JEE Mains Feb 1, 2024 Shift 1 Questions with solutions

Question 1

Which of the following reactions are disproportionation reactions?

(A)

\mathrm{Cu}^{+} \rightarrow \mathrm{Cu}^{2+}+\mathrm{Cu}

(B)

3 \mathrm{MnO}_4^{2-}+4 \mathrm{H}^{+} \longrightarrow 2 <br/><br/>\mathrm{MnO}_4^{-}+\mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O}

(C)

2 \mathrm{KMnO}_4 \longrightarrow \mathrm{K}_2 \mathrm{MnO}_4+\mathrm{MnO}_2+\mathrm{O}_2

(D)

2 \mathrm{MnO}_4^{-}+3 \mathrm{Mn}^{2+}+2 \mathrm{H}_2 \mathrm{O} \longrightarrow 5 \mathrm{MnO}_2+4 \mathrm{H}^{+}

Choose the correct answer from the options given below :

Options:

A)

(A), (B)

B)

(A), (D)

C)

(B), (C), (D)

D)

(A), (B), (C)

Question 2

Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following :

Options:

A)

\mathrm{q}=0, \Delta \mathrm{T}=0, \mathrm{w}=0

B)

\mathrm{q}=0, \Delta \mathrm{T} \neq 0, \mathrm{w}=0

C)

\mathrm{q} \neq 0, \Delta \mathrm{T}=0, \mathrm{w}=0

D)

\mathrm{q}=0, \Delta \mathrm{T}<0, \mathrm{w} \neq 0

Question 3

Identify

A

and

B

in the following sequence of reaction

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English

Options:

A)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English Option 1

B)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English Option 2

C)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English Option 3

D)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Aldehydes, Ketones and Carboxylic Acids Question 15 English Option 4

Question 4

Which of the following complex is homoleptic?

Options:

A)

\left[\mathrm{Ni}\left(\mathrm{NH}_3\right)_2 \mathrm{Cl}_2\right]

B)

\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right]^{+}

C)

\left[\mathrm{Fe}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right]^{+}

D)

\left[\mathrm{Ni}(\mathrm{CN})_4\right]^{2-}

Question 5

In Kjeldahl's method for estimation of nitrogen,

\mathrm{CuSO}_4

acts as :

Options:

A)

catalytic agent

B)

hydrolysis agent

C)

reducing agent

D)

oxidising agent

Question 6

Given below are two statements :

Statement (I) : Potassium hydrogen phthalate is a primary standard for standardisation of sodium hydroxide solution.

Statement (II) : In this titration phenolphthalein can be used as indicator.

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

Options:

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

Question 7

In case of isoelectronic species the size of

\mathrm{F}^{-}, \mathrm{Ne}

and

\mathrm{Na}^{+}

is affected by :

Options:

A)

Nuclear charge

(\mathrm{z})

B)

None of the factors because their size is the same

C)

Electron-electron interaction in the outer orbitals

D)

Principal quantum number (n)

Question 8

Ionic reactions with organic compounds proceed through :

(A) homolytic bond cleavage

(B) heterolytic bond cleavage

(C) free radical formation

(D) primary free radical

(E) secondary free radical

Choose the correct answer from the options given below :

Options:

A)

(A) only

B)

(B) only

C)

(C) only

D)

(D) and (E) only

Question 9

Which of the following compound will most easily be attacked by an electrophile?

Options:

A)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 29 English Option 1

B)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 29 English Option 2

C)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 29 English Option 3

D)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 29 English Option 4

Question 10

Given below are two statements :

Statement (I) : A solution of

\left[\mathrm{Ni}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{2+}

is green in colour.

Statement (II) : A solution of

\left[\mathrm{Ni}(\mathrm{CN})_4\right]^{2-}

is colourless.

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

Options:

A)

Statement I is incorrect but Statement II is correct

B)

Both Statement I and Statement II are correct

C)

Both Statement I and Statement II are incorrect

D)

Statement I is correct but Statement II is incorrect

Question 11

We have three aqueous solutions of

\mathrm{NaCl}

labelled as '

\mathrm{A}

', '

\mathrm{B}

' and '

\mathrm{C}

' with concentration

0.1 \mathrm{M}

,

0.01 \mathrm{M}

and

0.001 \mathrm{M}

, respectively. The value of van 't Hoff factor(i) for these solutions will be in the order :

Options:

A)

\mathrm{i}_{\mathrm{A}}<\mathrm{i}_{\mathrm{C}}<\mathrm{i}_{\mathrm{B}}

B)

\mathrm{i}_{\mathrm{A}}<\mathrm{i}_{\mathrm{B}}<\mathrm{i}_{\mathrm{C}}

C)

\mathrm{i}_{\mathrm{A}}>\mathrm{i}_{\mathrm{B}}>\mathrm{i}_{\mathrm{C}}

D)

\mathrm{i}_{\mathrm{A}}=\mathrm{i}_{\mathrm{B}}=\mathrm{i}_{\mathrm{C}}

Question 12

Given below are two statements :

Statement (I) : The

\mathrm{NH}_2

group in Aniline is ortho and para directing and a powerful activating group.

