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Jan 29, 2024

JEE Mains

Shift: 1

Total Questions Available: 90

Question 1

Identify the incorrect pair from the following :

Options:

A)

Fluoroapatite 3Ca3(PO4)2CaF2-3 \mathrm{Ca}_3\left(\mathrm{PO}_4\right)_2 \cdot \mathrm{CaF}_2

B)

Carnallite KClMgCl26H2O-\mathrm{KCl} \cdot \mathrm{MgCl}_2 \cdot 6 \mathrm{H}_2 \mathrm{O}

C)

Cryolite Na3AlF6-\mathrm{Na}_3 \mathrm{AlF}_6

D)

Fluorspar BF3-\mathrm{BF}_3

Question 2

Given below are two statements : one is labelled as Assertion A\mathbf{A} and the other is labelled as Reason R\mathbf{R} :

Assertion A : The first ionisation enthalpy decreases across a period.

Reason R\mathbf{R} : The increasing nuclear charge outweighs the shielding across the period.

In the light of the above statements, choose the most appropriate from the options given below :

Options:

A)

A\mathbf{A} is false but R\mathbf{R} is true

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 true but R\mathbf{R} is false

Question 3

The major product(P) in the following reaction is

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Alcohols, Phenols and Ethers Question 3 English

Options:

A)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Alcohols, Phenols and Ethers Question 3 English Option 1

B)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Alcohols, Phenols and Ethers Question 3 English Option 2

C)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Alcohols, Phenols and Ethers Question 3 English Option 3

D)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Alcohols, Phenols and Ethers Question 3 English Option 4

Question 4

Identify product A and product B :

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 6 English

Options:

A)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 6 English Option 1

B)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 6 English Option 2

C)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 6 English Option 3

D)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 6 English Option 4

Question 5

the arenium ion which is not involved in the bromination of Aniline is __________.

Options:

A)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Compounds Containing Nitrogen Question 6 English Option 1

B)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Compounds Containing Nitrogen Question 6 English Option 2

C)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Compounds Containing Nitrogen Question 6 English Option 3

D)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Compounds Containing Nitrogen Question 6 English Option 4

Question 6

Given below are two statements :

Statement I : The electronegativity of group 14 elements from Si\mathrm{Si} to Pb\mathrm{Pb}, gradually decreases.

Statement II : Group 14 contains non-metallic, metallic, as well as metalloid elements.

In the light of the above statements, choose the most appropriate from the options given below :

Options:

A)

Statement I is false but Statement II is true

B)

Both Statement I and Statement II are true

C)

Both Statement I and Statement II are false

D)

Statement I is true but Statement II is false

Question 7

Chlorine undergoes disproportionation in alkaline medium as shown below :

aCl2( g)+bOH(aq)cClO(aq)+dCl(aq)+eH2O(l)\mathrm{aCl}_{2(\mathrm{~g})}+\mathrm{b} \mathrm{OH}_{(\mathrm{aq})}^{-} \rightarrow \mathrm{c} \mathrm{ClO}_{(\mathrm{aq)}}^{-}+\mathrm{d} \mathrm{Cl}_{(\mathrm{aq})}^{-}+\mathrm{e} \mathrm{H}_2 \mathrm{O}_{(\mathrm{l})}

The values of a,b,ca, b, c and dd in a balanced redox reaction are respectively :

Options:

A)

3, 4, 4 and 2

B)

1, 2, 1 and 1

C)

2, 4, 1 and 3

D)

2, 2, 1 and 3

Question 8

In chromyl chloride test for confirmation of Cl\mathrm{Cl}^{-} ion, a yellow solution is obtained. Acidification of the solution and addition of amyl alcohol and 10% H2O210 \% \mathrm{~H}_2 \mathrm{O}_2 turns organic layer blue indicating formation of chromium pentoxide. The oxidation state of chromium in that is

Options:

A)

+6

B)

+5

C)

+3

D)

+10

Question 9

Type of amino acids obtained by hydrolysis of proteins is :

Options:

A)

γ\gamma

B)

β\beta

C)

α\alpha

D)

δ\delta

Question 10

The correct set of four quantum numbers for the valence electron of rubidium atom (Z=37)(\mathrm{Z}=37) is :

Options:

A)

5,1,1,+125,1,1,+\frac{1}{2}

B)

5,0,0,+125,0,0,+\frac{1}{2}

C)

5,0,1,+125,0,1,+\frac{1}{2}

D)

