\left[\mathrm{Mg}^{2+}\right]}
(C) Ecell =Ecell o−RT2 Fln⁡[Mg2+][Ag+]\mathrm{E}_{\text {cell }}=\mathrm{E}_{\text {cell }}^{\mathrm{o}}-\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Mg}^{2+}\right]}{\left[\mathrm{Ag}^{+}\right]}
(D) Ecell =Ecell o−RT2 Fln⁡[Ag+]2[Mg2+]\mathrm{E}_{\text {cell }}=\mathrm{E}_{\text {cell }}^{\mathrm{o}}-\frac{\mathrm{RT}}{2 \mathrm{~F}} \ln \frac{\left[\mathrm{Ag}^{+}\right]^{2}}{\left[\mathrm{Mg}^{2+}\right]}
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61Explain
The correct option with order of melting points of the pairs (Mn,Fe),(Tc,Ru)(\mathrm{Mn}, \mathrm{Fe}),(\mathrm{Tc}, \mathrm{Ru}) and (Re,Os)(\mathrm{Re}, \mathrm{Os}) is :
(A) Fe<Mn,Ru<Tc\mathrm{Fe}<\mathrm{Mn}, \mathrm{Ru}<\mathrm{Tc} and Re<Os\mathrm{Re}<\mathrm{Os}
(B) Mn<Fe,Tc<Ru\mathrm{Mn}<\mathrm{Fe}, \mathrm{Tc}<\mathrm{Ru} and Re<Os\mathrm{Re}<\mathrm{Os}
(C) Mn<Fe,Tc<Ru\mathrm{Mn}<\mathrm{Fe}, \mathrm{Tc}<\mathrm{Ru} and Os<Re\mathrm{Os}<\mathrm{Re}
(D) Fe<Mn,Ru<Tc\mathrm{Fe}<\mathrm{Mn}, \mathrm{Ru}<\mathrm{Tc} and Os<Re\mathrm{Os}<\mathrm{Re}
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62Explain
24 g of AX2\mathrm{AX}_{2} (molar mass 124 g mol−1124 \mathrm{~g} \mathrm{~mol}^{-1} ) is dissolved in 1 kg of water to form a solution with boiling point of 100.0156∘C100.0156^{\circ} \mathrm{C}, while 25.4 g of AY2\mathrm{AY}_{2} (molar mass 250 g mol−1250 \mathrm{~g} \mathrm{~mol}^{-1} ) in 2 kg of water constitutes a solution with a boiling point of 100.0260∘C100.0260^{\circ} \mathrm{C}. Kb(H2O)=0.52 K kg mol−1\mathrm{K}_{\mathrm{b}}\left(\mathrm{H}_{2} \mathrm{O}\right)=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1} Which of the following is correct?
(A) AX2\mathrm{AX}_{2} and AY2\mathrm{AY}_{2} (both) are completely unionised.
(B) AX2\mathrm{AX}_{2} and AY2\mathrm{AY}_{2} (both) are fully ionised.
(C) AX2\mathrm{AX}_{2} is completely unionised while AY2\mathrm{AY}_{2} is fully ionised.
(D) AX2\mathrm{AX}_{2} is completely ionised while AY2\mathrm{AY}_{2} is fully unionised.