Statement (II) : Aniline does not undergo Friedel-Craft's reaction (alkylation and acylation).

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

Options:

A)

Both Statement I and Statement II are correct

B)

Both Statement I and Statement II are incorrect

C)

Statement I is correct but Statement II is incorrect

D)

Statement I is incorrect but Statement II is correct

Question 13

According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron

(\lambda)

and momentum of electron

(p)

?

Options:

A)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Structure of Atom Question 8 English Option 1

B)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Structure of Atom Question 8 English Option 2

C)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Structure of Atom Question 8 English Option 3

D)

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Structure of Atom Question 8 English Option 4

Question 14

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

Assertion (A):

\mathrm{PH}_3

has lower boiling point than

\mathrm{NH}_3

.

Reason (R) : In liquid state

\mathrm{NH}_3

molecules are associated through vander Waal's forces, but

\mathrm{PH}_3

molecules are associated through hydrogen bonding.

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

Options:

A)

Both (A) and (R) are correct and (R) is the correct explanation of (A)

B)

(A) is not correct but (R) is correct

C)

(A) is correct but (R) is not correct

D)

Both

(\mathbf{A})

and

(\mathbf{R})

are correct but

(\mathbf{R})

is not the correct explanation of

(\mathbf{A})

Question 15

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

Assertion (A) : Haloalkanes react with KCN to form alkyl cyanides as a main product while with

\mathrm{AgCN}

form isocyanide as the main product.

Reason (R):

\mathrm{KCN}

and

\mathrm{AgCN}

both are highly ionic compounds.

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

Options:

A)

(A) is correct but

(\mathbf{R})

is not correct

B)

Both (A) and (R) are correct and (R) is the correct explanation of (A)

C)

(A) is not correct but (R) is correct

D)

Both (A) and (R) are correct but (R) is not the correct explanation of (A)

Question 16

If one strand of a DNA has the sequence ATGCTTCA, sequence of the bases in complementary strand is :

Options:

A)

CATTAGCT

B)

TACGAAGT

C)

ATGCGACT

D)

GTACTTAC

Question 17

In acidic medium,

\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7

shows oxidising action as represented in the half reaction:

\mathrm{Cr}_2 \mathrm{O}_7{ }^{2-}+\mathrm{XH}^{+}+\mathrm{Ye}^{\ominus} \rightarrow 2 \mathrm{~A}+\mathrm{ZH}_2 \mathrm{O}

\mathrm{X}, \mathrm{Y}, \mathrm{Z}

and

\mathrm{A}

are respectively are :

Options:

A)

14,7,6

and

\mathrm{Cr}^{3+}

B)

14,6,7

and

\mathrm{Cr}^{3+}

C)

8,4,6

and

\mathrm{Cr}_2 \mathrm{O}_3

D)

8,6,4

and

\mathrm{Cr}_2 \mathrm{O}_3

Question 18

Given below are two statements :

Statement (I) : Aminobenzene and aniline are same organic compounds.

Statement (II) : Aminobenzene and aniline are different organic compounds.

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

Options:

A)

Both Statement I and Statement II are correct

B)

Statement I is incorrect but Statement II is correct

C)

Statement I is correct but Statement II is incorrect

D)

Both Statement I and Statement II are incorrect

Question 19

Match List - I with List - II.

List I (Reactions)	List II (Reagents)
(A)	(I) CH₃MgBr, H₂O
(B) C₆H₅COC₆H₅ ⟶ C₆H₅CH=C₆H₅	(II) Zn(Hg) and conc. HCl
(C) C₆H₅CHO ⟶ C₆H₅CH(OH)CH₃	(III) NaBH₄, H⁺
(D)	(IV) DIBAL-H, H₂O

Choose the correct answer from the options given below :

Options:

A)

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

B)

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

C)

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

D)

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

Question 20

Arrange the bonds in order of increasing ionic character in the molecules.

