JEE Main 2019 — 10 Apr Shift 1 (Previous Year Paper)

Given below in the left column are different modes of communication using the kinds of waves given in the right column. A. Optical Fibre Communication — P. Ultrasound B. Radar — Q. Infrared Light C. Sonar — R. Microwaves D. Mobile Phones — S. Radio Waves From the options given below, find the most appropriate match between entries in the left and the right column.

1. A.A-Q, B-S, C-R, D-P
2. B.A-Q, B-S, C-P, D-R✓
3. C.A-S, B-Q, C-R, D-P
4. D.A-R, B-P, C-S, D-Q

Correct answer: B

In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line. One may conclude that:

1. A.$R(T)=\frac{R_0}{T^2}$
2. B.$R(T)=R_0e^{-T^2/T_0^2}$
3. C.$R(T)=R_0e^{-T_0^2/T^2}$✓
4. D.$R(T)=R_0e^{T^2/T_0^2}$

Correct answer: C

A particle of mass $m$ is moving along a trajectory given by $x=x_0+a\cos\omega_1 t$ $y=y_0+b\sin\omega_2 t$ The torque, acting on the particle about the origin, at $t=0$ is:

1. A.$+my_0a\omega_1^2\hat{k}$✓
2. B.$m(-x_0b+y_0a)\omega_1^2\hat{k}$
3. C.$-m(x_0b\omega_2^2-y_0a\omega_1^2)\hat{k}$
4. D.Zero

Correct answer: A

A ball is thrown upward with an initial velocity $V_0$ from the surface of the earth. The motion of the ball is affected by a drag force equal to $m\gamma v^2$ (where $m$ is mass of the ball, $v$ is its instantaneous velocity and $\gamma$ is a constant). Time taken by the ball to rise to its zenith is:

1. A.$\frac{1}{\sqrt{2\gamma g}}\tan^{-1}\left(\sqrt{\frac{2\gamma}{g}}V_0\right)$
2. B.$\frac{1}{\sqrt{\gamma g}}\ln\left(1+\sqrt{\frac{\gamma}{g}}V_0\right)$
3. C.$\frac{1}{\sqrt{\gamma g}}\tan^{-1}\left(\sqrt{\frac{\gamma}{g}}V_0\right)$✓
4. D.$\frac{1}{\sqrt{\gamma g}}\sin^{-1}\left(\sqrt{\frac{\gamma}{g}}V_0\right)$

Correct answer: C

Two particles, of masses $M$ and $2M$, moving, as shown, with speeds of 10 m/s and 5 m/s, collide elastically at the origin. After the collision, they move along the indicated directions with speeds $v_1$ and $v_2$, respectively. The values of $v_1$ and $v_2$ are nearly:

1. A.6.5 m/s and 3.2 m/s
2. B.3.2 m/s and 12.6 m/s
3. C.6.5 m/s and 6.3 m/s✓
4. D.3.2 m/s and 6.3 m/s

Correct answer: C

Two coaxial discs, having moments of inertia $I_1$ and $\frac{I_1}{2}$, are rotating with respective angular velocities $\omega_1$ and $\frac{\omega_1}{2}$, about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies, then $(E_f-E_i)$ is:

1. A.$-\frac{I_1\omega_1^2}{12}$
2. B.$-\frac{I_1\omega_1^2}{24}$✓
3. C.$\frac{I_1\omega_1^2}{6}$
4. D.$\frac{3}{8}I_1\omega_1^2$

Correct answer: B

A thin disc of mass $M$ and radius $R$ has mass per unit area $\sigma(r)=kr^2$ where $r$ is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is:

1. A.$\frac{MR^2}{6}$
2. B.$\frac{2MR^2}{3}$✓
3. C.$\frac{MR^2}{2}$
4. D.$\frac{MR^2}{3}$

Correct answer: B

The value of acceleration due to gravity at Earth's surface is $9.8\ \text{ms}^{-2}$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9\ \text{ms}^{-2}$, is close to: (Radius of earth $=6.4\times10^6$ m)

