JEE Main 2023 29th January Morning Shift Question Paper Question Paper with Solutions | JEE Main Mock Test | JeeMate

Q 1PhysicsSection A • MCQelectrostatics

+4 / -1

In a cuboid of dimension $2 \mathrm{~L} \times 2 \mathrm{~L} \times \mathrm{L}$ , a charge $q$ is placed at the center of the surface ' $\mathrm{S}$ ' having area of $4 \mathrm{~L}^{2}$ . The flux through the opposite surface to ' $\mathrm{S}$ ' is given by

A

\frac{q}{2 \epsilon_{0}}

B

\frac{q}{3 \epsilon_{0}}

C

\frac{q}{12 \epsilon_{0}}

D

\frac{q}{6 \in_{0}}

Q 2PhysicsSection A • MCQproperties of matter

+4 / -1

Surface tension of a soap bubble is $2.0 \times 10^{-2} \mathrm{Nm}^{-1}$ . Work done to increase the radius of soap bubble from $3.5 \mathrm{~cm}$ to $7 \mathrm{~cm}$ will be:

Take $\left[\pi=\frac{22}{7}\right]$

A

18 .48 \times 10^{-4} \mathrm{~J}

B

5.76 \times 10^{-4} \mathrm{~J}

C

0.72 \times 10^{-4} \mathrm{~J}

D

9.24 \times 10^{-4} \mathrm{~J}

Q 3PhysicsSection A • MCQgravitation

+4 / -1

Two particles of equal mass ' $m$ ' move in a circle of radius ' $r$ ' under the action of their mutual gravitational attraction. The speed of each particle will be :

A

\sqrt{\frac{G m}{4 r}}

B

\sqrt{\frac{G m}{2 r}}

C

\sqrt{\frac{G m}{r}}

D

\sqrt{\frac{4 G m}{r}}

Q 4PhysicsSection A • MCQunits and measurements

+4 / -1

Match List I with List II :

	List I (Physical Quantity)		List II (Dimensional Formula)
A.	Pressure gradient	I.	$\left[\mathrm{M}^{\circ} \mathrm{L}^{2} \mathrm{~T}^{-2}\right]$
B.	Energy density	II.	$\left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{~T}^{-2}\right]$
C.	Electric Field	III.	$\left[\mathrm{M}^{1} \mathrm{L}^{-2} \mathrm{~T}^{-2}\right]$
D.	Latent heat	IV.	$\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]$

Choose the correct answer from the options given below:

A

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

B

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

C

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

D

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

Q 5PhysicsSection B • Numericalproperties of matter

+4 / -1

A body cools from 60 $^\circ$ C to 40 $^\circ$ C in 6 minutes. If, temperature of surroundings is 10 $^\circ$ C. Then, after the next 6 minutes, its temperature will be ____________ $^\circ$ C.

Answer Format:Enter numerical value rounded to the nearest integer

Q 6PhysicsSection A • MCQheat and thermodynamics

+4 / -1

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

Assertion A: If $d Q$ and $d W$ represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics $d Q=d U-d W$ .

Reason R: First law of thermodynamics is based on law of conservation of energy.

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

A

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

B

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

C

A is correct but R is not correct

D

A is not correct but R is correct

Q 7PhysicsSection A • MCQelectronic devices

+4 / -1

Which one of the following statement is not correct in the case of light emitting diodes?

A. It is a heavily doped p-n junction.

B. It emits light only when it is forward biased.

C. It emits light only when it is reverse biased.

D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.

Choose the correct answer from the options given below:

A

B

C

C and D

D

C

Q 8PhysicsSection A • MCQmagnetics

+4 / -1

The magnitude of magnetic induction at mid point $\mathrm{O}$ due to current arrangement as shown in Fig will be

qImage-biot-savarts-law-and-magnetic-field-due-to-current-carrying-wire-magnetics-electricity-Jeemate

A

\frac{\mu_{0} I}{\pi a}

B

\frac{\mu_{0} I}{4 \pi a}

C

\frac{\mu_{0} I}{2 \pi a}

D

0

Q 9PhysicsSection B • Numericalrotational motion

+4 / -1

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be _______ ms $^{-1}$ .

Answer Format:Enter numerical value rounded to the nearest integer

Q 10PhysicsSection B • Numericalwaves

+4 / -1

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is _________ degree.

Answer Format:Enter numerical value rounded to the nearest integer

Q 11PhysicsSection A • MCQwork power and energy

+4 / -1

A stone is projected at angle $30^{\circ}$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be -

A

1 : 4

B

1 : 2

C

4 : 3

D

4 : 1

Q 12PhysicsSection A • MCQcommunication systems

+4 / -1

If the height of transmitting and receiving antennas are 80 m each, the maximum line of sight distance will be:

Given : Earth's radius $=6.4\times10^6$ m

A

28 km

B

64 km

C

32 km

D

36 km

Q 13PhysicsSection A • MCQcircular motion

+4 / -1

A car is moving on a horizontal curved road with radius 50 m. The approximate maximum speed of car will be, if friction between tyres and road is 0.34. [take g = 10 ms $^{-2}$ ]

A

3.4 ms

^{-1}

B

13 ms

^{-1}

C

22.4 ms

^{-1}

D

17 ms

^{-1}

Q 14PhysicsSection A • MCQelectromagnetic waves

+4 / -1

Which of the following are true?