\mathrm{LiF}

,

\mathrm{K}_2 \mathrm{O}, \mathrm{N}_2, \mathrm{SO}_2

and

\mathrm{ClF}_3

:

Options:

A)

\mathrm{N}_2<\mathrm{SO}_2<\mathrm{ClF}_3<\mathrm{K}_2 \mathrm{O}<\mathrm{LiF}

B)

\mathrm{ClF}_3<\mathrm{N}_2<\mathrm{SO}_2<\mathrm{K}_2 \mathrm{O}<\mathrm{LiF}

C)

\mathrm{LiF}<\mathrm{K}_2 \mathrm{O}<\mathrm{ClF}_3<\mathrm{SO}_2<\mathrm{N}_2

D)

\mathrm{N}_2<\mathrm{ClF}_3<\mathrm{SO}_2<\mathrm{K}_2 \mathrm{O}<\mathrm{LiF}

Numerical TypeQuestion 21

The number of white coloured salts, among the following is

(a)

\mathrm{SrSO}_4

(b)

\mathrm{Mg}\left(\mathrm{NH}_4\right) \mathrm{PO}_4

(c)

\mathrm{BaCrO}_4

(d)

\mathrm{Mn}(\mathrm{OH})_2

(e)

\mathrm{PbSO}_4

(f)

\mathrm{PbCrO}_4

(g)

\mathrm{AgBr}

(h)

\mathrm{PbI}_2

(i)

\mathrm{CaC}_2 \mathrm{O}_4

(j)

\left[\mathrm{Fe}(\mathrm{OH})_2\left(\mathrm{CH}_3 \mathrm{COO}\right)\right]

Numerical TypeQuestion 22

Total number of deactivating groups in aromatic electrophilic substitution reaction among the following is _______ .

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 28 English

JEE Main 2024 (Online) 1st February Morning Shift Chemistry - Basics of Organic Chemistry Question 28 English

Numerical TypeQuestion 23

The potential for the given half cell at

298 \mathrm{~K}

is (-) __________

\times 10^{-2} \mathrm{~V}

\begin{aligned} & 2 \mathrm{H}_{(\mathrm{aq})}^{+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{H}_2(\mathrm{~g}) \\\\ & {\left[\mathrm{H}^{+}\right]=1 \mathrm{M}, \mathrm{P}_{\mathrm{H}_2}=2 \mathrm{~atm}} \end{aligned}

(Given :

2.303 \mathrm{RT} / \mathrm{F}=0.06 \mathrm{~V}, \log 2=0.3

)

Numerical TypeQuestion 24

Among the following oxides of p-block elements, number of oxides having amphoteric nature is ________.

\mathrm{Cl}_2 \mathrm{O}_7, \mathrm{CO}, \mathrm{PbO}_2, \mathrm{~N}_2 \mathrm{O}, \mathrm{NO}, \mathrm{Al}_2 \mathrm{O}_3, \mathrm{SiO}_2, \mathrm{~N}_2 \mathrm{O}_5, \mathrm{SnO}_2

Numerical TypeQuestion 25

Consider the following reaction :

3 \mathrm{PbCl}_2+2\left(\mathrm{NH}_4\right)_3 \mathrm{PO}_4 \rightarrow \mathrm{Pb}_3\left(\mathrm{PO}_4\right)_2+6 \mathrm{NH}_4 \mathrm{Cl}

If

72 ~\mathrm{mmol}

of

\mathrm{PbCl}_2

is mixed with

50 ~\mathrm{mmol}

of

\left(\mathrm{NH}_4\right)_3 \mathrm{PO}_4

, then the amount of

\mathrm{Pb}_3\left(\mathrm{PO}_4\right)_2

formed is ________ mmol (nearest integer).

Numerical TypeQuestion 26

The number of molecules/ion/s having trigonal bipyramidal shape is _______.

\mathrm{PF}_5, \mathrm{BrF}_5, \mathrm{PCl}_5,\left[\mathrm{Pt} \mathrm{Cl}_4\right]^{2-}, \mathrm{BF}_3, \mathrm{Fe}(\mathrm{CO})_5

Numerical TypeQuestion 27

The lowest oxidation number of an atom in a compound

\mathrm{A}_2 \mathrm{B}

is -2 . The number of electrons in its valence shell is _______.

Numerical TypeQuestion 28

\mathrm{K}_{\mathrm{a}}

for

\mathrm{CH}_3 \mathrm{COOH}

is

1.8 \times 10^{-5}

and

\mathrm{K}_{\mathrm{b}}

for

\mathrm{NH}_4 \mathrm{OH}

is

1.8 \times 10^{-5}

. The

\mathrm{pH}

of ammonium acetate solution will be _________.

Numerical TypeQuestion 29

Number of optical isomers possible for 2-chlorobutane ________.

