JEE Main 2024 — 4 Apr Shift 2 (Previous Year Paper)

Applying the principle of homogeneity of dimensions, determine which one is correct, where T is time period, G is gravitational constant, M is mass, r is radius of orbit.

1. A.$T^2=\frac{4\pi^2r^2}{GM}$
2. B.$T^2=\frac{4\pi^2r}{GM^2}$
3. C.$T^2=\frac{4\pi^2r^3}{GM}$✓
4. D.$T^2=4\pi^2r^3$

Correct answer: C

A cyclist starts from the point P of a circular ground of radius 2 km and travels along its circumference to the point S. The displacement of a cyclist is:

1. A.$\sqrt8$ km✓
2. B.8 km
3. C.6 km
4. D.4 km

Correct answer: A

A 2 kg brick begins to slide over a surface which is inclined at an angle of 45° with respect to horizontal axis. The co-efficient of static friction between their surfaces is:

1. A.1.7
2. B.$\frac1{\sqrt3}$
3. C.0.5
4. D.1✓

Correct answer: D

A body of m kg slides from rest along the curve of vertical circle from point A to B in friction less path. The velocity of the body at B is: (given, R = 14 m, g = 10 m/s² and $\sqrt2=1.4$)

1. A.16.7 m/s
2. B.19.8 m/s
3. C.10.6 m/s
4. D.21.9 m/s✓

Correct answer: D

A 90 kg body placed at 2R distance from surface of earth experiences gravitational pull of: (R = Radius of earth, g = 10 m s$^{-2}$)

1. A.100 N✓
2. B.300 N
3. C.225 N
4. D.120 N

Correct answer: A

Correct formula for height of a satellite from earths surface is:

1. A.$\left(\frac{T^2R^2}{4\pi^2g}\right)^{1/3}-R$
2. B.$\left(\frac{T^2R^2g}{4\pi^2}\right)^{1/3}-R$✓
3. C.$\left(\frac{T^2R^2g}{4\pi^2}\right)^{-1/3}+R$
4. D.$\left(\frac{T^2R^2g}{4\pi}\right)^{1/2}-R$

Correct answer: B

Given below are two statements : Statement I : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well. Statement II : The rise of a liquid in a capillary tube does not depend on the inner radius of the tube. In the light of the above statements, choose the correct answer from the options given below:

1. A.Statement I is true but Statement II is false.✓
2. B.Statement I is false but Statement II is true.
3. C.Both Statement I and Statement II are false.
4. D.Both Statement I and Statement II are true.

Correct answer: A

A sample of gas at temperature T is adiabatically expanded to double its volume. Adiabatic constant for the gas is γ = 3/2. The work done by the gas in the process is: (μ = 1 mole)

1. A.$RT[1-2\sqrt2]$
2. B.$RT[\sqrt2-2]$
3. C.$RT[2-\sqrt2]$✓
4. D.$RT[2\sqrt2-1]$

Correct answer: C

The translational degrees of freedom $(f_t)$ and rotational degrees of freedom $(f_r)$ of $\text{CH}_4$ molecule are:

1. A.$f_t=2$ and $f_r=2$
2. B.$f_t=3$ and $f_r=3$✓
3. C.$f_t=3$ and $f_r=2$
4. D.$f_t=2$ and $f_r=3$

Correct answer: B

A charge q is placed at the center of one of the surface of a cube. The flux linked with the cube is:

1. A.$\frac{q}{2\epsilon_0}$✓
2. B.$\frac{q}{8\epsilon_0}$
3. C.Zero
4. D.$\frac{q}{4\epsilon_0}$

Correct answer: A

An electric bulb rated 50 W – 200 V is connected across a 100 V supply. The power dissipation of the bulb is:

1. A.25 W
2. B.12.5 W✓
3. C.50 W
4. D.100 W

Correct answer: B

The magnetic moment of a bar magnet is 0.5 A m². It is suspended in a uniform magnetic field of $8\times10^{-2}$ T. The work done in rotating it from its most stable to most unstable position is:

1. A.$8\times10^{-2}$ J✓
2. B.$4\times10^{-2}$ J
3. C.Zero
4. D.$16\times10^{-2}$ J

Correct answer: A

Match List I with List II List I: A. Purely capacitive circuit; B. Purely inductive circuit; C. LCR series at resonance; D. LCR series circuit List II: I. phasor diagram with I vertical and V horizontal (90° between them, I leading); II. I and V along the same line; III. V making angle θ with I; IV. phasor diagram with V vertical and I horizontal (90°, V leading) Choose the correct answer from the options given below:

1. A.A-I, B-IV, C-II, D-III✓
2. B.A-IV, B-I, C-II, D-III
3. C.A-IV, B-I, C-III, D-II
4. D.A-I, B-IV, C-III, D-II

Correct answer: A

Arrange the following in the ascending order of wavelength: A. Gamma rays $(\lambda_1)$ B. x - rays $(\lambda_2)$ C. Infrared waves $(\lambda_3)$ D. Microwaves $(\lambda_4)$ Choose the most appropriate answer from the options given below

1. A.$\lambda_4<\lambda_3<\lambda_1<\lambda_2$
2. B.$\lambda_2<\lambda_1<\lambda_4<\lambda_3$
3. C.$\lambda_1<\lambda_2<\lambda_3<\lambda_4$✓
4. D.$\lambda_4<\lambda_3<\lambda_2<\lambda_1$

Correct answer: C

The width of one of the two slits in a Young's double slit experiment is 4 times that of the other slit. The ratio of the maximum of the minimum intensity in the interference pattern is:

1. A.1 : 1
2. B.4 : 1
3. C.9 : 1✓
4. D.16 : 1

Correct answer: C

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Number of photons increases with increase in frequency of light. Reason R: Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below:

1. A.Both A and R are correct and R is the correct explanation of A.
2. B.Both A and R are correct and R is NOT the correct explanation of A.
3. C.A is not correct but R is correct.✓
4. D.A is correct but R is not correct.

Correct answer: C

According to Bohr's theory, the moment of momentum of an electron revolving in $4^{th}$ orbit of hydrogen atom is:

1. A.$\frac h\pi$
2. B.$\frac{h}{2\pi}$
3. C.$8\frac h\pi$
4. D.$2\frac h\pi$✓

Correct answer: D

Identify the logic gate given in the circuit:

1. A.NAND-gate
2. B.AND gate
3. C.NOR gate
4. D.OR- gate✓

Correct answer: D

A bus moving along a straight highway with speed of 72 km/h is brought to halt within 4s after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is ______ m.

Correct answer: 40

In a system two particles of masses $m_1=3$ kg and $m_2=2$ kg are placed at certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance 2 cm. In order to keep the center of mass of the system at the original position, the particle of mass $m_2$ should move towards the center of mass by the distance ______ cm.

Correct answer: 3

Mercury is filled in a tube of radius 2 cm up to a height of 30 cm. The force exerted by mercury on the bottom of the tube is ______ N. (Given, atmospheric pressure = $10^5$ N m$^{-2}$, density of mercury = $1.36\times10^4$ kg m$^{-3}$, g = 10 m s$^{-2}$, $\pi=\frac{22}{7}$)

Correct answer: 177

The displacement of a particle executing SHM is given by $x=10\sin\left(\omega t+\frac\pi3\right)$ m. The time period of motion is 3.14 s. The velocity of the particle at t = 0 is ______ m/s.

Correct answer: 10

A parallel plate capacitor of capacitance 12.5 pF is charged by a battery connected between its plates to potential difference of 12.0 V. The battery is now disconnected and a dielectric slab $(\epsilon_r=6)$ is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ______ $\times10^{-12}$ J.

Correct answer: 750

Two wires A and B are made up of the same material and have the same mass. Wire A has radius of 2.0 mm and wire B has radius of 4.0 mm. The resistance of wire B is 2Ω. The resistance of wire A is ______ Ω.

Correct answer: 32

Two parallel long current carrying wire separated by a distance 2r are shown in the figure. The ratio of magnetic field at A to the magnetic field produced at C is $\frac x7$. The value of x is ______.

