JEE Main 2024 — 5 Apr Shift 1 (Previous Year Paper)

The angle between vector $\vec Q$ and the resultant of $(2\vec Q+2\vec P)$ and $(2\vec Q-2\vec P)$ is :

1. A.$\tan^{-1}\frac{(2\vec Q-2\vec P)}{2\vec Q+2\vec P}$
2. B.$0^\circ$✓
3. C.$\tan^{-1}(P/Q)$
4. D.$\tan^{-1}(2Q/P)$

Correct answer: B

Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as 4.62 s, 4.632 s, 4.6 s and 4.64 s. The arithmetic mean of these readings in correct significant figure is :

1. A.5 s
2. B.4.623 s
3. C.4.6 s✓
4. D.4.62 s

Correct answer: C

If G be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{uG}$ :

1. A.pressure gradient per unit mass
2. B.Gravitational potential
3. C.Energy per unit mass
4. D.Force per unit mass✓

Correct answer: D

A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of 0.1 m s$^{-2}$. The action force of the system on the floor is equal to:

1. A.196 N
2. B.291 N✓
3. C.294 N
4. D.297 N

Correct answer: B

A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :

1. A.1 : 2
2. B.$\sqrt3:2$
3. C.2 : 1
4. D.1 : 1✓

Correct answer: D

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is $\sqrt{8/x}$. The value of x is :

1. A.51
2. B.34
3. C.17
4. D.67✓

Correct answer: D

A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_1=4$ s. $T_2$ would be it's time period if it is brought to a point which is at a height 2R from earth surface. Choose the correct relation [R = radius of earth] :

1. A.$2T_1=T_2$
2. B.$2T_1=3T_2$
3. C.$T_1=T_2$
4. D.$3T_1=2T_2$✓

Correct answer: D

In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of :

1. A.$10^{39}$✓
2. B.$10^{29}$
3. C.$10^{19}$
4. D.$10^{36}$

Correct answer: A

Match List I with List II : List I: (A) Kinetic energy of planet; (B) Gravitation Potential energy of sun-planet system; (C) Total mechanical energy of planet; (D) Escape energy at the surface of planet for unit mass object List II: (I) $-\frac{GMm}{a}$; (II) $\frac{GMm}{2a}$; (III) $\frac{Gm}{r}$; (IV) $-\frac{GMm}{2a}$ (Where a = radius of planet orbit, r = radius of planet, M = mass of Sun, m = mass of planet) Choose the correct answer from the options given below :

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

Correct answer: B

Given below are two statements : Statement I : When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be 0°. Statement II : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well. In the light of the above statement, choose the correct answer from the options given below.

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

Correct answer: C

The heat absorbed by a system in going through the given cyclic process is :

1. A.19.6 J
2. B.61.6 J✓
3. C.616 J
4. D.431.2 J

Correct answer: B

If the collision frequency of hydrogen molecules in a closed chamber at 27°C is Z, then the collision frequency of the same system at 127°C is :

1. A.$\frac{\sqrt3}{2}Z$
2. B.$\frac{2}{\sqrt3}Z$✓
3. C.$\frac34Z$
4. D.$\frac43Z$

Correct answer: B

In the given figure $R_1=10\ \Omega$, $R_2=8\ \Omega$, $R_3=4\ \Omega$ and $R_4=8\ \Omega$. Battery is ideal with emf 12 V. Equivalent resistant of the circuit and current supplied by battery are respectively :

1. A.10.5Ω and 1.14 A
2. B.12Ω and 1 A✓
3. C.10.5Ω and 1 A
4. D.12Ω and 11.4 A

Correct answer: B

In a co-axial straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero :

1. A.outside the cable✓
2. B.inside the outer conductor
3. C.inside the inner conductor
4. D.in between the two conductors

Correct answer: A

Two conducting circular loops A and B are placed in the same plane with their centers coinciding as shown in figure. The mutual inductance between them is :

1. A.$\frac{\mu_0\pi b^2}{2a}$
2. B.$\frac{\mu_0}{2\pi}\cdot\frac{b^2}{a}$
3. C.$\frac{\mu_0}{2\pi}\cdot\frac{a^2}{b}$
4. D.$\frac{\mu_0\pi a^2}{2b}$✓

Correct answer: D

An alternating voltage of amplitude 40 V and frequency 4 kHz is applied directly across the capacitor of 12 μF. The maximum displacement current between the plates of the capacitor is nearly :