A. Speed of light in vacuum is dependent on the direction of propagation.

B. Speed of light in a medium is independent of the wavelength of light.

C. The speed of light is independent of the motion of the source.

D. The speed of light in a medium is independent of intensity.

Choose the correct answer from the options given below:

A

B and D only

B

B and C only

C

A and C only

D

C and D only

Q 15PhysicsSection A • MCQlaws of motion

+4 / -1

A block of mass $m$ slides down the plane inclined at angle $30^{\circ}$ with an acceleration $\frac{g}{4}$ . The value of coefficient of kinetic friction will be:

A

\frac{2 \sqrt{3}-1}{2}

B

\frac{\sqrt{3}}{2}

C

\frac{1}{2 \sqrt{3}}

D

\frac{2 \sqrt{3}+1}{2}

Q 16PhysicsSection A • MCQdual nature of radiation

+4 / -1

The threshold wavelength for photoelectric emission from a material is 5500 $\mathop A\limits^o$ . Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a

A. 75 W infra-red lamp

B. 10 W infra-red lamp

C. 75 W ultra-violet lamp

D. 10 W ultra-violet lamp

Choose the correct answer from the options given below :

A

C only

B

A and D only

C

C and D only

D

B and C only

Q 17PhysicsSection B • Numericalwave optics

+4 / -1

As shown in the figure, three identical polaroids P $_1$ , P $_2$ and P $_3$ are placed one after another. The pass axis of P $_2$ and P $_3$ are inclined at angle of 60 $^\circ$ and 90 $^\circ$ with respect to axis of P $_1$ . The source S has an intensity of 256 $\frac{W}{m^2}$ . The intensity of light at point O is ____________ $\frac{W}{m^2}$ .

qImage-polarisation-wave-optics-optics-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 18PhysicsSection B • Numericalcurrent electricity

+4 / -1

In a metre bridge experiment the balance point is obtained if the gaps are closed by 2 $\Omega$ and 3 $\Omega$ . A shunt of X $\Omega$ is added to 3 $\Omega$ resistor to shift the balancing point by 22.5 cm. The value of X is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 19PhysicsSection A • MCQcurrent electricity

+4 / -1

Ratio of thermal energy released in two resistors R and 3R connected in parallel in an electric circuit is :

A

1 : 1

B

1 : 27

C

1 : 3

D

3 : 1

Q 20PhysicsSection B • Numericalmotion

+4 / -1

A tennis ball is dropped on to the floor from a height of 9.8 m. It rebounds to a height 5.0 m. Ball comes in contact with the floor for 0.2s. The average acceleration during contact is ___________ ms $^{-2}$ .

(Given g = 10 ms $^{-2}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 21PhysicsSection B • Numericalelectrostatics

+4 / -1

A point charge $q_1=4q_0$ is placed at origin. Another point charge $q_2=-q_0$ is placed at $x=12$ cm. Charge of proton is $q_0$ . The proton is placed on $x$ axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is ___________ cm.

Answer Format:Enter numerical value rounded to the nearest integer

Q 22PhysicsSection B • Numericalwork power and energy

+4 / -1

A 0.4 kg mass takes 8s to reach ground when dropped from a certain height 'P' above surface of earth. The loss of potential energy in the last second of fall is __________ J.

(Take g = 10 m/s $^2$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 23PhysicsSection B • Numericalatoms and nuclei

+4 / -1

A radioactive element $_{92}^{242}$ X emits two $\alpha$ -particles, one electron and two positrons. The product nucleus is represented by $_{\mathrm{P}}^{234}$ Y. The value of P is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 24PhysicsSection A • MCQelectromagnetic induction

+4 / -1

Find the mutual inductance in the arrangement, when a small circular loop of wire of radius ' $R$ ' is placed inside a large square loop of wire of side $L$ $(L \gg R)$ . The loops are coplanar and their centres coincide :

qImage-inductance-self-and-mutual-electromagnetic-induction-electricity-Jeemate

A

M=\frac{\sqrt{2} \mu_{0} R}{L^{2}}

B

M=\frac{2 \sqrt{2} \mu_{0} R}{L^{2}}

C

M=\frac{2 \sqrt{2} \mu_{0} R^{2}}{L}

D

M=\frac{\sqrt{2} \mu_{0} R^{2}}{L}

Q 25PhysicsSection A • MCQproperties of matter

+4 / -1

A bicycle tyre is filled with air having pressure of $270 ~\mathrm{kPa}$ at $27^{\circ} \mathrm{C}$ . The approximate pressure of the air in the tyre when the temperature increases to $36^{\circ} \mathrm{C}$ is

A

262 kPa

B

360 kPa

C

270 kPa

D

278 kPa

Q 26PhysicsSection A • MCQatoms and nuclei

+4 / -1

If a radioactive element having half-life of $30 \mathrm{~min}$ is undergoing beta decay, the fraction of radioactive element remains undecayed after $90 \mathrm{~min}$ . will be

A

\frac{1}{16}

B

\frac{1}{4}

C

\frac{1}{8}

D

\frac{1}{2}

Q 27PhysicsSection A • MCQwaves

+4 / -1

A person observes two moving trains, 'A' reaching the station and 'B' leaving the station with equal speed of $30 \mathrm{~m} / \mathrm{s}$ . If both trains emit sounds with frequency $300 \mathrm{~Hz}$ , (Speed of sound: $330 \mathrm{~m} / \mathrm{s}$ ) approximate difference of frequencies heard by the person will be:

A

10 Hz

B

55 Hz

C

80 Hz

D

33 Hz

Q 28PhysicsSection A • MCQmagnetics

+4 / -1

A single current carrying loop of wire carrying current I flowing in anticlockwise direction seen from +ve $\mathrm{z}$ direction and lying in $x y$ plane is shown in figure. The plot of $\hat{j}$ component of magnetic field (By) at a distance ' $a$ ' (less than radius of the coil) and on $y z$ plane vs $z$ coordinate looks like

qImage-biot-savarts-law-and-magnetic-field-due-to-current-carrying-wire-magnetics-electricity-Jeemate

A

B

C

D

Q 29PhysicsSection B • Numericalelectromagnetic induction

+4 / -1

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field B = 0.8 T. When released the radius of the loop starts shrinking at a constant rate of 2 cms $^{-1}$ . The induced emf in the loop at an instant when the radius of the loop is 10 cm will be __________ mV.

