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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Tuesday 28th 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 number of different 5 digit numbers greater than 50000 that can be formed using the digits 0,1 , 2,3,4,5,6,72,3,4,5,6,7, such that the sum of their first and last digits should not be more than 8 , is
(A) 4608
(B) 5720
(C) 5719
(D) 4607
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2Explain
Let ABCD be a trapezium whose vertices lie on the parabola y2=4x\mathrm{y}^{2}=4 \mathrm{x}. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length 254\frac{25}{4} and it passes through the point (1,0)(1,0), then the area of ABCD is :
(A) 754\frac{75}{4}
(B) 252\frac{25}{2}
(C) 1258\frac{125}{8}
(D) 758\frac{75}{8}
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3Explain
Two number k1\mathrm{k}_{1} and k2\mathrm{k}_{2} are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1+ik2,(i=−1)\mathrm{i}^{\mathrm{k}_{1}}+\mathrm{i}^{\mathrm{k}_{2}},(\mathrm{i}=\sqrt{-1}) is non-zero, equals
(A) 12\frac{1}{2}
(B) 14\frac{1}{4}
(C) 34\frac{3}{4}
(D) 23\frac{2}{3}
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4Explain
If f(x)=2x2x+2,x∈Rf(x)=\frac{2^{x}}{2^{x}+\sqrt{2}}, x \in R, then ∑k=181f(k82)\sum_{k=1}^{81} f\left(\frac{k}{82}\right) is equal to :
(A) 41
(B) 812\frac{81}{2}
(C) 82
(D) 81281 \sqrt{2}
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5Explain
Let f:R→R\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} be a function defined by f(x)=(2+3a)x2+(a+2a−1)x+b,a≠1f(x)=(2+3 a) x^{2}+\left(\frac{a+2}{a-1}\right) x+b, a \neq 1. If f(x+y)=f(x)+f(y)+1−27xyf(x+y)=f(x)+f(y)+1-\frac{2}{7} x y, then the value of 28∑i=15∣f(i)∣28 \sum_{i=1}^{5}|f(i)| is:
(A) 715
(B) 735
(C) 545
(D) 675
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6Explain
Let A(x,y,z)\mathrm{A}(\mathrm{x}, \mathrm{y}, \mathrm{z}) be a point in xy -plane, which is equidistant from three points (0,3,2),(2,0,3)(0,3,2),(2,0,3) and ( 0,0,10,0,1 ). Let B=(1,4,−1)\mathrm{B}=(1,4,-1) and C=(2,0,−2)\mathrm{C}=(2,0,-2). Then among the statements (S1) : △ABC\triangle \mathrm{ABC} is an isosceles right angled triangle and (S2) : the area of △ABC\triangle \mathrm{ABC} is 922\frac{9 \sqrt{2}}{2}.
(A) both are true
(B) only (S1) is true
(C) only (S2) is true
(D) both are false
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7Explain
The relation R={(x,y):x,y∈z\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{z} and x+y\mathrm{x}+\mathrm{y} is even }\} is :
(A) reflexive and transitive but not symmetric
(B) reflexive and symmetric but not transitive
(C) an equivalence relation
(D) symmetric and transitive but not reflexive
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8Explain
Let the equation of the circle, which touches x -axis at the point (a,0),a>0(\mathrm{a}, 0), \mathrm{a}>0 and cuts off an intercept of length bb on yy-axis be x2+y2−αx+βy+γ=0x^{2}+y^{2}-\alpha x+\beta y+\gamma=0. If the circle lies below x-axis, then the ordered pair ( 2a,b22 a, b^{2} ) is equal to:
(A) (α,β2+4γ)\left(\alpha, \beta^{2}+4 \gamma\right)
(B) (γ,β2−4α)\left(\gamma, \beta^{2}-4 \alpha\right)
(C) (γ,β2+4α)\left(\gamma, \beta^{2}+4 \alpha\right)
(D) (α,β2−4γ)\left(\alpha, \beta^{2}-4 \gamma\right)
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9Explain
Let <an><\mathrm{a}_{\mathrm{n}}> be a sequence such that a0=0,a1=12\mathrm{a}_{0}=0, \mathrm{a}_{1}=\frac{1}{2} and 2an+2=5an+1−3an,n=0,1,2,3,……2 a_{n+2}=5 a_{n+1}-3 a_{n}, n=0,1,2,3, \ldots \ldots Then ∑k=1100ak\sum_{k=1}^{100} a_{k} is equal to:
(A) 3a99−1003 a_{99}-100
(B) 3a100−1003 \mathrm{a}_{100}-100
(C) 3a100+1003 a_{100}+100
(D) 3a99+1003 \mathrm{a}_{99}+100
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10Explain
cos⁡(sin⁡−135+sin⁡−1513+sin⁡−13365)\cos \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{33}{65}\right) is equal to :
(A) 1
(B) 0
(C) 3365\frac{33}{65}
(D) 3265\frac{32}{65}
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11Explain
