JEE Main 2024 — 8 Apr Shift 1 (Previous Year Paper)

In an expression $a\times10^b$ :

1. A.b is order of magnitude for $a\ge5$
2. B.b is order of magnitude for $a\le5$✓
3. C.a is order of magnitude for $b\le5$
4. D.b is order of magnitude for $5<a\le10$

Correct answer: B

Young's modulus is determined by the equation given by $Y=49000\frac Ml\frac{\text{dyn}}{\text{cm}^2}$ where M is the mass and l is the extension of wire used in the experiment. Now error in Young modules (Y) is estimated by taking data from M − l plot in graph paper. The smallest scale divisions are 5 g and 0.02 cm along load axis and extension axis respectively. If the value of M and l are 500 g and 2 cm respectively then percentage error of Y is :

1. A.0.5%
2. B.2%✓
3. C.0.02%
4. D.0.2%

Correct answer: B

A clock has 75 cm, 60 cm long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel x distance more than the tip of minute hand. The value of x in meter is nearly (Take π = 3.14) :

1. A.140.5
2. B.118.9
3. C.139.4✓
4. D.220.0

Correct answer: C

A stationary particle breaks into two parts of masses $m_A$ and $m_B$ which move with velocities $v_A$ and $v_B$ respectively. The ratio of their kinetic energies $(K_B:K_A)$ is :

1. A.$v_B:v_A$✓
2. B.$m_B:m_A$
3. C.$m_Bv_B:m_Av_A$
4. D.1 : 1

Correct answer: A

Three bodies A, B and C have equal kinetic energies and their masses are 400 g, 1.2 kg and 1.6 kg respectively. The ratio of their linear momenta is :

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

Correct answer: B

A player caught a cricket ball of mass 150 g moving at a speed of 20 m/s. If the catching process is completed in 0.1 s, the magnitude of force exerted by the ball on the hand of the player is:

1. A.3 N
2. B.300 N
3. C.150 N
4. D.30 N✓

Correct answer: D

Two planets A and B having masses $m_1$ and $m_2$ move around the sun in circular orbits of $r_1$ and $r_2$ radii respectively. If angular momentum of A is L and that of B is 3L, the ratio of time period $\left(\frac{T_A}{T_B}\right)$ is:

1. A.$\left(\frac{r_2}{r_1}\right)^{3/2}$
2. B.$\frac1{27}\left(\frac{m_2}{m_1}\right)^3$✓
3. C.$27\left(\frac{m_1}{m_2}\right)^3$
4. D.$\left(\frac{r_1}{r_2}\right)^3$

Correct answer: B

Correct Bernoulli's equation is (symbols have their usual meaning) :

1. A.$P+mgh+\frac12mv^2=$ constant
2. B.$P+\rho gh+\frac12\rho v^2=$ constant✓
3. C.$P+\rho gh+\rho v^2=$ constant
4. D.$P+\frac12\rho gh+\frac12\rho v^2=$ constant

Correct answer: B

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio $\frac{V_a}{V_d}$ and the ratio $\frac{V_b}{V_c}$ is:

1. A.$\frac{V_a}{V_d}\ne\frac{V_b}{V_c}$
2. B.$\frac{V_a}{V_d}=\frac{V_b}{V_c}$✓
3. C.$\frac{V_a}{V_d}=\left(\frac{V_b}{V_c}\right)^{-1}$
4. D.$\frac{V_a}{V_d}=\left(\frac{V_b}{V_c}\right)^2$

Correct answer: B

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is:

1. A.$\frac75$
2. B.$\frac35$✓
3. C.$\frac53$
4. D.$\frac32$

Correct answer: B

Two charged conducting spheres of radii a and b are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

1. A.$\frac ab$✓
2. B.$\sqrt{ab}$
3. C.$\frac ba$
4. D.ab

Correct answer: A

In the given circuit, the terminal potential difference of the cell is :

1. A.2 V✓
2. B.3 V
3. C.4 V
4. D.1.5 V

Correct answer: A

Paramagnetic substances: A. align themselves along the directions of external magnetic field. B. attract strongly towards external magnetic field. C. has susceptibility little more than zero. D. move from a region of strong magnetic field to weak magnetic field. Choose the most appropriate answer from the options given below:

1. A.A, B, C Only
2. B.A, B, C, D
3. C.A, C Only✓
4. D.B, D Only

Correct answer: C

A LCR circuit is at resonance for a capacitor C, inductance L and resistance R. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