5,1,0,+125,1,0,+\frac{1}{2}

Question 11

The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is

Options:

A)

electromeric energy

B)

resonance energy

C)

ionization energy

D)

hyperconjugation energy

Question 12

Match List I with List II

List - I
(Substances)
List - II
(Element Present)
(A) Ziegler catalyst (I) Rhodium
(B) Blood Pigment (II) Cobalt
(C) Wilkinson catalyst (III) Iron
(D) Vitamin B12\mathrm{B_{12}} (IV) Titanium

Choose the correct answer from the options given below:

Options:

A)

A-IV, B-III, C-I, D-II

B)

A-II, B-IV, C-I, D-III

C)

A-III, B-II, C-IV, D-I

D)

A-II, B-III, C-IV, D-I

Question 13

The final product A formed in the following multistep reaction sequence is

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 4 English

Options:

A)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 4 English Option 1

B)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 4 English Option 2

C)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 4 English Option 3

D)

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 4 English Option 4

Question 14

Which of the following is not correct?

Options:

A)

ΔG\Delta \mathrm{G} is positive for a spontaneous reaction

B)

ΔG\Delta \mathrm{G} is positive for a non-spontaneous reaction

C)

ΔG\Delta \mathrm{G} is zero for a reversible reaction

D)

ΔG\Delta \mathrm{G} is negative for a spontaneous reaction

Question 15

The interaction between π\pi bond and lone pair of electrons present on an adjacent atom is responsible for

Options:

A)

Resonance effect

B)

Electromeric effect

C)

Hyperconjugation

D)

Inductive effect

Question 16

In which one of the following metal carbonyls, CO\mathrm{CO} forms a bridge between metal atoms?

Options:

A)

[Os3(CO)12]\left[\mathrm{Os}_3(\mathrm{CO})_{12}\right]

B)

[Ru3(CO)12]\left[\mathrm{Ru}_3(\mathrm{CO})_{12}\right]

C)

[Mn2(CO)10]\left[\mathrm{Mn}_2(\mathrm{CO})_{10}\right]

D)

[Co2(CO)8]\left[\mathrm{Co}_2(\mathrm{CO})_8\right]

Question 17

KMnO4\mathrm{KMnO}_4 decomposes on heating at 513 K513 \mathrm{~K} to form O2\mathrm{O}_2 along with

Options:

A)

K2MnO4 & Mn\mathrm{K}_2 \mathrm{MnO}_4 ~\& \mathrm{~Mn}

B)

MnO2 & K2O2\mathrm{MnO}_2 ~\& \mathrm{~K}_2 \mathrm{O}_2

C)

K2MnO4 & MnO2\mathrm{K}_2 \mathrm{MnO}_4 ~\& \mathrm{~MnO}_2

D)

Mn & KO2\mathrm{Mn} ~\& \mathrm{~KO}_2

Question 18

Appearance of blood red colour, on treatment of the sodium fusion extract of an organic compound with FeSO4\mathrm{FeSO}_4 in presence of concentrated H2SO4\mathrm{H}_2 \mathrm{SO}_4 indicates the presence of element/s

Options:

A)

Br

B)

S

C)

N and S

D)

N

Question 19

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R :

Assertion A : Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom.

Reason R : Phenols react with halogen acids violently.

In the light of the above statements, choose the most appropriate from the options given below :

Options:

A)

A is true but R is false

B)

Both A and R are true but R is NOT the correct explanation of A

C)

A is false but R is true

D)

Both A and R are true and R is the correct explanation of A

Question 20

In alkaline medium, MnO4\mathrm{MnO}_4^{-} oxidises I\mathrm{I}^{-} to

Options:

A)

I2I_2

B)

IO3\mathrm{IO}_3^{-}

C)

IO\mathrm{IO}^{-}

D)

IO4\mathrm{IO}_4^{-}

Numerical TypeQuestion 21

The mass of zinc produced by the electrolysis of zine sulphate solution with a steady current of 0.015 A0.015 \mathrm{~A} for 15 minutes is _________ ×104 g\times 10^{-4} \mathrm{~g}.