1. A.10 A
2. B.12 A✓
3. C.8 A
4. D.13 A

Correct answer: B

Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is :

1. A.both spherical and cylindrical
2. B.plane✓
3. C.spherical
4. D.cylindrical

Correct answer: B

Given below are two statements : Statement I : Figure shows the variation of stopping potential with frequency (ν) for the two photosensitive materials $M_1$ and $M_2$. The slope gives value of $\frac he$, where h is Planck's constant, e is the charge of electron. Statement II : $M_2$ will emit photoelectrons of greater kinetic energy for the incident radiation having same frequency. In the light of the above statements, choose the most appropriate answer from the options given below.

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

Correct answer: D

An electron rotates in a circle around a nucleus having positive charge Ze. Correct relation between total energy (E) of electron to its potential energy (U) is :

1. A.E = U
2. B.2E = U✓
3. C.2E = 3U
4. D.E = 2U

Correct answer: B

Following gates section is connected in a complete suitable circuit. For which of the following combination, bulb will glow (ON) :

1. A.A = 0, B = 0, C = 0, D = 1
2. B.A = 0, B = 1, C = 1, D = 1
3. C.A = 1, B = 0, C = 0, D = 0✓
4. D.A = 1, B = 1, C = 1, D = 0

Correct answer: C

A body moves on a frictionless plane starting from rest. If $S_n$ is distance moved between t = n − 1 and t = n and $S_{n-1}$ is distance moved between t = n − 2 and t = n − 1, then the ratio $\frac{S_{n-1}}{S_n}$ is $\left(1-\frac2x\right)$ for n = 10. The value of x is ______.

Correct answer: 19

Three blocks $M_1$, $M_2$, $M_3$ having masses 4 kg, 6 kg and 10 kg respectively are hanging from a smooth pully using rope 1, 2 and 3 as shown in figure. The tension in the rope 1, $T_1$ when they are moving upward with acceleration of 2 m s$^{-2}$ is ______ N (if g = 10 m/s²).

Correct answer: 240

The density and breaking stress of a wire are $6\times10^4$ kg/m³ and $1.2\times10^8$ N/m² respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $\frac13$rd of the value on the surface of earth. The maximum length of the wire with breaking is ______ m (take, g = 10 m/s²).

Correct answer: 600

Three capacitors of capacitances 25 μF, 30 μF and 45 μF are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is $\frac9xE$. The value of x is ______.

Correct answer: 86

The electric field between the two parallel plates of a capacitor of 1.5 μF capacitance drops to one third of its initial value in 6.6 μs when the plates are connected by a thin wire. The resistance of this wire is ______ Ω. (Given, log 3 = 1.1)

Correct answer: 4

In the experiment to determine the galvanometer resistance by half-deflection method, the plot of $1/\theta$ vs the resistance (R) of the resistance box is shown in the figure. The figure of merit of the galvanometer is ______ $\times10^{-1}$ A/division. [The source has emf 2V]

Correct answer: 5

A 2 A current carrying straight metal wire of resistance 1Ω, resistivity $2\times10^{-6}$ Ωm, area of cross-section 10 mm² and mass 500 g is suspended horizontally in mid air by applying a uniform magnetic field $\vec B$. The magnitude of B is ______ $\times10^{-1}$ T (given, g = 10 m/s²).

Correct answer: 5

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of 20 μF is ______ V.

Correct answer: 50

In Young's double slit experiment, carried out with light of wavelength 5000 Å, the distance between the slits is 0.3 mm and the screen is at 200 cm from the slits. The central maximum is at x = 0 cm. The value of x for third maxima is ______ mm.

Correct answer: 10

If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is ______ $\times10^{-2}$ MeV. (Given 1u = 931 MeV/c², atomic mass of helium = 4.002603 u)

Correct answer: 727

An organic compound has 42.1% carbon, 6.4% hydrogen and remainder is oxygen. If its molecular weight is 342, then its molecular formula is :

1. A.$\text{C}_{11}\text{H}_{18}\text{O}_{12}$
2. B.$\text{C}_{12}\text{H}_{20}\text{O}_{12}$
3. C.$\text{C}_{12}\text{H}_{22}\text{O}_{11}$✓
4. D.$\text{C}_{14}\text{H}_{20}\text{O}_{10}$