1. A.Zero
2. B.same
3. C.halved
4. D.double✓

Correct answer: D

Critical angle of incidence for a pair of optical media is 45°. The refractive indices of first and second media are in the ratio:

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

Correct answer: B

A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is: (Assume h = $6.63\times10^{-34}$ J s, $m_e=9.0\times10^{-31}$ kg and $m_p$ = 1836 times $m_e$)

1. A.$1:\sqrt{1836}$
2. B.$1:\frac1{1836}$
3. C.$1:\frac1{\sqrt{1836}}$
4. D.1 : 1836✓

Correct answer: D

Average force exerted on a non-reflecting surface at normal incidence is $2.4\times10^{-4}$ N. If 360 W/cm² is the light energy flux during span of 1 hour 30 minutes, Then the area of the surface is:

1. A.0.2 m²
2. B.20 m²
3. C.0.1 m²
4. D.0.02 m²✓

Correct answer: D

Binding energy of a certain nucleus is $18\times10^8$ J. How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus:

1. A.10 μg
2. B.20 μg✓
3. C.0.2 μg
4. D.2 μg

Correct answer: B

The output Y of following circuit for given inputs is :

1. A.$A\cdot B(A+B)$
2. B.0✓
3. C.$\bar A\cdot B$
4. D.$A\cdot B$

Correct answer: B

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to 1 mm. The main scale reading is 2 cm and second division of vernier scale coincides with a division on main scale. If mass of the sphere is 8.635 g, the density of the sphere is:

1. A.2.0 g/cm³✓
2. B.1.7 g/cm³
3. C.2.2 g/cm³
4. D.2.5 g/cm³

Correct answer: A

Three vectors $\vec{OP}$, $\vec{OQ}$ and $\vec{OR}$ each of magnitude A are acting as shown in figure. The resultant of the three vectors is $A\sqrt x$. The value of x is ______.

Correct answer: 3

A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate in $\frac n9$. The value of n is ______

Correct answer: 15

A liquid column of height 0.04 cm balances excess pressure of a soap bubble of certain radius. If density of liquid is $8\times10^3$ kg m$^{-3}$ and surface tension of soap solution is 0.28 N m$^{-1}$, then diameter of the soap bubble is ______ cm. (if g = 10 m s$^{-2}$)

Correct answer: 7

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$ then the value of a is ______

Correct answer: 16

An electric field, $\vec E=\frac{2\hat i+6\hat j+8\hat k}{\sqrt6}$ passes through the surface of 4 m² area having unit vector $\hat n=\left(\frac{2\hat i+\hat j+\hat k}{\sqrt6}\right)$. The electric flux for that surface is ______ V m.

Correct answer: 12

Resistance of a wire at 0°C, 100°C and t°C is found to be 10Ω, 10.2Ω and 10.95Ω respectively. The temperature t in Kelvin scale is ______

Correct answer: 748

An electron with kinetic energy 5 eV enters a region of uniform magnetic field of 3 μT perpendicular to its direction. An electric field E is applied perpendicular to the direction of velocity and magnetic field. The value of E, so that electron moves along the same path, is ______ N C$^{-1}$. (Given, mass of electron = $9\times10^{-31}$ kg, electric charge = $1.6\times10^{-19}$ C)

Correct answer: 4

A square loop PQRS having 10 turns, area $3.6\times10^{-3}$ m² and resistance 100Ω is slowly and uniformly being pulled out of a uniform magnetic field of magnitude B = 0.5 T as shown. Work done in pulling the loop out of the field in 1.0 s is ______ $\times10^{-6}$ J.

Correct answer: 3

A parallel beam of monochromatic light of wavelength 600 nm passes through single slit of 0.4 mm width. Angular divergence corresponding to second order minima would be ______ $\times10^{-3}$ rad.

Correct answer: 6

In an alpha particle scattering experiment distance of closest approach for the α particle is $4.5\times10^{-14}$ m. If target nucleus has atomic number 80, then maximum velocity of α - particle is ______ $\times10^5$ m/s approximately. ($\frac1{4\pi\epsilon_0}=9\times10^9$ SI unit, mass of α particle = $6.72\times10^{-27}$ kg)

Correct answer: 156

Combustion of glucose ($\text{C}_6\text{H}_{12}\text{O}_6$) produces $\text{CO}_2$ and water. The amount of oxygen (in g) required for the complete combustion of 900 g of glucose is : [Molar mass of glucose in g mol$^{-1}$ = 180]

1. A.480
2. B.800
3. C.960✓
4. D.32

Correct answer: C

Match List I with List II. List-I (Elements): A. Cl, S; B. Ge, As; C. Fr, Ra; D. F, O. List-II (Properties in their respective groups): I. Elements with highest electronegativity; II. Elements with largest atomic size; III. Elements which show properties of both metals and non-metal; IV. Elements with highest negative electron gain enthalpy.