Correct answer: 5

A rod of length 60 cm rotates with a uniform angular velocity 20 rad s$^{-1}$ about its perpendicular bisector, in a uniform magnetic filed 0.5 T. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is ______ V.

Correct answer: 0

A light ray is incident on a glass slab of thickness $4\sqrt3$ cm and refractive index $\sqrt2$. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is ______ cm. (Given sin 15° = 0.25)

Correct answer: 2

The disintegration energy Q for the nuclear fission of ${}^{235}\text{U}\to{}^{140}\text{Ce}+{}^{94}\text{Zr}+n$ is ______ MeV. Given atomic masses of ${}^{235}$U : 235.0439 u; ${}^{140}$Ce : 139.9054 u, ${}^{94}$Zr : 93.9063 u; n : 1.0086 u, Value of $c^2=931$ MeV/u

Correct answer: 208

The correct order of the first ionization enthalpy is

1. A.Al > Ga > Tl
2. B.Ga > Al > B
3. C.Tl > Ga > Al✓
4. D.B > Al > Ga

Correct answer: C

Given below are two statements : Statement I : The correct order of first ionization enthalpy values of Li, Na, F and Cl is Na < Li < Cl < F. Statement II : The correct order of negative electron gain enthalpy values of Li, Na, F and Cl is Na < Li < F < Cl. In the light of the above statements, choose the correct answer from the options given below :

1. A.Statement I is true but Statement II is false
2. B.Statement I is false but Statement II is true
3. C.Both Statement I and Statement II are false
4. D.Both Statement I and Statement II are true✓

Correct answer: D

The correct statement/s about Hydrogen bonding is/are A. Hydrogen bonding exists when H is covalently bonded to the highly electro negative atom. B. Intermolecular H bonding is present in o-nitro phenol C. Intramolecular H bonding is present in HF. D. The magnitude of H bonding depends on the physical state of the compound. E. H-bonding has powerful effect on the structure and properties of compounds Choose the correct answer from the options given below:

1. A.A, B, D only
2. B.A, D, E only✓
3. C.A only
4. D.A, B, C only

Correct answer: B

The number of species from the following that have pyramidal geometry around the central atom is ______. $\text{S}_2\text{O}_3^{2-}$, $\text{SO}_4^{2-}$, $\text{SO}_3^{2-}$, $\text{S}_2\text{O}_7^{2-}$

1. A.4
2. B.3
3. C.2
4. D.1✓

Correct answer: D

The equilibrium constant for the reaction $\text{SO}_3(g)\rightleftharpoons\text{SO}_2(g)+\frac12\text{O}_2(g)$ is $K_c=4.9\times10^{-2}$. The value of $K_c$ for the reaction given below is $2\text{SO}_2(g)+\text{O}_2(g)\rightleftharpoons2\text{SO}_3(g)$ is :

1. A.4.9
2. B.49
3. C.41.6
4. D.416✓

Correct answer: D

Correct order of stability of carbanion is - (a) cyclopropenyl anion (b) cyclobutyl anion (c) cyclopentenyl (allylic) anion (d) cyclopentadienyl anion

1. A.d > a > c > b
2. B.a > b > c > d
3. C.d > c > b > a✓
4. D.c > b > d > a

Correct answer: C

Common name of Benzene - 1, 2 - diol is -

1. A.catechol✓
2. B.o-cresol
3. C.quinol
4. D.resorcinol

Correct answer: A

The adsorbent used in adsorption chromatography is/are - A. silica gel B. alumina C. quick lime D. magnesia Choose the most appropriate answer from the options given below :

1. A.A only
2. B.B only
3. C.C and D only
4. D.A and B only✓

Correct answer: D

(2-Bromo-3-methylbutyl)benzene, $\text{C}_6\text{H}_5\text{CH}_2\text{-CHBr-CH(CH}_3)_2$ $\xrightarrow{\text{KOH(alc)},\ \Delta}$ major product "P". Product P is