1. A.$2.6\times10^6$ m✓
2. B.$6.4\times10^6$ m
3. C.$1.6\times10^6$ m
4. D.$9.0\times10^6$ m

Correct answer: A

The ratio of surface tensions of mercury and water is given to be 7.5 while the ratio of their densities is 13.6. Their contact angles, with glass, are close to 135° and 0°, respectively. It is observed that mercury gets depressed by an amount $h$ in a capillary tube of radius $r_1$, while water rises by the same amount $h$ in a capillary tube of radius $r_2$. The ratio, $(r_1/r_2)$, is then close to:

1. A.2/5✓
2. B.4/5
3. C.2/3
4. D.3/5

Correct answer: A

$n$ moles of an ideal gas with constant volume heat capacity $C_V$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is:

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

Correct answer: D

The displacement of a damped harmonic oscillator is given by $x(t)=e^{-0.1t}\cos(10\pi t+\varphi)$. Here $t$ is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:

1. A.4 s
2. B.27 s
3. C.7 s✓
4. D.13 s

Correct answer: C

A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in $\text{ms}^{-1}$, (Given speed of sound $=300$ m/s)

1. A.12, 18✓
2. B.16, 14
3. C.8, 18
4. D.12, 16

Correct answer: A

Figure shows charge ($q$) versus voltage ($V$) graph for series and parallel combination of two given capacitors. The capacitances are:

1. A.50 $\mu$F and 30 $\mu$F
2. B.20 $\mu$F and 30 $\mu$F
3. C.40 $\mu$F and 10 $\mu$F✓
4. D.60 $\mu$F and 40 $\mu$F

Correct answer: C

A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by 20°C is: [Given that $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$]

1. A.700 J
2. B.748 J✓
3. C.374 J
4. D.350 J

Correct answer: B

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $xy$-plane centred at origin. A point charge $q$ is moving towards the ring along the $z$-axis and has speed $v$ at $z=4a$. The minimum value of $v$ such that it crosses the origin is:

1. A.$\sqrt{\frac{2}{m}}\left(\frac{1}{15}\frac{q^2}{4\pi\epsilon_0 a}\right)^{1/2}$
2. B.$\sqrt{\frac{2}{m}}\left(\frac{2}{15}\frac{q^2}{4\pi\epsilon_0 a}\right)^{1/2}$✓
3. C.$\sqrt{\frac{2}{m}}\left(\frac{1}{5}\frac{q^2}{4\pi\epsilon_0 a}\right)^{1/2}$
4. D.$\sqrt{\frac{2}{m}}\left(\frac{4}{15}\frac{q^2}{4\pi\epsilon_0 a}\right)^{1/2}$

Correct answer: B

In the given circuit, an ideal voltmeter connected across the 10 $\Omega$ resistance reads 2 V. The internal resistance $r$, of each cell is:

1. A.1 $\Omega$
2. B.1.5 $\Omega$
3. C.0.5 $\Omega$✓
4. D.0 $\Omega$

Correct answer: C

A current of 5 A passes through a copper conductor (resistivity $=1.7\times10^{-8}\ \Omega$m) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is $1.1\times10^{-3}$ m/s.

1. A.1.5 m$^2$/Vs
2. B.1.8 m$^2$/Vs
3. C.1.0 m$^2$/Vs✓
4. D.1.3 m$^2$/Vs

Correct answer: C

A proton, an electron, and a Helium nucleus, have the same energy. They are in circular orbits in a plane due to magnetic field perpendicular to the plane. Let $r_p$, $r_e$ and $r_{He}$ be their respective radii, then,

1. A.$r_e<r_p<r_{He}$
2. B.$r_e>r_p>r_{He}$
3. C.$r_e>r_p=r_{He}$
4. D.$r_e<r_p=r_{He}$✓

Correct answer: D

Two wires A & B are carrying currents $I_1$ & $I_2$ as shown in the figure. The separation between them is $d$. A third wire C carrying a current $I$ is to be kept parallel to them at a distance $x$ from A such that the net force acting on it is zero. The possible values of $x$ are:

1. A.$x=\pm\frac{I_1d}{(I_1-I_2)}$✓
2. B.$x=\left(\frac{I_1}{I_1+I_2}\right)d$ and $x=\frac{I_2}{(I_1-I_2)}d$
3. C.$x=\left(\frac{I_1}{I_1-I_2}\right)d$ and $x=\frac{I_2}{(I_1+I_2)}d$
4. D.$x=\left(\frac{I_2}{I_1+I_2}\right)d$ and $x=\left(\frac{I_2}{I_1-I_2}\right)d$