Correct answer: C

The incorrect postulates of the Dalton's atomic theory are : (A) Atoms of different elements differ in mass. (B) Matter consists of divisible atoms. (C) Compounds are formed when atoms of different element combine in a fixed ratio. (D) All the atoms of given element have different properties including mass. (E) Chemical reactions involve reorganisation of atoms. Choose the correct answer from the options given below :

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

Correct answer: B

Given below are two statements : Statement I : In group 13, the stability of +1 oxidation state increases down the group. Statement II : The atomic size of gallium is greater than that of aluminium. In the light of the above statements, choose the most appropriate answer from the options given below :

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

Correct answer: D

The statement(s) that are correct about the species $\text{O}^{2-}$, $\text{F}^-$, $\text{Na}^+$ and $\text{Mg}^{2+}$. (A) All are isoelectronic (B) All have the same nuclear charge (C) $\text{O}^{2-}$ has the largest ionic radii (D) $\text{Mg}^{2+}$ has the smallest ionic radii Choose the most appropriate answer from the options given below :

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

Correct answer: C

The following reaction occurs in the Blast furnance where iron ore is reduced to iron metal $\text{Fe}_2\text{O}_{3(s)}+3\text{CO}_{(g)}\rightleftharpoons\text{Fe}_{(l)}+3\text{CO}_{2(g)}$. Using the Le-chatelier's principle, predict which one of the following will not disturb the equilibrium.

1. A.Addition of $\text{CO}_2$
2. B.Removal of $\text{CO}_2$
3. C.Addition of $\text{Fe}_2\text{O}_3$✓
4. D.Removal of CO

Correct answer: C

The number of neutrons present in the more abundant isotope of boron is 'x'. Amorphous boron upon heating with air forms a product, in which the oxidation state of boron is 'y'. The value of x + y is ______

1. A.3
2. B.9✓
3. C.4
4. D.6

Correct answer: B

Number of σ and π bonds present in ethylene molecule is respectively :

1. A.4 and 1
2. B.5 and 2
3. C.3 and 1
4. D.5 and 1✓

Correct answer: D

For the Compounds : (A) $\text{H}_3\text{C-CH}_2\text{-O-CH}_2\text{-CH}_2\text{-CH}_3$ (B) $\text{H}_3\text{C-CH}_2\text{-CH}_2\text{-CH}_2\text{-CH}_3$ (C) $\text{CH}_3\text{-CH}_2\text{-CO-CH}_2\text{-CH}_3$ (D) $\text{H}_3\text{C-CH(OH)-CH}_2\text{-CH}_2\text{-CH}_3$. The increasing order of boiling point is : Choose the correct answer from the options given below :

1. A.(D) < (C) < (A) < (B)
2. B.(B) < (A) < (C) < (D)✓
3. C.(A) < (B) < (C) < (D)
4. D.(B) < (A) < (D) < (C)

Correct answer: B

Given below are two statements: Statement I : Nitration of benzene involves the following step - $\text{H-}\overset{+}{\text{O}}(\text{H})\text{-NO}_2\rightleftharpoons\text{H}_2\text{O}+\overset{+}{\text{N}}\text{O}_2$. Statement II : Use of Lewis base promotes the electrophilic substitution of benzene. In the light of the above statements, choose the most appropriate answer from the options given below :

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

Correct answer: A

The reaction at cathode in the cells commonly used in clocks involves.

1. A.reduction of Mn from +7 to +2
2. B.reduction of Mn from +4 to +3✓
3. C.oxidation of Mn from +3 to +4
4. D.oxidation of Mn from +2 to +7

Correct answer: B

Molar ionic conductivities of divalent cation and anion are 57 S cm² mol$^{-1}$ and 73 S cm² mol$^{-1}$ respectively. The molar conductivity of solution of an electrolyte with the above cation and anion will be :

1. A.187 S cm² mol$^{-1}$
2. B.260 S cm² mol$^{-1}$
3. C.130 S cm² mol$^{-1}$✓
4. D.65 S cm² mol$^{-1}$

Correct answer: C

The metal that shows highest and maximum number of oxidation state is :

1. A.Fe
2. B.Mn✓
3. C.Co
4. D.Ti

Correct answer: B

Which one of the following complexes will exhibit the least paramagnetic behaviour? [Atomic number, Cr = 24, Mn = 25, Fe = 26, Co = 27]

1. A.$[\text{Cr(H}_2\text{O)}_6]^{2+}$
2. B.$[\text{Fe(H}_2\text{O)}_6]^{2+}$
3. C.$[\text{Co(H}_2\text{O)}_6]^{2+}$✓
4. D.$[\text{Mn(H}_2\text{O)}_6]^{2+}$

Correct answer: C

The correct order of ligands arranged in increasing field strength.