Numerical TypeQuestion 30

The ratio of

\frac{{ }^{14} \mathrm{C}}{{ }^{12} \mathrm{C}}

in a piece of wood is

\frac{1}{8}

part that of atmosphere. If half life of

{ }^{14} \mathrm{C}

is 5730 years, the age of wood sample is ________ years.

Question 31

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :

Options:

A)

\frac{2}{5}

B)

\frac{2}{7}

C)

\frac{1}{7}

D)

\frac{1}{5}

Question 32

The value of the integral

\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}

equals :

Options:

A)

\frac{\sqrt{2} \pi^2}{8}

B)

\frac{\sqrt{2} \pi^2}{16}

C)

\frac{\sqrt{2} \pi^2}{32}

D)

\frac{\sqrt{2} \pi^2}{64}

Question 33

If

\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}

and

\mathrm{X}=\mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{~A}

, then

\operatorname{det} \mathrm{X}

is equal to :

Options:

A)

243

B)

729

C)

27

D)

891

Question 34

If

\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}

and

\tan \mathrm{C}=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<\mathrm{A}, \mathrm{B}, \mathrm{C}<\frac{\pi}{2}

, then

\mathrm{A}+\mathrm{B}

is equal to :

Options:

A)

\mathrm{C}

B)

\pi-C

C)

2 \pi-C

D)

\frac{\pi}{2}-\mathrm{C}

Question 35

If

\mathrm{n}

is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then

\mathrm{n}

is equal to :

Options:

A)

47

B)

53

C)

51

D)

43

Question 36

Let

\mathrm{S}=|\mathrm{z} \in \mathrm{C}:| z-1 \mid=1

and

(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \mid

. Let

z_1, z_2 \in \mathrm{S}

be such that

\left|z_1\right|=\max\limits_{z \in s}|z|

and

\left|z_2\right|=\min\limits _{z \in S}|z|

. Then

\left|\sqrt{2} z_1-z_2\right|^2

equals :

Options:

A)

1

B)

4

C)

3

D)

2

Question 37

Let the median and the mean deviation about the median of 7 observation

170,125,230,190,210

, a, b be 170 and

\frac{205}{7}

respectively. Then the mean deviation about the mean of these 7 observations is :

Options:

A)

31

B)

28

C)

30

D)

32

Question 38

Let

\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}

and

\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \hat{i}) \times \hat{i}) \times \hat{i}

. Then

\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})

is equal to :

Options:

A)

-12

B)

-10

C)

-13

D)

-15

Question 39

Let

\mathbf{S}=\left\{x \in \mathbf{R}:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}

. Then the number of elements in

\mathrm{S}

is :

Options:

A)

4

B)

0

C)

2

D)

1

Question 40

The area enclosed by the curves

x y+4 y=16

and

x+y=6

is equal to :

Options:

A)

28-30 \log _{\mathrm{e}} 2

B)

30-28 \log _{\mathrm{e}} 2

C)

30-32 \log _{\mathrm{e}} 2

D)

32-30 \log _{\mathrm{e}} 2

Question 41

Let

f: \mathbf{R} \rightarrow \mathbf{R}

and

g: \mathbf{R} \rightarrow \mathbf{R}

be defined as

f(x)=\left\{\begin{array}{ll}\log _{\mathrm{e}} x, & x>0 \\ \mathrm{e}^{-x}, & x \leq 0\end{array}\right.

and

g(x)=\left\{\begin{array}{ll}x, & x \geqslant 0 \\ \mathrm{e}^x, & x<0\end{array}\right.

. Then, gof :

\mathbf{R} \rightarrow \mathbf{R}

is :

Options:

A)

one-one but not onto

B)

neither one-one nor onto

C)

onto but not one-one

D)

both one-one and onto

Question 42

If the system of equations

\begin{aligned} & 2 x+3 y-z=5 \\\\ & x+\alpha y+3 z=-4 \\\\ & 3 x-y+\beta z=7 \end{aligned}

has infinitely many solutions, then

13 \alpha \beta

is equal to :

Options:

A)

1110

B)

1120

C)

1210

D)

1220

Question 43

For

0<\theta<\pi / 2

, if the eccentricity of the hyperbola

x^2-y^2 \operatorname{cosec}^2 \theta=5

is

\sqrt{7}

times eccentricity of the

ellipse

x^2 \operatorname{cosec}^2 \theta+y^2=5

, then the value of

\theta

is :

Options:

A)

\frac{\pi}{6}

B)

\frac{5 \pi}{12}

C)

\frac{\pi}{3}

D)

\frac{\pi}{4}

Question 44

Let

y=y(x)

be the solution of the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x(x+y)^3-x(x+y)-1, y(0)=1

.