Correct answer: A

A transformer consisting of 300 turns in the primary and 150 turns in the secondary gives output power of 2.2 kW. If the current in the secondary coil is 10 A, then the input voltage and current in the primary coil are:

1. A.440 V and 20 A
2. B.220 V and 10 A
3. C.440 V and 5 A✓
4. D.220 V and 20 A

Correct answer: C

The electric field of a plane electromagnetic wave is given by $\vec{E}=E_0\hat{i}\cos(kz)\cos(\omega t)$ The corresponding magnetic field $\vec{B}$ is then given by:

1. A.$\vec{B}=\frac{E_0}{C}\hat{j}\sin(kz)\sin(\omega t)$✓
2. B.$\vec{B}=\frac{E_0}{C}\hat{j}\sin(kz)\cos(\omega t)$
3. C.$\vec{B}=\frac{E_0}{C}\hat{j}\cos(kz)\sin(\omega t)$
4. D.$\vec{B}=\frac{E_0}{C}\hat{k}\sin(kz)\cos(\omega t)$

Correct answer: A

One plano-convex and one plano-concave lens of same radius of curvature '$R$' but of different materials are joined side by side as shown in the figure. If the refractive index of the material of 1 is $\mu_1$ and that of 2 is $\mu_2$, then the focal length of the combination is:

1. A.$\frac{R}{\mu_1-\mu_2}$✓
2. B.$\frac{2R}{\mu_1-\mu_2}$
3. C.$\frac{R}{2(\mu_1-\mu_2)}$
4. D.$\frac{R}{2-(\mu_1-\mu_2)}$

Correct answer: A

A ray of light AO in vacuum is incident on a glass slab at angle 60° and refracted at angle 30° along OB as shown in the figure. The optical path length of light ray from A to B is:

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

Correct answer: C

In a photoelectric effect experiment the threshold wavelength of light is 380 nm. If the wavelength of incident light is 260 nm, the maximum kinetic energy of emitted electrons will be: Given $E\ (\text{in eV})=\frac{1237}{\lambda\ (\text{in nm})}$

1. A.3.0 eV
2. B.4.5 eV
3. C.15.1 eV
4. D.1.5 eV✓

Correct answer: D

Two radioactive materials A and B have decay constants $10\lambda$ and $\lambda$, respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of A to that of B will be $1/e$ after a time:

1. A.$\frac{1}{10\lambda}$
2. B.$\frac{1}{11\lambda}$
3. C.$\frac{11}{10\lambda}$
4. D.$\frac{1}{9\lambda}$✓

Correct answer: D

An npn transistor operates as a common emitter amplifier, with a power gain of 60 dB. The input circuit resistance is 100 $\Omega$ and the output load resistance is 10 k$\Omega$. The common emitter current gain $\beta$ is:

1. A.$6\times10^2$
2. B.60
3. C.$10^2$✓
4. D.$10^4$

Correct answer: C

A message signal of frequency 100 MHz and peak voltage 100 V is used to execute amplitude modulation on a carrier wave of frequency 300 GHz and peak voltage 400 V. The modulation index and difference between the two side band frequencies are:

1. A.0.25 ; $1\times10^8$ Hz
2. B.4 ; $2\times10^8$ Hz
3. C.0.25 ; $2\times10^8$ Hz✓
4. D.4 ; $1\times10^8$ Hz

Correct answer: C

In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure. Sl. No. 1: R = 1000 $\Omega$, $l$ = 60 cm Sl. No. 2: R = 100 $\Omega$, $l$ = 13 cm Sl. No. 3: R = 10 $\Omega$, $l$ = 1.5 cm Sl. No. 4: R = 1 $\Omega$, $l$ = 1.0 cm Which of the readings is inconsistent?