1. A.$\text{C}_6\text{H}_5\text{CH}_2\text{CH}_2\text{C(CH}_3\text{)=CH}_2$
2. B.$\text{C}_6\text{H}_5\text{C(=CH}_2\text{)CH(CH}_3)_2$
3. C.$\text{C}_6\text{H}_5\text{CH=CH-CH(CH}_3)_2$✓
4. D.$\text{C}_6\text{H}_5\text{CH}_2\text{CH=C(CH}_3)_2$

Correct answer: C

1-Methylcyclohexene $\xrightarrow{\text{"A"}}$ 6-oxoheptanal $\xrightarrow{\text{"B"}}$ sodium 5-formylpentanoate. In the above chemical reaction sequence "A" and "B" respectively are

1. A.$\text{H}_2\text{O}$, $\text{H}^+$ and $\text{KMnO}_4$
2. B.$\text{O}_3$, Zn/$\text{H}_2\text{O}$ and $\text{NaOH}_{(alc)}$/$\text{I}_2$✓
3. C.$\text{O}_3$, Zn/$\text{H}_2\text{O}$ and $\text{KMnO}_4$
4. D.$\text{H}_2\text{O}$, $\text{H}^+$ and $\text{NaOH}_{(alc)}$/$\text{I}_2$

Correct answer: B

For a strong electrolyte, a plot of molar conductivity against (concentration)$^{1/2}$ is a straight line, with a negative slope, the correct unit for the slope is

1. A.S cm² mol$^{-3/2}$ L$^{-1/2}$
2. B.S cm² mol$^{-3/2}$ L$^{1/2}$✓
3. C.S cm² mol$^{-3/2}$ L
4. D.S cm² mol$^{-1}$ L$^{1/2}$

Correct answer: B

Fuel cell, using hydrogen and oxygen as fuels, A. has been used in spaceship B. has as efficiency of 40% to produce electricity C. uses aluminum as catalysts D. is eco-friendry E. is actually a type of Galvanic cell only. Choose the correct answer from the options given below:

1. A.A, B, D, E only
2. B.A, D, E only✓
3. C.A, B, D only
4. D.A, B, C only

Correct answer: B

When $\text{MnO}_2$ and $\text{H}_2\text{SO}_4$ is added to a salt (A), the greenish yellow gas liberated as salt (A) is :

1. A.$\text{CaI}_2$
2. B.NaBr
3. C.$\text{KNO}_3$
4. D.$\text{NH}_4\text{Cl}$✓

Correct answer: D

A first row transition metal in its +2 oxidation state has a spin-only magnetic moment value of 3.86 BM. The atomic number of the metal is

1. A.26
2. B.25
3. C.23✓
4. D.22

Correct answer: C

If an iron (III) complex with the formula $[\text{Fe(NH}_3)_x(\text{CN})_y]^-$ has no electron in its $e_g$ orbital, then the value of x + y is

1. A.4
2. B.5
3. C.6✓
4. D.3

Correct answer: C

The number of unpaired d-electrons in $[\text{Co(H}_2\text{O)}_6]^{3+}$ is

1. A.2
2. B.1
3. C.0✓
4. D.4

Correct answer: C

$\text{CH}_3\text{-CH}_2\text{-CH}_2\text{-Br}+\text{NaOH}\xrightarrow{\text{C}_2\text{H}_5\text{OH}}$ Product 'A'; Product A $\xrightarrow{\text{H}_2\text{O}/\text{H}^+}$ Product "B"; Product A $\xrightarrow{\text{Diborane},\ \text{H}_2\text{O}/\text{H}_2\text{O}_2/\overset{-}{\text{O}}\text{H}}$ Product "C". Consider the above reactions, identify product B and product C.