(Atomic mass of zinc =65.4 amu=65.4 \mathrm{~amu})

Numerical TypeQuestion 22

A solution of H2SO4\mathrm{H}_2 \mathrm{SO}_4 is 31.4%H2SO431.4 \% \mathrm{H}_2 \mathrm{SO}_4 by mass and has a density of 1.25 g/mL1.25 \mathrm{~g} / \mathrm{mL}. The molarity of the H2SO4\mathrm{H}_2 \mathrm{SO}_4 solution is _________ M\mathrm{M} (nearest integer)

[Given molar mass of H2SO4=98 g mol1\mathrm{H}_2 \mathrm{SO}_4=98 \mathrm{~g} \mathrm{~mol}^{-1}]

Numerical TypeQuestion 23

JEE Main 2024 (Online) 29th January Morning Shift Chemistry - Hydrocarbons Question 5 English

Consider the given reaction. The total number of oxygen atom/s present per molecule of the product (P)(\mathrm{P}) is _________.

Numerical TypeQuestion 24

For the reaction N2O4( g)2NO2( g),Kp=0.492 atm\mathrm{N}_2 \mathrm{O}_{4(\mathrm{~g})} \rightleftarrows 2 \mathrm{NO}_{2(\mathrm{~g})}, \mathrm{K}_{\mathrm{p}}=0.492 \mathrm{~atm} at 300 K.Kc300 \mathrm{~K} . \mathrm{K}_{\mathrm{c}} for the reaction at same temperature is _________ ×102\times 10^{-2}.

(Given : R=0.082 L atm mol1 K1\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1})

Numerical TypeQuestion 25

The number of species from the following which are paramagnetic and with bond order equal to one is _________.

H2,He2+,O2+,N22,O22,F2,Ne2+,B2\mathrm{H}_2, \mathrm{He}_2^{+}, \mathrm{O}_2^{+}, \mathrm{N}_2^{2-}, \mathrm{O}_2^{2-}, \mathrm{F}_2, \mathrm{Ne}_2^{+}, \mathrm{B}_2

Numerical TypeQuestion 26

For a reaction taking place in three steps at same temperature, overall rate constant K=K1 K2 K3\mathrm{K}=\frac{\mathrm{K}_1 \mathrm{~K}_2}{\mathrm{~K}_3}. If Ea1,Ea2\mathrm{Ea}_1, \mathrm{Ea}_2 and Ea3\mathrm{Ea}_3 are 40, 50 and 60 kJ/mol60 \mathrm{~kJ} / \mathrm{mol} respectively, the overall Ea\mathrm{Ea} is ________ kJ/mol\mathrm{kJ} / \mathrm{mol}.

Numerical TypeQuestion 27

From the compounds given below, number of compounds which give positive Fehling's test is _________.

Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4nitrobenzaldehyde, cyclohexane carbaldehyde.

Numerical TypeQuestion 28

Number of compounds with one lone pair of electrons on central atom amongst following is _________.

O3,H2O,SF4,ClF3,NH3,BrF5,XeF4\mathrm{O}_3, \mathrm{H}_2 \mathrm{O}, \mathrm{SF}_4, \mathrm{ClF}_3, \mathrm{NH}_3, \mathrm{BrF}_5, \mathrm{XeF}_4

Numerical TypeQuestion 29

Number of compounds among the following which contain sulphur as heteroatom is ___________.

Furan, Thiophene, Pyridine, Pyrrole, Cysteine, Tyrosine

Numerical TypeQuestion 30

The osmotic pressure of a dilute solution is 7×105 Pa7 \times 10^5 \mathrm{~Pa} at 273 K273 \mathrm{~K}. Osmotic pressure of the same solution at 283 K283 \mathrm{~K} is _________ ×104Nm2\times 10^4 \mathrm{Nm}^{-2}.

Question 31

Suppose f(x)=(2x+2x)tanxtan1(x2x+1)(7x2+3x+1)3f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}. Then the value of f(0)f^{\prime}(0) is equal to

Options:

A)

π\pi

B)

π\sqrt{\pi}

C)

0

D)

π2\frac{\pi}{2}

Question 32

Let a,b\vec{a}, \vec{b} and c\vec{c} be three non-zero vectors such that b\vec{b} and c\vec{c} are non-collinear. If a+5b\vec{a}+5 \vec{b} is collinear with c,b+6c\vec{c}, \vec{b}+6 \vec{c} is collinear with a\vec{a} and a+αb+βc=0\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}, then α+β\alpha+\beta is equal to

Options:

A)

30

B)

-30

C)

-25

D)

35

Question 33

Let (5,a4)\left(5, \frac{a}{4}\right) be the circumcenter of a triangle with vertices A(a,2),B(a,6)\mathrm{A}(a,-2), \mathrm{B}(a, 6) and C(a4,2)C\left(\frac{a}{4},-2\right). Let α\alpha denote the circumradius, β\beta denote the area and γ\gamma denote the perimeter of the triangle. Then α+β+γ\alpha+\beta+\gamma is

Options:

A)

60

B)

62

C)

53

D)

30

Question 34

If α,π2<α<π2\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2} is the solution of 4cosθ+5sinθ=14 \cos \theta+5 \sin \theta=1, then the value of tanα\tan \alpha is

Options:

A)

101012\frac{10-\sqrt{10}}{12}

B)

10106\frac{\sqrt{10}-10}{6}

C)

101012\frac{\sqrt{10}-10}{12}

D)

10106\frac{10-\sqrt{10}}{6}

Question 35

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

Options:

A)

7

B)

6

C)

5

D)

4

Question 36

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

Options:

A)

511\frac{5}{11}

B)

56\frac{5}{6}

C)

16\frac{1}{6}

D)

611\frac{6}{11}

Question 37

In an A.P., the sixth term a6=2a_6=2. If the product a1a4a5a_1 a_4 a_5 is the greatest, then the common difference of the A.P. is equal to

Options:

A)

23\frac{2}{3}

B)

58\frac{5}{8}

C)

32\frac{3}{2}

D)

85\frac{8}{5}

Question 38

 Let A=[1000αβ0βα] and 2 A3=221 where α,βZ, Then a value of α is \text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }

Options:

A)

9

B)

17

C)

3

D)

5

Question 39

Let RR be a relation on Z×ZZ \times Z defined by (a,b)R(c,d)(a, b) R(c, d) if and only if adbca d-b c is divisible by 5. Then RR is

Options:

A)

Reflexive and transitive but not symmetric

B)

Reflexive and symmetric but not transitive

C)

Reflexive but neither symmetric nor transitive

D)

Reflexive, symmetric and transitive

Question 40

If f(x)={2+2x,1x<01x3,0x3;g(x)={x,3x0x,0<x1f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right., then range of (fog)(x)(f o g)(x) is

Options:

A)

[0,1)[0,1)

B)

[0,3)[0,3)

C)

(0,1](0,1]

D)

[0,1][0,1]

Question 41

Let OO be the origin and the position vectors of AA and BB be 2i^+2j^+k^2 \hat{i}+2 \hat{j}+\hat{k} and 2i^+4j^+4k^2 \hat{i}+4 \hat{j}+4 \hat{k} respectively. If the internal bisector of AOB\angle \mathrm{AOB} meets the line AB\mathrm{AB} at C\mathrm{C}, then the length of OCO C is

Options:

A)

3234\frac{3}{2} \sqrt{34}

B)

2331\frac{2}{3} \sqrt{31}

C)

2334\frac{2}{3} \sqrt{34}

D)

3231\frac{3}{2} \sqrt{31}

Question 42

In a ABC\triangle A B C, suppose y=xy=x is the equation of the bisector of the angle BB and the equation of the side ACA C is 2xy=22 x-y=2. If 2AB=BC2 A B=B C and the points AA and BB are respectively (4,6)(4,6) and (α,β)(\alpha, \beta), then α+2β\alpha+2 \beta is equal to

Options:

A)

42

B)

39

C)

48

D)

45

Question 43

For x(π2,π2)x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right), if y(x)=cosecx+sinxcosecxsecx+tanxsin2xdxy(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x, and \lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0 then y(π4)y\left(\frac{\pi}{4}\right) is equal to

Options:

A)

12tan1(12)-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)

B)

tan1(12)\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)

C)

12tan1(12)\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)

D)

12tan1(12)\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)

Question 44

Let A\mathrm{A} be a square matrix such that AAT=I\mathrm{AA}^{\mathrm{T}}=\mathrm{I}. Then 12A[(A+AT)2+(AAT)2]\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right] is equal to

Options:

A)

A2+AT\mathrm{A}^2+\mathrm{A}^{\mathrm{T}}

B)

A3+I\mathrm{A}^3+\mathrm{I}

C)

A3+AT\mathrm{A}^3+\mathrm{A}^{\mathrm{T}}

D)

A2+I\mathrm{A}^2+\mathrm{I}

Question 45

If z=122iz=\frac{1}{2}-2 i is such that z+1=αz+β(1+i),i=1|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1} and α,βR\alpha, \beta \in \mathbb{R}, then α+β\alpha+\beta is equal to

Options:

A)

2

B)

-4

C)

3

D)

-1

Question 46

Consider the function f:[12,1]Rf:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R} defined by f(x)=42x332x1f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1. Consider the statements

(I) The curve y=f(x)y=f(x) intersects the xx-axis exactly at one point.

(II) The curve y=f(x)y=f(x) intersects the xx-axis at x=cosπ12x=\cos \frac{\pi}{12}.

Then

Options:

A)

Both (I) and (II) are correct.

B)

Only (I) is correct.

C)

Both (I) and (II) are incorrect.

D)

Only (II) is correct.

Question 47

Let PQRP Q R be a triangle with R(1,4,2)R(-1,4,2). Suppose M(2,1,2)M(2,1,2) is the mid point of PQ\mathrm{PQ}. The distance of the centroid of PQR\triangle \mathrm{PQR} from the point of intersection of the lines x20=y2=z+31\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1} and x11=y+33=z+11\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1} is

Options:

A)

69

B)

99\sqrt{99}

C)

69\sqrt{69}

D)

9

Question 48

limxπ2(1(xπ2)2x3(π2)3cos(t13)dt)\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right) is equal to

Options:

A)

3π24\frac{3 \pi^2}{4}

B)

3π28\frac{3 \pi^2}{8}

C)

3π4\frac{3 \pi}{4}

D)

3π8\frac{3 \pi}{8}

Question 49

A function y=f(x)y=f(x) satisfies f(x)sin2x+sinx(1+cos2x)f(x)=0f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0 with condition f(0)=0f(0)=0. Then, f(π2)f\left(\frac{\pi}{2}\right) is equal to

Options:

A)

2

B)

1

C)

-1

D)

0

Question 50

If the value of the integral \int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2, then the value of aa is

Options:

A)

32-\frac{3}{2}

B)

3

C)

32\frac{3}{2}

D)

2

Numerical TypeQuestion 51

The area (in sq. units) of the part of the circle x2+y2=169x^2+y^2=169 which is below the line 5xy=135 x-y=13 is πα2β652+αβsin1(1213)\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right), where α,β\alpha, \beta are coprime numbers. Then α+β\alpha+\beta is equal to __________.

Numerical TypeQuestion 52

If the mean and variance of the data 65,68,58,44,48,45,60,α,β,6065,68,58,44,48,45,60, \alpha, \beta, 60 where α>β\alpha> \beta, are 56 and 66.2 respectively, then α2+β2\alpha^2+\beta^2 is equal to _________.

Numerical TypeQuestion 53

Equations of two diameters of a circle are 2x3y=52 x-3 y=5 and 3x4y=73 x-4 y=7. The line joining the points (227,4)\left(-\frac{22}{7},-4\right) and (17,3)\left(-\frac{1}{7}, 3\right) intersects the circle at only one point P(α,β)P(\alpha, \beta). Then, 17βα17 \beta-\alpha is equal to _________.

Numerical TypeQuestion 54

A line with direction ratios 2,1,22,1,2 meets the lines x=y+2=zx=y+2=z and x+2=2y=2zx+2=2 y=2 z respectively at the points P\mathrm{P} and Q\mathrm{Q}. If the length of the perpendicular from the point (1,2,12)(1,2,12) to the line PQ\mathrm{PQ} is ll, then l2l^2 is __________.

Numerical TypeQuestion 55

 If 11C12+11C23++11C910=nm with gcd(n,m)=1, then n+m is equal to \text { If } \frac{{ }^{11} C_1}{2}+\frac{{ }^{11} C_2}{3}+\ldots+\frac{{ }^{11} C_9}{10}=\frac{n}{m} \text { with } \operatorname{gcd}(n, m)=1 \text {, then } n+m \text { is equal to } _______.

Numerical TypeQuestion 56

If the points of intersection of two distinct conics x2+y2=4bx^2+y^2=4 b and x216+y2b2=1\frac{x^2}{16}+\frac{y^2}{b^2}=1 lie on the curve y2=3x2y^2=3 x^2, then 333 \sqrt{3} times the area of the rectangle formed by the intersection points is _________.

Numerical TypeQuestion 57

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is _________.

Numerical TypeQuestion 58

Let α,β\alpha, \beta be the roots of the equation x2x+2=0x^2-x+2=0 with Im(α)>Im(β)\operatorname{Im}(\alpha)>\operatorname{Im}(\beta). Then α6+α4+β45α2\alpha^6+\alpha^4+\beta^4-5 \alpha^2 is equal to ___________.