1. A.F$^-$ < Br$^-$ < I$^-$ < NH$_3$
2. B.Br$^-$ < F$^-$ < H$_2$O < NH$_3$✓
3. C.H$_2$O < $-$OH < CN$^-$ < NH$_3$
4. D.Cl$^-$ < $-$OH < Br$^-$ < CN$^-$

Correct answer: B

Given below are two statement: Statements I : Bromination of phenol in solvent with low polarity such as $\text{CHCl}_3$ or $\text{CS}_2$ requires Lewis acid catalyst. Statements II : The Lewis acid catalyst polarises the bromine to generate Br$^+$. In the light of the above statements, choose the correct answer from the options given below :

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

Correct answer: C

Identify compound (Z) in the following reaction sequence. Chlorobenzene + NaOH $\xrightarrow{623\text{ K},\ 300\text{ atm}}$ X $\xrightarrow{\text{HCl}}$ Y $\xrightarrow{\text{Conc. HNO}_3}$ Z

1. A.2,4,6-Trinitrophenol (picric acid)✓
2. B.2,4-Dinitrophenol
3. C.2-Nitrophenol
4. D.4-Nitrophenol

Correct answer: A

Identify 'A' in the following reaction: Butan-2-one $\xrightarrow{\text{(i) N}_2\text{H}_4\ \text{(ii) ethylene glycol/KOH}}$ 'A'

1. A.Butan-2-ol
2. B.Acetone hydrazone ($(\text{CH}_3)_2\text{C=N-NH}_2$)
3. C.Butan-2-one hydrazone
4. D.n-Butane✓

Correct answer: D

Which of the following gives a positive test with ninhydrin?

1. A.Starch
2. B.Egg albumin✓
3. C.Polyvinyl chloride
4. D.Cellulose

Correct answer: B

9.3 g of pure aniline is treated with bromine water at room temperature to give a white precipitate of the product 'P'. The mass of product 'P' obtaind is 26.4 g. The percentage yield is ______ %.

Correct answer: 80

The value of Rydberg constant $(R_H)$ is $2.18\times10^{-18}$ J. The velocity of electron having mass $9.1\times10^{-31}$ kg in Bohr's first orbit of hydrogen atom = ______ $\times10^5$ m s$^{-1}$ (nearest integer).

Correct answer: 22

In the lewis dot structure for $\text{NO}_2^-$, total number of valence electrons around nitrogen is ______

Correct answer: 8

The heat of combustion of solid benzoic acid at constant volume is −321.30 kJ at 27°C. The heat of combustion at constant pressure is (−321.30 − x R) kJ, the value of x is ______.

Correct answer: 150

An artificial cell is made by encapsulating 0.2M glucose solution within a semipermeable membrane. The osmotic pressure developed when the artificial cell is placed within a 0.05M solution of NaCl at 300 K is ______ $\times10^{-1}$ bar. (nearest integer). [Given : R = 0.083 L bar mol$^{-1}$ K$^{-1}$] Assume complete dissociation of NaCl

Correct answer: 25

During Kinetic study of reaction 2A + B → C + D, the following results were obtained : | | A [M] | B [M] | initial rate of formation of D | | I | 0.1 | 0.1 | $6.0\times10^{-3}$ | | II | 0.3 | 0.2 | $7.20\times10^{-2}$ | | III | 0.3 | 0.4 | $2.88\times10^{-1}$ | | IV | 0.4 | 0.1 | $2.40\times10^{-2}$ | Based on above data, overall order of the reaction is ______

Correct answer: 3

The spin-only magnetic moment value of the ion among $\text{Ti}^{2+}$, $\text{V}^{2+}$, $\text{Co}^{3+}$ and $\text{Cr}^{2+}$, that acts as strong oxidising agent in aqueous solution is ______ BM (Near integer). (Given atomic numbers : Ti : 22, V : 23, Cr : 24, Co : 27)

Correct answer: 5

The number of halobenzenes from the following that can be prepared by Sandmeyer's reaction is ______ (I) fluorobenzene (II) chlorobenzene (III) bromobenzene (IV) iodobenzene (V) astatobenzene

