JEE-MAIN EXAMINATION – APRIL 2024

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Thursday 04th April 2024, Time: 9:00 AM to 12:00 NOON

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JEE Main
Mathematics, Physics, Chemistry
Morning Session
3 hours

Paper Overview

90
Total Questions
0
Correct
0
Incorrect
90
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Complete Solutions

Q#ExplanationQuestionCorrectSolutionStatus
1Explain
Let f:RRf: \mathrm{R} \rightarrow \mathrm{R} be a function given by f(x)={1cos2xx2,x<0α,x=0, where α,βR. If β1cosxx,x>0f(x)=\left\{\begin{array}{lll}\frac{1-\cos 2 x}{x^{2}} & , & x<0 \\ \alpha & , & x=0, \text { where } \alpha, \beta \in R . \text { If } \\ \frac{\beta \sqrt{1-\cos x}}{x} & , & x>0\end{array}\right. ff is continuous at x=0\mathrm{x}=0, then α2+β2\alpha^{2}+\beta^{2} is equal to:
(A) 48
(B) 12
(C) 3
(D) 6
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2Explain
Three urns A,B\mathrm{A}, \mathrm{B} and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :
(A) 417\frac{4}{17}
(B) 518\frac{5}{18}
(C) 718\frac{7}{18}
(D) 516\frac{5}{16}
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3Explain
The vertices of a triangle are A(1,3),B(2,2)\mathrm{A}(-1,3), \mathrm{B}(-2,2) and C(3,1)\mathrm{C}(3,-1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :
(A) xy(2+2)=0x-y-(2+\sqrt{2})=0
(B) x+y(22)=0-x+y-(2-\sqrt{2})=0
(C) x+y(22)=0x+y-(2-\sqrt{2})=0
(D) x+y+(22)=0x+y+(2-\sqrt{2})=0
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4Explain
If the solution y=y(x)y=y(x) of the differential equation (x4+2x3+3x2+2x+2)dy(2x2+2x+3)dx=0\left(x^{4}+2 x^{3}+3 x^{2}+2 x+2\right) d y-\left(2 x^{2}+2 x+3\right) d x=0 satisfies y(1)=π4y(-1)=-\frac{\pi}{4}, then y(0)y(0) is equal to:
(A) π12-\frac{\pi}{12}
(B) 0
(C) π4\frac{\pi}{4}
(D) π2\frac{\pi}{2}
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5Explain
Let the sum of the maximum and the minimum values of the function f(x)=2x23x+82x2+3x+8f(x)=\frac{2 x^{2}-3 x+8}{2 x^{2}+3 x+8} be mn\frac{m}{n}, where gcd(m,n)=1\operatorname{gcd}(m, n)=1. Then m+nm+n is equal to:
(A) 182
(B) 217
(C) 195
(D) 201
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6Explain
One of the points of intersection of the curves y=1+3x2x2\mathrm{y}=1+3 \mathrm{x}-2 \mathrm{x}^{2} and y=1x\mathrm{y}=\frac{1}{\mathrm{x}} is (12,2)\left(\frac{1}{2}, 2\right). Let the area of the region enclosed by these curves be 124(5+m)nloge(1+5)\frac{1}{24}(\ell \sqrt{5}+m)-n \log _{e}(1+\sqrt{5}), where ,m,n\ell, m, n \in N . Then +m+n\ell+\mathrm{m}+\mathrm{n} is equal to
(A) 32
(B) 30
(C) 29
(D) 31
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7Explain
If the system of equations x+(2sinα)y+(2cosα)z=0x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 x+(cosα)y+(sinα)z=0x+(\cos \alpha) y+(\sin \alpha) z=0 x+(sinα)y(cosα)z=0x+(\sin \alpha) y-(\cos \alpha) z=0 has a non-trivial solution, then α(0,π2)\alpha \in\left(0, \frac{\pi}{2}\right) is equal to :
(A) 3π4\frac{3 \pi}{4}
(B) 7π24\frac{7 \pi}{24}
(C) 5π24\frac{5 \pi}{24}
(D) 11π24\frac{11 \pi}{24}
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8Explain
There are 5 points P1,P2,P3,P4,P5\mathrm{P}_{1}, \mathrm{P}_{2}, \mathrm{P}_{3}, \mathrm{P}_{4}, \mathrm{P}_{5} on the side AB , excluding A and B , of a triangle ABC . Similarly there are 6 points P6,P7,,P11\mathrm{P}_{6}, \mathrm{P}_{7}, \ldots, \mathrm{P}_{11} on the side BC and 7 points P12,P13,,P18\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18} on the side CA of the triangle. The number of triangles, that can be formed using the points P1,P2,,P18\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{18} as vertices, is :
(A) 776
(B) 751
(C) 796
(D) 771
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9Explain
Let f(x)={2,2x0x2,0<x2f(\mathrm{x})=\left\{\begin{array}{cc}-2, & -2 \leq \mathrm{x} \leq 0 \\ \mathrm{x}-2, & 0<\mathrm{x} \leq 2\end{array}\right. and h(x)=f(x)+f(x)\mathrm{h}(\mathrm{x})=\mathrm{f}(|\mathrm{x}|)+|\mathrm{f}(\mathrm{x})|. Then 22h(x)dx\int_{-2}^{2} h(x) d x is equal to :
(A) 2
(B) 4
(C) 1
(D) 6
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10Explain
The sum of all rational terms in the expansion of (215+513)15\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15} is equal to :
(A) 3133
(B) 633
(C) 931
(D) 6131
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11Explain
Let a unit vector which makes an angle of 6060^{\circ} with 2i^+2j^k^2 \hat{i}+2 \hat{j}-\hat{k} and an angle of 4545^{\circ} with i^k^\hat{i}-\hat{k} be C\vec{C}. Then C+(12i^+132j^23k^)\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right) is :
(A) 23i^+23j^+(12+223)k^-\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{\sqrt{2}}{3} \hat{\mathrm{j}}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{\mathrm{k}}
(B) 23i^+132j^12k^\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{1}{2} \hat{\mathrm{k}}
(C) (13+12)i^+(13132)j^+(13+23)k^\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{\mathrm{i}}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{\mathrm{j}}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{\mathrm{k}}
(D) 23i^12k^\frac{\sqrt{2}}{3} \hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{k}}
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12Explain
Let the first three terms 2,p2, p and qq, with q2q \neq 2, of a G.P. be respectively the 7th ,8th 7^{\text {th }}, 8^{\text {th }} and 13th 13^{\text {th }} terms of an A.P. If the 5th 5^{\text {th }} term of the G.P. is the nth n^{\text {th }} term of the A.P., then n is equal to
(A) 151
(B) 169
(C) 177
(D) 163
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13Explain
Let a,bRa, b \in R. Let the mean and the variance of 6 observations 3,4,7,6,a,b-3,4,7,-6, \mathrm{a}, \mathrm{b} be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is :
(A) 133\frac{13}{3}
(B) 163\frac{16}{3}
(C) 113\frac{11}{3}
(D) 143\frac{14}{3}
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14Explain
If 2 and 6 are the roots of the equation ax2+bx+1=0\mathrm{ax}^{2}+\mathrm{bx}+1=0, then the quadratic equation, whose roots are 12a+b\frac{1}{2 a+b} and 16a+b\frac{1}{6 a+b}, is :
(A) 2x2+11x+12=02 \mathrm{x}^{2}+11 \mathrm{x}+12=0
(B) 4x2+14x+12=04 x^{2}+14 x+12=0
(C) x2+10x+16=0\mathrm{x}^{2}+10 \mathrm{x}+16=0
(D) x2+8x+12=0x^{2}+8 x+12=0
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15Explain
Let α\alpha and β\beta be the sum and the product of all the non-zero solutions of the equation (zˉ)2+z=0,zC(\bar{z})^{2}+|z|=0, z \in C. Then 4(α2+β2)4\left(\alpha^{2}+\beta^{2}\right) is equal to:
(A) 6
(B) 4
(C) 8
(D) 2
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16Explain
Let the point, on the line passing through the points P(1,2,3)\mathrm{P}(1,-2,3) and Q(5,4,7)\mathrm{Q}(5,-4,7), farther from the origin and at a distance of 9 units from the point P , be (α,β,γ)(\alpha, \beta, \gamma). Then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to:
(A) 155
(B) 150
(C) 160
(D) 165
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17Explain
A square is inscribed in the circle x2+y210x6y+30=0x^{2}+y^{2}-10 x-6 y+30=0. One side of this square is parallel to y=x+3y=x+3. If (xi,yi)\left(x_{i}, y_{i}\right) are the vertices of the square, then (xi2+yi2)\sum\left(x_{i}^{2}+y_{i}^{2}\right) is equal to:
(A) 148
(B) 156
(C) 160
(D) 152
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18Explain
If the domain of the function sin1(3x222x19)+loge(3x28x+5x23x10)\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{e}\left(\frac{3 x^{2}-8 x+5}{x^{2}-3 x-10}\right) is (α,β]\quad(\alpha, \beta], then 3α+10β3 \alpha+10 \beta is equal to :
(A) 97
(B) 100
(C) 95
(D) 98
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19Explain
Let f(x)=x5+2ex/4f(\mathrm{x})=\mathrm{x}^{5}+2 \mathrm{e}^{\mathrm{x} / 4} for all xR\mathrm{x} \in \mathrm{R}. Consider a function g(x)\mathrm{g}(\mathrm{x}) such that (gof) (x)=x(\mathrm{x})=\mathrm{x} for all xR\mathrm{x} \in \mathrm{R}. Then the value of 8 g(2)8 \mathrm{~g}^{\prime}(2) is :
(A) 16
(B) 4
(C) 8
(D) 2
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20Explain
Let α(0,)\alpha \in(0, \infty) and A=[12α101012]A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]. If det(adj(2 AAT)adj(A2 AT))=28\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^{8}, then (det(A))2(\operatorname{det}(\mathrm{A}))^{2} is equal to :
(A) 1
(B) 49
(C) 16
(D) 36
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21Explain
If limx1(5x+1)1/3(x+5)1/3(2x+3)1/2(x+4)1/2=m5n(2n)2/3\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}, where gcd(m,n)=1\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1, then 8 m+12n8 \mathrm{~m}+12 \mathrm{n} is equal to \qquad
(100)
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22Explain
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m+nm+n is equal to \qquad
(45)
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23Explain
Let the solution y=y(x)y=y(x) of the differential equation dydxy=1+4sinx\frac{d y}{d x}-y=1+4 \sin x satisfy y(π)=1y(\pi)=1. Then y(π2)+10\mathrm{y}\left(\frac{\pi}{2}\right)+10 is equal to \qquad
(7)
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24Explain
If the shortest distance between the lines x+22=y+33=z54\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4} and x31=y23=z+42\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2} is 3835k\frac{38}{3 \sqrt{5}} \mathrm{k} \quad and 0k[x2]dx=αα,\quad \int_{0}^{\mathrm{k}}\left[\mathrm{x}^{2}\right] \mathrm{dx}=\alpha-\sqrt{\alpha}, \quad where denotes the greatest integer function, then 6α36 \alpha^{3} is equal to \qquad
(48)
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25Explain
Let A be a square matrix of order 2 such that A=2|\mathrm{A}|=2 and the sum of its diagonal elements is -3 . If the points (x,y)(x, y) satisfying A2+xA+yI=0A^{2}+x A+y I=0 lie on a hyperbola, whose transverse axis is parallel to the xx-axis, eccentricity is ee and the length of the latus rectum is \ell, then e4+4\mathrm{e}^{4}+\ell^{4} is equal to \qquad
(25)
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26Explain
Let a=1+2C23!+3C24!+4C25!+\mathrm{a}=1+\frac{{ }^{2} \mathrm{C}_{2}}{3!}+\frac{{ }^{3} \mathrm{C}_{2}}{4!}+\frac{{ }^{4} \mathrm{C}_{2}}{5!}+\ldots, b=1+1C0+1C11!+2C0+2C1+2C22!+3C0+3C1+3C2+3C33!+\mathrm{b}=1+\frac{{ }^{1} \mathrm{C}_{0}+{ }^{1} \mathrm{C}_{1}}{1!}+\frac{{ }^{2} \mathrm{C}_{0}+{ }^{2} \mathrm{C}_{1}+{ }^{2} \mathrm{C}_{2}}{2!}+\frac{{ }^{3} \mathrm{C}_{0}+{ }^{3} \mathrm{C}_{1}+{ }^{3} \mathrm{C}_{2}+{ }^{3} \mathrm{C}_{3}}{3!}+\ldots Then 2ba2\frac{2 b}{a^{2}} is equal to \qquad
(8)
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27Explain
Let A be a 3×33 \times 3 matrix of non-negative real elements such that A[111]=3[111]\mathrm{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]. Then the maximum value of det(A)\operatorname{det}(\mathrm{A}) is \qquad
(27)
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28Explain
Let the length of the focal chord PQP Q of the parabola y2=12xy^{2}=12 x be 15 units. If the distance of PQP Q from the origin is pp, then 10p210 p^{2} is equal to \qquad
(72)
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29Explain
Let ABC be a triangle of area 15215 \sqrt{2} and the vectors AB=i^+2j^7k^,BC=ai^+bj^+ck^\overrightarrow{\mathrm{AB}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-7 \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{\mathrm{i}}+\mathrm{b} \hat{\mathrm{j}}+\mathrm{c} \hat{\mathrm{k}} and AC=6i^+dj^2k^,d>0\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0. Then the square of the length of the largest side of the triangle ABC is
(54)
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30Explain
If 0π4sin2x1+sinxcosxdx=1aloge(a3)+πb3\int_{0}^{\frac{\pi}{4}} \frac{\sin ^{2} \mathrm{x}}{1+\sin \mathrm{x} \cos \mathrm{x}} \mathrm{dx}=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}, where a , bN\mathrm{b} \in \mathrm{N}, then a+b\mathrm{a}+\mathrm{b} is equal to \qquad
(8)
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31Explain
An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :
(A) the electron will be accelerated along the axis.
(B) the electron will continue to move with uniform velocity along the axis of the solenoid.
(C) the electron path will be circular about the axis.
(D) the electron will experience a force at 4545^{\circ} to the axis and execute a helical path.
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32Explain
The electric field in an electromagnetic wave is given by E=i^40cosω(tzc)NC1.\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}} 40 \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right) \mathrm{NC}^{-1} . \quad The magnetic field induction of this wave is (in SI unit):
(A) B=i^40ccosω(tzc)\vec{B}=\hat{i} \frac{40}{c} \cos \omega\left(t-\frac{z}{c}\right)
(B) B=j^40cosω(tzc)\vec{B}=\hat{j} 40 \cos \omega\left(t-\frac{z}{c}\right)
(C) B=k^40ccosω(tzc)\overrightarrow{\mathrm{B}}=\hat{\mathrm{k}} \frac{40}{\mathrm{c}} \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right)
(D) B=j^40ccosω(tzc)\vec{B}=\hat{j} \frac{40}{c} \cos \omega\left(t-\frac{z}{c}\right)
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33Explain
Which of the following nuclear fragments corresponding to nuclear fission between neutron (01n)\left({ }_{0}^{1} \mathrm{n}\right) and uranium isotope (92235U)\left({ }_{92}^{235} \mathrm{U}\right) is correct:
(A) 56144Ba+3689Kr+401n{ }_{56}^{144} \mathrm{Ba}+{ }_{36}^{89} \mathrm{Kr}+4{ }_{0}^{1} \mathrm{n}
(B) 56140Xe+3894Sr+301n{ }_{56}^{140} \mathrm{Xe}+{ }_{38}^{94} \mathrm{Sr}+3{ }_{0}^{1} \mathrm{n}
(C) 51153Sb+4199Nb+301n{ }_{51}^{153} \mathrm{Sb}+{ }_{41}^{99} \mathrm{Nb}+3{ }_{0}^{1} \mathrm{n}
(D) 56144Ba+3689Kr+301n{ }_{56}^{144} \mathrm{Ba}+{ }_{36}^{89} \mathrm{Kr}+3{ }_{0}^{1} \mathrm{n}
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34Explain
In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image ( v ) are Δu\Delta \mathrm{u} and Δv\Delta \mathrm{v}, respectively. The error in the measurement of the focal length of the convex lens will be :
(A) Δuu+Δvv\frac{\Delta u}{u}+\frac{\Delta v}{v}
(B) f2[Δuu2+Δvv2]f^{2}\left[\frac{\Delta u}{u^{2}}+\frac{\Delta v}{v^{2}}\right]
(C) 2f[Δuu+Δvv]2 f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]
(D) f[Δuu+Δvv]f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]
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35Explain
Given below are two statements : Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation P1P2=ρg(h2h1)P_{1}-P_{2}=\rho g\left(h_{2}-h_{1}\right) Statement II : In ventury tube shown 2gh=v12v222 \mathrm{gh}=v_{1}^{2}-v_{2}^{2} ![] In the light of the above statements, choose the most appropriate answer from the options given below.
(A) Both Statement I and Statement II are correct.
(B) Statement I is incorrect but Statement II is correct.
(C) Both Statement I and Statement II are incorrect.
(D) Statement I is correct but Statement II is incorrect.