Answer Format:Enter numerical value rounded to the nearest integer

Q 30PhysicsSection A • MCQwave optics

+4 / -1

In a Young's double slit experiment, two slits are illuminated with a light of wavelength $800 \mathrm{~nm}$ . The line joining $A_{1} P$ is perpendicular to $A_{1} A_{2}$ as shown in the figure. If the first minimum is detected at $P$ , the value of slits separation 'a' will be:

qImage-huygens-principle-and-interference-of-light-wave-optics-optics-Jeemate

The distance of screen from slits D = 5 cm

A

0.2 mm

B

0.5 mm

C

0.4 mm

D

0.1 mm

Q 31ChemistrySection A • MCQbiomolecules

+4 / -1

Number of cyclic tripeptides formed with 2 amino acids A and B is :

A

2

B

3

C

5

D

4

Q 32ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

Match List I with List II

	List I		List II
	Reaction		Reagents
(A)	Hoffmann Degradation	(I)	$\mathrm{Conc. KOH}$ , $\Delta$
(B)	Clemenson reduction	(II)	$\mathrm{CHCl_3,NaOH/H_3O}$ $^ \oplus$
(C)	Cannizaro reaction	(III)	$\mathrm{Br_2,NaOH}$
(D)	Reimer-Tiemann Reaction	(IV)	$\mathrm{Zn-Hg/HCl}$

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Official Correct Answer:Option (A)

Step-by-Step Solution:

	Reaction		Reagents
(A)	$\begin{aligned}  &  \text { Hoffmann } \\ &  \text { degradation }\end{aligned}$	$\longrightarrow$	$\mathrm{Br}_{2}, \mathrm{NaOH}$
(B)	$\begin{aligned}  &  \text { Clemenson } \\ &  \text { reduction }\end{aligned}$	$\longrightarrow$	$\mathrm{Zn}-\mathrm{Hg} / \mathrm{HCl}$
(C)	$\begin{aligned}  &  \text { Cannizaro- } \\ &  \text { reaction }\end{aligned}$	$\longrightarrow$	Conc. $\mathrm{KOH}, \Delta$
(D)	$\begin{aligned}  &  \text { Reimer-Tiemann } \\ &  \text { reaction }\end{aligned}$	$\longrightarrow$	$\begin{aligned}  &  \mathrm{CHCl}_{3}, \\ &  \mathrm{NaOH} / \mathrm{H}_{3} \mathrm{O}^{+}\end{aligned}$

\therefore

Correct match is : (A)-III, (B) -IV, (C) -I, (D)-II

Q 33ChemistrySection A • MCQchemistry in everyday life

+4 / -1

Match List I with List II.

	List I		List II
	Antimicrobials		Names
(A)	Narrow Spectrum Antibiotic	(I)	Furacin
(B)	Antiseptic	(II)	Sulphur dioxide
(C)	Disinfectants	(III)	Penicillin G
(D)	Broad spectrum antibiotic	(IV)	Chloramphenicol

Choose the correct answer from the options given below :

A

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

B

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

C

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

D

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

Q 34ChemistrySection A • MCQcompounds containing nitrogen

+4 / -1

The major product 'P' for the following sequence of reactions is :

qImage-aliphatic-amines-compounds-containing-nitrogen-organic-chemistry-Jeemate

A

B

C

D

Q 35ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

The bond dissociation energy is highest for

A

Cl

_2

B

I

_2

C

Br

_2

D

F

_2

Q 36ChemistrySection A • MCQenvironmental chemistry

+4 / -1

Correct statement about smog is :

A

NO

_2

is present in classical smog

B

Both NO

_2

and SO

_2

are present in classical smog

C

Photochemical smog has high concentration of oxidizing agents

D

Classical smog also has high concentration of oxidizing agents

Q 37ChemistrySection A • MCQs block elements

+4 / -1

The correct order of hydration enthalpies is

(A) K $^+$

(B) Rb $^+$

(C) Mg $^{2+}$

(D) Cs $^+$

(E) Ca $^{2+}$

Choose the correct answer from the options given below :

A

E > C > A > B > D

B

C > A > E > B > D

C

C > E > A > D > B

D

C > E > A > B > D

Q 38ChemistrySection A • MCQhydrogen

+4 / -1

Which of the given compounds can enhance the efficiency of hydrogen storage tank?

A

SiH

_4

B

Li/P

_4

C

NaNi

_5

D

Di-isobutylaluminium hydride

Q 39ChemistrySection A • MCQatomic structure

+4 / -1

The shortest wavelength of hydrogen atom in Lyman series is $\lambda$ . The longest wavelength is Balmer series of He $^+$ is

A

\frac{36\lambda}{5}

B

\frac{5}{9\lambda}

C

\frac{9\lambda}{5}

D

\frac{5\lambda}{9}

Q 40ChemistrySection A • MCQhaloalkanes and haloarenes

+4 / -1

Identify the correct order for the given property for following compounds.

qImage-haloalkanes-haloalkanes-and-haloarenes-organic-chemistry-Jeemate

Choose the correct answer from the option given below :

A

(A), (B) and (E) only

B

(A), (C) and (D) only

C

(A), (C) and (E) only

D

(B), (C) and (D) only

Q 41ChemistrySection A • MCQpractical organic chemistry

+4 / -1

Compound that will give positive Lassaigne's test for both nitrogen and halogen is :