Correct answer: 2

Consider the given chemical reaction sequence : Phenol $\xrightarrow{\text{Conc. H}_2\text{SO}_4}$ Product A $\xrightarrow{\text{Conc. HNO}_3}$ Product B. Total sum of oxygen atoms in Product A and Product B are ______

Correct answer: 14

In a borax bead test under hot condition, a metal salt (one from the given) is heated at point B of the flame, resulted in green colour salt bead. The spin-only magnetic moment value of the salt is ______ BM (Nearest integer) [Given atomic number of Cu = 29, Ni = 28, Mn = 25, Fe = 26]

Correct answer: 6

Consider the following two statements : Statement I : For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\frac{z_1}{|z_1|}+\frac{z_2}{|z_2|}\right|\le2(|z_1|+|z_2|)$, and Statement II : If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that $\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}$, then $\frac{a^2}{y-z}+\frac{b^2}{z-x}+\frac{c^2}{x-y}=1$. Between the above two statements,

1. A.Statement I is correct but Statement II is incorrect.✓
2. B.both Statement I and Statement II are correct.
3. C.both Statement I and Statement II are incorrect.
4. D.Statement I is incorrect but Statement II is correct.

Correct answer: A

If $\frac1{\sqrt1+\sqrt2}+\frac1{\sqrt2+\sqrt3}+\ldots+\frac1{\sqrt{99}+\sqrt{100}}=m$ and $\frac1{1\cdot2}+\frac1{2\cdot3}+\ldots+\frac1{99\cdot100}=n$, then the point (m, n) lies on the line

1. A.$11(x-1)-100(y-2)=0$
2. B.$11x-100y=0$✓
3. C.$11(x-2)-100(y-1)=0$
4. D.$11(x-1)-100y=0$

Correct answer: B

Suppose $\theta\in\left[0,\frac\pi4\right]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is equal to :

1. A.$\frac{4}{(3\sqrt6+2)}$
2. B.$\frac{6+\sqrt6}{(3\sqrt6+2)}$
3. C.$\frac{4}{(3\sqrt6-2)}$✓
4. D.$\frac{6-\sqrt6}{(3\sqrt6-2)}$

Correct answer: C

Let two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that △OPQ is an isosceles triangle and ∠POQ = 90°. If $l=OP^2+PQ^2+QO^2$, then the greatest integer less than or equal to l is :

1. A.42
2. B.46✓
3. C.44
4. D.48

Correct answer: B

If A(1, −1, 2), B(5, 7, −6), C(3, 4, −10) and D(−1, −4, −2) are the vertices of a quadrilateral ABCD, then its area is :

1. A.$48\sqrt7$
2. B.$12\sqrt{29}$✓
3. C.$24\sqrt7$
4. D.$24\sqrt{29}$

Correct answer: B

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

1. A.$2\sqrt2$
2. B.$4\sqrt2$
3. C.4✓
4. D.5

Correct answer: C

Let the line 2x + 3y − k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is $x^2+y^2-3x-2y=0$ and the length of the latus rectum of the ellipse $x^2+9y^2=k^2$ is $\frac mn$, where m and n are coprime, then 2m + n is equal to

1. A.11✓
2. B.10
3. C.12
4. D.13

Correct answer: A

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then $\left|A^TA(\text{adj}(2A))^{-1}(\text{adj}(4B))(\text{adj}(AB))^{-1}AA^T\right|$ is equal to :

1. A.108
2. B.32
3. C.81
4. D.64✓

Correct answer: D

If the system of equations $11x+y+\lambda z=-5$, $2x+3y+5z=3$, $8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

1. A.51
2. B.45
3. C.47✓
4. D.49

Correct answer: C

Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f : A → B, such that f(1) + f(3) = 14, is :

1. A.480
2. B.240✓
3. C.120
4. D.180

Correct answer: B

Let $f(x)=x^5+2x^3+3x+1$, $x\in\mathbb{R}$, and g(x) be a function such that g(f(x)) = x for all $x\in\mathbb{R}$. Then $\frac{g(7)}{g'(7)}$ is equal to :

1. A.14✓
2. B.42
3. C.7
4. D.1

Correct answer: A

If the function $f(x)=\frac{\sin3x+\alpha\sin x-\beta\cos3x}{x^3}$, $x\in\mathbb{R}$, is continuous at x = 0, then f(0) is equal to :