1. A.B = 1-Propanol C = 2-Propanol
2. B.B = C = 2-Propanol
3. C.B = 2-Propanol C = 1-Propanol✓
4. D.B = C = 1-Propanol

Correct answer: C

Match List I with List II List - I: A. α - Glucose and α - Galactose; B. α - Glucose and β - Glucose; C. α - Glucose and α - Fructose; D. α - Glucose and α - Ribose List - II: I. Functional isomers; II. Homologous; III. Anomers; IV. Epimers Choose the correct answer from the options given below:

1. A.A-IV, B-III, C-I, D-II✓
2. B.A-III, B-IV, C-I, D-II
3. C.A-IV, B-III, C-II, D-I
4. D.A-III, B-IV, C-II, D-I

Correct answer: A

The maximum number of orbitals which can be identified with n = 4 and $m_l=0$ is ______

Correct answer: 4

Number of compounds / species from the following with non-zero dipole moment is ______ $\text{BeCl}_2$, $\text{BCl}_3$, $\text{NF}_3$, $\text{XeF}_4$, $\text{CCl}_4$, $\text{H}_2\text{O}$, $\text{H}_2\text{S}$, HBr, $\text{CO}_2$, $\text{H}_2$, HCl

Correct answer: 5

Three moles of an ideal gas are compressed isothermally from 60 L to 20 L using constant pressure of 5 atm. Heat exchange Q for the compression is - ______ Lit. atm.

Correct answer: 200

The total number of 'sigma' and 'Pi' bonds in 2-oxohex-4-ynoic acid is ______

Correct answer: 18

2.7 kg of each of water and acetic acid are mixed. The freezing point of the solution will be $-x$°C. Consider the acetic acid does not dimerise in water, nor dissociates in water. x = ______ (nearest integer) [Given: Molar mass of water = 18 g mol$^{-1}$, acetic acid = 60 g mol$^{-1}$, $K_f$ $\text{H}_2\text{O}$ : 1.86 K kg mol$^{-1}$, $K_f$ acetic acid : 3.90 K kg mol$^{-1}$, freezing point : $\text{H}_2\text{O}$ = 273 K, acetic acid = 290 K]

Correct answer: 31

Consider the following reaction, the rate expression of which is given below $A+B\to C$, rate $=k[A]^{1/2}[B]^{1/2}$. The reaction is initiated by taking 1M concentration of A and B each. If the rate constant (k) is $4.6\times10^{-2}$ s$^{-1}$, then the time taken for A to become 0.1M is ______ sec. (nearest integer)

Correct answer: 50

A first row transition metal with highest enthalpy of atomisation, upon reaction with oxygen at high temperature forms oxides of formula $\text{M}_2\text{O}_n$ (where n = 3, 4, 5). The 'spin-only' magnetic moment value of the amphoteric oxide from the above oxides is ______ BM (near integer) (Given atomic number: Sc : 21, Ti : 22, V : 23, Cr : 24, Mn : 25, Fe : 26, Co : 27, Ni : 28, Cu : 29, Zn : 30)

Correct answer: 0

Phthalimide is made to undergo following sequence of reactions. Phthalimide $\xrightarrow{\text{(i) KOH (ii) Benzylchloride}}$ 'P'. Total number of π bonds present in product 'P' is/are ______

Correct answer: 8

From 6.55 g of aniline, the maximum amount of acetanilide that can be prepared will be ______ $\times10^{-1}$ g.

Correct answer: 95

Vanillin compound obtained from vanilla beans, has total sum of oxygen atoms and π electrons is ______

Correct answer: 11

The area (in sq. units) of the region $S=\{z\in\mathbb{C}:|z-1|\le2;\ (z+\bar z)+i(z-\bar z)\le2,\ \text{Im}(z)\ge0\}$ is

1. A.$\frac{7\pi}{3}$
2. B.$\frac{7\pi}{4}$
3. C.$\frac{17\pi}{8}$
4. D.$\frac{3\pi}{2}$✓

Correct answer: D

The value of $\frac{1\times2^2+2\times3^2+\ldots+100\times(101)^2}{1^2\times2+2^2\times3+\ldots+100^2\times101}$ is

1. A.$\frac{32}{31}$
2. B.$\frac{31}{30}$
3. C.$\frac{306}{305}$
4. D.$\frac{305}{301}$✓

Correct answer: D

Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the arithmetic mean of a, b and c is 8, then the cube of the geometric mean of a, b and c is

1. A.128
2. B.316
3. C.120✓
4. D.312

Correct answer: C

If the coefficients of $x^4$, $x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of n is:

1. A.7
2. B.21
3. C.28
4. D.14✓

Correct answer: D

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope −1. Then, a distance (in units) between the chord PQ and the chord MN is

1. A.$3-\sqrt2$✓
2. B.$\sqrt2+1$
3. C.$\sqrt2-1$
4. D.$2-\sqrt3$

Correct answer: A

Let PQ be a chord of the parabola $y^2=12x$ and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q?

1. A.(3, −3)
2. B.(2, −9)
3. C.$\left(\frac32,-16\right)$
4. D.$\left(\frac12,-20\right)$✓

Correct answer: D

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola H and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are 36π and 4π, respectively, then the length (in units) of latus rectum of H is

1. A.$\frac{14}{3}$
2. B.$\frac{28}{3}$✓
3. C.$\frac{11}{3}$
4. D.$\frac{10}{3}$

Correct answer: B

Let $f(x)=\int_0^x\left(t+\sin(1-e^t)\right)dt$, $x\in\mathbb{R}$. Then, $\lim_{x\to0}\frac{f(x)}{x^3}$ is equal to

1. A.$-\frac16$✓
2. B.$\frac23$
3. C.$-\frac23$
4. D.$\frac16$

Correct answer: A

If the mean of the following probability distribution of a random variable X: | X | 0 | 2 | 4 | 6 | 8 | | P(X) | a | 2a | a + b | 2b | 3b | is $\frac{46}{9}$, then the variance of the distribution is

1. A.$\frac{173}{27}$
2. B.$\frac{566}{81}$✓
3. C.$\frac{151}{27}$
4. D.$\frac{581}{81}$

Correct answer: B

Let a relation R on N × N be defined as: $(x_1,y_1)R(x_2,y_2)$ if and only if $x_1\le x_2$ or $y_1\le y_2$. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?

1. A.Both (I) and (II) are correct.
2. B.Only (II) is correct.
3. C.Neither (I) nor (II) is correct.
4. D.Only (I) is correct.✓

Correct answer: D

Let $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $B=I+\text{adj}(A)+(\text{adj}A)^2+\ldots+(\text{adj}A)^{10}$. Then, the sum of all the elements of the matrix B is:

1. A.−124
2. B.22
3. C.−88✓
4. D.−110

Correct answer: C

Given that the inverse trigonometric function assumes principal values only. Let x, y be any two real numbers in [−1, 1] such that $\cos^{-1}x-\sin^{-1}y=\alpha$, $-\frac\pi2\le\alpha\le\pi$. Then, the minimum value of $x^2+y^2+2xy\sin\alpha$ is

1. A.0✓
2. B.−1
3. C.$\frac12$
4. D.$-\frac12$

Correct answer: A

If the function $f(x)=\begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt2-\sqrt{1+\cos x}}, & x\ne0\\ a\log_e2\log_e3, & x=0\end{cases}$ is continuous at x = 0, then the value of $a^2$ is equal to

1. A.968
2. B.1152✓
3. C.746
4. D.1250

Correct answer: B

Let $f(x)=3\sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If α and β are respectively the minimum and the maximum values of f, then $\alpha^2+2\beta^2$ is equal to

1. A.42✓
2. B.38
3. C.24
4. D.44

Correct answer: A

If the value of the integral $\int_{-1}^{1}\frac{\cos\alpha x}{1+3^x}dx$ is $\frac2\pi$. Then, a value of α is

1. A.$\frac\pi3$
2. B.$\frac\pi6$
3. C.$\frac\pi4$
4. D.$\frac\pi2$✓

Correct answer: D

The area (in sq. units) of the region described by $\{(x,y):y^2\le2x,\text{ and }y\ge4x-1\}$ is

1. A.$\frac{11}{32}$
2. B.$\frac89$
3. C.$\frac{11}{12}$
4. D.$\frac9{32}$✓

Correct answer: D

Let y = y(x) be the solution of the differential equation $(x^2+4)^2dy+(2x^3y+8xy-2)dx=0$. If y(0) = 0, then y(2) is equal to

1. A.$\frac\pi{32}$✓
2. B.$2\pi$
3. C.$\frac\pi8$
4. D.$\frac\pi{16}$

Correct answer: A

Let $\vec a=\hat i+\hat j+\hat k$, $\vec b=2\hat i+4\hat j-5\hat k$ and $\vec c=x\hat i+2\hat j+3\hat k$, $x\in\mathbb{R}$. If $\vec d$ is the unit vector in the direction of $\vec b+\vec c$ such that $\vec a\cdot\vec d=1$, then $(\vec a\times\vec b)\cdot\vec c$ is equal to

1. A.11✓
2. B.3
3. C.9
4. D.6

Correct answer: A

For λ > 0, let θ be the angle between the vectors $\vec a=\hat i+\lambda\hat j-3\hat k$ and $\vec b=3\hat i-\hat j+2\hat k$. If the vectors $\vec a+\vec b$ and $\vec a-\vec b$ are mutually perpendicular, then the value of $(14\cos\theta)^2$ is equal to

1. A.50
2. B.40
3. C.25✓
4. D.20

Correct answer: C

Let P be the point of intersection of the lines $\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}$ and $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}$. Then, the shortest distance of P from the line 4x = 2y = z is

1. A.$\frac{5\sqrt{14}}{7}$
2. B.$\frac{3\sqrt{14}}{7}$✓
3. C.$\frac{\sqrt{14}}{7}$
4. D.$\frac{6\sqrt{14}}{7}$

Correct answer: B

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ______

Correct answer: 5626

Let $S=\{\sin^22\theta:(\sin^4\theta+\cos^4\theta)x^2+(\sin2\theta)x+(\sin^6\theta+\cos^6\theta)=0$ has real roots$\}$. If α and β be the smallest and largest elements of the set S, respectively, then $3((\alpha-2)^2+(\beta-1)^2)$ equals ______

Correct answer: 4

Consider a triangle ABC having the vertices A(1, 2), B(α, β) and C(γ, δ) and angles $\angle ABC=\frac\pi6$ and $\angle BAC=\frac{2\pi}{3}$. If the points B and C lie on the line y = x + 4, then $\alpha^2+\gamma^2$ is equal to ______

Correct answer: 14

Let A be a 2 × 2 symmetric matrix such that $A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}$ and the determinant of A be 1. If $A^{-1}=\alpha A+\beta I$, where I is an identity matrix of order 2 × 2, then α + β equals ______

Correct answer: 5

Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\frac{2x}{\sqrt{1+9x^2}}$. If the composition of f, $\underbrace{(f\circ f\circ\cdots\circ f)}_{10\text{ times}}(x)=\frac{2^{10}x}{\sqrt{1+9\alpha x^2}}$, then the value of $\sqrt{3\alpha+1}$ is equal to ______

Correct answer: 1024

Let $f:\mathbb{R}\to\mathbb{R}$ be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = −1, f(3) = 2 and f(4) = −2. Then, the minimum number of zeros of $(3f'f''+ff''')(x)$ is ______

Correct answer: 5

If $\int\operatorname{cosec}^5x\,dx=\alpha\cot x\operatorname{cosec}x\left(\operatorname{cosec}^2x+\frac32\right)+\beta\log_e\left|\tan\frac x2\right|+C$ where α, β ∈ R and C is the constant of integration, then the value of 8(α + β) equals ______

Correct answer: 1

Let y = y(x) be the solution of the differential equation $(x+y+2)^2dx=dy$, y(0) = −2. Let the maximum and minimum values of the function y = y(x) in $\left[0,\frac\pi3\right]$ be α and β, respectively. If $(3\alpha+\pi)^2+\beta^2=\gamma+\delta\sqrt3$, γ, δ ∈ Z, then γ + δ equals ______

Correct answer: 31

Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, −1). If the mirror image of the point A(2, 2, 2) in the line L is (α, β, γ), then α + β + 6γ is equal to ______

Correct answer: 6

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac13$ and $\frac23$ respectively. Let x be the number of matches that the team wins, and y be the number of matches that team loses. If the probability $P(|x-y|\le2)$ is p, then $3^9p$ equals ______

Correct answer: 8288