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63Explain
500 J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm . The final temperature and the change in internal energy respectively are : Given : R=8.3 J K−1 mol−1\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}
(A) 348 K and 300 J
(B) 378 K and 300 J
(C) 368 K and 500 J
(D) 378 K and 500 J
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64Explain
The reaction A2+B2→2AB\mathrm{A}_{2}+\mathrm{B}_{2} \rightarrow 2 \mathrm{AB} follows the mechanism A2⇌k1k−1 A+A\mathrm{A}_{2} \underset{\mathrm{k}_{-1}}{\stackrel{\mathrm{k}_{1}}{\rightleftharpoons}} \mathrm{~A}+\mathrm{A} (fast) A+B2→k2AB+B\mathrm{A}+\mathrm{B}_{2} \xrightarrow{\mathrm{k}_{2}} \mathrm{AB}+\mathrm{B} (slow) A+B→AB\mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB} (fast) The overall order of the reaction is :
(A) 1.5
(B) 3
(C) 2.5
(D) 2
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65Explain
If a0\mathrm{a}_{0} is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength ( λ\lambda ) of the electron present in the second orbit of hydrogen atom ? [ n : any integer]
(A) 2a0nπ\frac{2 a_{0}}{n \pi}
(B) 8πa0n\frac{8 \pi a_{0}}{n}
(C) 4πa0n\frac{4 \pi a_{0}}{n}
(D) 4nπa0\frac{4 n}{\pi a_{0}}
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66Explain
The product (P)(\mathrm{P}) formed in the following reaction is :
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67Explain
An element 'E' has the ionisation enthalpy value of 374 kJ mol−1374 \mathrm{~kJ} \mathrm{~mol}^{-1}. 'E' reacts with elements A, B, C and D with electron gain enthalpy values of −328,−349-328,-349, -325 and −295 kJ mol−1-295 \mathrm{~kJ} \mathrm{~mol}^{-1}, respectively. The correct order of the products EA, EB, EC and ED in terms of ionic character is :
(A) EB>EA>EC>ED\mathrm{EB}>\mathrm{EA}>\mathrm{EC}>\mathrm{ED}
(B) ED>EC>EA>EB\mathrm{ED}>\mathrm{EC}>\mathrm{EA}>\mathrm{EB}
(C) EA>EB>EC>ED\mathrm{EA}>\mathrm{EB}>\mathrm{EC}>\mathrm{ED}
(D) ED>EC>EB>EA\mathrm{ED}>\mathrm{EC}>\mathrm{EB}>\mathrm{EA}
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68Explain
Match List - I with List - II. Choose the correct answer form the options given below :
(A) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
(B) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)
(C) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
(D) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
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69Explain
The steam volatile compounds among the following are: Choose the correct answer from the options given below:
(A) (B) and (D) only
(B) (A) and (C) only
(C) (A) and (B) only
(D) (A),(B) and (C) only
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70Explain
Given below are two statements : Statement (I): The radii of isoelectronic species increases in the order. Mg2+<Na+<F−<O2−\mathrm{Mg}^{2+}<\mathrm{Na}^{+}<\mathrm{F}^{-}<\mathrm{O}^{2-} Statement (II) : The magnitude of electron gain enthalpy of halogen decreases in the order. Cl>F>Br>I\mathrm{Cl}>\mathrm{F}>\mathrm{Br}>\mathrm{I} In the light of the above statements, choose the most appropriate answer from the options given below :
(A) Statement I is incorrect but Statement II is correct
(B) Both Statement I and Statement II are incorrect
(C) Statement I is correct but Statement II is incorrect
(D) Both Statement I and Statement II are correct
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71Explain
Given below are some nitrogen containing compounds. Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume ____\_\_\_\_ mg of HCl . (Given molar mass in gmol−1C:12,H:1,O:16\mathrm{g} \mathrm{mol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16, Cl : 35.5)
(341)
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72Explain
The molar mass of the water insoluble product formed from the fusion of chromite ore (FeCr2O4)\left(\mathrm{FeCr}_{2} \mathrm{O}_{4}\right) with Na2CO3\mathrm{Na}_{2} \mathrm{CO}_{3} in presence of O2\mathrm{O}_{2} is ____\_\_\_\_ gmol−1\mathrm{g} \mathrm{mol}^{-1}.
(160)
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73Explain
The sum of sigma (σ)(\sigma) and pi⁡(π)\operatorname{pi}(\pi) bonds in Hex-1,3-dien-5-yne is ____\_\_\_\_ .
(15)
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74Explain
If A2 B\mathrm{A}_{2} \mathrm{~B} is 30%30 \% ionised in an aqueous solution, then the value of van't Hoff factor (i) is ____\_\_\_\_ ×10−1\times 10^{-1}.
(16)
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75Explain
1 mole of compound ' S ' will weigh ____\_\_\_\_ g. (Given molar mass in gmol−1C:12,H:1,O:16\mathrm{g} \mathrm{mol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16 )
(13)
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