Then,

\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2

equals :

Options:

A)

\frac{4}{4+\sqrt{\mathrm{e}}}

B)

\frac{3}{3-\sqrt{\mathrm{e}}}

C)

\frac{2}{1+\sqrt{\mathrm{e}}}

D)

\frac{1}{2-\sqrt{\mathrm{e}}}

Question 45

Let

f: \mathbf{R} \rightarrow \mathbf{R}

be defined as :

f(x)= \begin{cases}\frac{a-b \cos 2 x}{x^2} ; & x<0 \\\\ x^2+c x+2 ; & 0 \leq x \leq 1 \\\\ 2 x+1 ; & x>1\end{cases}

If

f

is continuous everywhere in

\mathbf{R}

and

m

is the number of points where

f

is NOT differential then

\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}

equals :

Options:

A)

1

B)

4

C)

3

D)

2

Question 46

Let

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a}>\mathrm{b}

be an ellipse, whose eccentricity is

\frac{1}{\sqrt{2}}

and the length of the latusrectum is

\sqrt{14}

. Then the square of the eccentricity of

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

is :

Options:

A)

3

B)

{7 \over 2}

C)

{3 \over 2}

D)

{5 \over 2}

Question 47

Let

3, a, b, c

be in A.P. and

3, a-1, b+1, c+9

be in G.P. Then, the arithmetic mean of

a, b

and

c

is :

Options:

A)

-4

B)

-1

C)

13

D)

11

Question 48

Let

C: x^2+y^2=4

and

C^{\prime}: x^2+y^2-4 \lambda x+9=0

be two circles. If the set of all values of

\lambda

so that the circles

\mathrm{C}

and

\mathrm{C}

intersect at two distinct points, is

\mathrm{R}-[\mathrm{a}, \mathrm{b}]

, then the point

(8 \mathrm{a}+12,16 \mathrm{~b}-20)

lies on the curve :

Options:

A)

x^2+2 y^2-5 x+6 y=3

B)

5 x^2-y=-11

C)

x^2-4 y^2=7

D)

6 x^2+y^2=42

Question 49

If

5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0

and

y=9 x^2 f(x)

, then

y

is strictly increasing in :

Options:

A)

\left(0, \frac{1}{\sqrt{5}}\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)

B)

\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)

C)

\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)

D)

\left(-\infty, \frac{1}{\sqrt{5}}\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)

Question 50

If the shortest distance between the lines

\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}

and

\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}

is 1 , then the sum of all possible values of

\lambda

is :

Options:

A)

0

B)

2 \sqrt{3}

C)

3 \sqrt{3}

D)

-2 \sqrt{3}

Numerical TypeQuestion 51

If

x=x(t)

is the solution of the differential equation

(t+1) \mathrm{d} x=\left(2 x+(t+1)^4\right) \mathrm{dt}, x(0)=2

, then,

x(1)

equals _________.

Numerical TypeQuestion 52

The number of elements in the set

\mathrm{S}=\{(x, y, z): x, y, z \in \mathbf{Z}, x+2 y+3 z=42, x, y, z \geqslant 0\}

equals __________.

Numerical TypeQuestion 53

If the Coefficient of

x^{30}

in the expansion of

\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0

is

\alpha

, then

|\alpha|

equals ___________.

Numerical TypeQuestion 54

Let

3,7,11,15, \ldots, 403

and

2,5,8,11, \ldots, 404

be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ___________.

Numerical TypeQuestion 55

Let

\{x\}

denote the fractional part of

x

and

f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0

. If

\mathrm{L}

and

\mathrm{R}

respectively denotes the left hand limit and the right hand limit of

f(x)

at

x=0

, then

\frac{32}{\pi^2}\left(\mathrm{~L}^2+\mathrm{R}^2\right)

is equal to ___________.

Numerical TypeQuestion 56

Let the line

\mathrm{L}: \sqrt{2} x+y=\alpha

pass through the point of the intersection

\mathrm{P}

(in the first quadrant) of the circle

x^2+y^2=3

and the parabola

x^2=2 y

. Let the line

\mathrm{L}

touch two circles

\mathrm{C}_1

and

\mathrm{C}_2

of equal radius

2 \sqrt{3}

. If the centres

Q_1

and

Q_2

of the circles

C_1

and

C_2

lie on the

y

-axis, then the square of the area of the triangle

\mathrm{PQ}_1 \mathrm{Q}_2

is equal to ___________.