Numerical TypeQuestion 59

Let f(x)=2xx2,xRf(x)=2^x-x^2, x \in \mathbb{R}. If mm and nn are respectively the number of points at which the curves y=f(x)y=f(x) and y=f(x)y=f^{\prime}(x) intersect the xx-axis, then the value of m+n\mathrm{m}+\mathrm{n} is ___________.

Numerical TypeQuestion 60

If the solution curve y=y(x)y=y(x) of the differential equation (1+y2)(1+logex)dx+xdy=0,x>0\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0 passes through the point (1,1)(1,1) and y(e)=αtan(32)β+tan(32)y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}, then α+2β\alpha+2 \beta is _________.

Question 61

The resistance R=VIR=\frac{V}{I} where V=(200±5)V\mathrm{V}=(200 \pm 5) \mathrm{V} and I=(20±0.2)AI=(20 \pm 0.2) \mathrm{A}, the percentage error in the measurement of R\mathrm{R} is :

Options:

A)

5.5%

B)

3%

C)

7%

D)

3.5%

Question 62

The de-Broglie wavelength of an electron is the same as that of a photon. If velocity of electron is 25%25 \% of the velocity of light, then the ratio of K.E. of electron and K.E. of photon will be:

Options:

A)

14\frac{1}{4}

B)

81\frac{8}{1}

C)

18\frac{1}{8}

D)

11\frac{1}{1}

Question 63

A convex mirror of radius of curvature 30 cm30 \mathrm{~cm} forms an image that is half the size of the object. The object distance is :

Options:

A)

-45 cm

B)

-15 cm

C)

45 cm

D)

15 cm

Question 64

The deflection in moving coil galvanometer falls from 25 divisions to 5 division when a shunt of 24Ω24 \Omega is applied. The resistance of galvanometer coil will be :

Options:

A)

48Ω48 \Omega

B)

100Ω100 \Omega

C)

96Ω96 \Omega

D)

12Ω12 \Omega

Question 65

A capacitor of capacitance 100μF100 \mu \mathrm{F} is charged to a potential of 12 V12 \mathrm{~V} and connected to a 6.4 mH6.4 \mathrm{~mH} inductor to produce oscillations. The maximum current in the circuit would be :

Options:

A)

2.0 A

B)

3.2 A

C)

1.5 A

D)

1.2 A

Question 66

At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as RR.)

Options:

A)

3RR2\frac{\sqrt{3} R-R}{2}

B)

R2\frac{R}{2}

C)

5RR2\frac{\sqrt{5} R-R}{2}

D)

5RR\sqrt{5} R-R

Question 67

A galvanometer having coil resistance 10Ω10 \Omega shows a full scale deflection for a current of 3 mA3 \mathrm{~mA}. For it to measure a current of 8 A8 \mathrm{~A}, the value of the shunt should be:

Options:

A)

3.75×103Ω3.75 \times 10^{-3} \Omega

B)

3×103Ω3 \times 10^{-3} \Omega

C)

4.85×103Ω4.85 \times 10^{-3} \Omega

D)

2.75×103Ω2.75 \times 10^{-3} \Omega

Question 68

A thermodynamic system is taken from an original state A\mathrm{A} to an intermediate state BB by a linear process as shown in the figure. It's volume is then reduced to the original value from B\mathrm{B} to C\mathrm{C} by an isobaric process. The total work done by the gas from AA to BB and BB to CC would be :

JEE Main 2024 (Online) 29th January Morning Shift Physics - Heat and Thermodynamics Question 8 English

Options:

A)

800 J

B)

2200 J

C)

33800 J

D)

1200 J

Question 69

Two charges of 5Q5 Q and 2Q-2 Q are situated at the points (3a,0)(3 a, 0) and (5a,0)(-5 a, 0) respectively. The electric flux through a sphere of radius '4a4 a' having center at origin is :

Options:

A)

2Qε0\frac{2 Q}{\varepsilon_0}

B)

7Qε0\frac{7 Q}{\varepsilon_0}

C)

3Qε0\frac{3 Q}{\varepsilon_0}

D)

5Qε0\frac{5 Q}{\varepsilon_0}

Question 70

A biconvex lens of refractive index 1.5 has a focal length of 20 cm20 \mathrm{~cm} in air. Its focal length when immersed in a liquid of refractive index 1.6 will be:

Options:

A)

-16 cm

B)

+16 cm

C)

+160 cm

D)

-160 cm

Question 71

A block of mass 100 kg100 \mathrm{~kg} slides over a distance of 10 m10 \mathrm{~m} on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction (inJ(\operatorname{in} J) is :

Options:

A)

3900

B)

4500

C)

4200

D)

4000

Question 72

Given below are two statements:

Statement I : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water.