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36Explain
The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are 8Ω8 \Omega and 10Ω10 \Omega respectively. After inserting in a hot bath of temperature 400C400^{\circ} \mathrm{C}, the resistance of platinum wire is :
(A) 2Ω2 \Omega
(B) 16Ω16 \Omega
(C) 8Ω8 \Omega
(D) 10Ω10 \Omega
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37Explain
A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is:
(A) GMmπ2 L2\frac{\mathrm{GMm} \pi}{2 \mathrm{~L}^{2}}
(B) 0
(C) GmMπ2 L2\frac{\mathrm{GmM} \pi^{2}}{\mathrm{~L}^{2}}
(D) 2GmMπL2\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^{2}}
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38Explain
On celcius scale the temperature of body increases by 40C40^{\circ} \mathrm{C}. The increase in temperature on Fahrenheit scale is:
(A) 70F70^{\circ} \mathrm{F}
(B) 68F68^{\circ} \mathrm{F}
(C) 72F72^{\circ} \mathrm{F}
(D) 75F75^{\circ} \mathrm{F}
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39Explain
An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D . Focal length of each of the convex lens is :
(A) 20 cm
(B) 50 cm
(C) 500 cm
(D) 25 cm
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40Explain
Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light ( I1<I2\mathrm{I}_{1}< \mathrm{I}_{2} ) of same wavelengths :
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41Explain
A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t . Which of the following curve best describes acceleration of the block with time :
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42Explain
If a rubber ball falls from a height hh and rebounds upto the height of h/2\mathrm{h} / 2. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :
(A) 50%,gh250 \%, \sqrt{\frac{\mathrm{gh}}{2}}
(B) 50%,gh50 \%, \sqrt{\mathrm{gh}}
(C) 40%,2gh40 \%, \sqrt{2 \mathrm{gh}}
(D) 50%,2gh50 \%, \sqrt{2 \mathrm{gh}}
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43Explain
The equation of stationary wave is : y=2asin(2πntλ)cos(2πxλ)y=2 a \sin \left(\frac{2 \pi n t}{\lambda}\right) \cos \left(\frac{2 \pi x}{\lambda}\right) Which of the following is NOT correct
(A) The dimensions of nt is [L]
(B) The dimensions of nn is [L1]\left[L^{-1}\right]
(C) The dimensions of n/λn / \lambda is [T][T]
(D) The dimensions of xx is [L][L]
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44Explain
A body travels 102.5 m in nth \mathrm{n}^{\text {th }} second and 115.0 m in (n+2)th (n+2)^{\text {th }} second. The acceleration is :
(A) 9 m/s29 \mathrm{~m} / \mathrm{s}^{2}
(B) 6.25 m/s26.25 \mathrm{~m} / \mathrm{s}^{2}
(C) 12.5 m/s212.5 \mathrm{~m} / \mathrm{s}^{2}
(D) 5 m/s25 \mathrm{~m} / \mathrm{s}^{2}
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45Explain
To measure the internal resistance of a battery, potentiometer is used. For R=10Ω\mathrm{R}=10 \Omega, the balance point is observed at =500 cm\ell=500 \mathrm{~cm} and for R=1Ω\mathrm{R}=1 \Omega the balance point is observed at =400 cm\ell=400 \mathrm{~cm}. The internal resistance of the battery is approximately :
(A) 0.2Ω0.2 \Omega
(B) 0.4Ω0.4 \Omega
(C) 0.1Ω0.1 \Omega
(D) 0.3Ω0.3 \Omega
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46Explain
An infinitely long positively charged straight thread has a linear charge density λCm1\lambda \mathrm{Cm}^{-1}. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :
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47Explain
The value of net resistance of the network as shown in the given figure is :
(A) (52)Ω\left(\frac{5}{2}\right) \Omega
(B) (154)Ω\left(\frac{15}{4}\right) \Omega
(C) 6Ω6 \Omega
(D) (3011)Ω\left(\frac{30}{11}\right) \Omega
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48Explain
P-T diagram of an ideal gas having three different densities ρ1,ρ2,ρ3\rho_{1}, \rho_{2}, \rho_{3} (in three different cases) is shown in the figure. Which of the following is correct :
(A) ρ2<ρ3\rho_{2}<\rho_{3}
(B) ρ1>ρ2\rho_{1}>\rho_{2}
(C) ρ1<ρ2\rho_{1}<\rho_{2}
(D) ρ1=ρ2=ρ3\rho_{1}=\rho_{2}=\rho_{3}
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49Explain
The co-ordinates of a particle moving in xyx-y plane are given by: x=2+4t,y=3t+8t2x=2+4 t, y=3 t+8 t^{2}. The motion of the particle is :
(A) non-uniformly accelerated.
(B) uniformly accelerated having motion along a straight line.
(C) uniform motion along a straight line.
(D) uniformly accelerated having motion along a parabolic path.
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50Explain
In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to :
(A) pure inductor.
(B) pure capacitor.
(C) pure resistor.
(D) combination of an inductor and capacitor. Choose the correct answer from the options given below :
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51Explain
An infinite plane sheet of charge having uniform surface charge density +σsC/m2+\sigma_{s} \mathrm{C} / \mathrm{m}^{2} is placed on x-y plane. Another infinitely long line charge having uniform linear charge density +λeC/m+\lambda_{\mathrm{e}} \mathrm{C} / \mathrm{m} is placed at z=4 m\mathrm{z}=4 \mathrm{~m} plane and parallel to y -axis. If the magnitude values σs=2λe\left|\sigma_{\mathrm{s}}\right|=2\left|\lambda_{\mathrm{e}}\right| then at point ( 0,0,20,0,2 ), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is πn:1\pi \sqrt{n}: 1. The value of nn is \qquad .