Let TrT_{r} be the rth r^{\text {th }} term of an A.P. If for some mm, Tm=125, T25=120\mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20} and 20∑r=125 Tr=1320 \sum_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13, then 5 m∑r=m2 m Tr5 \mathrm{~m} \sum_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}} is equal to:
(A) 112
(B) 126
(C) 98
(D) 142
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12Explain
If the image of the point (4,4,3)(4,4,3) in the line x−12=y−21=z−13\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3} is (α,β,γ)(\alpha, \beta, \gamma), then α+β+γ\alpha+\beta+\gamma is equal to
(A) 9
(B) 12
(C) 8
(D) 7
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13Explain
If ∫−π2π296x2cos⁡2x(1+ex)dx=π(απ2+β),α,β∈Z\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^{2} \cos ^{2} x}{\left(1+e^{x}\right)} d x=\pi\left(\alpha \pi^{2}+\beta\right), \alpha, \beta \in Z, then (α+β)2(\alpha+\beta)^{2} equals :
(A) 144
(B) 196
(C) 100
(D) 64
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14Explain
The sum of all local minimum values of the function f(x)={1−2x,x<−113(7+2∣x∣),−1≤x≤21118(x−4)(x−5),x>2f(x)=\left\{\begin{array}{cc}1-2 x, & x<-1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x>2\end{array}\right.
(A) 17172\frac{171}{72}
(B) 13172\frac{131}{72}
(C) 15772\frac{157}{72}
(D) 16772\frac{167}{72}
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15Explain
The sum, of the squares of all the roots of the equation x2+∣2x−3∣−4=0x^{2}+|2 x-3|-4=0, is :
(A) 3(3−2)3(3-\sqrt{2})
(B) 6(3−2)6(3-\sqrt{2})
(C) 6(2−2)6(2-\sqrt{2})
(D) 3(2−2)3(2-\sqrt{2})
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16Explain
Let for some function y=f(x),∫0xtf(t)dt=x2f(x)y=f(x), \int_{0}^{x} t f(t) d t=x^{2} f(x), x>0x>0 and f(2)=3f(2)=3. Then f(6)f(6) is equal to :
(A) 1
(B) 2
(C) 6
(D) 3
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17Explain
Let nCr−1=28,nCr=56{ }^{n} C_{r-1}=28,{ }^{n} C_{r}=56 and nCr+1=70{ }^{n} C_{r+1}=70. Let A(4cos⁡t,4sin⁡t),B(2sin⁡t,−2cos⁡t)\mathrm{A}(4 \cos t, 4 \sin t), \mathrm{B}(2 \sin t,-2 \cos t) and C(3r−n,r2−n−1)\mathrm{C}\left(3 \mathrm{r}-\mathrm{n}, \mathrm{r}^{2}-\mathrm{n}-1\right) be the vertices of a triangle ABC , where t is a parameter. If (3x−1)2+(3y)2=α(3 x-1)^{2}+(3 y)^{2}=\alpha, is the locus of the centroid of triangle ABC , then α\alpha equals :
(A) 20
(B) 8
(C) 6
(D) 18
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18Explain
Let OO be the origin, the point AA be z1=3+22iz_{1}=\sqrt{3}+2 \sqrt{2 i}, the point B(z2)B\left(z_{2}\right) be such that 3∣z2∣=∣z1∣\sqrt{3}\left|z_{2}\right|=\left|z_{1}\right| and arg⁡(z2)=arg⁡(z1)+π6\arg \left(z_{2}\right)=\arg \left(z_{1}\right)+\frac{\pi}{6}. Then
(A) area of triangle ABO is 113\frac{11}{\sqrt{3}}
(B) ABO is a scalene triangle
(C) area of triangle ABO is 114\frac{11}{4}
(D) ABO is an obtuse angled isosceles triangle
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19Explain
Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of x is :
(A) 28/7528 / 75
(B) 14/2514 / 25
(C) 26/7526 / 75
(D) 18/2518 / 25
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20Explain
The area (in sq. units) of the region is
(A) 803\frac{80}{3}
(B) 643\frac{64}{3}
(C) 173\frac{17}{3}
(D) 323\frac{32}{3}
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21Explain
Let M denote the set of all real matrices of order 3×33 \times 3 and let S={−3,−2,−1,1,2}\mathrm{S}=\{-3,-2,-1,1,2\}. Let S1={A=[aij]∈M:A=ATS_{1}=\left\{A=\left[a_{i j}\right] \in M: A=A^{T}\right. and aij∈S,∀i,j}\left.a_{i j} \in S, \forall i, j\right\} S2={A=[aij]∈M:A=−ATS_{2}=\left\{A=\left[a_{i j}\right] \in M: A=-A^{T}\right. and aij∈S,∀i,j}\left.a_{i j} \in S, \forall i, j\right\} S3={A=[aij]∈M:a11+a22+a33=0\mathrm{S}_{3}=\left\{\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{a}_{11}+\mathrm{a}_{22}+\mathrm{a}_{33}=0\right. and aij∈S,∀i,j}\left.\mathrm{a}_{\mathrm{ij}} \in \mathrm{S}, \forall \mathrm{i}, \mathrm{j}\right\} If n(S1∪ S2∪ S3)=125α\mathrm{n}\left(\mathrm{S}_{1} \cup \mathrm{~S}_{2} \cup \mathrm{~S}_{3}\right)=125 \alpha, then α\alpha equals.