A

\mathrm{NH_2OH.HCl}

B

\mathrm{N_2H_4.HCl}

C

\mathrm{CH_3NH_2.HCl}

D

\mathrm{NH_4Cl}

Q 42ChemistrySection A • MCQchemical bonding and molecular structure

+4 / -1

The magnetic behaviour of $\mathrm{Li_2O,Na_2O_2}$ and $\mathrm{KO_2}$ , respectively, are :

A

paramagnetic, paramagnetic and diamagnetic

B

paramagnetic, diamagnetic and paramagnetic

C

diamagnetic, paramagnetic and diamagnetic

D

diamagnetic, diamagnetic and paramagnetic

Q 43ChemistrySection B • Numericalchemical equilibrium

+4 / -1

Consider the following reaction approaching equilibrium at 27 $^\circ$ C and 1 atm pressure

$\mathrm{A+B}$ $\mathrel{\mathop{\kern0pt\rightleftharpoons} \limits_{{k_r} = {{10}^2}}^{{k_f} = {{10}^3}}}$ $\mathrm{C+D}$

The standard Gibb's energy change $\mathrm{(\Delta_r G^\theta)}$ at 27 $^\circ$ C is ( $-$ ) ___________ kJ mol $^{-1}$ (Nearest integer).

(Given : $\mathrm{R=8.3~J~K^{-1}~mol^{-1}}$ and $\mathrm{\ln 10=2.3}$ )

Answer Format:Enter numerical value rounded to the nearest integer

Q 44ChemistrySection B • Numericalionic equilibrium

+4 / -1

Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is $x\times10^{-1}$ . The value of $x$ is ___________ (Nearest integer).

Assume complete dissociation.

Answer Format:Enter numerical value rounded to the nearest integer

Q 45ChemistrySection B • Numericalbasics of organic chemistry

+4 / -1

Following chromatogram was developed by adsorption of compound 'A' on a 6 cm TLC glass plate. Retardation factor of the compound 'A' is _________ $\times 10^{-1}$ .

qImage-purification-of-organic-compounds-basics-of-organic-chemistry-organic-chemistry-Jeemate

Answer Format:Enter numerical value rounded to the nearest integer

Q 46ChemistrySection B • Numericalcoordination compounds

+4 / -1

The sum of bridging carbonyls in $\mathrm{W(CO)_6}$ and $\mathrm{Mn_2(CO)_{10}}$ is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 47ChemistrySection B • Numericalhydrocarbons

+4 / -1

17 mg of a hydrocarbon (M.F. $\mathrm{C_{10}H_{16}}$ ) takes up 8.40 mL of the H $_2$ gas measured at 0 $^\circ$ C and 760 mm of Hg. Ozonolysis of the same hydrocarbon yields

qImage-reaction-of-alkenes-hydrocarbons-organic-chemistry-Jeemate

The number of double bond/s present in the hydrocarbon is ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 48ChemistrySection A • MCQsurface chemistry

+4 / -1

Which of the following salt solutions would coagulate the colloid solution formed when $\mathrm{FeCl_3}$ is added to $\mathrm{NaOH}$ solution, at the fastest rate?

A

10 mL of 0.15 mol dm

^{-3}

CaCl

_2

B

10 mL of 0.2 mol dm

^{-3}

AlCl

_3

C

10 mL of 0.1 mol dm

^{-3}

Na

_2

SO

_4

D

10 mL of 0.1 mol dm

^{-3}

Ca

_3

(PO

_4

)

_2

Q 49ChemistrySection A • MCQp block elements

+4 / -1

"A" obtained by Ostwald's method involving air oxidation of NH $_3$ , upon further air oxidation produces "B", "B" on hydration forms an oxoacid of Nitrogen along with evolution of "A". The oxoacid also produces "A" and gives positive brown ring test.

Identify A and B, respectively :

A

\mathrm{N_2O_3,NO_2}

B

\mathrm{NO_2,N_2O_5}

C

\mathrm{NO_2,N_2O_4}

D

\mathrm{NO,NO_2}

Q 50ChemistrySection B • Numericalchemical bonding and molecular structure

+4 / -1

The number of molecules or ions from the following, which do not have odd number of electrons are _________.

(A) NO $_2$

(B) ICl $_4^ -$

(C) BrF $_3$

(D) ClO $_2$

(E) NO $_2^ +$

(F) NO

Answer Format:Enter numerical value rounded to the nearest integer

Q 51ChemistrySection A • MCQelectrochemistry

+4 / -1

The standard electrode potential $\mathrm{(M^{3+}/M^{2+})}$ for V, Cr, Mn & Co are $-$ 0.26 V, $-$ 0.41 V, + 1.57 V and + 1.97 V, respectively. The metal ions which can liberate $\mathrm{H_2}$ from a dilute acid are :

A

\mathrm{Mn^{2+}}

and

\mathrm{Co^{2+}}

B

\mathrm{V^{2+}}

and

\mathrm{Mn^{2+}}

C

\mathrm{V^{2+}}

and

\mathrm{Cr^{2+}}

D

\mathrm{Cr^{2+}}

and

\mathrm{Co^{2+}}

Q 52ChemistrySection B • Numericalchemical kinetics and nuclear chemistry

+4 / -1

For certain chemical reaction $X\to Y$ , the rate of formation of product is plotted against the time as shown in the figure. The number of $\mathrm{\underline {correct} }$ statement/s from the following is ___________.

qImage-different-methods-to-determine-order-of-reaction-chemical-kinetics-and-nuclear-chemistry-physical-chemistry-Jeemate

(A) Over all order of this reaction is one.

(B) Order of this reaction can't be determined.

(C) In region I and III, the reaction is of first and zero order respectively.

(D) In region-II, the reaction is of first order.

(E) In region-II, the order of reaction is in the range of 0.1 to 0.9.