Statement II : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water.

In the light of the above statements, choose the most appropriate from the options given below

Options:

A)

Both Statement I and Statement II are false

B)

Both Statement I and Statement II are true

C)

Statement I is true but Statement II is false

D)

Statement I is false but Statement II is true

Question 73

If the radius of curvature of the path of two particles of same mass are in the ratio 3:43: 4, then in order to have constant centripetal force, their velocities will be in the ratio of :

Options:

A)

1:31: \sqrt{3}

B)

2:32: \sqrt{3}

C)

3:2\sqrt{3}: 2

D)

3:1\sqrt{3}: 1

Question 74

Match List I with List II

List - I List - II
(A) Bdl=μoic+μoεodϕEdt\oint \vec{B} \cdot \overrightarrow{d l}=\mu_o i_c+\mu_o \varepsilon_o \frac{d \phi_E}{d t} (I) Gauss' law for electricity
(B) Edl=dϕBdt\oint \vec{E} \cdot \overrightarrow{d l}=\frac{d \phi_B}{d t} (II) Gauss' law for magnetism
(C) EdA=Qεo\oint \vec{E} \cdot \overrightarrow{d A}=\frac{Q}{\varepsilon_o} (III) Faraday law
(D) BdA=0\oint \vec{B} \cdot \overrightarrow{d A}=0 (IV) Ampere - Maxwell law

Choose the correct answer from the options given below:

Options:

A)

A-IV, B-III, C-I, D-II

B)

A-I, B-II, C-III, D-IV

C)

A-IV, B-I, C-III, D-II

D)

A-II, B-III, C-I, D-IV

Question 75

Two vessels AA and BB are of the same size and are at same temperature. A contains 1 g1 \mathrm{~g} of hydrogen and BB contains 1 g1 \mathrm{~g} of oxygen. PA\mathrm{P}_{\mathrm{A}} and PB\mathrm{P}_{\mathrm{B}} are the pressures of the gases in A\mathrm{A} and B\mathrm{B} respectively, then PAPB\frac{P_A}{P_B} is:

Options:

A)

4

B)

32

C)

8

D)

16

Question 76

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V3.0 \mathrm{~V}. What is the value of Iz\mathrm{I}_{\mathrm{z}} ?

JEE Main 2024 (Online) 29th January Morning Shift Physics - Semiconductor Question 4 English

Options:

A)

3.3 mA

B)

10 mA

C)

5.5 mA

D)

7 mA

Question 77

A body starts moving from rest with constant acceleration covers displacement S1S_1 in first (p1)(p-1) seconds and S2\mathrm{S}_2 in first pp seconds. The displacement S1+S2\mathrm{S}_1+\mathrm{S}_2 will be made in time :

Options:

A)

(2p+1)s(2 p+1) s

B)

(2p1)s(2 p-1) s

C)

(2p22p+1)s\left(2 p^2-2 p+1\right) s

D)

(2p22p+1)s\sqrt{\left(2 p^2-2 p+1\right)} s

Question 78

The electric current through a wire varies with time as I=I0+βtI=I_0+\beta t, where I0=20 AI_0=20 \mathrm{~A} and β=3 A/s\beta=3 \mathrm{~A} / \mathrm{s}. The amount of electric charge crossed through a section of the wire in 20 s20 \mathrm{~s} is :

Options:

A)

80 C

B)

800 C

C)

1000 C

D)

1600 C

Question 79

The potential energy function (in JJ ) of a particle in a region of space is given as U=(2x2+3y3+2z)U=\left(2 x^2+3 y^3+2 z\right). Here x,yx, y and zz are in meter. The magnitude of xx-component of force (in NN ) acting on the particle at point P(1,2,3)mP(1,2,3) \mathrm{m} is :

Options:

A)

4

B)

2

C)

8

D)

6

Question 80

The explosive in a Hydrogen bomb is a mixture of 1H2,1H3{ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3 and 3Li6{ }_3 \mathrm{Li}^6 in some condensed form. The chain reaction is given by

3Li6+0n12He4+1H31H2+1H32He4+0n1\begin{aligned} & { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \\ & { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1 \end{aligned}

During the explosion the energy released is approximately

[Given ; M(Li)=6.01690 amu,M(1H2)=2.01471 amu,M(2He4)=4.00388\mathrm{M}(\mathrm{Li})=6.01690 \mathrm{~amu}, \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \mathrm{~amu}, \mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388 amu\mathrm{amu}, and 1 amu=931.5 MeV]1 \mathrm{~amu}=931.5 \mathrm{~MeV}]

Options:

A)

22.22 MeV

B)

28.12 MeV

C)

16.48 MeV

D)

12.64 MeV

Numerical TypeQuestion 81

In a double slit experiment shown in figure, when light of wavelength 400 nm400 \mathrm{~nm} is used, dark fringe is observed at PP. If D=0.2 m\mathrm{D}=0.2 \mathrm{~m}, the minimum distance between the slits S1S_1 and S2S_2 is _________ mm\mathrm{mm}.

JEE Main 2024 (Online) 29th January Morning Shift Physics - Wave Optics Question 5 English

Numerical TypeQuestion 82

A cylinder is rolling down on an inclined plane of inclination 6060^{\circ}. It's acceleration during rolling down will be x3m/s2\frac{x}{\sqrt{3}} m / s^2, where x=x= ________ (use g=10 m/s2\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2).

Numerical TypeQuestion 83

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S\mathrm{S} having surface charge density +σ+\sigma. The electron at t=0t=0 is at a distance of 1 m1 \mathrm{~m} from SS and has a speed of 1 m/s1 \mathrm{~m} / \mathrm{s}. The maximum value of σ\sigma if the electron strikes SS at t=1 st=1 \mathrm{~s} is α[mϵ0e]Cm2\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{C}{m^2}, the value of α\alpha is ___________.

Numerical TypeQuestion 84

When a hydrogen atom going from n=2n=2 to n=1n=1 emits a photon, its recoil speed is x5 m/s\frac{x}{5} \mathrm{~m} / \mathrm{s}. Where x=x= ________. (Use, mass of hydrogen atom =1.6×1027 kg=1.6 \times 10^{-27} \mathrm{~kg})

Numerical TypeQuestion 85

A square loop of side 10 cm10 \mathrm{~cm} and resistance 0.7Ω0.7 \Omega is placed vertically in east-west plane. A uniform magnetic field of 0.20T0.20 T is set up across the plane in north east direction. The magnetic field is decreased to zero in 1 s1 \mathrm{~s} at a steady rate. Then, magnitude of induced emf is x×103 V\sqrt{x} \times 10^{-3} \mathrm{~V}. The value of xx is __________.

Numerical TypeQuestion 86

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm20 \mathrm{~cm} from its center is 1.5×105 T m1.5 \times 10^{-5} \mathrm{~T} \mathrm{~m}. The magnetic moment of the dipole is _________ A m2A \mathrm{~m}^2. (Given : μo4π=107TmA1\frac{\mu_o}{4 \pi}=10^{-7} \mathrm{Tm} A^{-1} )

Numerical TypeQuestion 87

A 16Ω16 \Omega wire is bend to form a square loop. A 9 V9 \mathrm{~V} battery with internal resistance 1Ω1 \Omega is connected across one of its sides. If a 4μF4 \mu F capacitor is connected across one of its diagonals, the energy stored by the capacitor will be x2μJ\frac{x}{2} \mu J, where x=x= _________

Numerical TypeQuestion 88

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is x8\frac{x}{8}, where x=x= _________.

Numerical TypeQuestion 89

A ball rolls off the top of a stairway with horizontal velocity uu. The steps are 0.1 m0.1 \mathrm{~m} high and 0.1 m0.1 \mathrm{~m} wide. The minimum velocity uu with which that ball just hits the step 5 of the stairway will be x ms1\sqrt{x} \mathrm{~ms}^{-1} where x=x= __________ [use g=10 m/s2\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 ].

Numerical TypeQuestion 90

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are 70 ms170 \mathrm{~ms}^{-1} and 65 ms165 \mathrm{~ms}^{-1} respectively. If the wing area is 2 m22 \mathrm{~m}^2, the lift of the wing is _________ NN.

(Given density of air =1.2 kg m3=1.2 \mathrm{~kg} \mathrm{~m}^{-3})