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JEE-MAIN EXAMINATION – JANUARY 2025

JEE-MAIN TEST PAPER WITH SOLUTION

Held on Friday 24th 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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0
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75
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Complete Solutions

Q#ExplanationQuestionCorrectSolutionStatus
1Explain
Let a⃗=i^+2j^+3k^,b⃗=3i^+j^−k^\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+\hat{j}-\hat{k} and c⃗\vec{c} be three vectors such that c⃗\vec{c} is coplanar with a⃗\vec{a} and b⃗\vec{b}. If the vector c⃗\vec{c} is perpendicular to b⃗\vec{b} and a⃗⋅c⃗=5\vec{a} \cdot \vec{c}=5, then ∣c→∣|\overrightarrow{\mathrm{c}}| is equal to
(A) 132\frac{1}{3 \sqrt{2}}
(B) 18
(C) 16
(D) 116\sqrt{\frac{11}{6}}
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2Explain
In I(m,n)=∫01xm−1(1−x)n−1dx,m,n>0I(m, n)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x, m, n>0, then I(9,14)+I(10,13)\mathrm{I}(9,14)+\mathrm{I}(10,13) is
(A) I (9,1)(9,1)
(B) I(19,27)\mathrm{I}(19,27)
(C) I(1,13)\mathrm{I}(1,13)
(D) I(9,13)I(9,13)
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3Explain
Let f:R−{0}→R\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R} be a function such that f(x)−6f(1x)=353x−52f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}. If the lim⁡x→0(1αx+f(x))=β;α,β∈R\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in \mathbb{R}, then α+2β\alpha+2 \beta is equal to
(A) 3
(B) 5
(C) 4
(D) 6
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4Explain
Let Sn=12+16+112+120+…\mathrm{S}_{\mathrm{n}}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is 2026 S2025\sqrt{2026 \mathrm{~S}_{2025}}, then the absolute difference between 20th 20^{\text {th }} and 15th 15^{\text {th }} terms of the A.P. is
(A) 25
(B) 90
(C) 20
(D) 45
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5Explain
Let f(x)=2x+2+1622x+1+2x+4+32f(x)=\frac{2^{x+2}+16}{2^{2 x+1}+2^{x+4}+32}. Then the value of 8(f(115)+f(215)+…+f(5915))8\left(\mathrm{f}\left(\frac{1}{15}\right)+\mathrm{f}\left(\frac{2}{15}\right)+\ldots+\mathrm{f}\left(\frac{59}{15}\right)\right) is equal to
(A) 118
(B) 92
(C) 102
(D) 108
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6Explain
If α\alpha and β\beta are the roots of the equation 2z2−3z−2i=02 \mathrm{z}^{2}-3 \mathrm{z}-2 \mathrm{i}=0, where i=−1\mathrm{i}=\sqrt{-1}, then 16. Re⁡(α19+β19+α11+β11α15+β15)⋅Im⁡(α19+β19+α11+β11α15+β15)\operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) is equal to
(A) 398
(B) 312
(C) 409
(D) 441
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7Explain
lim⁡x→0cosec⁡x(2cos⁡2x+3cos⁡x−cos⁡2x+sin⁡x+4)\lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^{2} x+3 \cos x}-\sqrt{\cos ^{2} x+\sin x+4}\right) is
(A) 0
(B) 125\frac{1}{2 \sqrt{5}}
(C) 115\frac{1}{\sqrt{15}}
(D) −125-\frac{1}{2 \sqrt{5}}
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8Explain
Let in a △ABC\triangle \mathrm{ABC}, the length of the side AC be 6 , the vertex BB be (1,2,3)(1,2,3) and the vertices A,CA, C lie on the line x−63=y−72=z−7−2\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}. Then the area (in sq. units) of △ABC\triangle \mathrm{ABC} is
(A) 42
(B) 21
(C) 56
(D) 17
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9Explain
Let y=y(x)y=y(x) be the solution of the differential equation (xy−5x21+x2)dx+(1+x2)dy=0\left(x y-5 x^{2} \sqrt{1+x^{2}}\right) d x+\left(1+x^{2}\right) d y=0, y(0)=0y(0)=0. Then y(3)y(\sqrt{3}) is equal to
(A) 532\frac{5 \sqrt{3}}{2}
(B) 143\sqrt{\frac{14}{3}}
(C) 222 \sqrt{2}
(D) 152\sqrt{\frac{15}{2}}
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10Explain
Let the product of the focal distances of the point (3,12)\left(\sqrt{3}, \frac{1}{2}\right) on the ellipse x2a2+y2b2=1,(a>b)\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b), be 74\frac{7}{4}. Then the absolute difference of the eccentricities of two such ellipses is