(16)
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52Explain
A hydrogen atom changes its state from n=3n=3 to n=2\mathrm{n}=2. Due to recoil, the percentage change in the wave length of emitted light is approximately 1×10n1 \times 10^{-n}. The value of nn is \qquad . [Given Rhc =13.6eV,hc=1242eVnm=13.6 \mathrm{eV}, \mathrm{hc}=1242 \mathrm{eV} \mathrm{nm}, h=6.6×1034 J\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J} s, mass of the hydrogen atom =1.6×1027 kg]\left.=1.6 \times 10^{-27} \mathrm{~kg}\right]
(7)
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53Explain
The magnetic field existing in a region is given by B=0.2(1+2x)k^T\overrightarrow{\mathrm{B}}=0.2(1+2 \mathrm{x}) \hat{\mathrm{k}} \mathrm{T}. A square loop of edge 50 cm carrying 0.5 A current is placed in xyx-y plane with its edges parallel to the xyx-y axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is \qquad mN.
(50)
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54Explain
A alternating current at any instant is given by i=[6+56sin(100πt+π3)]\mathrm{i}=\left[6+\sqrt{56} \sin \left(100 \pi \mathrm{t}+\frac{\pi}{3}\right)\right] A. The rms value of the current is \qquad A.
(8)
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55Explain
Twelve wires each having resistance 2Ω2 \Omega are joined to form a cube. A battery of 6 V emf is joined across point aa and cc. The voltage difference between e and f is \qquad V.
(1)
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56Explain
A soap bubble is blown to a diameter of 7 cm . 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne /cm/ \mathrm{cm} then the new radius is \qquad cm. Take :(π=227):\left(\pi=\frac{22}{7}\right).
(7)
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57Explain
Two wavelengths λ1\lambda_{1} and λ2\lambda_{2} are used in Young's double slit experiment λ1=450 nm\lambda_{1}=450 \mathrm{~nm} and λ2=650 nm\lambda_{2}=650 \mathrm{~nm}. The minimum order of fringe produced by λ2\lambda_{2} which overlaps with the fringe produced by λ1\lambda_{1} is nn. The value of nn is \qquad .
(9)
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58Explain
An elastic spring under tension of 3 N has a length Its length is b under tension 2 N . For its length (3a2b)(3 a-2 b), the value of tension will be \qquad N.
(5)
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59Explain
Two forces F1\overrightarrow{\mathrm{F}}_{1} and F2\overrightarrow{\mathrm{F}}_{2} are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between F1\overrightarrow{\mathrm{F}}_{1} and F2\overrightarrow{\mathrm{F}}_{2} is cos1(1n)\cos ^{-1}\left(\frac{1}{\mathrm{n}}\right). The value of n|\mathrm{n}| is \qquad .
(6)
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60Explain
A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights h1h_{1} and h2h_{2}, respectively, above the initial level. The ratio h1:h2h_{1}: h_{2} is n10\frac{n}{10}. The value of n is \qquad .
(7)
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61Explain
What pressure (bar) of H2\mathrm{H}_{2} would be required to make emf of hydrogen electrode zero in pure water at 25C25^{\circ} \mathrm{C} ?
(A) 101410^{-14}
(B) 10710^{-7}
(C) 1
(D) 0.5 \section*{NTA Ans. (3)}
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62Explain
The correct sequence of ligands in the order of decreasing field strength is :
(A) CO>H2O>F>S2\mathrm{CO}>\mathrm{H}_{2} \mathrm{O}>\mathrm{F}^{-}>\mathrm{S}^{2-}
(B) OH>F>NH3>CN{ }^{-} \mathrm{OH}>\mathrm{F}^{-}>\mathrm{NH}_{3}>\mathrm{CN}^{-}
(C) NCS>EDTA4>CN>CO\mathrm{NCS}^{-}>\mathrm{EDTA}^{4-}>\mathrm{CN}^{-}>\mathrm{CO}
(D) S2>OH>EDTA4>CO\mathrm{S}^{2-}>^{-} \mathrm{OH}>\mathrm{EDTA}^{4-}>\mathrm{CO}
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63Explain
Match List -I with List II: Choose the correct answer from the options given below:
1
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64Explain
What will be the decreasing order of basic strength of the following conjugate bases? OH,RO,CH3COO,Cl{ }^{-} \mathrm{OH}, \mathrm{R} \overline{\mathrm{O}}, \mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}, \mathrm{C} \overline{\mathrm{l}}
(A) C1>OH>RO>CH3COO\mathrm{C} \overline{1}>\mathrm{OH}>\mathrm{R} \overline{\mathrm{O}}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}
(B) RO>OH>CH3COO>C1\mathrm{R} \overline{\mathrm{O}}>{ }^{-} \mathrm{OH}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}>\mathrm{C} \overline{1}
(C) OH>RO>CH3COO>C1{ }^{-} \mathrm{OH}>\mathrm{R} \overline{\mathrm{O}}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}>\mathrm{C} \overline{1}
(D) Cl>RO>OH>CH3COO\mathrm{C} \overline{\mathrm{l}}>\mathrm{R} \overline{\mathrm{O}}>{ }^{-} \mathrm{OH}>\mathrm{CH}_{3} \mathrm{CO} \overline{\mathrm{O}}
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65Explain
In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to:
(A) prevent interference by phosphate ions
(B) decrease concentration of OH{ }^{-} \mathrm{OH} ions
(C) increase concentration of Cl\mathrm{Cl}^{-}ions
(D) increase concentration of NH4+\mathrm{NH}_{4}{ }^{+}ions
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66Explain
Identify(B) and (C) and how are(A) and (C) related?