(1613)
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22Explain
If α=1+∑r=16(−3)r−112C2r−1\alpha=1+\sum_{\mathrm{r}=1}^{6}(-3)^{\mathrm{r}-1}{ }^{12} \mathrm{C}_{2 \mathrm{r}-1}, then the distance of the point (12,3)(12, \sqrt{3}) form the line αx−3y+1=0\alpha x-\sqrt{3} y+1=0 is .....
(5)
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23Explain
Let a⃗=i^+j^+k^,b⃗=2i^+2j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+2 \hat{j}+\hat{k} and d⃗=a⃗×b⃗\vec{d}=\vec{a} \times \vec{b}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣,∣c⃗−2a⃗∣2=8\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-2 \vec{a}|^{2}=8 and the angle between d→\overrightarrow{\mathrm{d}} and c→\overrightarrow{\mathrm{c}} is π4\frac{\pi}{4}, then ∣10−3 b→⋅c→∣+∣d→×c→∣2|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^{2} is equal to .....
(6)
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24Explain
Let f(x)={3x,x<0min⁡{1+x+[x],x+2[x]},0≤x≤25,x>2f(x)=\left\{\begin{array}{cc}3 x, & x<0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x>2\end{array}\right. where [.] denotes greatest integer function. If α\alpha and β\beta are the number of points, where ff is not continuous and is not differentiable, respectively, then α+β\alpha+\beta equals.......
(5)
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25Explain
Let E1:x29+y24=1\mathrm{E}_{1}: \frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1 be an ellipse. Ellipses Ei\mathrm{E}_{\mathrm{i}} 's are constructed such that their centres and eccentricities are same as that of E1\mathrm{E}_{1}, and the length of minor axis of Ei\mathrm{E}_{\mathrm{i}} is the length of major axis of Ei+1(i≥1)\mathrm{E}_{\mathrm{i}+1}(\mathrm{i} \geq 1). If Ai\mathrm{A}_{\mathrm{i}} is the area of the ellipse Ei\mathrm{E}_{\mathrm{i}}, then 5π(∑i=1∞Ai)\frac{5}{\pi}\left(\sum_{\mathrm{i}=1}^{\infty} \mathrm{A}_{\mathrm{i}}\right), is equal to
(54)
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26Explain
Two capacitors C1\mathrm{C}_{1} and C2\mathrm{C}_{2} are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are U1U_{1} and U2U_{2}, respectively. Which of the given statements is true?
(A) C1>C2,U1>U2\mathrm{C}_{1}>\mathrm{C}_{2}, \mathrm{U}_{1}>\mathrm{U}_{2}
(B) C2>C1,U2<U1\mathrm{C}_{2}>\mathrm{C}_{1}, \mathrm{U}_{2}<\mathrm{U}_{1}
(C) C1>C2,U1<U2\mathrm{C}_{1}>\mathrm{C}_{2}, \mathrm{U}_{1}<\mathrm{U}_{2}
(D) C2>C1,U2>U1\mathrm{C}_{2}>\mathrm{C}_{1}, \mathrm{U}_{2}>\mathrm{U}_{1}
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27Explain
In the experiment for measurement of viscosity ' η\eta ' of given liquid with a ball having radius R , consider following statements. A.Graph between terminal velocity V and R will be a parabola B. The terminal velocities of different diameter balls are constant for a given liquid. C. Measurement of terminal velocity is dependent on the temperature. D. This experiment can be utilized to assess the density of a given liquid. E. If balls are dropped with some initial speed, the value of η\eta will change. Choose the correct answer from the options given below:
(A) ) B, D and E only
(B) A, C and D only
(C) C, D and E only
(D) A, B and E only
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28Explain
Consider following statements: A. Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface, of a liquid. B. As the temperature of liquid rises, the coefficient of viscosity increases. C. As the temperature of gas increases, the coefficient of viscosity increases. D. The onset of turbulence is determined by Reynold's number. E. In a steady flow two stream lines never intersect. Choose the correct answer from the options given below :
(A) A, D, E only
(B) C, D, E only
(C) B, C, D only
(D) A, B, C only
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29Explain
Three infinitely long wires with linear charge density λ\lambda are placed along the x -axis, y -axis and z axis respectively. Which of the following denotes an equipotential surface ?