1. A.1
2. B.2
3. C.3
4. D.4✓

Correct answer: D

The increasing order of the reactivity of the following compounds towards electrophilic aromatic substitution reactions is: (I) chlorobenzene, (II) toluene, (III) acetophenone

1. A.II < I < III
2. B.I < III < II
3. C.III < I < II✓
4. D.III < II < I

Correct answer: C

Major products of the following reaction are: Benzaldehyde ($C_6H_5CHO$) + HCHO $\xrightarrow{\text{(i) 50\% NaOH; (ii) }H_3O^+}$

1. A.$CH_3OH$ and benzoic acid ($C_6H_5COOH$)
2. B.HCOOH and benzyl alcohol ($C_6H_5CH_2OH$)✓
3. C.Benzyl alcohol ($C_6H_5CH_2OH$) and benzoic acid ($C_6H_5COOH$)
4. D.$CH_3OH$ and $HCO_2H$

Correct answer: B

Amylopectin is composed of:

1. A.$\alpha$-D-glucose, $C_1-C_4$ and $C_2-C_6$ linkages
2. B.$\beta$-D-glucose, $C_1-C_4$ and $C_1-C_6$ linkages
3. C.$\alpha$-D-glucose, $C_1-C_4$ and $C_1-C_6$ linkages✓
4. D.$\beta$-D-glucose, $C_1-C_4$ and $C_2-C_6$ linkages

Correct answer: C

The principle of column chromatography is:

1. A.Differential adsorption of the substances on the solid phase.✓
2. B.Differential absorption of the substances on the solid phase.
3. C.Capillary action.
4. D.Gravitational force.

Correct answer: A

Increasing rate of $S_N1$ reaction in the following compounds is: (A) 1-iodo-1-phenylethane; (B) 1-iodo-1-(3-methoxyphenyl)ethane; (C) 1-iodo-1-(4-methylphenyl)ethane; (D) 1-iodo-1-(4-methoxyphenyl)ethane

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

Correct answer: B

Ethylamine ($C_2H_5NH_2$) can be obtained from N-ethylphthalimide on treatment with:

1. A.$CaH_2$
2. B.$NaBH_4$
3. C.$H_2O$
4. D.$NH_2NH_2$✓

Correct answer: D

The major product of the following reaction is: 3-Methoxyphenyl–CH(CN)–O–CH(CH$_3$)$_2$ (the benzylic carbon carries a CN group and an isopropoxy group; the ring carries OCH$_3$ meta to it) $\xrightarrow[\Delta]{\text{HI (excess)}}$

1. A.3-Hydroxyphenyl–CH(OH)–CN (ring OH, benzylic OH)
2. B.3-Hydroxyphenyl–CH(I)–CN (ring OH, benzylic I)✓
3. C.3-Iodophenyl–CH(OH)–CN (ring I, benzylic OH)
4. D.3-Iodophenyl–CH(I)–CN (ring I, benzylic I)

Correct answer: B

The major product of the following reaction is: $CH_3-CH(CH_3)-CH(Br)-CH_3 \xrightarrow{CH_3OH}$

1. A.$CH_3-CH(CH_3)-CH(OCH_3)-CH_3$
2. B.$CH_3-C(CH_3)=CH-CH_3$
3. C.$CH_3-C(CH_3)(OCH_3)-CH_2CH_3$✓
4. D.$CH_3-CH(CH_3)-CH=CH_2$

Correct answer: C

Which of the following is a condensation polymer?

1. A.Buna - S
2. B.Nylon 6, 6✓
3. C.Teflon
4. D.Neoprene

Correct answer: B

The major product of the following reaction is: $CH_3CH(OH)CH_2CH_2NH_2 \xrightarrow[\text{triethylamine}]{\text{ethyl formate (1 equiv.)}}$

1. A.$CH_3CH(OCHO)CH_2CH_2NH_2$ (O-formyl ester)
2. B.$CH_3CH(OH)CH_2CH_2NHCHO$✓
3. C.$CH_3CH=CH-CH_2NH_2$
4. D.$CH_3-CH(OH)-CH=CH_2$

Correct answer: B

The isoelectronic set of ions is:

1. A.$N^{3-}$, $O^{2-}$, $F^-$ and $Na^+$✓
2. B.$Li^+$, $Na^+$, $O^{2-}$ and $F^-$
3. C.$N^{3-}$, $Li^+$, $Mg^{2+}$ and $O^{2-}$
4. D.$F^-$, $Li^+$, $Na^+$ and $Mg^{2+}$

Correct answer: A

Match the refining methods (Column I) with metals (Column II). Column I (Refining methods): (I) Liquation (II) Zone Refining (III) Mond Process (IV) Van Arkel Method Column II (Metals): (a) Zr (b) Ni (c) Sn (d) Ga