Numerical TypeQuestion 57

Let

\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|z+2-3 i| \leq 1\}

and

\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}

. Let in

\mathrm{P} \cap \mathrm{Q}

,

|z-3+2 i|

be maximum and minimum at

z_1

and

z_2

respectively. If

\left|z_1\right|^2+2\left|z_2\right|^2=\alpha+\beta \sqrt{2}

, where

\alpha, \beta

are integers, then

\alpha+\beta

equals _____________.

Numerical TypeQuestion 58

If

\int\limits_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \mathrm{~d} x}{\left(1+\mathrm{e}^{\sin x}\right)\left(1+\sin ^4 x\right)}=\alpha \pi+\beta \log _{\mathrm{e}}(3+2 \sqrt{2})

, where

\alpha, \beta

are integers, then

\alpha^2+\beta^2

equals :

Numerical TypeQuestion 59

Let the line of the shortest distance between the lines

\begin{aligned} & \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}) \text { and } \\\\ & \mathrm{L}_2: \overrightarrow{\mathrm{r}}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(\hat{i}+\hat{j}-\hat{k}) \end{aligned}

intersect

\mathrm{L}_1

and

\mathrm{L}_2

at

\mathrm{P}

and

\mathrm{Q}

respectively. If

(\alpha, \beta, \gamma)

is the mid point of the line segment

\mathrm{PQ}

, then

2(\alpha+\beta+\gamma)

is equal to ____________.

Numerical TypeQuestion 60

Let

A=\{1,2,3, \ldots, 20\}

. Let

R_1

and

R_2

two relation on

A

such that

R_1=\{(a, b): b

is divisible by

a\}

R_2=\{(a, b): a

is an integral multiple of

b\}

.

Then, number of elements in

R_1-R_2

is equal to _____________.

Question 61

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is 0.04 , the acceleration of the system in

\mathrm{ms}^{-2}

is :

(Consider that the string is massless and unstretchable and the pulley is also massless and frictionless) :

JEE Main 2024 (Online) 1st February Morning Shift Physics - Laws of Motion Question 8 English

JEE Main 2024 (Online) 1st February Morning Shift Physics - Laws of Motion Question 8 English

Options:

A)

1.2

B)

4

C)

3

D)

2

Question 62

If

\mathrm{R}

is the radius of the earth and the acceleration due to gravity on the surface of earth is

g=\pi^2 \mathrm{~m} / \mathrm{s}^2

, then the length of the second's pendulum at a height

\mathrm{h}=2 R

from the surface of earth will be, :

Options:

A)

\frac{1}{9} \mathrm{~m}

B)

\frac{8}{9} \mathrm{~m}

C)

\frac{2}{9} \mathrm{~m}

D)

\frac{4}{9} \mathrm{~m}

Question 63

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is :

Options:

A)

\frac{5}{11}

B)

\frac{10}{11}

C)

\frac{50}{11}

D)

\frac{1}{2}

Question 64

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :

Options:

A)

\frac{3}{2} \mathrm{R}

B)

\frac{7}{4} \mathrm{R}

C)

\frac{5}{2} \mathrm{R}

D)

\frac{9}{4} \mathrm{R}

Question 65

The dimensional formula of angular impulse is :

Options:

A)

\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-2}\right]

B)

\left[\mathrm{M} \mathrm{L}^{-2} \mathrm{~T}^{-1}\right]

C)

\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1}\right]

D)

\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-1}\right]

Question 66

A monochromatic light of wavelength

6000 ~\mathring{A}

is incident on the single slit of width

0.01 \mathrm{~mm}

. If the diffraction pattern is formed at the focus of the convex lens of focal length

20 \mathrm{~cm}

, the linear width of the central maximum is :

Options:

A)

12 \mathrm{~mm}

B)

24 \mathrm{~mm}

C)

60 \mathrm{~mm}

D)

120 \mathrm{~mm}

Question 67

A ball of mass

0.5 \mathrm{~kg}

is attached to a string of length

50 \mathrm{~cm}

. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is

400 \mathrm{~N}

. The maximum possible value of angular velocity of the ball in

\mathrm{rad} / \mathrm{s}

is, :

Options:

A)

1600

B)

20

C)

40

D)

1000

Question 68

The reading in the ideal voltmeter

(\mathrm{V})

shown in the given circuit diagram is :

JEE Main 2024 (Online) 1st February Morning Shift Physics - Current Electricity Question 23 English

Options:

A)

3 \mathrm{~V}

B)

10 \mathrm{~V}

C)

5 \mathrm{~V}

D)

0 \mathrm{~V}

Question 69

A galvanometer has a resistance of

50 ~\Omega

and it allows maximum current of

5 \mathrm{~mA}

. It can be converted into voltmeter to measure upto

100 \mathrm{~V}

by connecting in series a resistor of resistance :

Options:

A)

19500 \Omega

B)

5975 \Omega

C)

20050 \Omega

D)

19950 \Omega

Question 70

The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly :

Options:

A)

13.6 \mathrm{eV}

B)

1.5 \mathrm{eV}

C)

12.1 \mathrm{eV}

D)

1.9 \mathrm{eV}

Question 71

A parallel plate capacitor has a capacitance

\mathrm{C}=200~ \mathrm{pF}

. It is connected to

230 \mathrm{~V}

ac supply with an angular frequency

300~ \mathrm{rad} / \mathrm{s}

. The rms value of conduction current in the circuit and displacement current in the capacitor respectively are :

Options:

A)

14.3 ~\mu \mathrm{A}

and

143 ~\mu \mathrm{A}

B)

13.8 ~\mu \mathrm{A}

and

13.8 ~\mu \mathrm{A}

C)

13.8 ~\mu \mathrm{A}

and

138 ~\mu \mathrm{A}

D)

1.38 ~\mu \mathrm{A}

and

1.38 ~\mu \mathrm{A}

Question 72

The de Broglie wavelengths of a proton and an

\alpha

particle are

\lambda

and

2 \lambda

respectively. The ratio of the velocities of proton and

\alpha

particle will be :

Options:

A)

8: 1

B)

1: 2

C)

1: 8

D)

4: 1

Question 73

The radius

(\mathrm{r})

, length

(l)

and resistance

(\mathrm{R})

of a metal wire was measured in the laboratory as

\begin{aligned} & \mathrm{r}=(0.35 \pm 0.05) ~\mathrm{cm} \\\\ & \mathrm{R}=(100 \pm 10) ~\mathrm{ohm} \\\\ & l=(15 \pm 0.2)~ \mathrm{cm} \end{aligned}

The percentage error in resistivity of the material of the wire is :

Options:

A)

37.3 \%

B)

25.6 \%

C)

35.6 \%

D)

39.9 \%

Question 74

Two identical capacitors have same capacitance

C

. One of them is charged to the potential

V

and other to the potential

2 \mathrm{~V}

. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is :

Options:

A)

\frac{1}{4} \mathrm{CV}^2

B)

\frac{3}{4} \mathrm{CV}^2

C)

\frac{1}{2} \mathrm{CV}^2

D)

2 \mathrm{CV}^2

Question 75

A simple pendulum of length

1 \mathrm{~m}

has a wooden bob of mass

1 \mathrm{~kg}

. It is struck by a bullet of mass

10^{-2} \mathrm{~kg}

moving with a speed of

2 \times 10^2 \mathrm{~ms}^{-1}

. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use

\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2

)

Options:

A)

0.20 \mathrm{~m}

B)

0.40 \mathrm{~m}

C)

0.30 \mathrm{~m}

D)

0.35 \mathrm{~m}

Question 76

The pressure and volume of an ideal gas are related as

\mathrm{PV}^{\frac{3}{2}}=\mathrm{K}

(Constant). The work done when the gas is taken from state

A\left(P_1, V_1, T_1\right)

to state

B\left(P_2, V_2, T_2\right)

is :

Options:

A)

2\left(\mathrm{P}_2 \sqrt{\mathrm{V}_2}-\mathrm{P}_1 \sqrt{\mathrm{V}_1}\right)

B)

2\left(\sqrt{\mathrm{P}_1} \mathrm{~V}_1-\sqrt{\mathrm{P}_2} \mathrm{~V}_2\right)

C)

2\left(\mathrm{P}_2 \mathrm{~V}_2-\mathrm{P}_1 \mathrm{~V}_1\right)

D)

2\left(\mathrm{P}_1 \mathrm{~V}_1-\mathrm{P}_2 \mathrm{~V}_2\right)

Question 77

In series LCR circuit, the capacitance is changed from

C

to

4 C

. To keep the resonance frequency unchanged, the new inductance should be:

Options:

A)

increased by

2 \mathrm{~L}

B)

reduced by

\frac{1}{4} \mathrm{~L}

C)

reduced by

\frac{3}{4} \mathrm{~L}

D)

increased to

4 \mathrm{~L}

Question 78

A particle moving in a circle of radius

\mathrm{R}

with uniform speed takes time

\mathrm{T}

to complete one revolution.