(A) xy+yz+zx=x y+y z+z x= constant
(B) (x+y)(y+z)(z+x)=(x+y)(y+z)(z+x)= constant
(C) (x2+y2)(y2+z2)(z2+x2)=\left(x^{2}+y^{2}\right)\left(y^{2}+z^{2}\right)\left(z^{2}+x^{2}\right)= constant
(D) xyz=x y z= constant
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30Explain
A hemispherical vessel is completely filled with a liquid of refractive index μ\mu. A small coin is kept at the lowest point ( O ) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point E (at the level of the vessel) is
(A) 3\sqrt{3}
(B) 32\frac{3}{2}
(C) 2\sqrt{2}
(D) 32\frac{\sqrt{3}}{2}
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31Explain
Consider a long thin conducting wire carrying a uniform current I. A particle having mass "M" and charge "q" is released at a distance "a" from the wire with a speed vo\mathrm{v}_{\mathrm{o}} along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance xx from the wire. The value of x is [ μ0\mu_{0} is vacuum permeability]
(A) a[1−mvo2qμoI]a\left[1-\frac{m v_{o}}{2 q \mu_{o} I}\right]
(B) a2\frac{a}{2}
(C) a[1−mvoqμoI]\mathrm{a}\left[1-\frac{\mathrm{mv}_{\mathrm{o}}}{\mathrm{q} \mu_{\mathrm{o}} \mathrm{I}}\right]
(D) ae−4πmvoqμoI\mathrm{ae}^{\frac{-4 \pi \mathrm{mv}_{\mathrm{o}}}{q \mu_{\mathrm{o}} \mathrm{I}}}
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32Explain
A Carnot engine (E) is working between two temperatures 473 K and 273 K . In a new system two engines - engine E1E_{1} works between 473 K to 373 K and engine E2\mathrm{E}_{2} works between 373 K to 273 K . If η12\eta_{12}, η1\eta_{1} and η2\eta_{2} are the efficiencies of the engines E,E1\mathrm{E}, \mathrm{E}_{1} and E2\mathrm{E}_{2}, respectively, then
(A) η12<η1+η2\eta_{12}<\eta_{1}+\eta_{2}
(B) η12=η1η2\eta_{12}=\eta_{1} \eta_{2}
(C) η12=η1+η2\eta_{12}=\eta_{1}+\eta_{2}
(D) η12≥η1+η2\eta_{12} \geq \eta_{1}+\eta_{2}
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33Explain
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: A sound wave has higher speed in solids than gases. Reason R: Gases have higher value of Bulk modulus than solids. In the light of the above statements, choose the correct answer from the options given below
(A) Both A\mathbf{A} and R\mathbf{R} are true and R\mathbf{R} is the correct explanation of A\mathbf{A}
(B) A\mathbf{A} is false but R\mathbf{R} is true
(C) Both A\mathbf{A} and R\mathbf{R} are true but R\mathbf{R} is NOT the correct explanation of A\mathbf{A}
(D) A\mathbf{A} is true but R\mathbf{R} is false.
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34Explain
For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?
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35Explain
A bead of mass ' mm ' slides without friction on the wall of a vertical circular hoop of radius ' R ' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is ' RR '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' RR ', would be (spring constant is ' k ', g is acceleration due to gravity)
(A) 2gR+kR2m2 \sqrt{g R+\frac{k R^{2}}{m}}
(B) 2Rg+4kR2m\sqrt{2 R g+\frac{4 k R^{2}}{m}}
(C) 2Rg+kR2 m\sqrt{2 \mathrm{Rg}+\frac{\mathrm{kR}^{2}}{\mathrm{~m}}}
(D) 3Rg+kR2m\sqrt{3 R g+\frac{k R^{2}}{m}}
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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: In a central force field, the work done is independent of the path chosen Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate answer from the options given below
(A) A\mathbf{A} is true but R\mathbf{R} is false
(B) Both A\mathbf{A} and R\mathbf{R} are true but R\mathbf{R} is NOT the correct explanation of A\mathbf{A}
(C) Both A\mathbf{A} and R\mathbf{R} are true and R\mathbf{R} is the correct explanation of A\mathbf{A}
(D) A\mathbf{A} is false but R\mathbf{R} is true
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37Explain
Choose the correct nuclear process from the below options [p: proton, n: neutron, e−:\mathrm{e}^{-}:electron, e+:\mathrm{e}^{+}:positron, v:v: neutrino, vˉ:\bar{v}: antineutrino]
(A) n→p+e−+vˉn \rightarrow p+e^{-}+\bar{v}
(B) n→p+e−+vn \rightarrow p+e^{-}+v
(C) n→p+e++v‾\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{+}+\overline{\mathrm{v}}
(D) n→p+e++vn \rightarrow p+e^{+}+v
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38Explain
Which of the following circuits has the same output as that of the given circuit?
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39Explain
Find the equivalent resistance between two ends of the following circuit.
(A) rr
(B) r6\frac{r}{6}
(C) r9\frac{r}{9}
(D) r3\frac{r}{3}
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40Explain
A wire of resistance RR is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is
(A) 9/89 / 8
(B) 8/98 / 9
(C) 27/3227 / 32
(D) 32/2732 / 27
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41Explain
Due to presence of an em-wave whose electric component is given by E=100sin⁡(ωt−kx)NC−1\mathrm{E}=100 \sin (\omega \mathrm{t}-\mathrm{kx}) \mathrm{NC}^{-1}, a cylinder of length 200 cm holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as
(A) 25sin⁡(ωt−kx)NC−125 \sin (\omega \mathrm{t}-\mathrm{kx}) \mathrm{NC}^{-1}
(B) 200sin⁡(ωt−kx)NC−1200 \sin (\omega t-\mathrm{kx}) \mathrm{NC}^{-1}
(C) 400sin⁡(ωt−kx)NC−1400 \sin (\omega t-k x) N C^{-1}
(D) 50sin⁡(ωt−kx)NC−150 \sin (\omega t-k x) \mathrm{NC}^{-1}
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42Explain
A particle of mass ' mm ' and charge ' qq ' is fastened to one end 'A' of a massless string having equilibrium length ℓ\ell, whose other end is fixed at point ' OO '. The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the x -axis is
(A) 2qEℓm\sqrt{\frac{2 \mathrm{qE} \ell}{\mathrm{m}}}
(B) qEℓ4 m\sqrt{\frac{\mathrm{qE} \ell}{4 \mathrm{~m}}}
(C) qEℓm\sqrt{\frac{\mathrm{qE} \ell}{\mathrm{m}}}
(D) qEℓ2 m\sqrt{\frac{\mathrm{qE} \ell}{2 \mathrm{~m}}}
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43Explain
A proton of mass ' mpm_{p} ' has same energy as that of a photon of wavelength ' λ\lambda '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.
(A) 1c2Emp\frac{1}{c} \sqrt{\frac{2 E}{m_{p}}}
(B) 1cEmp\frac{1}{c} \sqrt{\frac{E}{m_{p}}}
(C) 1cE2mp\frac{1}{c} \sqrt{\frac{E}{2 m_{p}}}
(D) 12cEmp\frac{1}{2 c} \sqrt{\frac{E}{m_{p}}}
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44Explain
The centre of mass of a thin rectangular plate (fig x)x) with sides of length aa and bb, whose mass per unit area ( σ\sigma ) varies as σ=σ0Xab\sigma=\frac{\sigma_{0} \mathrm{X}}{\mathrm{ab}} (where σ0\sigma_{0} is a constant), would be
(A) (23a,b2)\left(\frac{2}{3} a, \frac{b}{2}\right)
(B) (23a,23b)\left(\frac{2}{3} a, \frac{2}{3} b\right)
(C) (a2,b2)\left(\frac{a}{2}, \frac{b}{2}\right)
(D) (13a,b2)\left(\frac{1}{3} a, \frac{b}{2}\right)
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45Explain
A thin prism P1P_{1} with angle 4∘4^{\circ} made of glass having refractive index 1.54, is combined with another thin prism P2\mathrm{P}_{2} made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism P2\mathrm{P}_{2} in degrees is
(A) 4
(B) 3
(C) 16/316 / 3
(D) 1.5
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46Explain
A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm , respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be X100\frac{\mathrm{X}}{100} where x is
(3)
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47Explain
The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n=\mathrm{n}= ____\_\_\_\_ . Consider all the bodies have equal masses.
(4)
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48Explain
In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of [ MaLbTc\mathrm{M}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}} ]. If b=3b=3, the value of cc is
(4)
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49Explain
Two iron solid discs of negligible thickness have radii R1R_{1} and R2R_{2} and moment of intertia I1I_{1} and I2I_{2}, respectively. For R2=2R1R_{2}=2 R_{1}, the ratio of I1I_{1} and I2I_{2} would be 1/x1 / x, where $x=
(16)
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50Explain
A double slit interference experiment performed with a light of wavelength 600 nm forms an interference fringe pattern on a screen with 10th 10^{\text {th }} bright fringe having its centre at a distance of 10 mm from the central maximum. Distance of the centre of the same 10th 10^{\text {th }} bright fringe from the central maximum when the source of light is replaced by another source of wavelength 660 nm would be ____\_\_\_\_ mm.
(11)
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51Explain
The incorrect decreasing order of atomic radii is :
(A) Mg>Al>C>O\mathrm{Mg}>\mathrm{Al}>\mathrm{C}>\mathrm{O}
(B) Al>B>N>F\mathrm{Al}>\mathrm{B}>\mathrm{N}>\mathrm{F}
(C) Be>Mg>Al>Si\mathrm{Be}>\mathrm{Mg}>\mathrm{Al}>\mathrm{Si}
(D) Si>P>Cl>F\mathrm{Si}>\mathrm{P}>\mathrm{Cl}>\mathrm{F}
3
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52Explain
Given below are two statements: Statement I: In the oxalic acid vs KMnO4\mathrm{KMnO}_{4} (in the presence of dil H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} ) titration the solution needs to be heated initially to 60∘C60^{\circ} \mathrm{C}, but no heating is required in Ferrous ammonium sulphate (FAS) vs KMnO4\mathrm{KMnO}_{4} titration (in the presence of dil H2SO4\mathrm{H}_{2} \mathrm{SO}_{4} ) Statement II : In oxalic acid vs KMnO4\mathrm{KMnO}_{4} titration, the initial formation of MnSO4\mathrm{MnSO}_{4} takes place at high temperature, which then acts as catalyst for further reaction. In the case of FAS vs KMnO4\mathrm{KMnO}_{4}, heating oxidizes Fe2+\mathrm{Fe}^{2+} into Fe3−\mathrm{Fe}^{3-} by oxygen of air and error may be introduced in the experiment. 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) Statement I is true but Statement II is false
(D) Both Statement I and Statement II are false
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53Explain
Match the List-I with List-II. Choose the correct answer from the options given below :
(A) A-II, B-III, C-IV, D-I
(B) A-II, B-III, C-I, D-IV Decomposition into gaseous product.
(C) A-III, B-IV, C-I, D-II Displacement of ' V ' by ' Ca ' atom.
(D) A-IV, B-I, C-II, D-III
1
Diagram Question
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54Explain
Given below are two statements : Statement I : EtEt◯N∼{ }_{\mathrm{Et}}^{\mathrm{Et}} \bigcirc^{\mathrm{N}} \sim Cl will undergo alkaline hydrolysis at a faster rate than . Statement II: EtEt◯N∼Cl{ }_{\mathrm{Et}}^{\mathrm{Et}} \bigcirc^{\mathrm{N}} \sim \mathrm{Cl}, intramolecular substitution takes place first by involving lone pair of electrons on nitrogen. In the light of the above statements, choose the most appropriate answer from the options given below : Rate of (a) is faster than rate of (b) because it is a intramolecular substitution.
(A) Both Statement I and Statement II are incorrect
(B) Statement I is incorrect but statement II is correct
(C) Both Statement I and Statement II are correct
(D) Statement I is correct but Statement II is incorrect
3
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55Explain
A weak acid HA has degree of dissociation xx. Which option gives the correct expression of pH=pKa\mathrm{pH}=\mathrm{pK}_{\mathrm{a}} )?
(A) log⁡(1+2x)\log (1+2 x)
(B) log⁡(1−xx)\log \left(\frac{1-x}{x}\right)
(C) 0
(D) log⁡(x1−x)\log \left(\frac{x}{1-x}\right)
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56Explain
Consider ' nn ' is the number of lone pair of electrons present in the equatorial position of the most stable structure of ClF3\mathrm{ClF}_{3}. The ions from the following with ' nn ' number of unpaired electrons are : A. V3+\mathrm{V}^{3+} B. Ti3+\mathrm{Ti}^{3+} C.Cu2+\mathrm{Cu}^{2+} D. Ni2+\mathrm{Ni}^{2+} E. Ti2+\mathrm{Ti}^{2+} Choose the correct answer from the options given below:
(A) A and C only
(B) A, D and E only
(C) B and C only
(D) B and D only
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57Explain
For a given reaction R -> P, t1/2 is related to [A]0 as given in table : Given : log 2 = 0.30 Which of the following is true ? A. The order of the reaction is 12. B.If [A]0 is 1M, then t1/2 is 200 10 min. C. The order of the reaction changes to 1 if the concentration of reactant changes from 0.100 M to 0.500 M. D. t1/2 is 800 min for [A]0 = 1.6 M Choose the correct answer from the options given below :
(A) A and C only
(B) A and B only
(C) A, B and D only
(D) C and D only
3
Diagram Question
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58Explain
A molecule ("P") on treatment with acid undergoes rearrangement and gives ("Q") ("Q") on ozonolysis followed by reflux under alkaline condition gives (" RR "). The structure of (" RR ") is given below :
2
Diagram Question
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59Explain
Ice and water are placed in a closed container at a pressure of 1 atm and temperature 273.15 K . If pressure of the system is increased 2 times, keeping temperature constant, then identify correct observation from following :
(A) Volume of system increases.