1. A.2
2. B.−2
3. C.4
4. D.−4✓

Correct answer: D

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then $(a+b)^2$ is equal to :

1. A.72✓
2. B.60
3. C.64
4. D.80

Correct answer: A

For the function $f(x)=\sin x+3x-\frac2\pi(x^2+x)$, where $x\in\left[0,\frac\pi2\right]$, consider the following two statements : (I) f is increasing in $\left(0,\frac\pi2\right)$. (II) f′ is decreasing in $\left(0,\frac\pi2\right)$. Between the above two statements,

1. A.only (II) is true.
2. B.only (I) is true.
3. C.neither (I) nor (II) is true.
4. D.both (I) and (II) are true✓

Correct answer: D

The value of $\int_{-\pi}^{\pi}\frac{2y(1+\sin y)}{1+\cos^2y}dy$ is :

1. A.$2\pi^2$
2. B.$\frac{\pi^2}{2}$
3. C.$\frac\pi2$
4. D.$\pi^2$✓

Correct answer: D

The integral $\int_0^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x}dx$ is equal to :

1. A.$3\pi-50\log_e2+20\log_e5$✓
2. B.$3\pi-25\log_e2+10\log_e5$
3. C.$3\pi-10\log_e(2\sqrt2)+10\log_e5$
4. D.$3\pi-30\log_e2+20\log_e5$

Correct answer: A

If y = y(x) is the solution of the differential equation $\frac{dy}{dx}+2y=\sin(2x)$, $y(0)=\frac34$, then $y\left(\frac\pi8\right)$ is equal to:

1. A.$e^{\pi/8}$
2. B.$e^{\pi/4}$
3. C.$e^{-\pi/4}$✓
4. D.$e^{-\pi/8}$

Correct answer: C

If the line $\frac{2-x}{3}=\frac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3\mu}=\frac{1-2y}{6}=\frac{5-z}{7}$, then $4\lambda+9\mu$ is equal to :

1. A.4
2. B.13
3. C.5
4. D.6✓

Correct answer: D

Let d be the distance of the point of intersection of the lines $\frac{x+6}{3}=\frac y2=\frac{z+1}{1}$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ from the point (7, 8, 9). Then $d^2+6$ is equal to :

1. A.69
2. B.78
3. C.72
4. D.75✓

Correct answer: D

The coefficients a, b, c in the quadratic equation $ax^2+bx+c=0$ are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is :

1. A.$\frac1{128}$
2. B.$\frac1{64}$✓
3. C.$\frac3{256}$
4. D.$\frac3{128}$

Correct answer: B

The number of ways of getting a sum 16 on throwing a dice four times is ______

Correct answer: 125

Let $a_1,a_2,a_3,\ldots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2k-1}^2-a_{2k}^2$. If $A_3=-153$, $A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$, then $a_{17}-A_7$ is equal to ______

Correct answer: 910

If the constant term in the expansion of $(1+2x-3x^3)\left(\frac32x^2-\frac1{3x}\right)^9$ is p, then 108p is equal to

Correct answer: 54

Suppose AB is a focal chord of the parabola $y^2=12x$ of length l and slope $m<\sqrt3$. If the distance of the chord AB from the origin is d, then $ld^2$ is equal to _______

Correct answer: 108

Let f be a differentiable function in the interval (0, ∞) such that f(1) = 1 and $\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$ for each x > 0. Then 2f(2) + 3f(3) is equal to _______

Correct answer: 24

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is $\sigma^2$, then $96\sigma^2$ is equal to ______

Correct answer: 56

The number of distinct real roots of the equation |x||x + 2| − 5|x + 1| − 1 = 0 is _______

Correct answer: 3

If $S=\{a\in\mathbb{R}:|2a-1|=3[a]+2\{a\}\}$, where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72\sum_{a\in S}a$ is equal to ______

Correct answer: 18

The area of the region enclosed by the parabolas $y=x^2-5x$ and $y=7x-x^2$ is

Correct answer: 72

Let $\vec a=\hat i-3\hat j+7\hat k$, $\vec b=2\hat i-\hat j+\hat k$ and $\vec c$ be a vector such that $(\vec a+2\vec b)\times\vec c=3(\vec c\times\vec a)$. If $\vec a\cdot\vec c=130$, then $\vec b\cdot\vec c$ is equal to _______

Correct answer: 30