If this particle is projected with the same speed at an angle

\theta

to the horizontal, the maximum height attained by it is equal to

4 R

. The angle of projection

\theta

is then given by :

Options:

A)

\sin ^{-1}\left[\frac{2 \mathrm{gT}^2}{\pi^2 \mathrm{R}}\right]^{\frac{1}{2}}

B)

\sin ^{-1}\left[\frac{\pi^2 \mathrm{R}}{2 \mathrm{gT}^2}\right]^{\frac{1}{2}}

C)

\cos ^{-1}\left[\frac{\pi \mathrm{R}}{2 \mathrm{gT}^2}\right]^{\frac{1}{2}}

D)

\cos ^{-1}\left[\frac{2 \mathrm{gT}^2}{\pi^2 \mathrm{R}}\right]^{\frac{1}{2}}

Question 79

With rise in temperature, the Young's modulus of elasticity :

Options:

A)

changes erratically

B)

increases

C)

decreases

D)

remains unchanged

Question 80

In the given circuit if the power rating of Zener diode is

10 \mathrm{~mW}

, the value of series resistance

R_s

to regulate the input unregulated supply is :

JEE Main 2024 (Online) 1st February Morning Shift Physics - Semiconductor Question 9 English

Options:

A)

10 \mathrm{k} \Omega

B)

10 \Omega

C)

1 \mathrm{k} \Omega

D)

{3\over7} \mathrm{k} \Omega

Numerical TypeQuestion 81

A rectangular loop of sides

12 \mathrm{~cm}

and

5 \mathrm{~cm}

, with its sides parallel to the

x

-axis and

y

-axis respectively, moves with a velocity of

5 \mathrm{~cm} / \mathrm{s}

in the positive

x

axis direction, in a space containing a variable magnetic field in the positive

z

direction. The field has a gradient of

10^{-3} \mathrm{~T} / \mathrm{cm}

along the negative

x

direction and it is decreasing with time at the rate of

10^{-3} \mathrm{~T} / \mathrm{s}

. If the resistance of the loop is

6 \mathrm{~m} \Omega

, the power dissipated by the loop as heat is __________

\times 10^{-9} \mathrm{~W}

.

Numerical TypeQuestion 82

A plane is in level flight at constant speed and each of its two wings has an area of

40 \mathrm{~m}^2

. If the speed of the air is

180 \mathrm{~km} / \mathrm{h}

over the lower wing surface and

252 \mathrm{~km} / \mathrm{h}

over the upper wing surface, the mass of the plane is ___________ kg.

(Take air density to be

1 \mathrm{~kg} \mathrm{~m}^{-3}

and

\mathrm{g}=10 \mathrm{~ms}^{-2}

)

Numerical TypeQuestion 83

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle

\theta

with each other. When suspended in water the angle remains the same. If density of the material of the sphere is

1.5 \mathrm{~g} / \mathrm{cc}

, the dielectric constant of water will be __________.

(Take density of water

=1 \mathrm{~g} / \mathrm{cc}

)

Numerical TypeQuestion 84

A particle is moving in one dimension (along

x

axis) under the action of a variable force. It's initial position was

16 \mathrm{~m}

right of origin. The variation of its position

(x)

with time

(t)

is given as

x=-3 t^3+18 t^2+16 t

, where

x

is in

\mathrm{m}

and

\mathrm{t}

is in

\mathrm{s}

.

The velocity of the particle when its acceleration becomes zero is _________

\mathrm{m} / \mathrm{s}

.

Numerical TypeQuestion 85

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is

\frac{1000}{x}

, where

x

is _______.

Numerical TypeQuestion 86

A regular polygon of 6 sides is formed by bending a wire of length

4 \pi

meter.

If an electric current of

4 \pi \sqrt{3}

A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be

x \times 10^{-7} \mathrm{~T}

.

The value of

x

is _________.

Numerical TypeQuestion 87

The distance between object and its 3 times magnified virtual image as produced by a convex lens is

20 \mathrm{~cm}

. The focal length of the lens used is __________

\mathrm{cm}

.

Numerical TypeQuestion 88

The identical spheres each of mass

2 \mathrm{M}

are placed at the corners of a right angled triangle with mutually perpendicular sides equal to

4 \mathrm{~m}

each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is

\frac{4 \sqrt{2}}{x}

, where the value of

x

is ___________ .

Numerical TypeQuestion 89

A tuning fork resonates with a sonometer wire of length

1 \mathrm{~m}

stretched with a tension of

6 \mathrm{~N}

. When the tension in the wire is changed to

54 \mathrm{~N}

, the same tuning fork produces 12 beats per second with it. The frequency of the tuning fork is ________________

\mathrm{Hz}

.

Numerical TypeQuestion 90

The current in a conductor is expressed as

I=3 t^2+4 t^3

, where

I

is in Ampere and

t

is in second. The amount of electric charge that flows through a section of the conductor during

t=1 \mathrm{~s}

to

t=2 \mathrm{~s}

is __________ C.