Answer Format:Enter numerical value rounded to the nearest integer

Q 53ChemistrySection A • MCQcoordination compounds

+4 / -1

Chiral complex from the following is :

Here en = ethylene diamine

A

\mathrm{cis-[PtCl_2(en)_2]^{2+}}

B

\mathrm{trans-[Co(NH_3)_4Cl_2]^{+}}

C

\mathrm{trans-[PtCl_2(en)_2]^{2+}}

D

\mathrm{cis-[PtCl_2(NH_3)_2]}

Q 54ChemistrySection A • MCQgaseous state

+4 / -1

For 1 mol of gas, the plot of pV vs. p is shown below. p is the pressure and V is the volume of the gas

qImage-real-gas-and-van-der-walls-equation-gaseous-state-physical-chemistry-Jeemate

What is the value of compressibility factor at point A ?

A

1+\frac{b}{V}

B

1-\frac{a}{RTV}

C

1+\frac{a}{RTV}

D

1-\frac{b}{V}

Q 55ChemistrySection B • Numericalelectrochemistry

+4 / -1

Following figure shows dependence of molar conductance of two electrolytes on concentration. $\Lambda \mathop m\limits^o$ is the limiting molar conductivity.

qImage-conductance-and-electrolysis-electrochemistry-physical-chemistry-Jeemate

The number of $\mathrm{\underline {incorrect} }$ statement(s) from the following is ___________

(A) $\Lambda \mathop m\limits^o$ for electrolyte A is obtained by extrapolation

(B) For electrolyte B, $\Lambda \mathop m\limits$ vs $\sqrt c$ graph is a straight line with intercept equal to $\Lambda \mathop m\limits^o$

(C) At infinite dilution, the value of degree of dissociation approaches zero for electrolyte B.

(D) $\Lambda \mathop m\limits^o$ for any electrolyte A and B can be calculated using $\lambda^\circ$ for individual ions

Answer Format:Enter numerical value rounded to the nearest integer

Q 56ChemistrySection B • Numericalchemical equilibrium

+4 / -1

Water decomposes at 2300 K

$\mathrm{H_2O(g)\to H_2(g)+\frac{1}{2}O_2(g)}$

The percent of water decomposing at 2300 K and 1 bar is ___________ (Nearest integer).

Equilibrium constant for the reaction is $2\times10^{-3}$ at 2300 K.

Answer Format:Enter numerical value rounded to the nearest integer

Q 57ChemistrySection A • MCQp block elements

+4 / -1

During the borax bead test with $\mathrm{CuSO_4}$ , a blue green colour of the bead was observed in oxidising flame due to the formation of :

A

CuO

B

Cu(BO

_2)_2

C

Cu

_3

B

_2

D

Cu

Q 58ChemistrySection A • MCQisolation of elements

+4 / -1

The reaction representing the Mond process for metal refining is _________.

A

\mathrm{ZnO+C\buildrel \Delta \over \longrightarrow Zn+CO}

B

\mathrm{2K[Au(CN)_2]+Zn\buildrel \Delta \over \longrightarrow K_2[Zn(CN)_4]+2Au}

C

\mathrm{Zr+2I_2 \buildrel \Delta \over \longrightarrow ZrI_4}

D

\mathrm{Ni+4CO\buildrel \Delta \over \longrightarrow Ni(CO)_4}

Q 59ChemistrySection B • Numericalsolutions

+4 / -1

Solid Lead nitrate is dissolved in 1 litre of water. The solution was found to boil at 100.15 $^\circ$ C. When 0.2 mol of NaCl is added to the resulting solution, it was observed that the solution froze at $-0.8^\circ$ C. The solubility product of PbCl $_2$ formed is __________ $\times$ 10 $^{-6}$ at 298 K. (Nearest integer)

Given : $\mathrm{K_b=0.5}$ K kg mol $^{-1}$ and $\mathrm{K_f=1.8}$ K kg mol $^{-1}$ . Assume molality to the equal to molarity in all cases.

Answer Format:Enter numerical value rounded to the nearest integer

Q 60ChemistrySection A • MCQalcohols phenols and ethers

+4 / -1

The increasing order of $\mathrm{pK_a}$ for the following phenols is

(A) 2, 4 - Dinitrophenol

(B) 4 - Nitrophenol

(C) 2, 4, 5 - Trimethylphenol

(D) Phenol

(E) 3-Chlorophenol

Choose the correct answer from the option given below :

A

(A), (E), (B), (D), (C)

B

(C), (E), (D), (B), (A)

C

(C), (D), (E), (B), (A)

D

(A), (B), (E), (D), (C)

Q 61MathematicsSection A • MCQquadratic equation and inequalities

+4 / -1

Let $\lambda \ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14{x^2} - 31x + 3\lambda = 0$ and $\alpha,\gamma$ be the roots of the equation $35{x^2} - 53x + 4\lambda = 0$ . Then ${{3\alpha } \over \beta }$ and ${{4\alpha } \over \gamma }$ are the roots of the equation

A

7{x^2} - 245x + 250 = 0

B

49{x^2} - 245x + 250 = 0

C

49{x^2} + 245x + 250 = 0

D

7{x^2} + 245x - 250 = 0

Official Correct Answer:Option (B)

Step-by-Step Solution:

14 x^{2}-31 x+3 \lambda=0

\begin{aligned} & \alpha+\beta=\frac{31}{14} \ldots .(1) \text { and } \alpha \beta=\frac{3 \lambda}{14}\quad...(2) \\ & 35 x^{2}-53 x+4 \lambda=0 \\ & \alpha+\gamma=\frac{53}{35} \ldots(3) \text { and } \alpha \gamma=\frac{4 \lambda}{35} \quad\ldots(4) \\ & \frac{(2)}{(4)} \Rightarrow \frac{\beta}{\gamma}=\frac{3 \times 35}{4 \times 14}=\frac{15}{8} \Rightarrow \beta=\frac{15}{8} \gamma \end{aligned}

(1)

-(3) \Rightarrow \beta-\gamma=\frac{31}{14}-\frac{53}{35}=\frac{155-106}{70}=\frac{7}{10}

\frac{15}{8} \gamma-\gamma=\frac{7}{10} \Rightarrow \gamma=\frac{4}{5}

\Rightarrow \beta=\frac{15}{8} \times \frac{4}{5}=\frac{3}{2}

\Rightarrow \alpha=\frac{31}{14}-\beta=\frac{31}{14}-\frac{3}{2}=\frac{5}{7}

\Rightarrow \lambda=\frac{14}{3} \alpha \beta=\frac{14}{3} \times \frac{5}{7} \times \frac{3}{2}=5

so, sum of roots

\frac{3 \alpha}{\beta}+\frac{4 \alpha}{\gamma}=\left(\frac{3 \alpha \gamma+4 \alpha \beta}{\beta \gamma}\right)

\begin{aligned} & =\frac{\left(3 \times \frac{4 \lambda}{35}+4 \times \frac{3 \lambda}{14}\right)}{\beta \gamma}=\frac{12 \lambda(14+35)}{14 \times 35 \beta \gamma} \\ & =\frac{49 \times 12 \times 5}{490 \times \frac{3}{2} \times \frac{4}{5}}=5 \end{aligned}

Product of roots

=\frac{3 \alpha}{\beta} \times \frac{4 \alpha}{\gamma}=\frac{12 \alpha^{2}}{\beta \gamma}=\frac{12 \times \frac{25}{49}}{\frac{3}{2} \times \frac{4}{5}}=\frac{250}{49}

So, required equation is

x^{2}-5 x+\frac{250}{49}=0

\Rightarrow 49 \mathrm{x}^{2}-245 \mathrm{x}+250=0

Q 62MathematicsSection B • Numericalsequences and series

+4 / -1

Let $a_1,a_2,a_3,...$ be a $GP$ of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1a_9+a_2a_4a_9+a_5+a_7$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 63MathematicsSection A • MCQdifferential equations

+4 / -1

Let $y=f(x)$ be the solution of the differential equation $y(x+1)dx-x^2dy=0,y(1)=e$ . Then $\mathop {\lim }\limits_{x \to {0^ + }} f(x)$ is equal to

A

{e^2}

B

0

C

{1 \over {{e^2}}}

D

{1 \over e}

Q 64MathematicsSection B • Numericalbinomial theorem

+4 / -1

If the co-efficient of $x^9$ in ${\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}$ and the co-efficient of $x^{-9}$ in ${\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}}$ are equal, then $(\alpha\beta)^2$ is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 65MathematicsSection A • MCQmathematical reasoning

+4 / -1

If $p,q$ and $r$ are three propositions, then which of the following combination of truth values of $p,q$ and $r$ makes the logical expression $\left\{ {(p \vee q) \wedge \left( {( \sim p) \vee r} \right)} \right\} \to \left( {( \sim q) \vee r} \right)$ false?

A

p = F,q = T,r = F

B

p = T,q = T,r = F

C

p = T,q = F,r = T

D

p = T,q = F,r = F

Official Correct Answer:Option (A)

Step-by-Step Solution:

	$\mathrm{p}$	$\mathrm{q}$	$\mathrm{r}$	$(p \vee q) \wedge((\sim p) \vee r)$	$\sim \mathrm{q} \vee \mathrm{r}$
$(1)$	$\mathrm{T}$	$\mathrm{F}$	$\mathrm{T}$	$\mathrm{T}$	$\mathrm{T}$
$(2)$	$\mathrm{T}$	$\mathrm{T}$	$\mathrm{F}$	$\mathrm{F}$	$\mathrm{F}$
$(3)$	$\mathrm{F}$	$\mathrm{T}$	$\mathrm{F}$	$\mathrm{T}$	$\mathrm{F}$
$(4)$	$\mathrm{T}$	$\mathrm{F}$	$\mathrm{F}$	$\mathrm{F}$	$\mathrm{T}$

So,

(p \vee q) \wedge(\sim q \vee r) \rightarrow(\sim p \vee r)

will be False.

Q 66MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

A light ray emits from the origin making an angle 30 $^\circ$ with the positive $x$ -axis. After getting reflected by the line $x+y=1$ , if this ray intersects $x$ -axis at Q, then the abscissa of Q is :

A

{2 \over {\left( {\sqrt 3 - 1} \right)}}

B

{2 \over {3 - \sqrt 3 }}

C

{{\sqrt 3 } \over {2\left( {\sqrt 3 + 1} \right)}}

D

{2 \over {3 + \sqrt 3 }}

Q 67MathematicsSection B • Numericalpermutations and combinations

+4 / -1

If all the six digit numbers $x_1\,x_2\,x_3\,x_4\,x_5\,x_6$ with $0< x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the $\mathrm{72^{th}}$ number is _____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 68MathematicsSection A • MCQcircle

+4 / -1

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3 x+10 y-15=0$ , intersect at the point $C$ . Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to :

A

\frac{2\sqrt{13}}{3}

B

\frac{3\sqrt{3}}{4}

C

\sqrt{13}

D

2\sqrt{13}

Q 69MathematicsSection A • MCQfunctions

+4 / -1

The domain of $f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}$ is

A

( - 1,\infty ) - \{ 3\}

B

\mathbb{R} - \{ - 1,3)

C

(2,\infty ) - \{ 3\}

D

\mathbb{R} - \{ 3\}

Q 70MathematicsSection A • MCQprobability

+4 / -1

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

A

\frac{1}{6}

B

\frac{2}{15}

C

\frac{5}{24}

D

0.08

Q 71MathematicsSection A • MCQdefinite integration

+4 / -1

Let $f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$ be a function which

satisfies $f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy}$ . then $(a+b)$ is equal to

A

- 2\pi (\pi + 2)

B

- \pi (\pi - 2)

C

- \pi (\pi + 2)

D

- 2\pi (\pi - 2)

Q 72MathematicsSection B • Numericalpermutations and combinations

+4 / -1

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 73MathematicsSection A • MCQcomplex numbers

+4 / -1

For two non-zero complex numbers $z_{1}$ and $z_{2}$ , if $\operatorname{Re}\left(z_{1} z_{2}\right)=0$ and $\operatorname{Re}\left(z_{1}+z_{2}\right)=0$ , then which of the following are possible?

A. $\operatorname{Im}\left(z_{1}\right)>0$ and $\operatorname{Im}\left(z_{2}\right) > 0$

B. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) > 0$

C. $\operatorname{Im}\left(z_{1}\right) > 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$

D. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$

Choose the correct answer from the options given below :

A

A and C

B

A and B

C

B and D

D

B and C

Q 74MathematicsSection A • MCQstatistics

+4 / -1

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to :

A

20

B

30

C

250

D

25

Official Correct Answer:Option (A)

Step-by-Step Solution:

3 rotten apples are mixed with 7 good apples.

$\therefore$ Total apples = 10

Among those 10 apples 4 are chosen randomly.

${x_i}$	${p_i}$	${p_i}{x_i}$	${p_i}{({x_i})^2}$
0	${{{}^7{C_4}} \over {{}^{10}{C_4}}} = {{35} \over {210}}$	0	0
1	${{{}^3{C_1} \times {}^7{C_3}} \over {{}^{10}{C_4}}} = {{105} \over {210}}$	${{105} \over {210}}$	${{105} \over {210}}$
2	${{{}^3{C_2} \times {}^7{C_2}} \over {{}^{10}{C_4}}} = {{63} \over {210}}$	${{126} \over {210}}$	${{252} \over {210}}$
3	${{{}^3{C_3} \times {}^7{C_1}} \over {{}^{10}{C_4}}} = {7 \over {210}}$	${{21} \over {210}}$	${{63} \over {210}}$

${x_i}$ = Number of rotten apples drawn.

${p_i}$ = Probability of rotten apple.

We know,

Mean $(\mu ) = \sum {{p_i}{x_i}}$

$= 0 + {{105} \over {210}} + {{126} \over {210}} + {{21} \over {210}}$

$= {{252} \over {210}} = {6 \over 5}$

Also,

Variance $({\sigma ^2}) = \left( {\sum {{p_i}{{({x_i})}^2}} } \right) - {\mu ^2}$

$= {{105} \over {210}} + {{252} \over {210}} + {{63} \over {210}} - {{36} \over {25}}$

$= {1 \over 2} + {{12} \over {10}} + {3 \over {10}} - {{36} \over {25}} = {{14} \over {25}}$

$\therefore$ $10(\mu^2 + {\sigma ^2})$

$= 10\left( {({6 \over 5})^2 + {{14} \over {25}}} \right)$

$= 10\left( {{{36 + 14} \over {25}}} \right)$

$= 10 \times {{50} \over {25}} = 20$

Q 75MathematicsSection B • Numericalfunctions

+4 / -1

Suppose $f$ is a function satisfying $f(x + y) = f(x) + f(y)$ for all $x,y \in N$ and $f(1) = {1 \over 5}$ . If $\sum\limits_{n = 1}^m {{{f(n)} \over {n(n + 1)(n + 2)}} = {1 \over {12}}}$ , then $m$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 76MathematicsSection A • MCQmatrices and determinants

+4 / -1

Let $\alpha$ and $\beta$ be real numbers. Consider a 3 $\times$ 3 matrix A such that $A^2=3A+\alpha I$ . If $A^4=21A+\beta I$ , then

A

\alpha=1

B

\alpha=4

C

\beta=8

D

\beta=-8

Q 77MathematicsSection A • MCQarea under the curves

+4 / -1

Let $\Delta$ be the area of the region $\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$ . Then ${1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)$ is equal to

A

2\sqrt 3 - {1 \over 3}

B

2\sqrt 3 - {2 \over 3}

C

\sqrt 3 - {4 \over 3}

D

\sqrt 3 - {2 \over 3}

Q 78MathematicsSection B • Numericalbinomial theorem

+4 / -1

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x)^n$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 79MathematicsSection A • MCQvector algebra

+4 / -1

If the vectors $\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$ , $\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k$ and $\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k$ are coplanar and the projection of $\overrightarrow a$ on the vector $\overrightarrow b$ is $\sqrt {54}$ units, then the sum of all possible values of $\lambda + \mu$ is equal to :

A

24

B

0

C

18

D

6

Q 80MathematicsSection A • MCQarea under the curves

+4 / -1

Let $[x]$ denote the greatest integer $\le x$ . Consider the function $f(x) = \max \left\{ {{x^2},1 + [x]} \right\}$ . Then the value of the integral $\int\limits_0^2 {f(x)dx}$ is

A

{{5 + 4\sqrt 2 } \over 3}

B

{{4 + 5\sqrt 2 } \over 3}

C

{{8 + 4\sqrt 2 } \over 3}

D

{{1 + 5\sqrt 2 } \over 3}

Q 81MathematicsSection A • MCQfunctions

+4 / -1

Let $f:R \to R$ be a function such that $f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$ . Then

A

f(x)

is many-one in

( - \infty , - 1)

B

f(x)

is one-one in

( - \infty ,\infty )

C

f(x)

is one-one in

[1,\infty )

but not in

( - \infty ,\infty )

D

f(x)

is many-one in

(1,\infty )

Q 82MathematicsSection B • Numericalvector algebra

+4 / -1

Let $\overrightarrow a$ , $\overrightarrow b$ and $\overrightarrow c$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be $\overrightarrow a - \overrightarrow b + \overrightarrow c ,\lambda \overrightarrow a - 3\overrightarrow b + 4\overrightarrow c , - \overrightarrow a + 2\overrightarrow b - 3\overrightarrow c$ and $2\overrightarrow a - 4\overrightarrow b + 6\overrightarrow c$ respectively. If $\overrightarrow {AB} ,\overrightarrow {AC}$ and $\overrightarrow {AD}$ are coplanar, then $\lambda$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 83MathematicsSection A • MCQarea under the curves

+4 / -1

Let $A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$ and

$B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^{2}}\right\}\right\} \text {. }$ .

Then the ratio of the area of A to the area of B is

A

\frac{\pi}{\pi+1}

B

\frac{\pi-1}{\pi+1}

C

\frac{\pi}{\pi-1}

D

\frac{\pi+1}{\pi-1}

Q 84MathematicsSection B • Numerical3d geometry

+4 / -1

Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point (1, 27, 7) be $c$ . Then $a^2+b^2+c^2$ is equal to __________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 85MathematicsSection A • MCQmatrices and determinants

+4 / -1

Consider the following system of equations

$\alpha x+2y+z=1$

$2\alpha x+3y+z=1$

$3x+\alpha y+2z=\beta$

for some $\alpha,\beta\in \mathbb{R}$ . Then which of the following is NOT correct.

A

It has a solution for all

\alpha\ne-1

and

\beta=2

B

It has no solution if

\alpha=-1

and

\beta\ne2

C

It has no solution for

\alpha=-1

and for all

\beta \in \mathbb{R}

D

It has no solution for

\alpha=3

and for all

\beta\ne2

Q 86MathematicsSection A • MCQlimits continuity and differentiability

+4 / -1

Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.$

Then $\mathop {\lim }\limits_{x \to 2{p^ + }} [f(x)]$ , where $\left[ . \right]$ denotes greatest integer function, is

A

2

B

1

C

0

D

-1

Q 87MathematicsSection B • Numericaldifferentiation

+4 / -1

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$ . If $f'(0)=2$ , then $|f(-2)|$ is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 88MathematicsSection B • Numerical3d geometry

+4 / -1

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha)$ and the other two vertices lie on the line ${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$ . For $\alpha \in \mathbb{Z}$ , if the area of $\Delta ABC$ is 21 sq. units and the line segment $BC$ has length $2\sqrt{21}$ units, then $\alpha^2$ is equal to ___________.

Answer Format:Enter numerical value rounded to the nearest integer

Q 89MathematicsSection A • MCQstraight lines and pair of straight lines

+4 / -1

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2 x=2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is :

A

\frac{10}{\sqrt{3}}

B

2 \sqrt{3}

C

3 \sqrt{3}

D

\frac{8}{\sqrt{3}}

Q 90MathematicsSection A • MCQtrigonometric ratios and identities

+4 / -1

Let $f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$ and $S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$ . If $4\beta = \sum\limits_{\theta \in S} \theta$ , then $f(\beta )$ is equal to

A

\frac{9}{8}

B

\frac{3}{2}

C

\frac{5}{4}

D

\frac{11}{8}

Official Correct Answer:Option (C)

Step-by-Step Solution:

f(\theta)=3\left(\sin ^{4}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{4}(3 x+\theta)\right)-2\left(1-\sin ^{2} 2 \theta\right)

S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}

\Rightarrow \mathrm{f}(\theta)=3\left(\cos ^{4} \theta+\sin ^{4} \theta\right)-2 \cos ^{2} 2 \theta

\Rightarrow \mathrm{f}(\theta)=3\left(1-\frac{1}{2} \sin ^{2} 2 \theta\right)-2 \cos ^{2} 2 \theta

\Rightarrow \mathrm{f}(\theta)=3-\frac{3}{2} \sin ^{2} 2 \theta-2 \cos ^{2} \theta

=\frac{3}{2}-\frac{1}{2} \cos ^{2} 2 \theta=\frac{3}{2}-\frac{1}{2}\left(\frac{1+\cos 4 \theta}{2}\right)

f(\theta)=\frac{5}{4}-\frac{\cos 4 \theta}{4}

f^{\prime}(\theta)=\sin 4 \theta

\Rightarrow f^{\prime}(\theta)=\sin 4 \theta=-\frac{\sqrt{3}}{2}

\Rightarrow 4 \theta=\mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{3}

\Rightarrow \theta=\frac{\mathrm{n} \pi}{4}+(-1)^{\mathrm{n}} \frac{\pi}{12}

\begin{aligned} & \Rightarrow \theta=\frac{\pi}{12},\left(\frac{\pi}{4}-\frac{\pi}{12}\right),\left(\frac{\pi}{2}+\frac{\pi}{12}\right),\left(\frac{3 \pi}{4}-\frac{\pi}{12}\right) \\ & \Rightarrow 4 \beta=\frac{\pi}{4}+\frac{\pi}{2}+\frac{3 \pi}{4}=\frac{3 \pi}{2} \\ & \Rightarrow \beta=\frac{3 \pi}{8} \Rightarrow f(\beta)=\frac{5}{4}-\frac{\cos \frac{3 \pi}{2}}{4}=\frac{5}{4} \end{aligned}

JEE Main 2023 29th January Morning Shift Question Paper with Solutions