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

Correct answer: C

Match List I with List II. List-I (Molecule): A. $\text{NH}_3$; B. $\text{BrF}_5$; C. $\text{PCl}_5$; D. $\text{CH}_4$. List-II (Shape): I. Square pyramid; II. Tetrahedral; III. Trigonal pyramidal; IV. Trigonal bipyramidal.

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

Correct answer: B

Match List I with List II. List-I: A. $\text{Fe}_4[\text{Fe(CN)}_6]_3\cdot x\text{H}_2\text{O}$; B. $[\text{Fe(CN)}_5\text{NOS}]^{4-}$; C. $[\text{Fe(SCN)}]^{2+}$; D. $(\text{NH}_4)_3\text{PO}_4\cdot12\text{MoO}_3$. List-II (colour): I. Violet; II. Blood Red; III. Prussian Blue; IV. Yellow.

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

Correct answer: A

Given below are two statements: Statement I: $\text{N(CH}_3)_3$ and $\text{P(CH}_3)_3$ can act as ligands to form transition metal complexes. Statement II: As N and P are from same group, the nature of bonding of $\text{N(CH}_3)_3$ and $\text{P(CH}_3)_3$ is always same with transition metals. 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

For the given hypothetical reactions, the equilibrium constants are as follows : X ⇌ Y; $K_1=1.0$. Y ⇌ Z; $K_2=2.0$. Z ⇌ W; $K_3=4.0$. The equilibrium constant for the reaction X ⇌ W is

1. A.6.0
2. B.12.0
3. C.7.0
4. D.8.0✓

Correct answer: D

Among the following halogens $\text{F}_2$, $\text{Cl}_2$, $\text{Br}_2$ and $\text{I}_2$. Which can undergo disproportionation reactions?

1. A.$\text{F}_2$, $\text{Cl}_2$ and $\text{Br}_2$
2. B.$\text{F}_2$ and $\text{Cl}_2$
3. C.Only $\text{I}_2$
4. D.$\text{Cl}_2$, $\text{Br}_2$ and $\text{I}_2$✓

Correct answer: D

Thiosulphate reacts differently with iodine and bromine in the reactions given below: $2\text{S}_2\text{O}_3^{2-}+\text{I}_2\to\text{S}_4\text{O}_6^{2-}+2\text{I}^-$; $\text{S}_2\text{O}_3^{2-}+5\text{Br}_2+5\text{H}_2\text{O}\to2\text{SO}_4^{2-}+4\text{Br}^-+10\text{H}^+$. Which of the following statement justifies the above dual behaviour of thiosulphate?

1. A.Bromine is a stronger oxidant than iodine✓
2. B.Thiosulphate undergoes oxidation by bromine and reduction by iodine in these reaction
3. C.Bromine is a weaker oxidant than iodine
4. D.Bromine undergoes oxidation and iodine undergoes reduction in these reactions

Correct answer: A

Give below are two statements: One is labelled as Assertion A and the other is labelled as Reason R: Assertion A: The stability order of +1 oxidation state of Ga, In and Tl is Ga < In < Tl. Reason R: The inert pair effect stabilizes the lower oxidation state down the group. In the light of the above statements, choose the correct answer from the options given below:

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

Correct answer: C

Given below are two statements: Statement I: IUPAC name of Compound A (benzene with Cl at C-1 and NO$_2$ at C-3 and C-5... as drawn: Cl para to nothing, NO$_2$ groups at 2,4-positions relative) is 4-chloro-1,3-dinitrobenzene. Statement II: IUPAC name of Compound B (aniline bearing CH$_3$ ortho and C$_2$H$_5$ para to NH$_2$) is 4-ethyl-2-methylaniline. 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 incorrect.
2. B.Both Statement I and Statement II are correct.
3. C.Statement I is correct but Statement II is incorrect.
4. D.Statement I is incorrect but Statement II is correct.✓

Correct answer: D

In the given compound $\text{CH}_3\text{-CH(CH}_3\text{)-CH}_2\text{-CH(CH}_3\text{)-CH}_3$ (as drawn), the number of 2° carbon atom/s is ______.

1. A.Four
2. B.Two
3. C.One✓
4. D.Three

Correct answer: C

Iron (III) catalyses the reaction between iodide and persulphate ions, in which A. $\text{Fe}^{3+}$ oxidises the iodide ion B. $\text{Fe}^{3+}$ oxidises the persulphate ion C. $\text{Fe}^{2+}$ reduces the iodide ion D. $\text{Fe}^{2+}$ reduces the persulphate ion. Choose the most appropriate answer from the options given below:

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

Correct answer: D

Number of Complexes with even number of electrons in $t_{2g}$ orbitals is - $[\text{Fe(H}_2\text{O)}_6]^{2+}$, $[\text{Co(H}_2\text{O)}_6]^{2+}$, $[\text{Co(H}_2\text{O)}_6]^{3+}$, $[\text{Cu(H}_2\text{O)}_6]^{2+}$, $[\text{Cr(H}_2\text{O)}_6]^{2+}$

1. A.2
2. B.3✓
3. C.1
4. D.5

Correct answer: B

An octahedral complex with the formula $\text{CoCl}_3\cdot n\text{NH}_3$ upon reaction with excess of $\text{AgNO}_3$ solution gives 2 moles of AgCl. Consider the oxidation state of Co in the complex is 'x'. The value of "x + n" is ______

1. A.6
2. B.8✓
3. C.3
4. D.5

Correct answer: B

Which among the following compounds will undergo fastest $S_N2$ reaction.

1. A.(Bromomethyl)cyclobutane✓
2. B.(1-Bromoethyl)cyclobutane
3. C.1-Bromo-1-methylcyclobutane
4. D.(2-Bromopropan-2-yl)cyclobutane

Correct answer: A

Identify the major products A and B respectively in the following set of reactions. 1-Methylcyclohexan-1-ol: $\xrightarrow{\text{Conc. H}_2\text{SO}_4,\ \Delta}$ A; $\xrightarrow{\text{CH}_3\text{COCl, Pyridine}}$ B

1. A.A = 1-methylcyclohexene, B = 1-methyl-cyclohexanol ring acetylated on carbon (as drawn)
2. B.A = methylenecyclohexane, B = 1-acetyl-1-methylcyclohexane
3. C.A = methylenecyclohexane, B = C-acetylated alcohol (as drawn)
4. D.A = 1-methylcyclohexene, B = 1-methylcyclohexyl acetate✓

Correct answer: D

Identify the product (P) in the following reaction: Cyclopentanecarboxylic acid $\xrightarrow{\text{i) Br}_2/\text{Red P ii) H}_2\text{O}}$ (P)

1. A.1-Bromocyclopentane-1-carboxylic acid✓
2. B.1-Bromocyclopentane-1-carbaldehyde
3. C.Cyclopentanecarbonyl bromide
4. D.2-Bromocyclopentane-1-carboxylic acid

Correct answer: A

Match List I with List II. List-I (Name of the test): A. Borax bead test; B. Charcoal cavity test; C. Cobalt nitrate test; D. Flame test. List-II (Reaction sequence involved) [M is metal]: I. $\text{MCO}_3\to\text{MO}\xrightarrow{\text{Co(NO}_3)_2}\text{CoO}\cdot\text{MO}$; II. $\text{MCO}_3\to\text{MCl}_2\to\text{M}^{2+}$; III. $\text{MSO}_4\xrightarrow{\text{Na}_2\text{B}_4\text{O}_7,\Delta}\text{M(BO}_2)_2\to\text{MBO}_2\to\text{M}$; IV. $\text{MSO}_4\xrightarrow{\text{Na}_2\text{CO}_3,\Delta}\text{MCO}_3\to\text{MO}\to\text{M}$.

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

Correct answer: B

For the given Fischer projection (D-glucose: CHO; H–OH; HO–H; H–OH; H–OH; CH$_2$OH), the incorrect statement regarding the given structure is

1. A.can be oxidized to a dicarboxylic acid with $\text{Br}_2$ water✓
2. B.will coexist in equilibrium with 2 other cyclic structure
3. C.despite the presence of -CHO does not give Schiff's test
4. D.has 4 asymmetric carbon atom

Correct answer: A

A hypothetical electromagnetic wave is show below (the distance between a crest and the next zero crossing, a quarter wavelength, is 1.5 pm). The frequency of the wave is $x\times10^{19}$ Hz. x = ______ (nearest integer)

Correct answer: 5

Number of molecules from the following which are exceptions to octet rule is ______ $\text{CO}_2$, $\text{NO}_2$, $\text{H}_2\text{SO}_4$, $\text{BF}_3$, $\text{CH}_4$, $\text{SiF}_4$, $\text{ClO}_2$, $\text{PCl}_5$, $\text{BeF}_2$, $\text{C}_2\text{H}_6$, $\text{CHCl}_3$, $\text{CBr}_4$

Correct answer: 6

Consider the figure provided. 1 mol of an ideal gas is kept in a cylinder, fitted with a piston, at the position A, at 18°C. If the piston is moved to position B, keeping the temperature unchanged, then 'x' L atm work is done in this reversible process. x = ______ L atm. (nearest integer) [Given : Absolute temperature = °C + 273.15, R = 0.08206 L atm mol$^{-1}$ K$^{-1}$]

Correct answer: 55

The number of optical isomers in following compound is: (a tricyclic perhydroanthracene-type skeleton bearing Br and CH$_3$ on adjacent ring-fusion carbons, as drawn)

Correct answer: 32

A solution containing 10 g of an electrolyte $\text{AB}_2$ in 100 g of water boils at 100.52°C. The degree of ionization of the electrolyte (α) is ______ $\times10^{-1}$. (nearest integer) [Given : Molar mass of $\text{AB}_2$ = 200 g mol$^{-1}$, $K_b$ = 0.52 K kg mol$^{-1}$, boiling point of water = 100°C; $\text{AB}_2$ ionises as $\text{AB}_2\to\text{A}^{2+}+2\text{B}^-$]

Correct answer: 5

Consider the following reaction A + B → C. The time taken for A to become 1/4th of its initial concentration is twice the time taken to become 1/2 of the same. Also, when the change of concentration of B is plotted against time, the resulting graph gives a straight line with a negative slope and a positive intercept on the concentration axis. The overall order of the reaction is ______

Correct answer: 1

The 'spin only' magnetic moment value of $\text{MO}_4^{2-}$ is ______ BM. (Where M is a metal having least metallic radii among Sc, Ti, V, Cr, Mn and Zn). (Given atomic number: Sc = 21, Ti = 22, V = 23, Cr = 24, Mn = 25 and Zn = 30)

Correct answer: 0

Major product B of the following reaction has ______ π-bond. Ethylbenzene $\xrightarrow{\text{KMnO}_4-\text{KOH},\ \Delta}$ (A) $\xrightarrow{\text{HNO}_3/\text{H}_2\text{SO}_4}$ (B)

Correct answer: 5

If 279 g of aniline is reacted with one equivalent of benzenediazonium chloride, the maximum amount of aniline yellow formed will be ______ g. (nearest integer) (consider complete conversion).

Correct answer: 591

Number of amine compounds from the following giving solids which are soluble in NaOH upon reaction with Hinsberg's reagent is ______

Correct answer: 5

The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^x+48=0$ is :

1. A.$1+\log_8(6)$✓
2. B.$1+\log_6(8)$
3. C.$\log_8(6)$
4. D.$\log_8(4)$

Correct answer: A

Let z be a complex number such that |z + 2| = 1 and $\text{Im}\left(\frac{z+1}{z+2}\right)=\frac15$. Then the value of |Re(z + 2)| is

1. A.$\frac{2\sqrt6}{5}$✓
2. B.$\frac{24}{5}$
3. C.$\frac{1+\sqrt6}{5}$
4. D.$\frac{\sqrt6}{5}$

Correct answer: A

If $\sin x=-\frac35$, where $\pi<x<\frac{3\pi}{2}$, then $80(\tan^2x-\cos x)$ is equal to

1. A.108
2. B.109✓
3. C.18
4. D.19

Correct answer: B

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio 2 : 1, then $(\alpha+\beta)+4(r_1^2+r_2^2)$ equals

1. A.125
2. B.130✓
3. C.110
4. D.145

Correct answer: B

Let [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f:A\to\mathbb{Z}$ be the function $f(x)=\left[\log_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from A to the range of f is

1. A.25
2. B.24
3. C.20
4. D.120✓

Correct answer: D

For the function f(x) = (cos x) − x + 1, x ∈ R, between the following two statements (S1) f(x) = 0 for only one value of x in [0, π]. (S2) f(x) is decreasing in $\left[0,\frac\pi2\right]$ and increasing in $\left[\frac\pi2,\pi\right]$.

1. A.Both (S1) and (S2) are correct.
2. B.Both (S1) and (S2) are incorrect.
3. C.Only (S2) is correct.
4. D.Only (S1) is correct.✓

Correct answer: D

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is

1. A.1
2. B.2✓
3. C.0
4. D.3

Correct answer: B

The value of k ∈ N for which the integral $I_n=\int_0^1(1-x^k)^ndx$, n ∈ N, satisfies $147I_{20}=148I_{21}$ is

1. A.14
2. B.8
3. C.10
4. D.7✓

Correct answer: D

Let f(x) be a positive function such that the area bounded by y = f(x), y = 0 from x = 0 to x = a > 0 is $e^{-a}+4a^2+a-1$. Then the differential equation, whose general solution is $y=c_1f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

1. A.$(8e^x-1)\frac{d^2y}{dx^2}+\frac{dy}{dx}=0$
2. B.$(8e^x-1)\frac{d^2y}{dx^2}-\frac{dy}{dx}=0$
3. C.$(8e^x+1)\frac{d^2y}{dx^2}-\frac{dy}{dx}=0$
4. D.$(8e^x+1)\frac{d^2y}{dx^2}+\frac{dy}{dx}=0$✓

Correct answer: D

The set of all α, for which the vectors $\vec a=\alpha t\hat i+6\hat j-3\hat k$ and $\vec b=t\hat i-2\hat j-2\alpha t\hat k$ are inclined at an obtuse angle for all t ∈ R, is

1. A.$\left(-\frac43,1\right)$
2. B.[0, 1)
3. C.$\left(-\frac43,0\right]$✓
4. D.(−2, 0]

Correct answer: C

Let P(x, y, z) be a point in the first octant, whose projection in the xy-plane is the point Q. Let OP = γ; the angle between OQ and the positive x-axis be θ; and the angle between OP and the positive z-axis be φ, where O is the origin. Then the distance of P from the x-axis is

1. A.$\gamma\sqrt{1-\sin^2\phi\cos^2\theta}$✓
2. B.$\gamma\sqrt{1-\sin^2\theta\cos^2\phi}$
3. C.$\gamma\sqrt{1+\cos^2\phi\sin^2\theta}$
4. D.$\gamma\sqrt{1+\cos^2\theta\sin^2\phi}$

Correct answer: A

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac34$ times their greatest possible product, is $\frac mn$, where gcd(m, n) = 1, then n − m equals

1. A.10✓
2. B.9
3. C.11
4. D.8

Correct answer: A

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______

Correct answer: 36

Let the positive integers be written in the form: row 1: 1; row 2: 2, 3; row 3: 4, 5, 6; row 4: 7, 8, 9, 10; .... If the $k^{th}$ row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is ______

Correct answer: 103

Let $\alpha=\sum_{r=0}^n(4r^2+2r+1)\,{}^nC_r$ and $\beta=\left(\sum_{r=0}^n\frac{{}^nC_r}{r+1}\right)+\frac{1}{n+1}$. If $140<\frac{2\alpha}{\beta}<281$, then the value of n is ______

Correct answer: 5

If the orthocentre of the triangle formed by the lines 2x + 3y − 1 = 0, x + 2y − 1 = 0 and ax + by − 1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (−6, −8), then the value of |a − b| is ______

Correct answer: 16

The value of $\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos2x}\sqrt[3]{\cos3x}\ldots\sqrt[10]{\cos10x}}{x^2}\right)$ is

Correct answer: 55

Let $A=\begin{bmatrix}2&-1\\1&1\end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then n is equal to ______

Correct answer: 7

If the range of $f(\theta)=\frac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, θ ∈ R is [α, β], then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac\alpha\beta$, is equal to ______

Correct answer: 96

Let the area of the region enclosed by the curve y = min{sin x, cos x} and the x axis between x = −π to x = π be A. Then $A^2$ is equal to ______

Correct answer: 16

Let $\vec a=9\hat i-13\hat j+25\hat k$, $\vec b=3\hat i+7\hat j-13\hat k$ and $\vec c=17\hat i-2\hat j+\hat k$ be three given vectors. If $\vec r$ is a vector such that $\vec r\times\vec a=(\vec b+\vec c)\times\vec a$ and $\vec r\cdot(\vec b-\vec c)=0$, then $\frac{|593\vec r+67\vec a|^2}{(593)^2}$ is equal to ______

Correct answer: 569

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and yellow balls. If $\bar X$ and $\bar Y$ are the means of X and Y respectively, then $7\bar X+4\bar Y$ is equal to ______

Correct answer: 17