1. A.(I)-(c); (II)-(a); (III)-(b); (IV)-(d)
2. B.(I)-(c); (II)-(d); (III)-(b); (IV)-(a)✓
3. C.(I)-(b); (II)-(c); (III)-(d); (IV)-(a)
4. D.(I)-(b); (II)-(d); (III)-(a); (IV)-(c)

Correct answer: B

The synonym for water gas when used in the production of methanol is:

1. A.fuel gas
2. B.natural gas
3. C.laughing gas
4. D.syn gas✓

Correct answer: D

The alloy used in the construction of aircrafts is:

1. A.Mg - Al✓
2. B.Mg - Sn
3. C.Mg - Zn
4. D.Mg - Mn

Correct answer: A

The oxoacid of sulphur that does not contain bond between sulphur atoms is:

1. A.$H_2S_2O_4$
2. B.$H_2S_4O_6$
3. C.$H_2S_2O_3$
4. D.$H_2S_2O_7$✓

Correct answer: D

The correct order of catenation is:

1. A.Si > Sn > C > Ge
2. B.Ge > Sn > Si > C
3. C.C > Si > Ge $\approx$ Sn✓
4. D.C > Sn > Si $\approx$ Ge

Correct answer: C

Consider the hydrated ions of $Ti^{2+}$, $V^{2+}$, $Ti^{3+}$, and $Sc^{3+}$. The correct order of their spin-only magnetic moments is:

1. A.$Sc^{3+}<Ti^{3+}<Ti^{2+}<V^{2+}$✓
2. B.$Ti^{3+}<Ti^{2+}<Sc^{3+}<V^{2+}$
3. C.$Sc^{3+}<Ti^{3+}<V^{2+}<Ti^{2+}$
4. D.$V^{2+}<Ti^{2+}<Ti^{3+}<Sc^{3+}$

Correct answer: A

Three complexes, $[CoCl(NH_3)_5]^{2+}$ (I), $[Co(NH_3)_5H_2O]^{3+}$ (II) and $[Co(NH_3)_6]^{3+}$ (III) absorb light in the visible region. The correct order of the wavelength of light absorbed by them is:

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

Correct answer: B

The species that can have a trans-isomer is: (en = ethane-1, 2-diamine, ox = oxalate)

1. A.$[Pt(en)Cl_2]$
2. B.$[Zn(en)Cl_2]$
3. C.$[Cr(en)_2(ox)]^+$
4. D.$[Pt(en)_2Cl_2]^{2+}$✓

Correct answer: D

The regions of the atmosphere, where clouds form and where we live, respectively, are:

1. A.Troposphere and Troposphere✓
2. B.Troposphere and Stratosphere
3. C.Stratosphere and Troposphere
4. D.Stratosphere and Stratosphere

Correct answer: A

At 300 K and 1 atmospheric pressure, 10 mL of a hydrocarbon required 55 mL of $O_2$ for complete combustion, and 40 mL of $CO_2$ is formed. The formula of the hydrocarbon is:

1. A.$C_4H_6$✓
2. B.$C_4H_8$
3. C.$C_4H_{10}$
4. D.$C_4H_7Cl$

Correct answer: A

Consider the following table: Gas A: $a=642.32$ kPa dm$^6$ mol$^{-1}$, $b=0.05196$ dm$^3$ mol$^{-1}$ Gas B: $a=155.21$, $b=0.04136$ Gas C: $a=431.91$, $b=0.05196$ Gas D: $a=155.21$, $b=0.4382$ $a$ and $b$ are van der Waals constants. The correct statement about the gases is:

1. A.Gas C will occupy lesser volume than gas A; gas B will be more compressible than gas D
2. B.Gas C will occupy more volume than gas A; gas B will be lesser compressible than gas D
3. C.Gas C will occupy more volume than gas A; gas B will be more compressible than gas D✓
4. D.Gas C will occupy lesser volume than gas A; gas B will be lesser compressible than gas D

Correct answer: C

The graph between $|\psi|^2$ and $r$ (radial distance) is shown below. This represents:

1. A.1s orbital
2. B.2s orbital✓
3. C.3s orbital
4. D.2p orbital

Correct answer: B

During the change of $O_2$ to $O_2^-$, the incoming electron goes to the orbital:

