JEE Main 2018 — 15 Apr Morning (online CBT) (Previous Year Paper)

The relative error in the determination of the surface area of a sphere is $\alpha$. Then the relative error in the determination of its volume is :

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

Correct answer: A

The velocity-time graphs of a car and a scooter are shown in the figure. (i) The difference between the distance travelled by the car and the scooter in 15 s and (ii) the time at which the car will catch up with the scooter are, respectively.

1. A.112.5 m and 22.5 s✓
2. B.337.5 m and 25 s
3. C.112.5 m and 15 s
4. D.225.5 m and 10 s

Correct answer: A

A given object takes n times more time to slide down a $45^\circ$ rough inclined plane as it takes to slide down a perfectly smooth $45^\circ$ incline. The coefficient of kinetic friction between the object and the incline is :

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

Correct answer: B

An automobile, travelling at 40 km/h, can be stopped at a distance of 40 m by applying brakes. If the same automobile is travelling at 80 km/h, the minimum stopping distance, in metres, is (assume no skidding) :

1. A.75 m
2. B.100 m
3. C.150 m
4. D.160 m✓

Correct answer: D

A particle is oscillating on the X-axis with an amplitude 2 cm about the point $x_0 = 10$ cm, with a frequency $\omega$. A concave mirror of focal length 5 cm is placed at the origin (see figure). Identify the correct statements. (A) The image executes periodic motion. (B) The image executes non-periodic motion. (C) The turning points of the image are asymmetric w.r.t. the image of the point at $x = 10$ cm. (D) The distance between the turning points of the oscillation of the image is $\frac{100}{21}$ cm.

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

Correct answer: B

A force of 40 N acts on a point B at the end of an L-shaped object, as shown in the figure. The angle $\theta$ that will produce maximum moment of the force about point A is given by :

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

Correct answer: B

A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius $\frac{R}{2}$, and the other mass, in a circular orbit of radius $\frac{3R}{2}$. The difference between the final and initial total energies is :

1. A.$-\frac{GMm}{2R}$
2. B.$+\frac{GMm}{6R}$
3. C.$\frac{GMm}{2R}$
4. D.$-\frac{GMm}{6R}$✓

Correct answer: D

Take the mean distance of the moon and the sun from the earth to be $0.4\times10^6$ km and $150\times10^6$ km respectively. Their masses are $8\times10^{22}$ kg and $2\times10^{30}$ kg respectively. The radius of the earth is 6400 km. Let $\Delta F_1$ be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and $\Delta F_2$ be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then, the number closest to $\frac{\Delta F_1}{\Delta F_2}$ is :

1. A.2✓
2. B.$10^{-2}$
3. C.0.6
4. D.6

Correct answer: A

A thin uniform tube is bent into a circle of radius r in the vertical plane. Equal volumes of two immiscible liquids, whose densities are $\rho_1$ and $\rho_2$ $(\rho_1 > \rho_2)$, fill half the circle. The angle $\theta$ between the radius vector passing through the common interface and the vertical is :

1. A.$\theta = \tan^{-1}\pi\left(\frac{\rho_1}{\rho_2}\right)$
2. B.$\theta = \tan^{-1}\frac{\pi}{2}\left(\frac{\rho_2}{\rho_1}\right)$
3. C.$\theta = \tan^{-1}\left[\frac{\pi}{2}\left(\frac{\rho_1-\rho_2}{\rho_1+\rho_2}\right)\right]$✓
4. D.$\theta = \tan^{-1}\frac{\pi}{2}\left(\frac{\rho_1+\rho_2}{\rho_1-\rho_2}\right)$

Correct answer: C

A Carnot's engine works as a refrigerator between 250 K and 300 K. It receives 500 cal heat from the reservoir at the lower temperature. The amount of work done in each cycle to operate the refrigerator is :

1. A.420 J✓
2. B.772 J
3. C.2100 J
4. D.2520 J

Correct answer: A

One mole of an ideal monoatomic gas is compressed isothermally in a rigid vessel to double its pressure at room temperature, $27^\circ$C. The work done on the gas will be :

1. A.300 R
2. B.300 R ln 6
3. C.300 R ln 2✓
4. D.300 R ln 7

Correct answer: C

A body of mass M and charge q is connected to a spring of spring constant k. It is oscillating along x-direction about its equilibrium position, taken to be at $x = 0$, with an amplitude A. An electric field E is applied along the x-direction. Which of the following statements is correct ?

1. A.The new equilibrium position is at a distance $\frac{qE}{2k}$ from $x = 0$.
2. B.The total energy of the system is $\frac{1}{2}m\omega^2A^2 + \frac{1}{2}\frac{q^2E^2}{k}$.
3. C.The total energy of the system is $\frac{1}{2}m\omega^2A^2 - \frac{1}{2}\frac{q^2E^2}{k}$.✓
4. D.The new equilibrium position is at a distance $\frac{2qE}{k}$ from $x = 0$.

Correct answer: C

A tuning fork vibrates with frequency 256 Hz and gives one beat per second with the third normal mode of vibration of an open pipe. What is the length of the pipe ? (Speed of sound in air is 340 m s$^{-1}$)

1. A.220 cm
2. B.190 cm
3. C.180 cm
4. D.200 cm✓

Correct answer: D

A charge Q is placed at a distance $a/2$ above the centre of the square surface of edge a as shown in the figure. The electric flux through the square surface is :

1. A.$\frac{Q}{\epsilon_0}$
2. B.$\frac{Q}{2\epsilon_0}$
3. C.$\frac{Q}{3\epsilon_0}$
4. D.$\frac{Q}{6\epsilon_0}$✓

Correct answer: D

The equivalent capacitance between A and B in the circuit given below, is :

1. A.2.4 $\mu$F✓
2. B.4.9 $\mu$F
3. C.3.6 $\mu$F
4. D.5.4 $\mu$F

Correct answer: A

In the given circuit all resistances are of value R ohm each. The equivalent resistance between A and B is :

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

Correct answer: A

In a meter bridge, as shown in the figure, it is given that resistance Y = 12.5 $\Omega$ and that the balance is obtained at a distance 39.5 cm from end A (by Jockey J). After interchanging the resistances X and Y, a new balance point is found at a distance $l_2$ from end A. What are the values of X and $l_2$ ?

1. A.8.16 $\Omega$ and 60.5 cm✓
2. B.19.15 $\Omega$ and 39.5 cm
3. C.8.16 $\Omega$ and 39.5 cm
4. D.19.15 $\Omega$ and 60.5 cm

Correct answer: A

A Helmholtz coil has a pair of loops, each with N turns and radius R. They are placed coaxially at distance R and the same current I flows through the loops in the same direction. The magnitude of magnetic field at P, midway between the centres A and C, is given by [Refer to figure given below] :

1. A.$\frac{8N\mu_0 I}{5^{1/2}R}$
2. B.$\frac{8N\mu_0 I}{5^{3/2}R}$✓
3. C.$\frac{4N\mu_0 I}{5^{1/2}R}$
4. D.$\frac{4N\mu_0 I}{5^{3/2}R}$

Correct answer: B

The B-H curve for a ferromagnet is shown in the figure. The ferromagnet is placed inside a long solenoid with 1000 turns/cm. The current that should be passed in the solenoid to demagnetise the ferromagnet completely is :

1. A.1 mA✓
2. B.2 mA
3. C.20 $\mu$A
4. D.40 $\mu$A

Correct answer: A

An ideal capacitor of capacitance 0.2 $\mu$F is charged to a potential difference of 10 V. The charging battery is then disconnected. The capacitor is then connected to an ideal inductor of self inductance 0.5 mH. The current at a time when the potential difference across the capacitor is 5 V, is :

1. A.0.34 A
2. B.0.25 A
3. C.0.17 A✓
4. D.0.15 A

Correct answer: C

A monochromatic beam of light has a frequency $\nu = \frac{3}{2\pi}\times10^{12}$ Hz and is propagating along the direction $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$. It is polarized along the $\hat{k}$ direction. The acceptable form for the magnetic field is :

1. A.$\frac{E_0}{C}\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos\left[10^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)\cdot\vec{r} - (3\times10^{12})t\right]$✓
2. B.$\frac{E_0}{C}\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos\left[10^4\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cdot\vec{r} - (3\times10^{12})t\right]$
3. C.$\frac{E_0}{C}\hat{k}\cos\left[10^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)\cdot\vec{r} + (3\times10^{12})t\right]$
4. D.$\frac{E_0}{C}\left(\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\right)\cos\left[10^4\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)\cdot\vec{r} + (3\times10^{12})t\right]$

Correct answer: A

A planoconvex lens becomes an optical system of 28 cm focal length when its plane surface is silvered and illuminated from left to right as shown in Fig-A. If the same lens is instead silvered on the curved surface and illuminated from other side as in Fig. B, it acts like an optical system of focal length 10 cm. The refractive index of the material of lens is :

1. A.1.50
2. B.1.55✓
3. C.1.75
4. D.1.51

Correct answer: B

Light of wavelength 550 nm falls normally on a slit of width $22.0\times10^{-5}$ cm. The angular position of the second minima from the central maximum will be (in radians) :

1. A.$\frac{\pi}{12}$
2. B.$\frac{\pi}{8}$
3. C.$\frac{\pi}{6}$✓
4. D.$\frac{\pi}{4}$

Correct answer: C

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are $\lambda_1$ and $\lambda_2$, their de Broglie wavelength in the frame of reference attached to their centre of mass is :

1. A.$\lambda_{CM} = \lambda_1 = \lambda_2$
2. B.$\lambda_{CM} = \frac{2\lambda_1\lambda_2}{\sqrt{\lambda_1^2+\lambda_2^2}}$✓
3. C.$\frac{1}{\lambda_{CM}} = \frac{1}{\lambda_1}+\frac{1}{\lambda_2}$
4. D.$\lambda_{CM} = \left(\frac{\lambda_1+\lambda_2}{2}\right)$

Correct answer: B

The energy required to remove the electron from a singly ionized Helium atom is 2.2 times the energy required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is :

1. A.20 eV
2. B.34 eV
3. C.79 eV✓
4. D.109 eV

Correct answer: C

A solution containing active cobalt ${}^{60}_{27}Co$ having activity of 0.8 $\mu$Ci and decay constant $\lambda$ is injected in an animal's body. If 1 cm$^3$ of blood is drawn from the animal's body after 10 hrs of injection, the activity found was 300 decays per minute. What is the volume of blood that is flowing in the body ? (1 Ci = $3.7\times10^{10}$ decays per second and at t = 10 hrs $e^{-\lambda t} = 0.84$)

1. A.6 liters
2. B.7 liters
3. C.4 liters
4. D.5 liters✓

Correct answer: D

In a common emitter configuration with suitable bias, it is given that $R_L$ is the load resistance and $R_{BE}$ is small signal dynamic resistance (input side). Then, voltage gain, current gain and power gain are given, respectively, by : $\beta$ is current gain, $I_B$, $I_C$ and $I_E$ are respectively base, collector and emitter currents.

1. A.$\beta\frac{R_L}{R_{BE}}$, $\frac{\Delta I_C}{\Delta I_B}$, $\beta^2\frac{R_L}{R_{BE}}$✓
2. B.$\beta\frac{R_L}{R_{BE}}$, $\frac{\Delta I_E}{\Delta I_B}$, $\beta^2\frac{R_L}{R_{BE}}$
3. C.$\beta^2\frac{R_L}{R_{BE}}$, $\frac{\Delta I_C}{\Delta I_E}$, $\beta^2\frac{R_L}{R_{BE}}$
4. D.$\beta^2\frac{R_L}{R_{BE}}$, $\frac{\Delta I_C}{\Delta I_B}$, $\beta\frac{R_L}{R_{BE}}$

Correct answer: A

The number of amplitude modulated broadcast stations that can be accommodated in a 300 kHz band width for the highest modulating frequency 15 kHz will be :

1. A.20
2. B.15
3. C.10✓
4. D.8

Correct answer: C

A uniform rod AB is suspended from a point X, at a variable distance x from A, as shown. To make the rod horizontal, a mass m is suspended from its end A. A set of (m, x) values is recorded. The appropriate variables that give a straight line, when plotted, are :

1. A.m, x
2. B.m, $\frac{1}{x}$✓
3. C.m, $\frac{1}{x^2}$
4. D.m, $x^2$

Correct answer: B

In a screw gauge, 5 complete rotations of the screw cause it to move a linear distance of 0.25 cm. There are 100 circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of 4 main scale divisions and 30 circular scale divisions. Assuming negligible zero error, the thickness of the wire is :

1. A.0.4300 cm
2. B.0.2150 cm✓
3. C.0.3150 cm
4. D.0.0430 cm

Correct answer: B

A sample of $NaClO_3$ is converted by heat to NaCl with a loss of 0.16 g of oxygen. The residue is dissolved in water and precipitated as AgCl. The mass of AgCl (in g) obtained will be : (Given : Molar mass of AgCl = 143.5 g mol$^{-1}$)

1. A.0.35
2. B.0.41
3. C.0.48✓
4. D.0.54

Correct answer: C

Which of the following arrangements shows the schematic alignment of magnetic moments of antiferromagnetic substance ?

1. A.$\uparrow\ \uparrow\ \uparrow\ \uparrow\ \uparrow\ \uparrow$
2. B.$\uparrow\ \downarrow\ \downarrow\ \downarrow\ \downarrow\ \uparrow$
3. C.$\uparrow\ \downarrow\ \uparrow\ \downarrow\ \uparrow\ \downarrow$✓
4. D.$\uparrow\ \uparrow\ \downarrow\ \uparrow\ \uparrow\ \downarrow$

Correct answer: C

The minimum volume of water required to dissolve 0.1 g lead(II) chloride to get a saturated solution ($K_{sp}$ of $PbCl_2 = 3.2\times10^{-8}$; atomic mass of Pb = 207 u) is :

1. A.0.36 L
2. B.17.98 L
3. C.0.18 L✓
4. D.1.798 L

Correct answer: C

In which of the following reactions, an increase in the volume of the container will favour the formation of products ?

1. A.$2NO_2(g) \rightleftharpoons 2NO(g) + O_2(g)$✓
2. B.$H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$
3. C.$4NH_3(g) + 5O_2(g) \rightleftharpoons 4NO(g) + 6H_2O(l)$
4. D.$3O_2(g) \rightleftharpoons 2O_3(g)$

Correct answer: A

When an electric current is passed through acidified water, 112 mL of hydrogen gas at N.T.P. was collected at the cathode in 965 seconds. The current passed, in ampere, is :

1. A.1.0✓
2. B.0.5
3. C.0.1
4. D.2.0

Correct answer: A

$N_2O_5$ decomposes to $NO_2$ and $O_2$ and follows first order kinetics. After 50 minutes, the pressure inside the vessel increases from 50 mmHg to 87.5 mmHg. The pressure of the gaseous mixture after 100 minute at constant temperature will be :

1. A.175.0 mmHg
2. B.116.25 mmHg
3. C.136.25 mmHg
4. D.106.25 mmHg✓

Correct answer: D

For which of the following reactions, $\Delta H$ is equal to $\Delta U$ ?

1. A.$N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$
2. B.$2HI(g) \rightarrow H_2(g) + I_2(g)$✓
3. C.$2NO_2(g) \rightarrow N_2O_4(g)$
4. D.$2SO_2(g) + O_2(g) \rightarrow 2SO_3(g)$

Correct answer: B

Which of the following statements about colloids is False ?

1. A.Freezing point of colloidal solution is lower than true solution at same concentration of a solute.✓
2. B.Colloidal particles can pass through ordinary filter paper.
3. C.When silver nitrate solution is added to potassium iodide solution, a negatively charged colloidal solution is formed.
4. D.When excess of electrolyte is added to colloidal solution, colloidal particle will be precipitated.

Correct answer: A

Ejection of the photoelectron from metal in the photoelectric effect experiment can be stopped by applying 0.5 V when the radiation of 250 nm is used. The work function of the metal is :

1. A.4 eV
2. B.4.5 eV✓
3. C.5 eV
4. D.5.5 eV

Correct answer: B

An ideal gas undergoes a cyclic process as shown in Figure. $\Delta U_{BC} = -5$ kJ mol$^{-1}$, $q_{AB} = 2$ kJ mol$^{-1}$, $W_{AB} = -5$ kJ mol$^{-1}$, $W_{CA} = 3$ kJ mol$^{-1}$. Heat absorbed by the system during process CA is :

1. A.$-5$ kJ mol$^{-1}$
2. B.$+5$ kJ mol$^{-1}$✓
3. C.18 kJ mol$^{-1}$
4. D.$-18$ kJ mol$^{-1}$

Correct answer: B

For $Na^+$, $Mg^{2+}$, $F^-$ and $O^{2-}$; the correct order of increasing ionic radii is :

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

Correct answer: C

In the molecular orbital diagram for the molecular ion, $N_2^+$, the number of electrons in the $\sigma_{2p}$ molecular orbital is :

1. A.0
2. B.1✓
3. C.2
4. D.3

Correct answer: B

Which of the following is a Lewis acid ?

1. A.$PH_3$
2. B.$B(CH_3)_3$✓
3. C.NaH
4. D.$NF_3$

Correct answer: B

In graphite and diamond, the percentage of p-characters of the hybrid orbitals in hybridisation are respectively :

1. A.33 and 25
2. B.33 and 75
3. C.50 and 75
4. D.67 and 75✓

Correct answer: D

A white sodium salt dissolves readily in water to give a solution which is neutral to litmus. When silver nitrate solution is added to the aforementioned solution, a white precipitate is obtained which does not dissolve in dil. nitric acid. The anion is :

1. A.$CO_3^{2-}$
2. B.$SO_4^{2-}$
3. C.$Cl^-$✓
4. D.$S^{2-}$

Correct answer: C

Identify the pair in which the geometry of the species is T-shape and square-pyramidal, respectively :

1. A.$ClF_3$ and $IO_4^-$
2. B.$ICl_2^-$ and $ICl_5$
3. C.$XeOF_2$ and $XeOF_4$✓
4. D.$IO_3^-$ and $IO_2F_2^-$

Correct answer: C

The correct combination is :

1. A.$[Ni(CN)_4]^{2-}$ — tetrahedral ; $[Ni(CO)_4]$ — paramagnetic
2. B.$[NiCl_4]^{2-}$ — paramagnetic ; $[Ni(CO)_4]$ — tetrahedral✓
3. C.$[NiCl_4]^{2-}$ — square-planar ; $[Ni(CN)_4]^{2-}$ — paramagnetic
4. D.$[NiCl_4]^{2-}$ — diamagnetic ; $[Ni(CO)_4]$ — square-planar

Correct answer: B

$H - N \overset{(I)}{- - -} N \overset{(II)}{- - -} N$ In hydrogen azide (above) the bond orders of bonds (I) and (II) are :

1. A.(I) $< 2$ ; (II) $> 2$✓
2. B.(I) $> 2$ ; (II) $< 2$
3. C.(I) $> 2$ ; (II) $> 2$
4. D.(I) $< 2$ ; (II) $< 2$

Correct answer: A

The decreasing order of bond angles in $BF_3$, $NH_3$, $PF_3$ and $I_3^-$ is :

1. A.$I_3^- > NH_3 > PF_3 > BF_3$
2. B.$I_3^- > BF_3 > NH_3 > PF_3$✓
3. C.$BF_3 > I_3^- > PF_3 > NH_3$
4. D.$BF_3 > NH_3 > PF_3 > I_3^-$

Correct answer: B

Xenon hexafluoride on partial hydrolysis produces compounds 'X' and 'Y'. Compounds 'X' and 'Y' and the oxidation state of Xe are respectively :

1. A.$XeO_2(+4)$ and $XeO_3(+6)$
2. B.$XeOF_4(+6)$ and $XeO_3(+6)$
3. C.$XeO_2F_2(+6)$ and $XeO_2(+4)$
4. D.$XeOF_4(+6)$ and $XeO_2F_2(+6)$✓

Correct answer: D

The IUPAC name of the following compound is : [bond-line structure: $CH_3CH_2-CH(CH_2CH_3)-C(CH_3)=CH-CH_3$]

1. A.4-methyl-3-ethylhex-4-ene
2. B.3-ethyl-4-methylhex-4-ene
3. C.4-ethyl-3-methylhex-2-ene✓
4. D.4,4-diethyl-3-methylbut-2-ene

Correct answer: C

Which of the following is the correct structure of Adenosine ?

1. A.Purine base with $-NH_2$ at C-6 (adenine), ribose attached at N-9 (the five-membered imidazole-ring nitrogen)✓
2. B.Purine base with $-NH_2$ at C-6, ribose attached at N-3 (a six-membered-ring nitrogen)
3. C.Purine base with $-NH_2$ at C-2, ribose attached at N-9 (imidazole-ring nitrogen)
4. D.Purine base with $-NH_2$ at C-2, ribose attached at N-3 (a six-membered-ring nitrogen)

Correct answer: A

The main reduction product of the following compound with $NaBH_4$ in methanol is : [structure: a cyclohexene ring carrying a ketone at C-1 and, on the adjacent carbon C-2, a $-CH_2-C(=O)NMe_2$ side chain; the ring C=C is between C-3 and C-4]

1. A.Ring ketone reduced to $-OH$ and the amide carbonyl also reduced (side chain drawn as an amine, $-CH_2-NMe_2$ type, with no C=O); ring C=C retained
2. B.Ring ketone reduced to $-OH$, amide $-CH_2C(=O)NMe_2$ retained, ring C=C also reduced (saturated cyclohexanol)
3. C.Ring ketone reduced to $-OH$, amide $-CH_2C(=O)NMe_2$ retained, ring C=C retained✓
4. D.Ketone and amide retained, ring C=C reduced (saturated cyclohexanone)

Correct answer: C

The increasing order of nitration of the following compounds is : (a) aniline $C_6H_5NH_2$ (b) chlorobenzene $C_6H_5Cl$ (c) anisole $C_6H_5OCH_3$ (d) toluene $C_6H_5CH_3$

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

Correct answer: D

The correct match between items of List-I and List-II is : List-I: (A) Coloured impurity (B) Mixture of o-nitrophenol and p-nitrophenol (C) Crude Naphtha (D) Mixture of glycerol and sugars. List-II: (P) Steam distillation (Q) Fractional distillation (R) Charcoal treatment (S) Distillation under reduced pressure

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

Correct answer: C

The copolymer formed by addition polymerization of styrene and acrylonitrile in the presence of peroxide is :

1. A.$\left[-CH_2-CH(C_6H_5)-CH(CN)-CH_2-\right]_n$
2. B.$\left[-C(C_6H_5)(CH_3)-CH(CN)-CH_2-\right]_n$
3. C.$\left[-CH(C_6H_5)-CH_2-CH_2-CH(CN)-\right]_n$ (phenyl and CN groups on the terminal CH carbons of the repeat unit)
4. D.$\left[-CH_2-CH(C_6H_5)-CH_2-CH(CN)-\right]_n$✓

Correct answer: D

Which of the following will most readily give the dehydrohalogenation product ?

1. A.$CH_2=CH-C(Br)=CH-CH_3$ (3-bromopenta-1,3-diene)
2. B.Bromobenzene, $C_6H_5Br$
3. C.1-Bromocyclohexene
4. D.$CH_2=CH-C(Br)=CH-C_6H_5$✓

Correct answer: D

Which of the following will not exist in zwitter ionic form at pH = 7 ?

1. A.2-Aminocyclohexane-1-carboxylic acid ($NH_2$ and $COOH$ on adjacent ring carbons)
2. B.2-Aminocyclohexane-1-sulphonic acid ($NH_2$ and $SO_3H$ on adjacent ring carbons)
3. C.2-Aminobenzenesulphonic acid (orthanilic acid)
4. D.N-Acetylalanine, $CH_3C(=O)-NH-CH(CH_3)-CO_2H$✓

Correct answer: D

The major product of the following reaction is : 3-methoxyphenol $\xrightarrow[\text{(ii) } AlCl_3\text{ (anhyd.)}]{\text{(i) } ClCH_2CH_2COCl}$

1. A.7-Methoxychroman-2-one (7-methoxy-3,4-dihydrocoumarin)✓
2. B.7-Methoxychroman-4-one
3. C.5-Methoxychroman-2-one
4. D.5-Methoxychroman-4-one

Correct answer: A

The reagent(s) required for the following conversion are : [3-(ethoxycarbonyl)-5-cyanocyclohexane-1-carboxylic acid, i.e. a cyclohexane bearing $-CO_2Et$, $-CO_2H$ and $-CN$ at alternate carbons] $\rightarrow$ [cyclohexane bearing $-CH_2OH$ (from $-CO_2H$), $-CO_2H$ (from $-CO_2Et$) and $-CHO$ (from $-CN$)]

1. A.(i) $B_2H_6$ (ii) $SnCl_2/HCl$ (iii) $H_3O^+$✓
2. B.(i) $LiAlH_4$ (ii) $H_3O^+$
3. C.(i) $B_2H_6$ (ii) DIBAL-H (iii) $H_3O^+$
4. D.(i) $NaBH_4$ (ii) Raney Ni/$H_2$ (iii) $H_3O^+$

Correct answer: A

Consider the following two binary relations on the set $A = \{a, b, c\}$ : $R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$. Then :

1. A.both $R_1$ and $R_2$ are not symmetric.
2. B.$R_1$ is not symmetric but it is transitive.
3. C.$R_2$ is symmetric but it is not transitive.✓
4. D.both $R_1$ and $R_2$ are transitive.

Correct answer: C

If $\lambda \in \mathbb{R}$ is such that the sum of the cubes of the roots of the equation, $x^2 + (2-\lambda)x + (10-\lambda) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :

1. A.$4\sqrt{2}$
2. B.$2\sqrt{5}$✓
3. C.$2\sqrt{7}$
4. D.20

Correct answer: B

The set of all $\alpha \in \mathbb{R}$, for which $w = \frac{1+(1-8\alpha)z}{1-z}$ is a purely imaginary number, for all $z \in \mathbb{C}$ satisfying $|z| = 1$ and $\text{Re}\,z \ne 1$, is :

1. A.an empty set
2. B.$\{0\}$✓
3. C.$\left\{0, \frac{1}{4}, -\frac{1}{4}\right\}$
4. D.equal to $\mathbb{R}$

Correct answer: B

Let A be a matrix such that $A\cdot\begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}$ is a scalar matrix and $|3A| = 108$. Then $A^2$ equals :

1. A.$\begin{bmatrix}4 & -32\\ 0 & 36\end{bmatrix}$
2. B.$\begin{bmatrix}36 & 0\\ -32 & 4\end{bmatrix}$
3. C.$\begin{bmatrix}4 & 0\\ -32 & 36\end{bmatrix}$
4. D.$\begin{bmatrix}36 & -32\\ 0 & 4\end{bmatrix}$✓

Correct answer: D

Let S be the set of all real values of k for which the system of linear equations $x + y + z = 2$, $2x + y - z = 3$, $3x + 2y + kz = 4$ has a unique solution. Then S is :

1. A.an empty set
2. B.equal to $\{0\}$
3. C.equal to $\mathbb{R}$
4. D.equal to $\mathbb{R} - \{0\}$✓

Correct answer: D

n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is :

1. A.6
2. B.7✓
3. C.8
4. D.9

Correct answer: B

If n is the degree of the polynomial, $\left[\frac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]^8 + \left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]^8$ and m is the coefficient of $x^n$ in it, then the ordered pair (n, m) is equal to :