(A) 3−2232\frac{3-2 \sqrt{2}}{3 \sqrt{2}}
(B) 1−32\frac{1-\sqrt{3}}{\sqrt{2}}
(C) 3−2223\frac{3-2 \sqrt{2}}{2 \sqrt{3}}
(D) 1−223\frac{1-2 \sqrt{2}}{\sqrt{3}}
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11Explain
A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before BB throws a sum of 8 , and B wins if he throws a sum of 8 before A throws a sum of 5 . The probability, that A wins if A makes the first throw, is
(A) 917\frac{9}{17}
(B) 919\frac{9}{19}
(C) 817\frac{8}{17}
(D) 819\frac{8}{19}
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12Explain
Consider the region R={(x,y):x≤y≤9−113x2,x≥0}R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3} x^{2}, x \geq 0\right\}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is :
(A) 625111\frac{625}{111}
(B) 730119\frac{730}{119}
(C) 567121\frac{567}{121}
(D) 821123\frac{821}{123}
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13Explain
The area of the region {(x,y):x2+4x+2≤y≤∣x+2∣}\left\{(x, y): x^{2}+4 x+2 \leq y \leq|x+2|\right\} is equal to
(A) 7
(B) 24/524 / 5
(C) 20/320 / 3
(D) 5
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14Explain
For a statistical data x1,x2,…,x10x_{1}, x_{2}, \ldots, x_{10} of 10 values, a student obtained the mean as 5.5 and ∑i=110xi2=371\sum_{\mathrm{i}=1}^{10} \mathrm{x}_{\mathrm{i}}^{2}=371. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
(A) 7
(B) 4
(C) 9
(D) 5
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15Explain
Let circle CC be the image of x2+y2−2x+4y−4=0x^{2}+y^{2}-2 x+4 y-4=0 in the line 2x−3y+5=02 x-3 y+5=0 and AA be the point on CC such that OA is parallel to x -axis and A lies on the right hand side of the centre O of C . If B(α,β)\mathrm{B}(\alpha, \beta), with β<4\beta<4, lies on C such that the length of the arc AB is (1/6)th (1 / 6)^{\text {th }} of the perimeter of C , then β−3α\beta-\sqrt{3} \alpha is equal to
(A) 3
(B) 3+33+\sqrt{3}
(C) 4−34-\sqrt{3}
(D) 4
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16Explain
For some n≠10n \neq 10, let the coefficients of the 5th ,6th 5^{\text {th }}, 6^{\text {th }} and 7th 7^{\text {th }} terms in the binomial expansion of (1+x)n+4(1+x)^{n+4} be in A.P. Then the largest coefficient in the expansion of (1+x)n+4(1+x)^{n+4} is :
(A) 70
(B) 35
(C) 20
(D) 10
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17Explain
The product of all the rational roots of the equation (x2−9x+11)2−(x−4)(x−5)=3\left(\mathrm{x}^{2}-9 \mathrm{x}+11\right)^{2}-(\mathrm{x}-4)(\mathrm{x}-5)=3, is equal to :
(A) 14
(B) 7
(C) 28
(D) 21
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18Explain
Let the line passing through the points ( −1,2,1-1,2,1 ) and parallel to the line x−12=y+13=z4\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4} intersect the line x+23=y−32=z−41\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1} at the point PP. Then the distance of P from the point Q(4,−5,1)\mathrm{Q}(4,-5,1) is :
(A) 5
(B) 10
(C) 565 \sqrt{6}
(D) 555 \sqrt{5}
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19Explain
Let the lines 3x−4y−α=0,8x−11y−33=03 x-4 y-\alpha=0,8 x-11 y-33=0, and 2x−3y+λ=02 x-3 y+\lambda=0 be concurrent. If the image of the point (1,2)(1,2) in the line 2x−3y+λ=02 \mathrm{x}-3 \mathrm{y}+\lambda=0 is (5713,−4013)\left(\frac{57}{13}, \frac{-40}{13}\right), then ∣αλ∣|\alpha \lambda| is equal to :
(A) 84
(B) 91
(C) 113
(D) 101
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20Explain
If the system of equations 2x−y+z=42 x-y+z=4 5x+λy+3z=125 x+\lambda y+3 z=12 100x−47y+μz=212100 x-47 y+\mu z=212, has infinitely many solutions, then μ−2λ\mu-2 \lambda is equal to
(A) 56
(B) 59
(C) 55
(D) 57
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21Explain
Let f be a differentiable function such that 2(x+2)2f(x)−3(x+2)2=10∫0x(t+2)f(t)dt2(x+2)^{2} f(x)-3(x+2)^{2}=10 \int_{0}^{x}(t+2) f(t) d t, x≥0x \geq 0. Then f(2)f(2) is equal to ____\_\_\_\_ .