3
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67Explain
One of the commonly used electrode is calomel electrode. Under which of the following categories calomel electrode comes ?
(A) Metal - Insoluble Salt - Anion electrodes
(B) Oxidation - Reduction electrodes
(C) Gas - Ion electrodes
(D) Metal ion - Metal electrodes
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68Explain
Number of complexes from the following with even number of unpaired "d" electrons is \qquad . [V(H2O)6]3+,[Cr(H2O)6]2+,[Fe(H2O)6]3+\left[\mathrm{V}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}, \quad\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}, \quad\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}, [Ni(H2O)6]3+,[Cu(H2O)6]2+\left[\mathrm{Ni}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+},\left[\mathrm{Cu}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+} [Given atomic numbers: V=23,Cr=24,Fe=26\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Fe}=26, Ni=28,Cu=29\mathrm{Ni}=28, \mathrm{Cu}=29 ]
(A) 2
(B) 4
(C) 5
(D) 1
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69Explain
Which one of the following molecules has maximum dipole moment?
(A) NF3\mathrm{NF}_{3}
(B) CH4\mathrm{CH}_{4}
(C) NH3\mathrm{NH}_{3}
(D) PF5\mathrm{PF}_{5}
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70Explain
Number of molecules/ions from the following in which the central atom is involved in sp3\mathrm{sp}^{3} hybridization is \qquad . NO3,BCl3,ClO2,ClO3\mathrm{NO}_{3}^{-}, \mathrm{BCl}_{3}, \mathrm{ClO}_{2}^{-}, \mathrm{ClO}_{3}
(A) 2
(B) 4
(C) 3
(D) 1
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71Explain
Which among the following is incorrect statement?
(A) Electromeric effect dominates over inductive effect
(B) The electromeric effect is, temporary effect
(C) The organic compound shows electromeric effect in the presence of the reagent only
(D) Hydrogen ion (H+)\left(\mathrm{H}^{+}\right)shows negative electromeric effect
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72Explain
Given below are two statements : Statement I: Acidity of α\alpha-hydrogens of aldehydes and ketones is responsible for Aldol reaction. Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product. In the light of above statements, choose the most appropriate answer from the options given below.
(A) Both Statement I and Statement II are correct.
(B) Both Statement I and Statement II are incorrect.
(C) Statement I is incorrect but Statement II is correct.
(D) Statement I is correct but Statement II is incorrect.
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73Explain
Which of the following nitrogen containing compound does not give Lassaigne's test ?
(A) Phenyl hydrazine
(B) Glycene
(C) Urea
(D) Hydrazine
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74Explain
Which of the following is the correct structure of L-Glucose?
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75Explain
The element which shows only one oxidation state other than its elemental form is :
(A) Cobalt
(B) Scandium
(C) Titanium
(D) Nickel
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76Explain
Identify the product in the following reaction :
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77Explain
Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is \qquad . B, C, N, S, O, F, P, Al, Si
(A) 7
(B) 5
(C) 6
(D) 3
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78Explain
The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is : (Given : Molar Mass Na:23\mathrm{Na}: 23 and Cl:35.5gmol1\mathrm{Cl}: 35.5 \mathrm{gmol}^{-1} )
(A) 20
(B) 0.2
(C) 2
(D) 4
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79Explain
Identify the correct set of reagents or reaction conditions ' XX ' and ' YY ' in the following set of transformation.
(A) X=\mathrm{X}= conc.alc. NaOH,80C,Y=Br2/CHCl3\mathrm{NaOH}, 80^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{Br}_{2} / \mathrm{CHCl}_{3}
(B) X=\mathrm{X}= dil.aq. NaOH,20C,Y=HBr\mathrm{NaOH}, 20^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{HBr} /acetic acid
(C) X=\mathrm{X}= conc.alc. NaOH,80C,Y=HBr\mathrm{NaOH}, 80^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{HBr} /acetic acid
(D) X=\mathrm{X}= dil.aq. NaOH,20C,Y=Br2/CHCl3\mathrm{NaOH}, 20^{\circ} \mathrm{C}, \mathrm{Y}=\mathrm{Br}_{2} / \mathrm{CHCl}_{3}
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80Explain
The correct order of first ionization enthalpy values of the following elements is :
(A) O
(B) N
(C) Be
(D) F (E) B Choose the correct answer from the options given below:
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81Explain
The enthalpy of formation of ethane (C2H6)\left(\mathrm{C}_{2} \mathrm{H}_{6}\right) from ethylene by addition of hydrogen where the bondenergies of CH,CC,HH\mathrm{C}-\mathrm{H}, \mathrm{C}-\mathrm{C}, \mathrm{H}-\mathrm{H} are 414 kJ,347 kJ414 \mathrm{~kJ}, 347 \mathrm{~kJ}, 615 kJ and 435 kJ respectively is - \qquad kJ .
(125)
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82Explain
The number of correct reaction(s) among the following is \qquad .
1
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83Explain
Xg of ethylamine is subjected to reaction with NaNO2/HCl\mathrm{NaNO}_{2} / \mathrm{HCl} followed by water; evolved dinitrogen gas which occupied 2.24 L volume at STP. X is \qquad ×101 g\times 10^{-1} \mathrm{~g}.
(45)
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84Explain
The de-Broglie's wavelength of an electron in the 4th 4^{\text {th }} orbit is \qquad πa0.(a0=\pi \mathrm{a}_{0} .\left(\mathrm{a}_{0}=\right. Bohr's radius ))
(8)
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85Explain
Only 2 mL of KMnO4\mathrm{KMnO}_{4} solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid ( 2 M ) in acidic medium. The molarity of KMnO4\mathrm{KMnO}_{4} solution should be \qquad M.
(50)
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86Explain
Consider the following reaction MnO2+KOH+O2 A+H2O\mathrm{MnO}_{2}+\mathrm{KOH}+\mathrm{O}_{2} \rightarrow \mathrm{~A}+\mathrm{H}_{2} \mathrm{O}. Product 'A' in neutral or acidic medium disproportionate to give products ' B ' and ' C ' along with water. The sum of spin-only magnetic moment values of B and C is \qquad BM. (nearest integer) (Given atomic number of Mn is 25 )
(4)
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87Explain
Consider the following transformation involving first order elementary reaction in each step at constant temperature as shown below. Some details of the above reaction are listed below. If the overall rate constant of the above transformation ( k ) is given as k=k1k2k3\mathrm{k}=\frac{\mathrm{k}_{1} \mathrm{k}_{2}}{\mathrm{k}_{3}} and the overall activation energy ( Ea\mathrm{E}_{\mathrm{a}} ) is 400 kJ mol1400 \mathrm{~kJ} \mathrm{~mol}^{-1}, then the value of Ea3E a_{3} is \qquad kJmol1\mathrm{kJ} \mathrm{mol}^{-1} (nearest integer)
(100)
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88Explain
2.5 g of a non-volatile, non-electrolyte is dissolved in 100 g of water at 25C25^{\circ} \mathrm{C}. The solution showed a boiling point elevation by 2C2^{\circ} \mathrm{C}. Assuming the solute concentration in negligible with respect to the solvent concentration, the vapour pressure of the resulting aqueous solution is \qquad mm of Hg (nearest integer) [Given : Molal boiling point elevation constant of water (Kb)=0.52K. kg mol1\left(\mathrm{K}_{\mathrm{b}}\right)=0.52 \mathrm{K} .\mathrm{~kg} \mathrm{~mol}^{-1}, 1 atm pressure =760 mm=760 \mathrm{~mm} of Hg , molar mass of water =18 g mol1=18 \mathrm{~g} \mathrm{~mol}^{-1} ]
(707)
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89Explain
The number of different chain isomers for C7H16\mathrm{C}_{7} \mathrm{H}_{16} is \qquad .
(9)
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90Explain
Number of molecules/species from the following having one unpaired electron is \qquad . O2,O21,NO,CN1,O22\mathrm{O}_{2}, \mathrm{O}_{2}^{-1}, \mathrm{NO}, \mathrm{CN}^{-1}, \mathrm{O}_{2}^{2-}
(2)
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