1. A.$(24, (10)^8)$
2. B.$(8, 5(10)^4)$
3. C.$(12, (20)^4)$✓
4. D.$(12, 8(10)^4)$

Correct answer: C

If $x_1, x_2, \ldots, x_n$ and $\frac{1}{h_1}, \frac{1}{h_2}, \ldots, \frac{1}{h_n}$ are two A.P.s such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$, then $x_5 \cdot h_{10}$ equals :

1. A.2560✓
2. B.2650
3. C.3200
4. D.1600

Correct answer: A

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

1. A.$(-\infty, -10]$
2. B.$(-10, 0)$
3. C.$(0, 10)$✓
4. D.$[10, \infty)$

Correct answer: C

If $f(x) = \begin{vmatrix}\cos x & x & 1\\ 2\sin x & x^2 & 2x\\ \tan x & x & 1\end{vmatrix}$, then $\lim_{x\to0}\frac{f'(x)}{x}$

1. A.does not exist.
2. B.exists and is equal to 2.
3. C.exists and is equal to 0.
4. D.exists and is equal to $-2$.✓

Correct answer: D

If $x^2 + y^2 + \sin y = 4$, then the value of $\frac{d^2y}{dx^2}$ at the point $(-2, 0)$ is :

1. A.$-34$✓
2. B.$-32$
3. C.4
4. D.$-2$

Correct answer: A

Let $S = \{(\lambda, \mu) \in \mathbb{R}\times\mathbb{R} : f(t) = (|\lambda|e^{|t|} - \mu)\cdot\sin(2|t|),\ t \in \mathbb{R}$, is a differentiable function$\}$. Then S is a subset of :

1. A.$\mathbb{R}\times[0, \infty)$✓
2. B.$[0, \infty)\times\mathbb{R}$
3. C.$\mathbb{R}\times(-\infty, 0)$
4. D.$(-\infty, 0)\times\mathbb{R}$

Correct answer: A

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm$^2$) of this cone is :

1. A.$6\sqrt{2}\pi$
2. B.$6\sqrt{3}\pi$
3. C.$8\sqrt{2}\pi$
4. D.$8\sqrt{3}\pi$✓

Correct answer: D

If $f\left(\frac{x-4}{x+2}\right) = 2x + 1$, $\left(x \in \mathbb{R} - \{1, -2\}\right)$, then $\int f(x)\,dx$ is equal to : (where C is a constant of integration)

1. A.$12\log_e|1-x| + 3x + C$
2. B.$-12\log_e|1-x| - 3x + C$✓
3. C.$12\log_e|1-x| - 3x + C$
4. D.$-12\log_e|1-x| + 3x + C$

Correct answer: B

The area (in sq. units) of the region $\{x \in \mathbb{R} : x \ge 0, y \ge 0, y \ge x - 2 \text{ and } y \le \sqrt{x}\}$, is :

1. A.$\frac{13}{3}$
2. B.$\frac{8}{3}$
3. C.$\frac{10}{3}$✓
4. D.$\frac{5}{3}$

Correct answer: C

The value of the integral $\int_{-\pi/2}^{\pi/2}\sin^4 x\left(1 + \log\left(\frac{2+\sin x}{2-\sin x}\right)\right)dx$ is :

1. A.0
2. B.$\frac{3}{4}$
3. C.$\frac{3}{8}\pi$✓
4. D.$\frac{3}{16}\pi$

Correct answer: C

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 2y = f(x)$, where $f(x) = \begin{cases}1, & x \in [0, 1]\\ 0, & \text{otherwise}\end{cases}$. If $y(0) = 0$, then $y\left(\frac{3}{2}\right)$ is :

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

Correct answer: D

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, $x + y = 5$ and $x = 4$. Then area of $\Delta ABC$ (in sq. units) is :

1. A.12
2. B.4
3. C.5
4. D.9✓

Correct answer: D

A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $y - 4x + 3 = 0$, then its radius is equal to :