JEE Main 2020 — 7 Jan Shift 1 (Previous Year Paper)

A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant 'b', the correct equivalence would be:

1. A.$L \leftrightarrow m,\ C \leftrightarrow k,\ R \leftrightarrow b$
2. B.$L \leftrightarrow m,\ C \leftrightarrow \frac{1}{k},\ R \leftrightarrow b$✓
3. C.$L \leftrightarrow k,\ C \leftrightarrow b,\ R \leftrightarrow m$
4. D.$L \leftrightarrow \frac{1}{b},\ C \leftrightarrow \frac{1}{m},\ R \leftrightarrow \frac{1}{k}$

Correct answer: B

As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be:

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

Correct answer: A

A 60 HP electric motor lifts an elevator having a maximum total load capacity of 2000 kg. If the frictional force on the elevator is 4000 N, the speed of the elevator at full load is close to: (1 HP = 746 W, $g = 10\ \mathrm{m\,s^{-2}}$)

1. A.$2.0\ \mathrm{m\,s^{-1}}$
2. B.$1.9\ \mathrm{m\,s^{-1}}$✓
3. C.$1.7\ \mathrm{m\,s^{-1}}$
4. D.$1.5\ \mathrm{m\,s^{-1}}$

Correct answer: B

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The center of mass of the system is at a point:

1. A.0.9 cm right and 2.0 cm above 1 kg mass✓
2. B.2.0 cm right and 0.9 cm above 1 kg mass
3. C.0.6 cm right and 2.0 cm above 1 kg mass
4. D.1.5 cm right and 1.2 cm above 1 kg mass

Correct answer: A

The radius of gyration of a uniform rod of length $l$, about an axis passing through a point $\frac{l}{4}$ away from the centre of the rod, and perpendicular to it, is:

1. A.$\frac{1}{4}l$
2. B.$\sqrt{\frac{3}{8}}\,l$
3. C.$\sqrt{\frac{7}{48}}\,l$✓
4. D.$\frac{1}{8}l$

Correct answer: C

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it ejects a rocket of mass $\frac{m}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is (G is the gravitational constant; M is the mass of the earth):

1. A.$5m\left(u^2 - \frac{119}{200}\frac{GM}{R}\right)$✓
2. B.$\frac{m}{20}\left(u^2 + \frac{113}{200}\frac{GM}{R}\right)$
3. C.$\frac{3m}{8}\left(u + \sqrt{\frac{5GM}{6R}}\right)^2$
4. D.$\frac{m}{20}\left(u - \sqrt{\frac{2GM}{3R}}\right)^2$

Correct answer: A

Two moles of an ideal gas with $\frac{C_P}{C_V} = \frac{5}{3}$ are mixed with 3 moles of another ideal gas with $\frac{C_P}{C_V} = \frac{4}{3}$. The value of $\frac{C_P}{C_V}$ for the mixture is:

1. A.1.50
2. B.1.47
3. C.1.42✓
4. D.1.45

Correct answer: C

A litre of dry air at STP expands adiabatically to a volume of 3 litres. If $\gamma = 1.40$, the work done by air is: ($3^{1.4} = 4.6555$) [Take air to be an ideal gas]

1. A.48 J
2. B.60.7 J
3. C.90.5 J✓
4. D.100.8 J

Correct answer: C

Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section 1.0 mm$^2$) is 90 m s$^{-1}$. If the Young's modulus of wire is $16\times10^{11}$ N m$^{-2}$, the extension of wire over its natural length is:

1. A.0.03 mm✓
2. B.0.02 mm
3. C.0.01 mm
4. D.0.04 mm

Correct answer: A

Two infinite planes each with uniform surface charge density $+\sigma$ are kept in such a way that the angle between them is 30°. The electric field in the region shown between them is given by:

1. A.$\frac{\sigma}{2\epsilon_0}\left[\left(1-\frac{\sqrt3}{2}\right)\hat y - \frac{\hat x}{2}\right]$✓
2. B.$\frac{\sigma}{\epsilon_0}\left[\left(1+\frac{\sqrt3}{2}\right)\hat y + \frac{\hat x}{2}\right]$
3. C.$\frac{\sigma}{2\epsilon_0}\left[\left(1+\sqrt3\right)\hat y + \frac{\hat x}{2}\right]$
4. D.$\frac{\sigma}{2\epsilon_0}\left[\left(1+\sqrt3\right)\hat y - \frac{\hat x}{2}\right]$

Correct answer: A

A parallel plate capacitor has plates of area A separated by distance 'd' between them. It is filled with a dielectric which has a dielectric constant that varies as $k(x) = K(1+\alpha x)$ where 'x' is the distance measured from one of the plates. If $(\alpha d) \ll 1$, the total capacitance of the system is best given by the expression:

1. A.$\frac{A\epsilon_0 K}{d}\left(1+\frac{\alpha^2d^2}{2}\right)$
2. B.$\frac{AK\epsilon_0}{d}(1+\alpha d)$
3. C.$\frac{A\epsilon_0 K}{d}\left(1+\left(\frac{\alpha d}{2}\right)^2\right)$
4. D.$\frac{AK\epsilon_0}{d}\left(1+\frac{\alpha d}{2}\right)$✓

Correct answer: D

The current $I_1$ (in A) flowing through 1 Ω resistor in the following circuit is:

1. A.0.4
2. B.0.5
3. C.0.2✓
4. D.0.25

Correct answer: C

Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by $\phi_i$. The magnetic flux through the area of the circular coil area is given by $\phi_0$. Which of the following option is correct?

1. A.$\phi_i = \phi_0$
2. B.$\phi_i < \phi_0$
3. C.$\phi_i > \phi_0$
4. D.$\phi_i = -\phi_0$✓

Correct answer: D

A long solenoid of radius R carries a time (t)-dependent current $I(t) = I_0t(1-t)$. A ring of radius 2R is placed coaxially near its middle. During the time interval $0 \le t \le 1$, the induced current ($I_R$) and the induced EMF ($V_R$) in the ring change as:

1. A.Direction of $I_R$ remains unchanged and $V_R$ is maximum at $t = 0.5$
2. B.At $t = 0.5$ direction of $I_R$ reverses and $V_R$ is zero✓
3. C.Direction of $I_R$ remains unchanged and $V_R$ is zero at $t = 0.25$
4. D.At $t = 0.25$ direction of $I_R$ reverses and $V_R$ is maximum

Correct answer: B

If the magnetic field in a plane electromagnetic wave is given by $\vec B = 3\times10^{-8}\sin(1.6\times10^3x + 48\times10^{10}t)\,\hat j$ T, then what will be expression for electric field?

1. A.$\vec E = (3\times10^{-8}\sin(1.6\times10^3x + 48\times10^{10}t)\,\hat j\ \mathrm{V/m})$
2. B.$\vec E = (9\sin(1.6\times10^3x + 48\times10^{10}t)\,\hat k\ \mathrm{V/m})$✓
3. C.$\vec E = (3\times10^{-8}\sin(1.6\times10^3x + 48\times10^{10}t)\,\hat i\ \mathrm{V/m})$
4. D.$\vec E = (60\sin(1.6\times10^3x + 48\times10^{10}t)\,\hat k\ \mathrm{V/m})$

Correct answer: B

If we need a magnification of 375 from a compound microscope of tube length 150 mm and an objective of focal length 5 mm, the focal length of the eye-piece, should be close to:

1. A.12 mm
2. B.22 mm✓
3. C.33 mm
4. D.2 mm

Correct answer: B

A polarizer - analyser set is adjusted such that the intensity of light coming out of the analyser is just 10% of the original intensity. Assuming that the polarizer - analyser set does not absorb any light, the angle by which the analyser need to be rotated further to reduce the output intensity to be zero, is:

1. A.90°
2. B.18.4°✓
3. C.71.6°
4. D.45°

Correct answer: B

The time period of revolution of electron in its ground state orbit in a hydrogen atom is $1.6\times10^{-16}$ s. The frequency of revolution of the electron in its first excited state (in s$^{-1}$) is:

1. A.$6.2\times10^{15}$
2. B.$1.6\times10^{14}$
3. C.$7.8\times10^{14}$✓
4. D.$5.6\times10^{12}$

Correct answer: C

Which of the following gives a reversible operation?

1. A.Option 1 (drawn): a single-input gate with an output bubble — a NOT gate✓
2. B.Option 2 (drawn): a two-input AND-shaped gate with an output bubble — a NAND gate
3. C.Option 3 (drawn): a two-input OR gate
4. D.Option 4 (drawn): a two-input OR-shaped gate with an output bubble — a NOR gate

Correct answer: A

Visible light of wavelength $6000\times10^{-8}$ cm falls normally on a single slit and produces a diffraction pattern. It is found that the second diffraction minimum is at 60° from the central maximum. If the first minimum is produced at $\theta_1$, then $\theta_1$ is close to:

1. A.45°
2. B.30°
3. C.25°✓
4. D.20°

Correct answer: C

A particle (m = 1 kg) slides down a frictionless track (AOC) starting from rest at a point A (height 2 m). After reaching C, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $g = 10\ \mathrm{m\,s^{-2}}$) ________.

Correct answer: 10

A non-isotropic solid metal cube has coefficients of linear expansion as: $5\times10^{-5}$/°C along the x-axis and $5\times10^{-6}$/°C along the y and the z-axis. If the coefficient of volume expansion of the solid is $C\times10^{-6}$/°C then the value of C is ________.

Correct answer: 60

A Carnot engine operates between two reservoirs of temperatures 900 K and 300 K. The engine performs 1200 J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is ________.

Correct answer: 600

A loop ABCDEFA of straight edges has six corner points A(0, 0, 0), B(5, 0, 0), C(5, 5, 0), D(0, 5, 0), E(0, 5, 5) and F(0, 0, 5). The magnetic field in this region is $\vec B = (3\hat i + 4\hat k)$ T. The quantity of flux through the loop ABCDEFA (in Wb) is ________.

Correct answer: 175

A beam of electromagnetic radiation of intensity $6.4\times10^{-5}$ W/cm$^2$ is comprised of wavelength, $\lambda = 310$ nm. It falls normally on a metal (work function $\varphi = 2$ eV) of surface area of 1 cm$^2$. If one in $10^3$ photons ejects an electron, total number of electrons ejected in 1 s is $10^x$. (hc = 1240 eV nm, 1 eV = $1.6\times10^{-19}$ J), then x is ________.

Correct answer: 11

At 35°C, the vapour pressure of CS$_2$ is 512 mm Hg and that of acetone is 344 mm Hg. A solution of CS$_2$ in acetone has a total vapour pressure of 600 mm Hg. The false statement amongst the following is:

1. A.heat must be absorbed in order to produce the solution at 35°C
2. B.a mixture of 100 mL CS$_2$ and 100 mL acetone has a volume < 200 mL✓
3. C.CS$_2$ and acetone are less attracted to each other than to themselves
4. D.Raoult's law is not obeyed by this system

Correct answer: B

Given that the standard potentials (E°) of Cu$^{2+}$/Cu and Cu$^+$/Cu are 0.34 V and 0.522 V respectively, the E° of Cu$^{2+}$/Cu$^+$ is:

1. A.0.182 V
2. B.−0.182 V
3. C.+0.158 V✓
4. D.−0.158 V

Correct answer: C

The dipole moments of CCl$_4$, CHCl$_3$ and CH$_4$ are in the order:

1. A.CH$_4$ < CCl$_4$ < CHCl$_3$
2. B.CCl$_4$ < CH$_4$ < CHCl$_3$
3. C.CH$_4$ = CCl$_4$ < CHCl$_3$✓
4. D.CHCl$_3$ < CH$_4$ = CCl$_4$

Correct answer: C

The number of orbitals associated with quantum numbers $n = 5$, $m_s = +\frac{1}{2}$ is:

1. A.50
2. B.15
3. C.25✓
4. D.11

Correct answer: C

The relative strength of interionic/intermolecular forces in decreasing order is:

1. A.ion-dipole > ion-ion > dipole-dipole
2. B.ion-ion > ion-dipole > dipole-dipole✓
3. C.ion-dipole > dipole-dipole > ion-ion
4. D.dipole-dipole > ion-dipole > ion-ion

Correct answer: B

Amongst the following statements, that which was not proposed by Dalton was:

1. A.matter consists of indivisible atoms.
2. B.all the atoms of a given element have identical properties including identical mass. Atoms of different elements differ in mass.
3. C.when gases combine or reproduced in a chemical reaction they do so in a simple ratio by volume provided all gases are at the same T & P.✓
4. D.chemical reactions involve reorganization of atoms. These are neither created nor destroyed in a chemical reaction.

Correct answer: C

The electron gain enthalpy (in kJ/mol) of fluorine, chlorine, bromine and iodine, respectively, are:

1. A.−333, −349, −325 and −296✓
2. B.−349, −333, −325 and −296
3. C.−333, −325, −349 and −296
4. D.−296, −325, −333 and −349

Correct answer: A

The purest form of commercial iron is:

1. A.wrought iron✓
2. B.pig iron
3. C.cast iron
4. D.scrap iron and pig iron

Correct answer: A

In comparison to the zeolite process for the removal of permanent hardness, the synthetic resins method is:

1. A.less efficient as it exchanges only anions
2. B.more efficient as it can exchange both cations as well as anions✓
3. C.more efficient as it can exchange only cations
4. D.less efficient as the resins cannot be regenerated

Correct answer: B

Oxidation number of potassium in K$_2$O, K$_2$O$_2$ and KO$_2$, respectively, is:

1. A.+1, +2 and +4
2. B.+1, +4 and +2
3. C.+2, +1 and $+\frac{1}{2}$
4. D.+1, +1 and +1✓

Correct answer: D

The atomic radius of Ag is closest to:

1. A.Ni
2. B.Cu
3. C.Hg
4. D.Au✓

Correct answer: D

The theory that can completely/properly explain the nature of bonding in [Ni(CO)$_4$] is:

1. A.Werner's theory
2. B.Valence bond theory
3. C.Crystal field theory
4. D.Molecular orbital theory✓

Correct answer: D

The IUPAC name of the complex [Pt(NH$_3$)$_2$Cl(NH$_2$CH$_3$)]Cl is:

1. A.Bisammine(methanamine)chlorido platinum(II)chloride
2. B.Diammine(methanamine)chlorido platinum(II)chloride
3. C.Diamminechlorido(methanamine) platinum(II)chloride✓
4. D.Diamminechlorido(aminomethane) platinum(II)chloride

Correct answer: C

Consider the following reaction: N,N-dimethylaniline + $\mathrm{Na^{+}\,{}^{-}O_3S{-}C_6H_4{-}N_2^{+}Cl^{-}}$ (p-sulphonatobenzenediazonium chloride) $\xrightarrow{\mathrm{OH^-}}$ 'X'. The product 'X' is used:

1. A.as food grade colourant
2. B.in protein estimation as an alternative to ninhydrin
3. C.in acid base titration as an indicator✓
4. D.in laboratory test for phenols

Correct answer: C

The increasing order of pK$_b$ for the following compounds will be: (A) NH$_2$–CH=NH, (B) the bicyclic amidine shown in the figure, (C) CH$_3$NHCH$_3$

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

Correct answer: D

A solution of m-chloroaniline, m-chlorophenol and m-chlorobenzoic acid in ethyl acetate was extracted initially with a saturated solution of NaHCO$_3$ to give fraction A. The left over organic phase was extracted with dilute NaOH solution to give fraction B. The final organic layer was labelled as fraction C. Fractions A, B and C, contain respectively:

1. A.m-chloroaniline, m-chlorobenzoic acid and m-chlorophenol
2. B.m-chlorobenzoic acid, m-chloroaniline and m-chlorophenol
3. C.m-chlorobenzoic acid, m-chlorophenol and m-chloroaniline✓
4. D.m-chlorophenol, m-chlorobenzoic acid and m-chloroaniline

Correct answer: C

Match the following: (i) Riboflavin, (ii) Thiamine, (iii) Pyridoxine, (iv) Ascorbic acid with (a) Beriberi, (b) Scurvy, (c) Cheilosis, (d) Convulsions

1. A.(i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)✓
2. B.(i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
3. C.(i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
4. D.(i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)

Correct answer: A

What is the product of following reaction? Hex-3-ynal $\xrightarrow{\text{(i) NaBH}_4\ \text{(ii) PBr}_3\ \text{(iii) Mg/ether}\ \text{(iv) CO}_2/\text{H}_3\text{O}^+}$ ?

1. A.Hept-4-enoic acid drawn as a zig-zag chain with a C=C double bond, ending in COOH
2. B.Heptanoic acid (saturated zig-zag chain ending in COOH)
3. C.Oct-4-enoic-type chain: longer zig-zag chain with a C=C double bond ending in COOH
4. D.Hept-4-ynoic acid: CH3CH2–C≡C–CH2CH2–COOH (zig-zag with a triple bond, ending in COOH)✓

Correct answer: D

1-methyl ethylene oxide when treated with an excess of HBr produces:

1. A.Br–CH=CH–CH3 (1-bromopropene, drawn as skeletal structure)
2. B.CH2=C(Br)–CH3 (2-bromopropene)
3. C.BrCH2–CH(Br)–CH3 (1,2-dibromopropane)✓
4. D.(CH3)2CBr2 (2,2-dibromopropane)

Correct answer: C

Consider the following reactions: (a) (CH$_3$)$_3$CCH(OH)CH$_3$ $\xrightarrow{\text{conc. H}_2\text{SO}_4}$ (b) (CH$_3$)$_2$CHCH(Br)CH$_3$ $\xrightarrow{\text{alc. KOH}}$ (c) (CH$_3$)$_2$CHCH(Br)CH$_3$ $\xrightarrow{(\text{CH}_3)_3\text{O}^-\text{K}^+}$ (d) (CH$_3$)$_2$C(OH)–CH$_2$–CHO $\xrightarrow{\Delta}$. Which of these reaction(s) will not produce Saytzeff product?

1. A.(d) only
2. B.(b) and (d)
3. C.(a), (c) and (d)
4. D.(c) only✓

Correct answer: D

During the nuclear explosion, one of the products is $^{90}$Sr with half life of 6.93 years. If 1 μg of $^{90}$Sr was absorbed in the bones of a newly born baby in place of Ca, how much time, in years, is required to reduce it by 90% if it is not lost metabolically ________.

Correct answer: 23.03

Two solutions, A and B, each of 100 L was made by dissolving 4 g of NaOH and 9.8 g of H$_2$SO$_4$ in water, respectively. The pH of the resultant solution obtained from mixing 40 L of solution A and 10 L of solution B is ________.

Correct answer: 10.6

For the reaction; A(l) → 2B(g), ΔU = 2.1 kcal, ΔS = 20 cal K$^{-1}$ at 300 K. Hence ΔG in kcal is ________.

Correct answer: -2.7

Chlorine reacts with hot and concentrated NaOH and produces compounds (X) and (Y). Compound (X) gives white precipitate with silver nitrate solution. The average bond order between Cl and O atoms in (Y) is ________.

Correct answer: 1.67

The number of chiral carbons in chloramphenicol is ________.

Correct answer: 2

If $g(x) = x^2 + x - 1$ and $(g\circ f)(x) = 4x^2 - 10x + 5$, then $f\left(\frac{5}{4}\right)$ is equal to:

1. A.$-\frac{1}{2}$✓
2. B.$\frac{1}{2}$
3. C.$-\frac{3}{2}$
4. D.$\frac{3}{2}$

Correct answer: A

If $\mathrm{Re}\left(\frac{z-1}{2z+i}\right) = 1$, where $z = x + iy$, then the point $(x, y)$ lies on a:

1. A.straight line whose slope is $-\frac{2}{3}$.
2. B.straight line whose slope is $\frac{3}{2}$.
3. C.circle whose centre is at $\left(-\frac{1}{2}, -\frac{3}{2}\right)$.
4. D.circle whose diameter is $\frac{\sqrt5}{2}$.✓

Correct answer: D

Let $\alpha$ be a root of the equation $x^2 + x + 1 = 0$ and the matrix $A = \frac{1}{\sqrt3}\begin{bmatrix}1 & 1 & 1\\ 1 & \alpha & \alpha^2\\ 1 & \alpha^2 & \alpha^4\end{bmatrix}$, then the matrix $A^{31}$ is equal to:

1. A.$A$
2. B.$A^2$
3. C.$A^3$✓
4. D.$I_3$

Correct answer: C

If the system of linear equations $2x + 2ay + az = 0$, $2x + 3by + bz = 0$, $2x + 4cy + cz = 0$, where $a, b, c \in \mathbb{R}$ are non-zero and distinct; has a non-zero solution, then:

1. A.$a + b + c = 0$
2. B.$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.✓
3. C.a, b, c are in G.P.
4. D.a, b, c are in A.P.

Correct answer: B

Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is:

1. A.$6!$
2. B.$\frac{1}{2}(6!)$
3. C.$\frac{5}{2}(6!)$✓
4. D.$5^6$

Correct answer: C

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $-\frac{1}{2}$, then the greatest number amongst them is:

1. A.7
2. B.16✓
3. C.$\frac{21}{2}$
4. D.27

Correct answer: B

The greatest positive integer k, for which $49^k + 1$ is a factor of the sum $49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1$, is:

1. A.32
2. B.60
3. C.63✓
4. D.65

Correct answer: C

Let $x^k + y^k = a^k$, $(a, k > 0)$ and $\frac{dy}{dx} + \left(\frac{y}{x}\right)^{\frac13} = 0$, then k is:

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

Correct answer: A

Let the function, $f: [-7, 0] \to \mathbb{R}$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f'(x) \le 2$, for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval:

1. A.$(-\infty, 11]$
2. B.$[-3, 11]$
3. C.$[-6, 20]$
4. D.$(-\infty, 20]$✓

Correct answer: D

If $f(a + b + 1 - x) = f(x)$, for all x, where a and b are fixed positive real numbers, then $\frac{1}{a+b}\int_a^b x\,(f(x) + f(x+1))\,dx$ is equal to:

1. A.$\int_{a-1}^{b-1} f(x)\,dx$
2. B.$\int_{a-1}^{b-1} f(x+1)\,dx$
3. C.$\int_{a+1}^{b+1} f(x)\,dx$✓
4. D.$\int_{a+1}^{b+1} f(x+1)\,dx$

Correct answer: C

The area of the region, enclosed by the circle $x^2 + y^2 = 2$ which is not common to the region bounded by the parabola $y^2 = x$ and the straight line $y = x$, is:

1. A.$\frac{1}{6}(24\pi - 1)$
2. B.$\frac{1}{6}(12\pi - 1)$✓
3. C.$\frac{1}{3}(6\pi - 1)$
4. D.$\frac{1}{3}(12\pi - 1)$

Correct answer: B

If $y = y(x)$ is the solution of the differential equation, $e^y\left(\frac{dy}{dx} - 1\right) = e^x$ such that $y(0) = 0$, then $y(1)$ is equal to:

1. A.$1 + \log_e 2$✓
2. B.$\log_e 2$
3. C.$2 + \log_e 2$
4. D.$2e$

Correct answer: A

If $y = mx + 4$ is a tangent to both the parabolas, $y^2 = 4x$ and $x^2 = 2by$, then b is equal to:

1. A.−128✓
2. B.128
3. C.−64
4. D.−32

Correct answer: A

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:

1. A.$\frac{3}{\sqrt2}$
2. B.$2\sqrt3$
3. C.$3\sqrt2$✓
4. D.$\sqrt3$

Correct answer: C

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is:

1. A.(6, 5, −2)✓
2. B.(4, 3, 2)
3. C.(6, 5, 2)
4. D.(3, 4, −2)

Correct answer: A

A vector $\vec a = \alpha\hat i + 2\hat j + \beta\hat k$ $(\alpha, \beta \in \mathbb{R})$ lies in the plane of the vectors, $\vec b = \hat i + \hat j$ and $\vec c = \hat i - \hat j + 4\hat k$. If $\vec a$ bisects the angle between $\vec b$ and $\vec c$, then:

1. A.$\vec a\cdot\hat i + 1 = 0$
2. B.$\vec a\cdot\hat k + 2 = 0$✓
3. C.$\vec a\cdot\hat i + 3 = 0$
4. D.$\vec a\cdot\hat k + 4 = 0$

Correct answer: B

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1)\tan^2x - \sqrt2\cdot\lambda\tan x = (1-k)$, where $k (\ne -1)$ and $\lambda$ are real numbers. If $\tan^2(\alpha + \beta) = 50$, then a value of $\lambda$ is:

1. A.5
2. B.10✓
3. C.$10\sqrt2$
4. D.$5\sqrt2$

Correct answer: B

An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value k when k consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value −1. Then the expected value of X, is:

1. A.$\frac{3}{16}$
2. B.$-\frac{3}{16}$
3. C.$-\frac{1}{8}$
4. D.$\frac{1}{8}$✓

Correct answer: D

If $y(\alpha) = \sqrt{2\left(\frac{\tan\alpha + \cot\alpha}{1 + \tan^2\alpha}\right) + \frac{1}{\sin^2\alpha}}$, $\alpha \in \left(\frac{3\pi}{4}, \pi\right)$, then $\frac{dy}{d\alpha}$ at $\alpha = \frac{5\pi}{6}$ is:

1. A.$\frac{4}{3}$
2. B.$-\frac{1}{4}$
3. C.−4
4. D.4✓

Correct answer: D

The logical statement $(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$ is equivalent to:

1. A.p
2. B.q
3. C.~p✓
4. D.~q