(19)
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22Explain
If for some α,β;α≤β,α+β=8\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8 and sec⁡2(tan⁡−1α)+cosec⁡2(cot⁡−1β)=36\sec ^{2}\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} \beta\right)=36, then α2+β\alpha^{2}+\beta is ____\_\_\_\_ .
(14)
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23Explain
The number of 3-digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is
(125)
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24Explain
Let be a 3×33 \times 3 matrix such that XTAX=O\mathrm{X}^{\mathrm{T}} \mathrm{AX}=\mathrm{O} for all nonzero 3×13 \times 1 matrices X=[xyz]X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]. If A[111]=[14−5],A[121]=[04−8]\mathrm{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}1 \\ 4 \\ -5\end{array}\right], \mathrm{A}\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 4 \\ -8\end{array}\right], and det⁡(adj⁡(2( A+I)))=2α3β5γ,α,β,γ,∈N\operatorname{det}(\operatorname{adj}(2(\mathrm{~A}+\mathrm{I})))=2^{\alpha} 3^{\beta} 5^{\gamma}, \alpha, \beta, \gamma, \in \mathrm{N}, then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is
(44)
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25Explain
Let S={p1,p2……,p10}\mathrm{S}=\left\{\mathrm{p}_{1}, \mathrm{p}_{2} \ldots \ldots, \mathrm{p}_{10}\right\} be the set of first ten prime numbers. Let A=S∪P\mathrm{A}=\mathrm{S} \cup \mathrm{P}, where P is the set of all possible products of distinct element of S . Then the number of all ordered pairs (x,y),x∈S(x, y), x \in S, y∈Ay \in A, such that xx divides yy, is ____\_\_\_\_ .
(5120)
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26Explain
Consider a parallel plate capacitor of area A (of each plate) and separation 'd' between the plates. If E is the electric field and ε0\varepsilon_{0} is the permittivity of free space between the plates, then potential energy stored in the capacitor is :-
(A) 12ε0E2Ad\frac{1}{2} \varepsilon_{0} E^{2} A d
(B) 34ε0E2Ad\frac{3}{4} \varepsilon_{0} E^{2} A d
(C) 14ε0E2Ad\frac{1}{4} \varepsilon_{0} E^{2} A d
(D) ε0E2Ad\varepsilon_{0} \mathrm{E}^{2} \mathrm{Ad}
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27Explain
What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5 D ? ['D' stands for dioptre]
(A) 0.04
(B) 0.40
(C) 0.1
(D) 0.01
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28Explain
An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density 1000 kg/m31000 \mathrm{~kg} / \mathrm{m}^{3}. If the pressure inside the bubble is 2100 N/m22100 \mathrm{~N} / \mathrm{m}^{2} greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use g=10 m/s2\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2} )
(A) 0.02
(B) 0.1
(C) 0.25
(D) 0.05
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29Explain
For an experimental expression y=32.3×112527.4\mathrm{y}=\frac{32.3 \times 1125}{27.4}, where all the digits are significant. Then to report the value of yy we should write :-
(A) y=1326.2y=1326.2
(B) y=1326.19y=1326.19
(C) y=1326.186y=1326.186
(D) y=1330y=1330
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30Explain
During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000A˚2000 \AA and it becomes 6000A˚6000 \AA when the electron jumps from state BB to state CC. Then the wavelength of the radiation emitted during the transition of electrons from state AA to state BB is :-
(A) 3000A˚3000 \AA
(B) 6000A˚6000 \AA
(C) 4000A˚4000 \AA
(D) 2000A˚∘2000 \AA{ }^{\circ}
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31Explain
Consider the following statements : A. The junction area of solar cell is made very narrow compared to a photo diode. B. Solar cells are not connected with any external bias. C. LED is made of lightly doped p-n junction. D. Increase of forward current results in continuous increase of LED light intensity. E. LEDs have to be connected in forward bias for emission of light.
(A) B, D, E Only
(B) A, C Only
(C) A, C, E Only
(D) B, E Only
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32Explain
The amount of work done to break a big water drop of radius ' R ' into 27 small drops of equal radius is 10 J . The work done required to break the same big drop into 64 small drops of equal radius will be :-
(A) 15 J
(B) 10 J
(C) 20 J
(D) 5 J
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33Explain
An object of mass ' mm ' is projected from origin in a vertical xy plane at an angle 45∘45^{\circ} with the x -axis with an initial velocity v0\mathrm{v}_{0}. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ gg is acceleration due to gravity]
(A) mv0322 g\frac{\mathrm{mv}_{0}^{3}}{2 \sqrt{2} \mathrm{~g}} along negative z -axis
(B) mv0322 g\frac{\mathrm{mv}_{0}^{3}}{2 \sqrt{2} \mathrm{~g}} along positive z -axis
(C) mv0342 g\frac{\mathrm{mv}_{0}^{3}}{4 \sqrt{2} \mathrm{~g}} along positive z -axis
(D) mv0342 g\frac{\mathrm{mv}_{0}^{3}}{4 \sqrt{2} \mathrm{~g}} along negative z -axis
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34Explain
The Young's double slit interference experiment is performed using light consisting of 480 nm and 600 nm wavelengths to form interference patterns. The least number of the bright fringes of 480 nm light that are required for the first coincidence with the bright fringes formed by 600 nm light is :-
(A) 4
(B) 8
(C) 6
(D) 5
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35Explain
A car of mass ' mm ' moves on a banked road having radius ' rr ' and banking angle θ\theta. To avoid slipping from banked road, the maximum permissible speed of the car is v0\mathrm{v}_{0}. The coefficient of friction μ\mu between the wheels of the car and the banked road is :-
(A) μ=v02+rgtan⁡θrg−v02tan⁡θ\mu=\frac{\mathrm{v}_{0}^{2}+\mathrm{rg} \tan \theta}{\mathrm{rg}-\mathrm{v}_{0}^{2} \tan \theta}
(B) μ=v02+rgtan⁡θrg+v02tan⁡θ\mu=\frac{v_{0}^{2}+r g \tan \theta}{r g+v_{0}^{2} \tan \theta}
(C) μ=v02−rgtan⁡θrg+v02tan⁡θ\mu=\frac{\mathrm{v}_{0}^{2}-\mathrm{rg} \tan \theta}{\mathrm{rg}+\mathrm{v}_{0}^{2} \tan \theta}
(D) μ=v02−rgtan⁡θrg−v02tan⁡θ\mu=\frac{\mathrm{v}_{0}^{2}-\mathrm{rg} \tan \theta}{\mathrm{rg}-\mathrm{v}_{0}^{2} \tan \theta}
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36Explain
A uniform solid cylinder of mass ' mm ' and radius ' rr ' rolls along an inclined rough plane of inclination 45∘45^{\circ}. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be :-
(A) 12 g\frac{1}{\sqrt{2}} \mathrm{~g}
(B) 132 g\frac{1}{3 \sqrt{2}} \mathrm{~g}
(C) 2g3\frac{\sqrt{2} g}{3}
(D) 2g\sqrt{2} g
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37Explain
A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2. When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length 0.2 m . The radius of curvature of the curved surface of the lens is :-
(A) 0.15 m
(B) 0.10 m
(C) 0.20 m
(D) 0.25 m
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38Explain
A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then Dd\frac{\mathrm{D}}{\mathrm{d}} is :-
(A) 154\frac{15}{4}
(B) 25
(C) 10
(D) 165\frac{16}{5}
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39Explain
A satellite is launched into a circular orbit of radius ' R ' around the earth. A second statellite is launched into an orbit of radius 1.03 R . The time period of revolution of the second satellite is larger than the first one approximately by :-
(A) 3%3 \%
(B) 4.5%4.5 \%
(C) 9%9 \%
(D) 2.5%2.5 \%
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40Explain
A plano-convex lens having radius of curvature of first surface 2 cm exhibits focal length of f1\mathrm{f}_{1} in air. Another plano-convex lens with first surface radius of curvature 3 cm has focal length of f2f_{2} when it is immersed in a liquid of refractive index 1.2. If both the lenses are made of same glass of refractive index 1.5 , the ratio of f1f_{1} and f2f_{2} will be :-
(A) 3:53: 5
(B) 1:31: 3
(C) 1:21: 2
(D) 2:32: 3
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41Explain
An alternating current is given by I=IAsin⁡ωt+IBcos⁡ωt\mathrm{I}=\mathrm{I}_{\mathrm{A}} \sin \omega \mathrm{t}+\mathrm{I}_{\mathrm{B}} \cos \omega \mathrm{t}. The r.m.s. current will be :-
(A) IA2+IB2\sqrt{I_{A}^{2}+I_{B}^{2}}
(B) IA2+IB22\frac{\sqrt{I_{A}^{2}+I_{B}^{2}}}{2}
(C) IA2+IB22\sqrt{\frac{I_{A}^{2}+I_{B}^{2}}{2}}
(D) ∣IA+IB∣2\frac{\left|I_{A}+I_{B}\right|}{\sqrt{2}}
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42Explain
An electron of mass ' mm ' with an initial velocity v→=v0i^(v0>0)\overrightarrow{\mathrm{v}}=\mathrm{v}_{0} \hat{\mathrm{i}}\left(\mathrm{v}_{0}>0\right) enters an electric field E→=−E0k^\overrightarrow{\mathrm{E}}=-\mathrm{E}_{0} \hat{\mathrm{k}}. If the initial de Broglie wavelength is λ0\lambda_{0}, the value after time t would be :-
(A) λ01+e2E02t2m2v02\frac{\lambda_{0}}{\sqrt{1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}
(B) λ01−e2E02t2m2v02\frac{\lambda_{0}}{\sqrt{1-\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}
(C) λ0\lambda_{0}
(D) λ01+e2E02t2m2v02\lambda_{0} \sqrt{1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}
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43Explain
A parallel plate capacitor was made with two rectangular plates, each with a length of l=3 cml=3 \mathrm{~cm} and breath of b=1 cmb=1 \mathrm{~cm}. The distance between the plates is 3μ m3 \mu \mathrm{~m}. Out of the following, which are the ways to increase the capacitance by a factor of 10 ? Choose the correct answer from the options given below :
(A) l=30 cm, b=1 cm, d=1μ ml=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}
(B) l=3 cm, b=1 cm, d=30μ ml=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}
(C) l=6 cm, b=5 cm, d=3μ ml=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}
(D) l=1 cm, b=1 cm, d=10μ ml=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m} E. l=5 cm, b=2 cm, d=1μ ml=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}
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44Explain
A force F=α+βx2F=\alpha+\beta x^{2} acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m . If the constant α=1 N\alpha=1 \mathrm{~N} then β\beta will be
(A) 15 N/m215 \mathrm{~N} / \mathrm{m}^{2}
(B) 10 N/m210 \mathrm{~N} / \mathrm{m}^{2}
(C) 12 N/m212 \mathrm{~N} / \mathrm{m}^{2}
(D) 8 N/m28 \mathrm{~N} / \mathrm{m}^{2}
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45Explain
An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature. Choose the correct answer from the options given below :-
(A) The work done by gas during the process is zero.
(B) The heat added to gas is different from change in its internal energy.
(C) The volume of the gas is increased.
(D) The internal energy of the gas is increased. E. The process is isochoric (constant volume process)
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46Explain
A square loop of sides a=1 ma=1 \mathrm{~m} is held normally in front of a point charge q=1C\mathrm{q}=1 \mathrm{C}. The flux of the electric field through the shaded region is 5p×1ε0Nm2C\frac{5}{\mathrm{p}} \times \frac{1}{\varepsilon_{0}} \frac{\mathrm{Nm}^{2}}{\mathrm{C}}, where the value of p is ____\_\_\_\_ . []
(48)
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47Explain
The least count of a screw guage is 0.01 mm . If the pitch is increased by 75%75 \% and number of divisions on the circular scale is reduced by 50%50 \%, the new least count will be ____\_\_\_\_ ×10−3 mm\times 10^{-3} \mathrm{~mm}.
(35)
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48Explain
A wire of resistance 9Ω9 \Omega is bent to form an equilateral triangle. Then the equivalent resistance across any two vertices will be ____\_\_\_\_ ohm.
(2)
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49Explain
A current of 5A exists in a square loop of side 12 m\frac{1}{\sqrt{2}} \mathrm{~m}. Then the magnitude of the magnetic field B at the centre of the square loop will be p×10−6 T\mathrm{p} \times 10^{-6} \mathrm{~T}. where, value of p is ____\_\_\_\_ . [Take μ0=4π×10−7 T mA−1\mu_{0}=4 \pi \times 10^{-7} \mathrm{~T} \mathrm{~mA}^{-1} ].
(8)
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50Explain
The temperature of 1 mole of an ideal monoatomic gas is increased by 50∘C50^{\circ} \mathrm{C} at constant pressure. The total heat added and change in internal energy are E1E_{1} and E2E_{2}, respectively. If E1E2=X9\frac{E_{1}}{E_{2}}=\frac{X}{9} then the value of xx is ____\_\_\_\_ .
(15)
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51Explain
For the given cell Fe2+(aq)+Ag+(aq)→Fe3+(aq)+Ag(s)\mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{Ag}^{+}(\mathrm{aq}) \rightarrow \mathrm{Fe}^{3+}(\mathrm{aq})+\mathrm{Ag}(\mathrm{s}) The standard cell potential of the above reaction is Given :
(A) x+y−zx+y-z
(B) x+2y−3zx+2 y-3 z
(C) y−2xy-2 x
(D) x+2yx+2y
2
Diagram Question
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52Explain
Following are the four molecules "P", "Q", "R" and "S". Which one among the four molecules will react with H−Br(aq)\mathrm{H}-\mathrm{Br}(\mathrm{aq}) at the fastest rate ? []
(A) S
(B) Q
(C) R
(D) P
2
Diagram Question
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53Explain
One mole of the octahedral complex compound Co(NH3)5Cl3\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}_{3} gives 3 moles of ions on dissolution in water. One mole of the same complex reacts with excess of AgNO3\mathrm{AgNO}_{3} solution to yield two moles of AgCl(s)\mathrm{AgCl}_{(\mathrm{s})}. The structure of the complex is :
(A) [Co(NH3)5Cl]Cl2\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}\right] \mathrm{Cl}_{2}
(B) [Co(NH3)4Cl]⋅Cl2⋅NH3\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}\right] \cdot \mathrm{Cl}_{2} \cdot \mathrm{NH}_{3}
(C) [Co(NH3)4Cl2]Cl.NH3\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right] \mathrm{Cl} . \mathrm{NH}_{3}
(D) [Co(NH3)3Cl3].2NH3\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3} \mathrm{Cl}_{3}\right] .2 \mathrm{NH}_{3}
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54Explain
Which one of the carbocations from the following is most stable?
2
Diagram Question
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55Explain
Which of the following linear combination of atomic orbitals will lead to formation of molecular orbitals in homonuclear diatomic molecules [internuclear axis in z-direction] ? A.2pz2 \mathrm{p}_{\mathrm{z}} and 2px2 \mathrm{p}_{\mathrm{x}}, B. 2s and 2px2 \mathrm{p}_{\mathrm{x}}, C. 3dxy3 d_{x y} and 3dx2−y23 d_{x^{2}-y^{2}}, D. 2s and 2pz2 \mathrm{p}_{\mathrm{z}} E. 2pz2 p_{z} and 3dx2−y23 d_{x^{2}-y^{2}}
(A) E Only
(B) A and B Only
(C) D Only
(D) C and D Only
3
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56Explain
Which of the following ions is the strongest oxidizing agent ? (Atomic Number of Ce=58,Eu=63, Tb=65,Lu=71]\mathrm{Ce}=58, \mathrm{Eu}=63, \mathrm{~Tb}=65, \mathrm{Lu}=71]
(A) Lu3+\mathrm{Lu}^{3+}
(B) Eu2+\mathrm{Eu}^{2+}
(C) Tb4+\mathrm{Tb}^{4+}
(D) Ce3+\mathrm{Ce}^{3+}
3
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57Explain
Ksp for Cr(OH)3\mathrm{Cr}(\mathrm{OH})_{3} is 1.6×10−301.6 \times 10^{-30}. What is the molar solubility of this salt in water?
(A) 1.6×10−30274\sqrt[4]{\frac{1.6 \times 10^{-30}}{27}}
(B) 1.8×10−3027\frac{1.8 \times 10^{-30}}{27}
(C) 1.8×10−305\sqrt[5]{1.8 \times 10^{-30}}
(D) 1.6×10−302\sqrt[2]{1.6 \times 10^{-30}}
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58Explain
Let us consider an endothermic reaction which is non-spontaneous at the freezing point of water. However, the reaction is spontaneous at boiling point of water. Choose the correct option.
(A) Both ΔH\Delta \mathrm{H} and ΔS\Delta \mathrm{S} are ( +ve )
(B) ΔH\Delta \mathrm{H} is (-ve) but ΔS\Delta \mathrm{S} is ( +ve )
(C) ΔH\Delta \mathrm{H} is ( +ve ) but ΔS\Delta \mathrm{S} is ( -ve )
(D) Both ΔH\Delta \mathrm{H} and ΔS\Delta \mathrm{S} are (-ve)
1
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59Explain
Given below are two statements I and II. Statement I : Dumas method is used for estimation of "Nitrogen" in an organic compound. Statement II : Dumas method involves the formation of ammonium sulphate by heating the organic compound with conc H2SO4\mathrm{H}_{2} \mathrm{SO}_{4}. In the light of the above statements, choose the correct answer from the options given below
(A) Both Statement I and Statement II are true
(B) Statement I is false but Statement II is true
(C) Both Statement I and Statement II are false
(D) Statement I is true but Statement II is false
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60Explain
Which of the following Statements are NOT true about the periodic table? A.The properties of elements are function of atomic weights. B.The properties of elements are function of atomic numbers. C.Elements having similar outer electronic configuration are arranged in same period. D.An element's location reflects the quantum numbers of the last filled orbital. E. The number of elements in a period is same as the number of atomic orbitals available in energy level that is being filled. Choose the correct answer from the options given below:
(A) A, C and E Only
(B) D and E Only
(C) A and E Only
(D) B, C and E Only
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61Explain
The carbohydrates "Ribose" present in DNA, is A. A pentose sugar B. present in pyranose from C. in "D" configuration D. a reducing sugar, when free E. in α\alpha-anomeric form Choose the correct answer from the options given below :
(A) A, C and D Only
(B) A, B and E Only
(C) B, D and E Only
(D) A, D and E Only
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62Explain
Preparation of potassium permanganate from MnO2\mathrm{MnO}_{2} involves two step process in which the 1st 1^{\text {st }} step is a reaction with KOH and KNO3\mathrm{KNO}_{3} to produce
(A) K4[Mn(OH)6]\mathrm{K}_{4}\left[\mathrm{Mn}(\mathrm{OH})_{6}\right]
(B) K3MnO4\mathrm{K}_{3} \mathrm{MnO}_{4}
(C) KMnO4\mathrm{KMnO}_{4}
(D) K2MnO4\mathrm{K}_{2} \mathrm{MnO}_{4}
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63Explain
The large difference between the melting and boiling points of oxygen and sulphur may be explained on the basis of
(A) Atomic size
(B) Atomicity
(C) Electronegativity
(D) Electron gain enthalpy
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64Explain
For a reaction, N2O5( g)→2NO2( g)+12O2( g)\mathrm{N}_{2} \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{NO}_{2(\mathrm{~g})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})} in a constant volume container, no products were present initially. The final pressure of the system when 50%50 \% of reaction gets completed is
(A) 7/27 / 2 times of initial pressure
(B) 5 times of initial pressure
(C) 5/25 / 2 times of initial pressure
(D) 7/4 times of initial pressure
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65Explain
Which of the following arrangements with respect to their reactivity in nucleophilic addition reaction is correct?
(A) benzaldehyde < acetophenone < p-nitrobenzaldehyde < p-tolualdehyde
(B) acetophenone < benzaldehyde < p-tolualdehyde < p-nitrobenzaldehyde
(C) acetophenone << p-tolualdehyde < benzaldehyde < p-nitrobenzaldehyde
(D) p-nitrobenzaldehyde < benzaldehyde << p-tolualdehyde < acetophenone
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66Explain
Aman has been asked to synthesise the molecule(x). He thought of preparing the molecule using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory. He thought of performing ozonolysis reaction on alkene to produce a dicarbonyl compound followed by aldol reaction to prepare "x". Predict the suitable alkene that can lead to the formation of "x".
1
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67Explain
Consider the given plots of vapour pressure (VP) vs temperature (T/K) Which amongst the following options is correct graphical representation showing ΔTf\Delta \mathrm{T}_{\mathrm{f}}, depression in the freezing point of solvent in a solution?
3
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68Explain
Which of the following statement is true with respect to H2O,NH3\mathrm{H}_{2} \mathrm{O}, \mathrm{NH}_{3} and CH4\mathrm{CH}_{4} ? A. The central atoms of all the molecules are sp3s p^{3} hybridized. B. The H−O−H,H−N−H\mathrm{H}-\mathrm{O}-\mathrm{H}, \mathrm{H}-\mathrm{N}-\mathrm{H} and H−C−H\mathrm{H}-\mathrm{C}-\mathrm{H} angles in the above molecules are 104.5∘,107.5∘104.5^{\circ}, 107.5^{\circ} and 109.5∘109.5^{\circ} respectively. C. The increasing order of dipole moment is CH4<NH3<H2O\mathrm{CH}_{4}<\mathrm{NH}_{3}<\mathrm{H}_{2} \mathrm{O}. D. Both H2O\mathrm{H}_{2} \mathrm{O} and NH3\mathrm{NH}_{3} are Lewis acids and CH4\mathrm{CH}_{4} is a Lewis base E. A solution of NH3\mathrm{NH}_{3} in H2O\mathrm{H}_{2} \mathrm{O} is basic. In this solution NH3\mathrm{NH}_{3} and H2O\mathrm{H}_{2} \mathrm{O} act as Lowry-Bronsted acid and base respectively. Choose the correct answer from the options given below :
(A) A, B and C Only
(B) C, D and E Only
(C) A, D and E Only
(D) A, B, C and E Only
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69Explain
Given below are two statements : Statement-I : The conversion proceeds well in the less polar medium. [] Statement-II : The conversion proceeds well in the more polar medium. [] <smiles>[R]N([Cl+])[N+]([R])[R]</smiles> In the light of the above statements, choose the correct answer from the options given below.
(A) Both statement I and statement II are true
(B) Both statement I and statement II are false
(C) Statement I is false but statement II is true
(D) Statement I is true but statement II is false
1
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70Explain
The product (A) formed in the following reaction sequence is : []
2
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71Explain
37.8 g N2O537.8 \mathrm{~g} \mathrm{~N}_{2} \mathrm{O}_{5} was taken in a 1 L reaction vessel and allowed to undergo the following reaction at 500 K 2 N2O5( g)→2 N2O4( g)+O2( g)2 \mathrm{~N}_{2} \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})} The total pressure at equilibrium was found to be 18.65 bar. Then, Kp=\mathrm{Kp}= ____\_\_\_\_ ×10−2\times 10^{-2} [nearest integer] Assume N2O5\mathrm{N}_{2} \mathrm{O}_{5} to behave ideally under these conditions Given : R=0.082\mathrm{R}=0.082 bar Lmol−1 K−1\mathrm{L} \mathrm{mol}^{-1} \mathrm{~K}^{-1}
(962)
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72Explain
Standard entropies of X2,Y2X_{2}, Y_{2} and XY5X Y_{5} are 70,50 and 110 J K−1 mol−1110 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} respectively. The temperature in Kelvin at which the reaction 12X2+52Y2→XY5ΔH−=−35 kJ mol−1\frac{1}{2} \mathrm{X}_{2}+\frac{5}{2} \mathrm{Y}_{2} \rightarrow \mathrm{XY}_{5} \Delta \mathrm{H}^{-}=-35 \mathrm{~kJ} \mathrm{~mol}^{-1} Will be at equilibrium is ____\_\_\_\_ (Nearest integer)
(700)
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73Explain
Xg of benzoic acid on reaction with aq. NaHCO3\mathrm{NaHCO}_{3} release CO2\mathrm{CO}_{2} that occupied 11.2 L volume at STP . XX is ____\_\_\_\_ g.
(61)
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74Explain
Among the following cations, the number of cations which will give characteristic precipitate in their identification tests with K4[Fe(CN)6]\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] is : Cu2+,Fe3+,Ba2+,Ca2+,NH4+,Mg2+,Zn2+\mathrm{Cu}^{2+}, \mathrm{Fe}^{3+}, \mathrm{Ba}^{2+}, \mathrm{Ca}^{2+}, \mathrm{NH}_{4}{ }^{+}, \mathrm{Mg}^{2+}, \mathrm{Zn}^{2+}
(3)
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75Explain
Consider the following reaction occurring in the blast furnace. Fe3O4( s)+4CO(g)→3Fe(l)+4CO2( g)\mathrm{Fe}_{3} \mathrm{O}_{4(\mathrm{~s})}+4 \mathrm{CO}_{(\mathrm{g})} \rightarrow 3 \mathrm{Fe}_{(\mathrm{l})}+4 \mathrm{CO}_{2(\mathrm{~g})} ' xx ' kg of iron is produced when 2.32×103 kgFe3O42.32 \times 10^{3} \mathrm{~kg} \mathrm{Fe}_{3} \mathrm{O}_{4} and 2.8×102 kgCO2.8 \times 10^{2} \mathrm{~kg} \mathrm{CO} are brought together in the furnace. The value of ' xx ' is ____\_\_\_\_ . (nearest integer) \{Given : Molar mass of Fe3O4=232 g mol−1\mathrm{Fe}_{3} \mathrm{O}_{4}=232 \mathrm{~g} \mathrm{~mol}^{-1} Molar mass of CO=28 g mol−1\mathrm{CO}=28 \mathrm{~g} \mathrm{~mol}^{-1} Molar mass of Fe=56 g mol−1}\left.\mathrm{Fe}=56 \mathrm{~g} \mathrm{~mol}^{-1}\right\}
(420)
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