(B) Liquid phase disappears completely.
(C) The amount of ice decreases.
(D) The solid phase (ice) disappears completely.
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60Explain
The molecules having square pyramidal geometry are
(A) BrF5&XeOF4\mathrm{BrF}_{5} \& \mathrm{XeOF}_{4}
(B) SbF5&XeOF4\mathrm{SbF}_{5} \& \mathrm{XeOF}_{4}
(C) SbF5&PCl5\mathrm{SbF}_{5} \& \mathrm{PCl}_{5}
(D) BrF5&PCl5\mathrm{BrF}_{5} \& \mathrm{PCl}_{5}
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61Explain
The metal ion whose electronic configuration is not affected by the nature of the ligand and which gives a violet colour in non-luminous flame under hot condition in borax bead test is
(A) Ti3+\mathrm{Ti}^{3+}
(B) Ni2+\mathrm{Ni}^{2+}
(C) Mn2+\mathrm{Mn}^{2+}
(D) Cr3+\mathrm{Cr}^{3+}
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62Explain
Both acetaldehyde and acetone (individually) undergo which of the following reactions? A. Iodoform Reaction B. Cannizaro Reaction C. Aldol condensation D. Tollen's Test E. Clemmensen Reduction Choose the correct answer from the options given below :
(A) A, B and D only
(B) A, C and E only
(C) C and E only
(D) B, C and D only
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63Explain
In a multielectron atom, which of the following orbitals described by three quantum numbers with have same energy in absence of electric and magnetic fields? A. n=1,ℓ=0, mℓ=0\mathrm{n}=1, \ell=0, \mathrm{~m}_{\ell}=0 B. n=2,ℓ=0, mℓ=0\mathrm{n}=2, \ell=0, \mathrm{~m}_{\ell}=0 C. n=3,ℓ=1, mℓ=1\mathrm{n}=3, \ell=1, \mathrm{~m}_{\ell}=1 D. n=3,ℓ=2, mℓ=1\mathrm{n}=3, \ell=2, \mathrm{~m}_{\ell}=1 E. n=3,ℓ=2, mℓ=0\mathrm{n}=3, \ell=2, \mathrm{~m}_{\ell}=0 Choose the correct answer from the options given below :
(A) A and B only
(B) B and C only
(C) C and D only
(D) D and E only
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64Explain
The products A and B in the following reactions, respectively are
(A) CH3−CH2−CH2−ONO,CH3−CH2−CH2−NC\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{ONO}, \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NC}
(B) CH3−CH2−CH2−ONO,CH3−CH2−CH2−CN\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{ONO}, \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{CN}
(C) CH3−CH2−CH2−NO2,CH3−CH2−CH2−CN\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NO}_{2}, \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{CN}
(D) CH3−CH2−CH2−NO2,CH3−CH2−CH2−NC\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NO}_{2}, \mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NC}
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65Explain
What is the freezing point depression constant of a solvent, 50 g of which contain 1 g non volatile solute (molar mass 256 g mol−1256 \mathrm{~g} \mathrm{~mol}^{-1} ) and the decrease in freezing point is 0.40 K ?
(A) 5.12 K kg mol−15.12 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}
(B) 4.43 K kg mol−14.43 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}
(C) 1.86 K kg mol−11.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}
(D) 3.72 K kg mol−13.72 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}
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66Explain
Consider the following elements In,Tl,Al,Pb,Sn\mathrm{In}, \mathrm{Tl}, \mathrm{Al}, \mathrm{Pb}, \mathrm{Sn} and Ge . The most stable oxidation states of elements with highest and lowest first ionisation enthalpies, respectively, are
(A) +2 and +3
(B) +4 and +3
(C) +4 and +1
(D) +1 and +4
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67Explain
The correct order of stability of following carbocations is :
(A) A>B>C>D\mathrm{A}>\mathrm{B}>\mathrm{C}>\mathrm{D}
(B) B>C>A>D\mathrm{B}>\mathrm{C}>\mathrm{A}>\mathrm{D}
(C) C >> B >> A >> D
(D) C >> A >> B >> D
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68Explain
The compounds that produce CO2\mathrm{CO}_{2} with aqueous NaHCO3\mathrm{NaHCO}_{3} solution are : Choose the correct answer from the options given below :
(A) A and C only
(B) A, B and E only
(C) A, C and D only
(D) A and B only
3
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69Explain
Which of the following oxidation reactions are carried out by both K2Cr2O7\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} and KMnO4\mathrm{KMnO}_{4} in acidic medium ? A. I−→I2\mathrm{I}^{-} \rightarrow \mathrm{I}_{2} B. S2−→S\mathrm{S}^{2-} \rightarrow \mathrm{S} C. Fe2+→Fe3+\mathrm{Fe}^{2+} \rightarrow \mathrm{Fe}^{3+} D. I−→IO3−\mathrm{I}^{-} \rightarrow \mathrm{IO}_{3}{ }^{-} E. S2O32−→SO42−\mathrm{S}_{2} \mathrm{O}_{3}{ }^{2-} \rightarrow \mathrm{SO}_{4}{ }^{2-} Choose the correct answer from the options given below:
(A) B, C and D only
(B) A, D and E only
(C) A, B and C only
(D) C, D and E only
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70Explain
Given below are two statements : Statement I : D-glucose pentaacetate reacts with 2, 4-dinitrophenylhydrazine. Statement II : Starch, on heating with concentrated sulfuric acid at 100∘C100^{\circ} \mathrm{C} and 2−32-3 atmosphere pressure produces glucose. In the light of the above statements, choose the correct answer from the options given below
(A) Both Statement I and Statement II are false
(B) Statement I is false but Statement II is true
(C) Statement I is true but Statement II is false
(D) Both Statement I and Statement II are true
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71Explain
Given below is the plot of the molar conductivity vs  concentration \sqrt{\text { concentration }} for KCl in aqueous solution. If, for the higher concentration of KCl solution, the resistance of the conductivity cell is 100Ω100 \Omega, then the resistance of the same cell with the dilute solution is ' x ' Ω\Omega. The value of x is ____\_\_\_\_ (Nearest integer)
(150)
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72Explain
Quantitative analysis of an organic compound (X) shows following \% composition. C: 14.5% Cl: 64.46% H: 1.8% (Empirical formula mass of the compound (X)(\mathrm{X}) is ____\_\_\_\_ ×10−1\times 10^{-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,Cl:35.5\mathrm{O}: 16, \mathrm{Cl}: 35.5 )
(1655)
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73Explain
The molarity of a 70%70 \% (mass/mass) aqueous solution of a monobasic acid ( X ) is ____\_\_\_\_ M(Nearest integer) [Given : Density of aqueous solution of ( X ) is 1.25 g mL−11.25 \mathrm{~g} \mathrm{~mL}^{-1} Molar mass of the acid is 70 g mol−170 \mathrm{~g} \mathrm{~mol}^{-1} ]
(125)
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74Explain
Consider the following sequence of reactions : Chlorobenzene 11.25 mg of chlorobenzene will produce ____\_\_\_\_ ×10−1mg\times 10^{-1} \mathrm{mg} of product B. (Consider the reactions result in complete conversion.) [Given molar mass of C,H,O,N\mathrm{C}, \mathrm{H}, \mathrm{O}, \mathrm{N} and Cl as 12,1 , 16,14 and 35.5 g mol−135.5 \mathrm{~g} \mathrm{~mol}^{-1} respectively]
(93)
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75Explain
The formation enthalpies, ΔHf⊖\Delta \mathrm{H}_{\mathrm{f}}^{\ominus} for H(g)\mathrm{H}_{(\mathrm{g})} and O(g)\mathrm{O}_{(\mathrm{g})} are 220.0 and 250.0 kJ mol−1250.0 \mathrm{~kJ} \mathrm{~mol}^{-1}, respectively, at 298.15 K , and ΔHf−\Delta \mathrm{H}_{\mathrm{f}}^{-}for H2O(g)\mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})} is −242.0 kJ mol−1-242.0 \mathrm{~kJ} \mathrm{~mol}^{-1} at the same temperature. The average bond enthalpy of the O−H\mathrm{O}-\mathrm{H} bond in water at 298.15 K is ____\_\_\_\_ kJmol−1\mathrm{kJ} \mathrm{mol}^{-1} (nearest integer).
(466)
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