1. A.$\sigma^*2p_z$
2. B.$\pi^*2p_x$✓
3. C.$\pi 2p_x$
4. D.$\pi 2p_y$

Correct answer: B

A process will be spontaneous at all temperatures if:

1. A.$\Delta H>0$ and $\Delta S>0$
2. B.$\Delta H<0$ and $\Delta S<0$
3. C.$\Delta H<0$ and $\Delta S>0$✓
4. D.$\Delta H>0$ and $\Delta S<0$

Correct answer: C

At room temperature, a dilute solution of urea is prepared by dissolving 0.60 g of urea in 360 g of water. If the vapour pressure of pure water at this temperature is 35 mmHg, lowering of vapour pressure will be: (molar mass of urea $=60\ \text{g mol}^{-1}$)

1. A.0.027 mmHg
2. B.0.031 mmHg
3. C.0.017 mmHg✓
4. D.0.028 mmHg

Correct answer: C

Consider the following statements (a) The pH of a mixture containing 400 mL of 0.1 M $H_2SO_4$ and 400 mL of 0.1 M NaOH will be approximately 1.3. (b) Ionic product of water is temperature dependent. (c) A monobasic acid with $K_a=10^{-5}$ has a pH = 5. The degree of dissociation of this acid is 50%. (d) The Le Chatelier's principle is not applicable to common-ion effect. The correct statements are:

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

Correct answer: A

Consider the statements S1 and S2: S1: Conductivity always increases with decrease in the concentration of electrolyte. S2: Molar conductivity always increases with decrease in the concentration of electrolyte. The correct option among the following is:

1. A.Both S1 and S2 are correct
2. B.Both S1 and S2 are wrong
3. C.S1 is wrong and S2 is correct✓
4. D.S1 is correct and S2 is wrong

Correct answer: C

A bacterial infection in an internal wound grows as $N'(t)=N_0\exp(t)$, where the time $t$ is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as $\frac{dN}{dt}=-5N^2$. What will be the plot of $\frac{N_0}{N}$ vs. $t$ after 1 hour?

1. A.Graph 1: $N_0/N$ first decreases then rises (a U-shaped curve) with $t$
2. B.Graph 2: $N_0/N$ increases linearly with $t$ (straight line with positive slope)✓
3. C.Graph 3: $N_0/N$ rises slowly and then abruptly steeply (two line segments)
4. D.Graph 4: $N_0/N$ decreases linearly with $t$ (straight line with negative slope)

Correct answer: B

A gas undergoes physical adsorption on a surface and follows the given Freundlich adsorption isotherm equation $\frac{x}{m}=kp^{0.5}$ Adsorption of the gas increases with:

1. A.Increase in p and increase in T
2. B.Increase in p and decrease in T✓
3. C.Decrease in p and increase in T
4. D.Decrease in p and decrease in T

Correct answer: B

Let $f(x)=x^2$, $x\in\mathbf{R}$. For any $A\subseteq\mathbf{R}$, define $g(A)=\{x\in\mathbf{R}: f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true?

1. A.$f(g(S))\neq f(S)$
2. B.$f(g(S))=S$
3. C.$g(f(S))\neq S$
4. D.$g(f(S))=g(S)$✓

Correct answer: D

If $a>0$ and $z=\frac{(1+i)^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to:

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

Correct answer: A

If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2+x\sin\theta-2\sin\theta=0$, $\theta\in\left(0,\frac{\pi}{2}\right)$, then $\frac{\alpha^{12}+\beta^{12}}{(\alpha^{-12}+\beta^{-12})(\alpha-\beta)^{24}}$ is equal to:

1. A.$\frac{2^6}{(\sin\theta+8)^{12}}$
2. B.$\frac{2^{12}}{(\sin\theta-8)^6}$
3. C.$\frac{2^{12}}{(\sin\theta+8)^{12}}$✓
4. D.$\frac{2^{12}}{(\sin\theta-4)^{12}}$

Correct answer: C

If $\Delta_1=\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$ and $\Delta_2=\begin{vmatrix} x & \sin2\theta & \cos2\theta \\ -\sin2\theta & -x & 1 \\ \cos2\theta & 1 & x \end{vmatrix}$, $x\neq0$; then for all $\theta\in\left(0,\frac{\pi}{2}\right)$:

1. A.$\Delta_1-\Delta_2=-2x^3$
2. B.$\Delta_1+\Delta_2=-2x^3$✓
3. C.$\Delta_1+\Delta_2=-2(x^3+x-1)$
4. D.$\Delta_1-\Delta_2=x(\cos2\theta-\cos4\theta)$

Correct answer: B

If the system of linear equations $x+y+z=5$ $x+2y+2z=6$ $x+3y+\lambda z=\mu$, $(\lambda,\mu\in\mathbf{R})$, has infinitely many solutions, then the value of $\lambda+\mu$ is:

1. A.12
2. B.10✓
3. C.9
4. D.7

Correct answer: B

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is:

1. A.48
2. B.60✓
3. C.72
4. D.36

Correct answer: B

If $a_1, a_2, a_3, \ldots, a_n$ are in A.P. and $a_1+a_4+a_7+\ldots+a_{16}=114$, then $a_1+a_6+a_{11}+a_{16}$ is equal to:

1. A.38
2. B.64
3. C.76✓
4. D.98

Correct answer: C

$\frac{3\times1^3}{1^2}+\frac{5\times(1^3+2^3)}{1^2+2^2}+\frac{7\times(1^3+2^3+3^3)}{1^2+2^2+3^2}+\ldots$ upto $10^{\text{th}}$ term, is:

1. A.600
2. B.620
3. C.660✓
4. D.680

Correct answer: C

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1+ax+bx^2)(1-3x)^{15}$ in powers of $x$, then the ordered pair $(a,b)$ is equal to:

1. A.(−21, 714)
2. B.(−54, 315)
3. C.(28, 861)
4. D.(28, 315)✓

Correct answer: D

If $\lim_{x\to1}\frac{x^4-1}{x-1}=\lim_{x\to k}\frac{x^3-k^3}{x^2-k^2}$, then $k$ is:

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

Correct answer: B

If $f(x)=\begin{cases}\frac{\sin(p+1)x+\sin x}{x}, & x<0\\ q, & x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}}, & x>0\end{cases}$ is continuous at $x=0$, then the ordered pair $(p,q)$ is equal to:

1. A.$\left(-\frac{3}{2},\frac{1}{2}\right)$✓
2. B.$\left(\frac{5}{2},\frac{1}{2}\right)$
3. C.$\left(-\frac{1}{2},\frac{3}{2}\right)$
4. D.$\left(-\frac{3}{2},-\frac{1}{2}\right)$

Correct answer: A

Let $f:\mathbf{R}\to\mathbf{R}$ be differentiable at $c\in\mathbf{R}$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c$, $g$ is:

1. A.differentiable if $f'(c)\neq0$
2. B.differentiable if $f'(c)=0$✓
3. C.not differentiable if $f'(c)=0$
4. D.not differentiable

Correct answer: B

Let $f(x)=e^x-x$ and $g(x)=x^2-x$, $\forall x\in\mathbf{R}$. Then the set of all $x\in\mathbf{R}$, where the function $h(x)=(f\circ g)(x)$ is increasing, is:

1. A.$\left[\frac{-1}{2},0\right]\cup[1,\infty)$
2. B.$\left[0,\frac{1}{2}\right]\cup[1,\infty)$✓
3. C.$\left[-1,\frac{-1}{2}\right]\cup\left[\frac{1}{2},\infty\right)$
4. D.$[0,\infty)$

Correct answer: B

If $\int\frac{dx}{(x^2-2x+10)^2}=A\left(\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2x+10}\right)+C$ where $C$ is a constant of integration, then:

1. A.$A=\frac{1}{81}$ and $f(x)=3(x-1)$
2. B.$A=\frac{1}{27}$ and $f(x)=9(x-1)$
3. C.$A=\frac{1}{54}$ and $f(x)=3(x-1)$✓
4. D.$A=\frac{1}{54}$ and $f(x)=9(x-1)^2$

Correct answer: C

The value of $\int_0^{2\pi}[\sin2x(1+\cos3x)]\,dx$, where $[t]$ denotes the greatest integer function, is:

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

Correct answer: D

$\lim_{n\to\infty}\left(\frac{(n+1)^{1/3}}{n^{4/3}}+\frac{(n+2)^{1/3}}{n^{4/3}}+\ldots+\frac{(2n)^{1/3}}{n^{4/3}}\right)$ is equal to: