JEE Main 2024 — 6 Apr Shift 2 (Previous Year Paper)

Given below are two statements : Statement (I) : Dimensions of specific heat is $[L^2T^{-2}K^{-1}]$. Statement (II) : Dimensions of gas constant is $[ML^2T^{-1}K^{-1}]$. 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 correct but statement (II) is incorrect✓
3. C.Both statement (I) and statement (II) are incorrect
4. D.Statement (I) is incorrect but statement (II) is correct

Correct answer: B

A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in $t_1$. If it is projected vertically downwards from the same point with the same speed, it reaches the ground in $t_2$. Time required to reach the ground, if it is dropped from the top of the tower, is :

1. A.$\sqrt{t_1t_2}$✓
2. B.$\sqrt{t_1+t_2}$
3. C.$\sqrt{t_1-t_2}$
4. D.$\sqrt{\frac{t_1}{t_2}}$

Correct answer: A

A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg. The branch pulls the chain by a force equal to (if g = 10 m/s²) :

1. A.100 N
2. B.200 N
3. C.300 N✓
4. D.150 N

Correct answer: C

A car of 800 kg is taking turn on a banked road of radius 300 m and angle of banking 30°. If coefficient of static friction is 0.2 then the maximum speed with which car can negotiate the turn safely: (g = 10 m/s², $\sqrt3=1.73$)

1. A.264 m/s
2. B.51.4 m/s✓
3. C.70.4 m/s
4. D.102.8 m/s

Correct answer: B

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be:

1. A.6%
2. B.600%
3. C.60%
4. D.500%✓

Correct answer: D

Assuming the earth to be a sphere of uniform mass density, a body weighed 300 N on the surface of earth. How much it would weigh at R/4 depth under surface of earth ?

1. A.75 N
2. B.300 N
3. C.375 N
4. D.225 N✓

Correct answer: D

Pressure inside a soap bubble is greater than the pressure outside by an amount : (given : R = Radius of bubble S = Surface tension of bubble)

1. A.$\frac{2S}{R}$
2. B.$\frac{4R}{S}$
3. C.$\frac SR$
4. D.$\frac{4S}{R}$✓

Correct answer: D

A total of 48 J heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by 2°C. The work done by the gas is: Given, R = 8.3 J K$^{-1}$ mol$^{-1}$.

1. A.24.9 J
2. B.72.9 J
3. C.48 J
4. D.23.1 J✓

Correct answer: D

Energy of 10 non rigid diatomic molecules at temperature T is :

1. A.$70K_BT$
2. B.$35K_BT$✓
3. C.$\frac72RT$
4. D.$35RT$

Correct answer: B

Two identical conducting spheres P and S with charge Q on each, repel each other with a force 16 N. A third identical uncharged conducting sphere R is successively brought in contact with the two spheres. The new force of repulsion between P and S is :

1. A.1 N
2. B.6 N✓
3. C.12 N
4. D.4 N

Correct answer: B

The number of electrons flowing per second in the filament of a 110 W bulb operating at 220 V is : (Given e = $1.6\times10^{-19}$ C)

1. A.$6.25\times10^{17}$
2. B.$1.25\times10^{19}$
3. C.$6.25\times10^{18}$
4. D.$31.25\times10^{17}$✓

Correct answer: D

Match List-I with List-II (Y vs X, shape of graph): (A) Y = magnetic susceptibility, X = magnetising field; (B) Y = magnetic field, X = distance from centre of a current carrying wire for x < a (where a = radius of wire); (C) Y = magnetic field, X = distance from centre of a current carrying wire for x > a; (D) Y = magnetic field inside solenoid, X = distance from centre. List-II: (I) straight line through origin; (II) constant then falling steeply near the end; (III) horizontal line; (IV) decreasing curve (hyperbola-like).

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

Correct answer: D

In a coil, the current changes from −2 A to +2 A in 0.2 s and induces an emf of 0.1 V. The self inductance of the coil is :

1. A.4 mH
2. B.1 mH
3. C.5 mH✓
4. D.2.5 mH

Correct answer: C

In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division = 49 MSD; 20 divisions on main scale in each cm. For mark on paper MSR = 8.45 cm, VC = 26. For mark on paper seen through slab MSR = 7.12 cm, VC = 41. For powder particle on the top surface of the glass slab MSR = 4.05 cm, VC = 1. (MSR = Main Scale Reading, VC = Vernier Coincidence) Refractive index of the glass slab is :

1. A.1.52
2. B.1.35
3. C.1.42✓
4. D.1.24

Correct answer: C

For the thin convex lens, the radii of curvature are at 15 cm and 30 cm respectively. The focal length the lens is 20 cm. The refractive index of the material is :

1. A.1.2
2. B.1.8
3. C.1.5✓
4. D.1.4

Correct answer: C

When UV light of wavelength 300 nm is incident on the metal surface having work function 2.13 eV, electron emission takes place. The stopping potential is: (Given hc = 1240 eV nm)

1. A.1.5 V
2. B.4.1 V
3. C.2 V✓
4. D.4 V

Correct answer: C

The longest wavelength associated with Paschen series is : (Given $R_H=1.097\times10^7$ SI unit)

1. A.$3.646\times10^{-6}$ m
2. B.$1.876\times10^{-6}$ m✓
3. C.$2.973\times10^{-6}$ m
4. D.$1.094\times10^{-6}$ m

Correct answer: B

The acceptor level of a p-type semiconductor is 6eV. The maximum wavelength of light which can create a hole would be : Given hc = 1242 eV nm.

1. A.414 nm
2. B.103.5 nm
3. C.207 nm✓
4. D.407 nm

Correct answer: C

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $4^{th}$ division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is 0.04 mm then how many main scale divisions are there in 1 cm ?

1. A.10
2. B.5
3. C.20✓
4. D.40

Correct answer: C

A particle moves in a straight line so that its displacement x at any time t is given by $x^2=1+t^2$. Its acceleration at any time t is $x^{-n}$ where n = ______

Correct answer: 3

Three balls of masses 2 kg, 4 kg and 6 kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of intertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ______ kg m².

Correct answer: 4

A wire of cross sectional area A, modulus of elasticity $2\times10^{11}$ N m$^{-2}$ and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle it sags lower from its original position making angle $\theta=\frac{1}{100}$ radian on the points of support. The value of A is ______ $\times10^{-4}$ m² (consider x << L). (given : g = 10 m/s²)

Correct answer: 1

Two open organ pipes of lengths 60 cm and 90 cm resonate at $6^{th}$ and $5^{th}$ harmonics respectively. The difference of frequencies for the given modes is ______ Hz. (Velocity of sound in air = 333 m/s)

Correct answer: 740

A capacitor of 10 μF capacitance whose plates are separated by 10 mm through air and each plate has area 4 cm² is now filled equally with two dielectric media of $K_1=2$, $K_2=3$ respectively as shown in figure. If new force between the plates is 8 N. The supply voltage is ______ $\times10^{-4}$ V.

Correct answer: 80

In the given figure an ammeter A consists of a 240Ω coil connected in parallel to a 10Ω shunt. The reading of the ammeter is ______ mA.

Correct answer: 160

A coil having 100 turns, area of $5\times10^{-3}$ m², carrying current of 1 mA is placed in uniform magnetic field of 0.20 T such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through 90° is ______ μJ.

Correct answer: 100

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is 25 nF. If resistance of 200Ω and 100 mH inductor is being used in the given circuit. The frequency of ac source is ______ $\times10^3$ Hz. (given π² = 10)

Correct answer: 10

Two coherent monochromatic light beams of intensities I and 4I are superimposed. The difference between maximum and minimum possible intensities in the resulting beam is xI. The value of x is ______.

Correct answer: 8

In Franck-Hertz experiment, the first dip in the current-voltage graph for hydrogen is observed at 10.2 V. The wavelength of light emitted by hydrogen atom when excited to the first excitation level is ______ nm. (Given hc = 1245 eV nm, e = $1.6\times10^{-19}$ C).

Correct answer: 122

Molality (m) of 3M aqueous solution of NaCl is : (Given : Density of solution = 1.25 g mL$^{-1}$, Molar mass in g mol$^{-1}$ : Na – 23, Cl – 35.5)

1. A.1.9 m
2. B.3.85 m
3. C.2.79 m✓
4. D.2.90 m

Correct answer: C

The ratio $\frac{K_P}{K_C}$ for the reaction : $\text{CO}_{(g)}+\frac12\text{O}_{2(g)}\rightleftharpoons\text{CO}_{2(g)}$ is :

1. A.$\frac1{\sqrt{RT}}$✓
2. B.$(RT)^{1/2}$
3. C.RT
4. D.1

Correct answer: A

Match List - I with List - II. List - I (Reaction): (A) $\text{N}_{2(g)}+\text{O}_{2(g)}\to2\text{NO}_{(g)}$; (B) $2\text{Pb(NO}_3)_{2(s)}\to2\text{PbO}_{(s)}+4\text{NO}_{2(g)}+\text{O}_{2(g)}$; (C) $2\text{Na}_{(s)}+2\text{H}_2\text{O}_{(l)}\to2\text{NaOH}_{(aq)}+\text{H}_{2(g)}$; (D) $2\text{NO}_{2(g)}+2\,^-\text{OH}_{(aq)}\to\text{NO}_{2(aq)}^-+\text{NO}_{3(aq)}^-+\text{H}_2\text{O}_{(l)}$. List - II (Type of redox reaction): (I) Decomposition (II) Displacement (III) Disproportionation (IV) Combination.

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

Correct answer: C

The correct statement among the following, for a "chromatography" purification method is :

1. A.Organic compounds run faster than solvent in the thin layer chromatographic plate.
2. B.$R_f$ is an integral value.
3. C.$R_f$ of a polar compound is smaller than that of a non-polar compound.✓
4. D.Non-polar compounds are retained at top and polar compounds come down in column chromatography.

Correct answer: C

The incorrect statement regarding the geometrical isomers of 2-butene is :

1. A.cis-2-butene and trans-2-butene are not interconvertible at room temperature.
2. B.cis-2-butene and trans-2-butene are stereoisomers.
3. C.cis-2-butene has less dipole moment than trans-2-butene.✓
4. D.trans-2-butene is more stable than cis-2-butene.

Correct answer: C

How can an electrochemical cell be converted into an electrolytic cell?

1. A.Applying an external opposite potential lower than $E^0$ cell.
2. B.Reversing the flow of ions in salt bridge.
3. C.Applying an external opposite potential greater than $E^0$ cell.✓
4. D.Exchanging the electrodes at anode and cathode.

Correct answer: C

Arrange the following elements in the increasing order of number of unpaired electrons in it. (A) Sc (B) Cr (C) V (D) Ti (E) Mn Choose the correct answer from the options given below :

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

Correct answer: C

The correct IUPAC name of $[\text{PtBr}_2(\text{PMe}_3)_2]$ is :

1. A.dibromodi(trimethylphosphine)platinum(II)
2. B.bis(trimethylphosphine)dibromoplatinum(II)
3. C.dibromobis(trimethylphosphine)platinum(II)✓
4. D.bis[bromo(trimethylphosphine)]platinum(II)

Correct answer: C

Given below are two statements : Statement I : $\text{PF}_5$ and $\text{BrF}_5$ both exhibit $sp^3d$ hybridisation. Statement II : Both $\text{SF}_6$ and $[\text{Co(NH}_3)_6]^{3+}$ exhibit $sp^3d^2$ hybridisation. 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 true
4. D.Both Statement I and Statement II are false✓

Correct answer: D

Match List - I with List - II. List - I: (A) $\text{TiCl}_4$ (B) $[\text{FeO}_4]^{2-}$ (C) $[\text{FeCl}_4]^-$ (D) $[\text{CoCl}_4]^{2-}$. List - II (d-electron distribution): (I) $e^2,t_2^0$ (II) $e^4,t_2^3$ (III) $e^0,t_2^0$ (IV) $e^2,t_2^3$.

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

Correct answer: C

For (I) toluene, (II) benzene, (III) anisole, (IV) benzotrifluoride ($\text{C}_6\text{H}_5\text{CF}_3$), the correct arrangement for decreasing order of electrophilic substitution for above compounds is :

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

Correct answer: A

Anisole $\xrightarrow{\text{HNO}_3,\ \text{H}_2\text{SO}_4}$ 'A' $\xrightarrow{\text{Br}_2\text{ (excess), Fe}}$ 'B'. The major products formed A and B respectively are:

1. A.4-Nitroanisole and 2,6-dibromo-4-nitroanisole✓
2. B.2-Nitroanisole and 4-bromo-2-nitroanisole
3. C.4-Nitroanisole and 2-bromo-4-nitroanisole
4. D.2-Nitroanisole and 2,4-dibromo-6-nitroanisole

Correct answer: A

Aniline $\xrightarrow{\text{(i) NaNO}_2+\text{HCl (ii) Cu}_2\text{Cl}_2\ \text{(iii) NaOH, 623 K, 300 atm (iv) H}^+}$ (A). Identify the product (A) in the following reaction.

1. A.4-Chlorophenol
2. B.4-Chloroaniline
3. C.4-Aminophenol
4. D.Phenol✓

Correct answer: D

During the detection of acidic radical present in a salt, a student gets a pale yellow precipitate soluble with difficulty in $\text{NH}_4\text{OH}$ solution when sodium carbonate extract was first acidified with dil. $\text{HNO}_3$ and then $\text{AgNO}_3$ solution was added. This indicates presence of :

1. A.Br$^-$✓
2. B.I$^-$
3. C.Cl$^-$
4. D.$\text{CO}_3^{2-}$

Correct answer: A

Match List - I with List - II. List - I: (A) Li (B) Na (C) Rb (D) Cs. List - II: (I) 589.2 (II) 455.5 (III) 670.8 (IV) 780.0

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

Correct answer: D

The incorrect statements regarding enzymes are : (A) Enzymes are biocatalysts. (B) Enzymes are non-specific and can catalyse different kinds of reactions. (C) Most Enzymes are globular proteins. (D) Enzyme - oxidase catalyses the hydrolysis of maltose into glucose. Choose the correct answer from the option given below :

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

Correct answer: B

Consider the following reactions $\text{NiS}+\text{HNO}_3+\text{HCl}\to A+\text{NO}+\text{S}+\text{H}_2\text{O}$; $A+\text{NH}_4\text{OH}+$ dimethylglyoxime ($\text{H}_3\text{C-C(=N-OH)-C(=N-OH)-CH}_3$) $\to B+\text{NH}_4\text{Cl}+\text{H}_2\text{O}$. The number of protons that do not involve in hydrogen bonding in the product B is ______.

Correct answer: 12

For hydrogen atom, energy of an electron in first excited state is −3.4 eV, K.E. of the same electron of hydrogen atom is x eV. Value of x is ______ $\times10^{-1}$ eV. (Nearest integer)

Correct answer: 34

An amine (X) is prepared by ammonolysis of benzyl chloride. On adding p-toluenesulphonyl chloride to it the solution remains clear. Molar mass of the amine (X) formed is ______ g mol$^{-1}$. (Given molar mass in g mol$^{-1}$ C : 12, H : 1, O : 16, N : 14)

Correct answer: 287

For the reaction at 298 K, 2A + B → C. ΔH = 400 kJ mol$^{-1}$ and ΔS = 0.2 kJ mol$^{-1}$ K$^{-1}$. The reaction will become spontaneous above ______ K.

Correct answer: 2000

Consider the two different first order reactions given below A + B → C (Reaction 1) P → Q (Reaction 2). The ratio of the half life of Reaction 1 : Reaction 2 is 5 : 2. If $t_1$ and $t_2$ represent the time taken to complete 2/3$^{rd}$ and 4/5$^{th}$ of Reaction 1 and Reaction 2, respectively, then the value of the ratio $t_1:t_2$ is ______ $\times10^{-1}$ (nearest integer). [Given : log$_{10}$(3) = 0.477 and log$_{10}$(5) = 0.699]

Correct answer: 17

Among $\text{VO}_2^+$, $\text{MnO}_4^-$ and $\text{Cr}_2\text{O}_7^{2-}$, the spin-only magnetic moment value of the species with least oxidising ability is ______ BM (Nearest integer). (Given atomic member V = 23, Mn = 25, Cr = 24)

Correct answer: 0

Number of carbocations from the following that are not stabilized by hyperconjugation is ______. (2-methylbutan-2-yl cation), (tert-butyl), (tert-butyl) $\overset{+}{\text{C}}\text{H}_3$ as drawn, cyclopentadienyl-type cation (as drawn), $\overset{+}{\text{C}}\text{H}_2\text{-OCH}_3$, isopropyl-type cation, $(\text{CH}_3)_2\text{N-}\overset{+}{\text{C}}\text{H}_2$

Correct answer: 5

When 'x' $\times10^{-2}$ mL methanol (molar mass = 32 g; density = 0.792 g/cm³) is added to 100 mL water (density = 1 g/cm³), the following diagram is obtained. x = ______ (nearest integer). [Given : Molal freezing point depression constant of water at 273.15 K is 1.86 K kg mol$^{-1}$]

Correct answer: 543

Total number of species from the following with central atom utilising $sp^2$ hybrid orbitals for bonding is ______. $\text{NH}_3$, $\text{SO}_2$, $\text{SiO}_2$, $\text{BeCl}_2$, $\text{C}_2\text{H}_2$, $\text{C}_2\text{H}_4$, $\text{BCl}_3$, HCHO, $\text{C}_6\text{H}_6$, $\text{BF}_3$, $\text{C}_2\text{H}_4\text{Cl}_2$

Correct answer: 6

Phenetole ($\text{C}_6\text{H}_5\text{OC}_2\text{H}_5$) $\xrightarrow{\text{HNO}_3,\ \text{H}_2\text{SO}_4}$ P (major product) $\xrightarrow{2\text{Br}_2,\ \text{Fe}}$ Q (major product). The ratio of number of oxygen atoms to bromine atoms in the product Q is ______ $\times10^{-1}$.

Correct answer: 15

If $z_1,z_2$ are two distinct complex number such that $\left|\frac{z_1-2z_2}{\frac12-z_1\bar z_2}\right|=2$, then

1. A.$z_1$ lies on a circle of radius $\frac12$ and $z_2$ lies on a circle of radius 1.
2. B.both $z_1$ and $z_2$ lie on the same circle.
3. C.either $z_1$ lies on a circle of radius $\frac12$ or $z_2$ lies on a circle of radius 1.
4. D.either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac12$.✓

Correct answer: D

Let 0 ≤ r ≤ n. If ${}^{n+1}C_{r+1}:{}^nC_r:{}^{n-1}C_{r-1}=55:35:21$, then 2n + 5r is equal to:

1. A.50✓
2. B.62
3. C.55
4. D.60

Correct answer: A

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then :

1. A.$P^2=6\sqrt3Q$
2. B.$P^2=36\sqrt3Q$✓
3. C.$P=36\sqrt3Q^2$
4. D.$P^2=72\sqrt3Q$

Correct answer: B

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to:

1. A.150✓
2. B.180
3. C.160
4. D.125

Correct answer: A

If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, −2), B(8, 3) and C(h, k), then the point C lies on the circle:

1. A.$x^2+y^2-61=0$
2. B.$x^2+y^2-52=0$
3. C.$x^2+y^2-65=0$✓
4. D.$x^2+y^2-74=0$

Correct answer: C

If the locus of the point, whose distances from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is $ax^2+by^2+cxy+dx+ey+170=0$, then the value of $a^2+2b+3c+4d+e$ is equal to :

1. A.37✓
2. B.437
3. C.−27
4. D.5

Correct answer: A

$\lim_{n\to\infty}\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1))\cdot1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)}$ is equal to :

1. A.$\frac23$
2. B.$\frac13$✓
3. C.$\frac34$
4. D.$\frac12$

Correct answer: B

Let A = {1, 2, 3, 4, 5}. Let R be a relation on A defined by xRy if and only if 4x ≤ 5y. Let m be the number of elements in R and n be the minimum number of elements from A × A that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

1. A.25✓
2. B.24
3. C.26
4. D.23

Correct answer: A

Let $f(x)=\frac{1}{7-\sin5x}$ be a function defined on R. Then the range of the function f(x) is equal to :

1. A.$\left[\frac17,\frac16\right]$
2. B.$\left[\frac18,\frac15\right]$
3. C.$\left[\frac17,\frac15\right]$
4. D.$\left[\frac18,\frac16\right]$✓

Correct answer: D

Suppose for a differentiable function h, h(0) = 0, h(1) = 1 and h′(0) = h′(1) = 2. If $g(x)=h(e^x)e^{h(x)}$, then g′(0) is equal to:

1. A.5
2. B.4✓
3. C.8
4. D.3

Correct answer: B

If $\int\frac{1}{a^2\sin^2x+b^2\cos^2x}dx=\frac1{12}\tan^{-1}(3\tan x)+$ constant, then the maximum value of a sin x + b cos x, is :

1. A.$\sqrt{40}$✓
2. B.$\sqrt{41}$
3. C.$\sqrt{39}$
4. D.$\sqrt{42}$

Correct answer: A

If the area of the region $\left\{(x,y):\frac{a}{x^2}\le y\le\frac1x,\ 1\le x\le2,\ 0<a<1\right\}$ is $(\log_e2)-\frac17$ then the value of 7a − 3 is equal to:

1. A.0
2. B.2
3. C.−1✓
4. D.1

Correct answer: C

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

1. A.$\frac{18}{25}$
2. B.$\frac{12}{25}$✓
3. C.$\frac6{25}$
4. D.$\frac4{25}$

Correct answer: B

Let α, β be roots of $x^2+\sqrt2x-8=0$. If $U_n=\alpha^n+\beta^n$, then $\frac{U_{10}+\sqrt2U_9}{2U_8}$ is equal to ______

Correct answer: 4

If $S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}$, x ≠ 0, and $(60)^2S(60)=a(b)^b+b$, where a, b ∈ N, then (a + b) equal to ______

Correct answer: 3660

In a triangle ABC, BC = 7, AC = 8, AB = α ∈ N and $\cos A=\frac23$. If $49\cos(3C)+42=\frac mn$, where gcd(m, n) = 1, then m + n is equal to ______

Correct answer: 39

If the system of equations $2x+7y+\lambda z=3$, $3x+2y+5z=4$, $x+\mu y+32z=-1$ has infinitely many solutions, then (λ − μ) is equal to ______

Correct answer: 38

Let [t] denote the greatest integer less than or equal to t. Let $f:[0,\infty)\to\mathbb{R}$ be a function defined by $f(x)=\left[\frac x2+3\right]-[\sqrt x]$. Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then $\sum_{a\in S}a$ is equal to ______

Correct answer: 17

Let [t] denote the largest integer less than or equal to t. If $\int_0^3\left([x^2]+\left[\frac{x^2}{2}\right]\right)dx=a+b\sqrt2-\sqrt3-\sqrt5+c\sqrt6-\sqrt7$, where a, b, c ∈ Z, then a + b + c is equal to ______

Correct answer: 23

If the solution y(x) of the given differential equation $(e^y+1)\cos x\,dx+e^y\sin x\,dy=0$ passes through the point $\left(\frac\pi2,0\right)$, then the value of $e^{y\left(\frac\pi6\right)}$ is equal to ______

Correct answer: 3

If the shortest distance between the lines $\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the largest possible value of |λ| is equal to ______

Correct answer: 43

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of X is $\frac mn$, where gcd(m, n) = 1, then n − m is equal to ______

Correct answer: 71