JEE Main 2019 — 8 Apr Shift 1 (Previous Year Paper)

In SI units, the dimensions of $\sqrt{\dfrac{\epsilon_0}{\mu_0}}$ is:

1. A.$AT^2M^{-1}L^{-1}$
2. B.$A^2T^3M^{-1}L^{-2}$✓
3. C.$A^{-1}TML^3$
4. D.$AT^{-3}ML^{3/2}$

Correct answer: B

Ship A is sailing towards north-east with velocity $\vec{v} = 30\hat{i} + 50\hat{j}$ km/hr where $\hat{i}$ points east and $\hat{j}$, north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. A will be at minimum distance from B in:

1. A.4.2 hrs.
2. B.2.2 hrs.
3. C.2.6 hrs.✓
4. D.3.2 hrs.

Correct answer: C

Four particles A, B, C and D with masses $m_A = m$, $m_B = 2m$, $m_C = 3m$ and $m_D = 4m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

1. A.Zero
2. B.$a(\hat{i} + \hat{j})$
3. C.$\dfrac{a}{5}(\hat{i} + \hat{j})$
4. D.$\dfrac{a}{5}(\hat{i} - \hat{j})$✓

Correct answer: D

A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is:

1. A.5 J
2. B.6.5 J✓
3. C.4 J
4. D.2.5 J

Correct answer: B

A thin circular plate of mass M and radius R has its density varying as $\rho(r) = \rho_0 r$ with $\rho_0$ as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = aMR^2$. The value of the coefficient a is:

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

Correct answer: A

Four identical particles of mass M are located at the corners of a square of side 'a'. What should be their speed if each of them revolves under the influence of others' gravitational field in a circular orbit circumscribing the square?

1. A.$1.21\sqrt{\dfrac{GM}{a}}$
2. B.$1.35\sqrt{\dfrac{GM}{a}}$
3. C.$1.41\sqrt{\dfrac{GM}{a}}$
4. D.$1.16\sqrt{\dfrac{GM}{a}}$✓

Correct answer: D

A boy's catapult is made of rubber cord which is 42 cm long, with 6 mm diameter of cross-section and of negligible mass. The boy keeps a stone weighing 0.02 kg on it and stretches the cord by 20 cm by applying a constant force. When released, the stone flies off with a velocity of 20 m s$^{-1}$. Neglect the change in the area of cross-section of the cord while stretched. The Young's modulus of rubber is closest to:

1. A.$10^6$ N m$^{-2}$✓
2. B.$10^8$ N m$^{-2}$
3. C.$10^4$ N m$^{-2}$
4. D.$10^3$ N m$^{-2}$

Correct answer: A

Water from a pipe is coming at a rate of 100 liters per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of: (density of water = 1000 kg/m$^3$, coefficient of viscosity of water = 1 mPa s)

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

Correct answer: A

Two identical beakers A and B contain equal volumes of two different liquids at 60°C each and left to cool down. Liquid in A has density of $8 \times 10^2$ kg/m$^3$ and specific heat of 2000 J kg$^{-1}$ K$^{-1}$ while liquid in B has density of $10^3$ kg m$^{-3}$ and specific heat of 4000 J kg$^{-1}$ K$^{-1}$. Which of the following best describes their temperature versus time graph schematically? (assume the emissivity of both the beakers to be the same)

1. A.Graph: T vs t starting at 60°C; curve A lies above curve B (B cools faster)
2. B.Graph: T vs t starting at 60°C; curve B lies above curve A (A cools faster)✓
3. C.Graph: T vs t starting at 60°C; a single common curve labelled "A and B"
4. D.Graph: T vs t starting at 60°C; A and B both fall steeply, A slightly above B, both levelling off quickly

Correct answer: B

A thermally insulated vessel contains 150 g of water at 0°C. Then the air from the vessel is pumped out adiabatically. A fraction of water turns into ice and the rest evaporates at 0°C itself. The mass of evaporated water will be closest to: (Latent heat of vaporization of water = $2.10 \times 10^6$ J kg$^{-1}$ and Latent heat of Fusion of water = $3.36 \times 10^5$ J kg$^{-1}$)

1. A.150 g
2. B.20 g✓
3. C.35 g
4. D.130 g

Correct answer: B

A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that $g = 3.1\pi$ m s$^{-2}$, what will be the tensile stress that would be developed in the wire?

1. A.$5.2\times10^6$ N m$^{-2}$
2. B.$3.1\times10^6$ N m$^{-2}$✓
3. C.$6.2\times10^6$ N m$^{-2}$
4. D.$4.8\times10^6$ N m$^{-2}$

Correct answer: B

A wire of length 2L, is made by joining two wires A and B of same length but different radii r and 2r and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire A is p and that in B is q then the ratio p : q is:

1. A.1 : 2✓
2. B.3 : 5
3. C.4 : 9
4. D.1 : 4

Correct answer: A

The bob of a simple pendulum has mass 2 g and a charge of 5.0 μC. It is at rest in a uniform horizontal electric field of intensity 2000 V/m. At equilibrium, the angle that the pendulum makes with the vertical is: (take g = 10 m/s$^2$)

1. A.$\tan^{-1}(0.5)$✓
2. B.$\tan^{-1}(5.0)$
3. C.$\tan^{-1}(2.0)$
4. D.$\tan^{-1}(0.2)$

Correct answer: A

A solid conducting sphere, having a charge Q, is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of $-4Q$, the new potential difference between the same two surfaces is:

1. A.V✓
2. B.2V
3. C.−2V
4. D.4V

Correct answer: A

Voltage rating of a parallel plate capacitor is 500 V. Its dielectric can withstand a maximum electric field of $10^6$ V/m. The plate area is $10^{-4}$ m$^2$. What is the dielectric constant if the capacitance is 15 pF? (given $\epsilon_0 = 8.86\times10^{-12}$ C$^2$/N m$^2$)

1. A.6.2
2. B.3.8
3. C.4.5
4. D.8.5✓

Correct answer: D

For the circuit shown, with $R_1 = 1.0\ \Omega$, $R_2 = 2.0\ \Omega$, $E_1 = 2$ V and $E_2 = E_3 = 4$ V, the potential difference between the points 'a' and 'b' is approximately (in V):

1. A.3.7
2. B.3.3✓
3. C.2.7
4. D.2.3

Correct answer: B

A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic field $B\hat{i}$. The torque on the coil due to the magnetic field is:

1. A.$B\pi r^2 IN$✓
2. B.$\dfrac{B\pi r^2 I}{N}$
3. C.Zero
4. D.$\dfrac{Br^2 I}{\pi N}$

Correct answer: A

An alternating voltage $v(t) = 220\sin 100\pi t$ volt is applied to a purely resistive load of 50 Ω. The time taken for the current to rise from half of the peak value to the peak value is:

1. A.2.2 ms
2. B.3.3 ms✓
3. C.5 ms
4. D.7.2 ms

Correct answer: B

A thin strip 10 cm long is on a U shaped wire of negligible resistance and it is connected to a spring of spring constant 0.5 N m$^{-1}$ (see figure). The assembly is kept in a uniform magnetic field of 0.1 T. If the strip is pulled from its equilibrium position and released, the number of oscillations it performs before its amplitude decreases by a factor of e is N. If the mass of the strip is 50 grams, its resistance 10 Ω and air drag negligible, N will be close to:

1. A.1000
2. B.5000✓
3. C.10000
4. D.50000

Correct answer: B

A 20 Henry inductor coil is connected to a 10 ohm resistance in series as shown in figure. The time at which rate of dissipation of energy (Joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor, is:

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

Correct answer: B

A plane electromagnetic wave travels in free space along the x-direction. The electric field component of the wave at a particular point of space and time is E = 6 V m$^{-1}$ along y-direction. Its corresponding magnetic field component, B would be:

1. A.$2\times10^{-8}$ T along z-direction✓
2. B.$6\times10^{-8}$ T along x-direction
3. C.$2\times10^{-8}$ T along y-direction
4. D.$6\times10^{-8}$ T along z-direction

Correct answer: A

An upright object is placed at a distance of 40 cm in front of a convergent lens of focal length 20 cm. A convergent mirror of focal length 10 cm is placed at a distance of 60 cm on the other side of the lens. The position and size of the final image will be:

1. A.40 cm from the convergent mirror, same size as the object
2. B.20 cm from the convergent mirror, same size as the object✓
3. C.20 cm from the convergent mirror, twice the size of the object
4. D.40 cm from the convergent lens, twice the size of the object

Correct answer: B

In an interference experiment the ratio of amplitudes of coherent waves is $\dfrac{a_1}{a_2} = \dfrac{1}{3}$. The ratio of maximum and minimum intensities of fringes will be:

1. A.2
2. B.4✓
3. C.9
4. D.18

Correct answer: B

In figure, the optical fiber is $l = 2$ m long and has a diameter of $d = 20$ μm. If a ray of light is incident on one end of the fiber at angle $\theta_1 = 40°$, the number of reflections it makes before emerging from the other end is close to: (refractive index of fiber is 1.31 and $\sin 40° = 0.64$)

1. A.57000✓
2. B.66000
3. C.55000
4. D.45000

Correct answer: A

Two particles move at right angle to each other. Their de Broglie wavelengths are $\lambda_1$ and $\lambda_2$ respectively. The particles suffer perfectly inelastic collision. The de Broglie wavelength $\lambda$, of the final particle, is given by:

1. A.$\lambda = \dfrac{\lambda_1 + \lambda_2}{2}$
2. B.$\lambda = \sqrt{\lambda_1\lambda_2}$
3. C.$\dfrac{2}{\lambda} = \dfrac{1}{\lambda_1} + \dfrac{1}{\lambda_2}$
4. D.$\dfrac{1}{\lambda^2} = \dfrac{1}{\lambda_1^2} + \dfrac{1}{\lambda_2^2}$✓

Correct answer: D

Radiation coming from transitions n = 2 to n = 1 of hydrogen atoms fall on He$^+$ ions in n = 1 and n = 2 states. The possible transition of helium ions as they absorb energy from the radiation is:

1. A.$n = 2 \rightarrow n = 5$
2. B.$n = 1 \rightarrow n = 4$
3. C.$n = 2 \rightarrow n = 3$
4. D.$n = 2 \rightarrow n = 4$✓

Correct answer: D

The reverse breakdown voltage of a Zener diode is 5.6 V in the given circuit. The current $I_z$ through the Zener is:

1. A.15 mA
2. B.10 mA✓
3. C.7 mA
4. D.17 mA

Correct answer: B

The wavelength of the carrier waves in a modern optical fiber communication network is close to:

1. A.1500 nm✓
2. B.900 nm
3. C.2400 nm
4. D.600 nm

Correct answer: A

A 200 Ω resistor has a certain color code. If one replaces the red color by green in the code, the new resistance will be:

1. A.100 Ω
2. B.300 Ω
3. C.400 Ω
4. D.500 Ω✓

Correct answer: D

An organic compound neither reacts with neutral ferric chloride solution nor with Fehling solution. It however, reacts with Grignard reagent and gives positive iodoform test. The compound is:

1. A.2-Hydroxyacetophenone (benzene ring bearing $-COCH_3$ and an ortho $-OH$)
2. B.2-Methoxybenzaldehyde (benzene ring bearing $-OCH_3$ and an ortho $-CHO$)
3. C.1-(2-Methoxyphenyl)propan-1-one (benzene ring bearing $-COC_2H_5$ and an ortho $-OCH_3$)
4. D.1-[2-(1-Hydroxyethyl)phenyl]propan-1-one (benzene ring bearing $-CH(OH)CH_3$ and an ortho $-COC_2H_5$)✓

Correct answer: D

Maltose on treatment with dilute HCl gives:

1. A.D-Galactose
2. B.D-Glucose and D-Fructose
3. C.D-Fructose
4. D.D-Glucose✓

Correct answer: D

Coupling of benzene diazonium chloride with 1-naphthol in alkaline medium will give:

1. A.4-(Phenylazo)-1-naphthol (azo group at C-4, para to the OH)✓
2. B.2-(Phenylazo)-1-naphthol (azo group at C-2, ortho to the OH)
3. C.3-(Phenylazo)-1-naphthol (azo group at C-3, meta to the OH)
4. D.5-(Phenylazo)-1-naphthol (azo group on the other ring, at C-5)

Correct answer: A

The major product of the following reaction is: phthalic anhydride + chlorobenzene $\xrightarrow[\text{(ii) } H_2O]{\text{(i) } AlCl_3,\ \text{heat}}$

1. A.2-Chloroanthraquinone
2. B.2-(4-Chlorobenzoyl)benzoic acid (Cl para to the carbonyl)✓
3. C.2-(3-Chlorobenzoyl)benzoic acid (Cl meta to the carbonyl)
4. D.1-Chloroanthraquinone

Correct answer: B

The major product of the following reaction is: 3-methoxystyrene (benzene ring bearing $-OCH_3$ and a meta $-CH=CH_2$) $\xrightarrow[\text{heat}]{\text{Conc. HBr (excess)}}$

1. A.3-(2-Bromoethyl)phenol (ring bearing $-OH$ and meta $-CH_2CH_2Br$)
2. B.3-(1-Bromoethyl)phenol (ring bearing $-OH$ and meta $-CHBrCH_3$)✓
3. C.1-Bromo-3-(2-bromoethyl)benzene (ring bearing $-Br$ and meta $-CH_2CH_2Br$)
4. D.1-Bromo-3-(1-bromoethyl)benzene (ring bearing $-Br$ and meta $-CHBrCH_3$)

Correct answer: B

The major product of the following reaction is: $C_6H_5COCH_2Br$ (phenacyl bromide) $\xrightarrow[\text{MeOH, 25°C}]{NaBH_4}$

1. A.2-Phenyloxirane (styrene oxide)✓
2. B.$C_6H_5CH(OH)CH_2Br$
3. C.$C_6H_5CH(OH)CH_2OMe$
4. D.$C_6H_5CH=CHOMe$

Correct answer: A

Which of the following amines can be prepared by Gabriel phthalimide reaction?

1. A.neo-pentylamine
2. B.n-butylamine✓
3. C.t-butylamine
4. D.triethylamine

Correct answer: B

The IUPAC name of the following compound is: $H_3C-CH(CH_3)-CH(OH)-CH_2-COOH$

1. A.3-Hydroxy-4-methylpentanoic acid✓
2. B.4-Methyl-3-hydroxypentanoic acid
3. C.4,4-Dimethyl-3-hydroxybutanoic acid
4. D.2-Methyl-3-hydroxypentan-5-oic acid

Correct answer: A

An organic compound 'X' showing the following solubility profile is: water - insoluble; 5% HCl - insoluble; 10% NaOH - soluble; 10% NaHCO$_3$ - insoluble.

1. A.Oleic acid
2. B.m-Cresol✓
3. C.o-Toluidine
4. D.Benzamide

Correct answer: B

In the following compounds, the decreasing order of basic strength will be:

1. A.$(C_2H_5)_2NH > C_2H_5NH_2 > NH_3$✓
2. B.$NH_3 > C_2H_5NH_2 > (C_2H_5)_2NH$
3. C.$(C_2H_5)_2NH > NH_3 > C_2H_5NH_2$
4. D.$C_2H_5NH_2 > NH_3 > (C_2H_5)_2NH$

Correct answer: A

The size of the iso-electronic species Cl$^-$, Ar and Ca$^{2+}$ is affected by:

1. A.Principal quantum number of valence shell
2. B.nuclear charge✓
3. C.azimuthal quantum number of valence shell
4. D.electron-electron interaction in the outer orbitals

Correct answer: B

With respect to an ore, Ellingham diagram helps to predict the feasibility of its

1. A.Thermal reduction✓
2. B.Electrolysis
3. C.Zone refining
4. D.Vapour phase refining

Correct answer: A

100 mL of a water sample contains 0.81 g of calcium bicarbonate and 0.73 g of magnesium bicarbonate. The hardness of this water sample expressed in terms of equivalents of CaCO$_3$ is: (molar mass of calcium bicarbonate is 162 g mol$^{-1}$ and magnesium bicarbonate is 146 g mol$^{-1}$)

1. A.1,000 ppm
2. B.10,000 ppm✓
3. C.100 ppm
4. D.5,000 ppm

Correct answer: B

The correct order of hydration enthalpies of alkali metal ions is:

1. A.$Li^+ > Na^+ > K^+ > Rb^+ > Cs^+$✓
2. B.$Li^+ > Na^+ > K^+ > Cs^+ > Rb^+$
3. C.$Na^+ > Li^+ > K^+ > Rb^+ > Cs^+$
4. D.$Na^+ > Li^+ > K^+ > Cs^+ > Rb^+$

Correct answer: A

Diborane (B$_2$H$_6$) reacts independently with O$_2$ and H$_2$O to produce, respectively:

1. A.$H_3BO_3$ and $B_2O_3$
2. B.$HBO_2$ and $H_3BO_3$
3. C.$B_2O_3$ and $H_3BO_3$✓
4. D.$B_2O_3$ and $[BH_4]^-$

Correct answer: C

The lanthanide ion that would show colour is:

1. A.$Gd^{3+}$
2. B.$Sm^{3+}$✓
3. C.$La^{3+}$
4. D.$Lu^{3+}$

Correct answer: B

The following ligand is:

1. A.bidentate
2. B.tridentate
3. C.tetradentate✓
4. D.hexadentate

Correct answer: C

The correct order of the spin-only magnetic moment of metal ions in the following low-spin complexes, $[V(CN)_6]^{4-}$, $[Fe(CN)_6]^{4-}$, $[Ru(NH_3)_6]^{3+}$, and $[Cr(NH_3)_6]^{2+}$, is:

1. A.$V^{2+} > Cr^{2+} > Ru^{3+} > Fe^{2+}$✓
2. B.$V^{2+} > Ru^{3+} > Cr^{2+} > Fe^{2+}$
3. C.$Cr^{2+} > Ru^{3+} > Fe^{2+} > V^{2+}$
4. D.$Cr^{2+} > V^{2+} > Ru^{3+} > Fe^{2+}$

Correct answer: A

Which is wrong with respect to our responsibility as a human being to protect our environment?

1. A.Setting up compost tin in gardens.
2. B.Using plastic bags.✓
3. C.Restricting the use of vehicles
4. D.Avoiding the use of floodlighted facilities.

Correct answer: B

Assertion: Ozone is destroyed by CFCs in the upper stratosphere. Reason: Ozone holes increase the amount of UV radiation reaching the earth.

1. A.Assertion and reason are correct, but the reason is not the explanation for the assertion.✓
2. B.Assertion is false, but the reason is correct.
3. C.Assertion and reason are both correct, and the reason is the correct explanation for the assertion.
4. D.Assertion and reason are incorrect.

Correct answer: A

In order to oxidise a mixture of one mole of each of $FeC_2O_4$, $Fe_2(C_2O_4)_3$, $FeSO_4$ and $Fe_2(SO_4)_3$ in acidic medium, the number of moles of KMnO$_4$ required is:

1. A.1
2. B.1.5
3. C.2✓
4. D.3

Correct answer: C

Element 'B' forms ccp structure and 'A' occupies half of the octahedral voids, while oxygen atoms occupy all the tetrahedral voids. The structure of bimetallic oxide is:

1. A.$A_2BO_4$
2. B.$AB_2O_4$✓
3. C.$A_4B_2O$
4. D.$A_2B_2O$

Correct answer: B

The quantum number of four electrons are given below: I. $n=4, l=2, m_l=-2, m_s=-\frac{1}{2}$; II. $n=3, l=2, m_l=1, m_s=+\frac{1}{2}$; III. $n=4, l=1, m_l=0, m_s=+\frac{1}{2}$; IV. $n=3, l=1, m_l=1, m_s=-\frac{1}{2}$. The correct order of their increasing energies will be:

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

Correct answer: D

For silver, $C_p$ (J K$^{-1}$ mol$^{-1}$) $= 23 + 0.01T$. If the temperature (T) of 3 moles of silver is raised from 300 K to 1000 K at 1 atm pressure, the value of $\Delta H$ will be close to:

1. A.13 kJ
2. B.21 kJ
3. C.16 kJ
4. D.62 kJ✓

Correct answer: D

Which one of the following equations does not correctly represent the first law of thermodynamics for the given processes involving an ideal gas? (Assume non-expansion work is zero)

1. A.Isothermal process: $q = -w$
2. B.Cyclic process: $q = -w$
3. C.Isochoric process: $\Delta U = q$
4. D.Adiabatic process: $\Delta U = -w$✓

Correct answer: D

The vapour pressures of pure liquids A and B are 400 and 600 mmHg, respectively at 298 K. On mixing the two liquids, the sum of their initial volumes is equal to the volume of the final mixture. The mole fraction of liquid B is 0.5 in the mixture. The vapour pressure of the final solution, the mole fractions of components A and B in vapour phase, respectively are:

1. A.500 mmHg, 0.4, 0.6✓
2. B.450 mmHg, 0.5, 0.5
3. C.500 mmHg, 0.5, 0.5
4. D.450 mmHg, 0.4, 0.6

Correct answer: A

If solubility product of $Zr_3(PO_4)_4$ is denoted by $K_{sp}$ and its molar solubility is denoted by S, then which of the following relation between S and $K_{sp}$ is correct?

1. A.$S = \left(\dfrac{K_{sp}}{144}\right)^{1/6}$
2. B.$S = \left(\dfrac{K_{sp}}{929}\right)^{1/9}$
3. C.$S = \left(\dfrac{K_{sp}}{6912}\right)^{1/7}$✓
4. D.$S = \left(\dfrac{K_{sp}}{216}\right)^{1/7}$

Correct answer: C

Given that $E^\ominus_{O_2/H_2O} = +1.23$ V; $E^\ominus_{S_2O_8^{2-}/SO_4^{2-}} = 2.05$ V; $E^\ominus_{Br_2/Br^-} = +1.09$ V; $E^\ominus_{Au^{3+}/Au} = +1.4$ V. The strongest oxidizing agent is:

1. A.$S_2O_8^{2-}$✓
2. B.$Au^{3+}$
3. C.$Br_2$
4. D.$O_2$

Correct answer: A

For the reaction $2A + B \rightarrow C$, the values of initial rate at different reactant concentrations are given in the table below. The rate law for the reaction is: [A] (mol L$^{-1}$), [B] (mol L$^{-1}$), Initial rate (mol L$^{-1}$ s$^{-1}$): (i) 0.05, 0.05, 0.045; (ii) 0.10, 0.05, 0.090; (iii) 0.20, 0.10, 0.72.

1. A.Rate $= k[A][B]$
2. B.Rate $= k[A]^2[B]$
3. C.Rate $= k[A][B]^2$✓
4. D.Rate $= k[A]^2[B]^2$

Correct answer: C

Adsorption of a gas follows Freundlich adsorption isotherm. x is the mass of the gas adsorbed on mass m of the adsorbent. The plot of $\log\frac{x}{m}$ versus $\log p$ is shown in the given graph. $\frac{x}{m}$ is proportional to:

1. A.$p^2$
2. B.$p^{3/2}$
3. C.$p^{2/3}$✓
4. D.$p^3$

Correct answer: C

If $f(x) = \log_e\left(\dfrac{1-x}{1+x}\right)$, $|x| < 1$, then $f\left(\dfrac{2x}{1+x^2}\right)$ is equal to:

1. A.$2f(x^2)$
2. B.$-2f(x)$
3. C.$2f(x)$✓
4. D.$(f(x))^2$

Correct answer: C

If $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$, then the least value of n for which $\left(\dfrac{\alpha}{\beta}\right)^n = 1$ is:

1. A.5
2. B.4✓
3. C.3
4. D.2

Correct answer: B

The sum of the solutions of the equation $|\sqrt{x} - 2| + \sqrt{x}(\sqrt{x} - 4) + 2 = 0$, $(x > 0)$ is equal to:

1. A.4
2. B.9
3. C.10✓
4. D.12

Correct answer: C

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $(\alpha \in \mathbb{R})$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then a value of $\alpha$ is:

1. A.0
2. B.$\dfrac{\pi}{64}$✓
3. C.$\dfrac{\pi}{32}$
4. D.$\dfrac{\pi}{16}$

Correct answer: B

The greatest value of $c \in \mathbb{R}$ for which the system of linear equations $x - cy - cz = 0$, $cx - y + cz = 0$, $cx + cy - z = 0$ has a non-trivial solution, is:

1. A.−1
2. B.0
3. C.$\dfrac{1}{2}$✓
4. D.2

Correct answer: C

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is:

1. A.160
2. B.162
3. C.175
4. D.180✓

Correct answer: D

The sum of the co-efficients of all even degree terms in x in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^6 + \left(x - \sqrt{x^3 - 1}\right)^6$, $(x > 1)$ is equal to:

1. A.24✓
2. B.26
3. C.29
4. D.32

Correct answer: A

The sum of all natural numbers 'n' such that $100 < n < 200$ and H.C.F.$(91, n) > 1$ is:

1. A.3203
2. B.3221
3. C.3121✓
4. D.3303

Correct answer: C

The sum of the series $2\cdot{}^{20}C_0 + 5\cdot{}^{20}C_1 + 8\cdot{}^{20}C_2 + 11\cdot{}^{20}C_3 + \ldots + 62\cdot{}^{20}C_{20}$ is equal to:

1. A.$2^{24}$
2. B.$2^{25}$✓
3. C.$2^{23}$
4. D.$2^{26}$

Correct answer: B

$\displaystyle\lim_{x\to 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$ equals:

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

Correct answer: A

If $2y = \left(\cot^{-1}\left(\dfrac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)\right)^2$, $x \in \left(0, \dfrac{\pi}{2}\right)$ then $\dfrac{dy}{dx}$ is equal to:

1. A.$\dfrac{\pi}{6} - x$
2. B.$\dfrac{\pi}{3} - x$
3. C.$x - \dfrac{\pi}{6}$✓
4. D.$2x - \dfrac{\pi}{3}$

Correct answer: C

Let $f: [0, 2] \rightarrow \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$, for all $x \in (0, 2)$. If $\phi(x) = f(x) + f(2 - x)$, then $\phi$ is:

1. A.increasing on (0, 1) and decreasing on (1, 2).
2. B.decreasing on (0, 1) and increasing on (1, 2).✓
3. C.increasing on (0, 2)
4. D.decreasing on (0, 2)

Correct answer: B

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in \mathbb{R}$, then:

1. A.$S_1 = \{-2, 0\};\ S_2 = \{1\}$
2. B.$S_1 = \{-2, 1\};\ S_2 = \{0\}$✓
3. C.$S_1 = \{-1\};\ S_2 = \{0, 2\}$
4. D.$S_1 = \{-2\};\ S_2 = \{0, 1\}$

Correct answer: B

$\displaystyle\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}}\,dx$ is equal to: (where c is a constant of integration.)

1. A.$x + 2\sin x + \sin 2x + c$✓
2. B.$x + 2\sin x + 2\sin 2x + c$
3. C.$2x + \sin x + \sin 2x + c$
4. D.$2x + \sin x + 2\sin 2x + c$

Correct answer: A

If $f(x) = \dfrac{2 - x\cos x}{2 + x\cos x}$ and $g(x) = \log_e x$, $(x > 0)$ then the value of the integral $\displaystyle\int_{-\pi/4}^{\pi/4} g(f(x))\,dx$ is:

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

Correct answer: C

The area (in sq. units) of the region $A = \{(x, y) \in \mathbb{R}\times\mathbb{R} \mid 0 \le x \le 3,\ 0 \le y \le 4,\ y \le x^2 + 3x\}$ is:

1. A.8
2. B.$\dfrac{26}{3}$
3. C.$\dfrac{59}{6}$✓
4. D.$\dfrac{53}{6}$

Correct answer: C

Let $y = y(x)$ be the solution of the differential equation, $(x^2 + 1)^2\dfrac{dy}{dx} + 2x(x^2 + 1)y = 1$ such that $y(0) = 0$. If $\sqrt{a}\,y(1) = \dfrac{\pi}{32}$, then the value of 'a' is:

1. A.$\dfrac{1}{16}$✓
2. B.$\dfrac{1}{2}$
3. C.1
4. D.$\dfrac{1}{4}$

Correct answer: A

Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of $\Delta AOP$ is 4, is:

1. A.$9x^2 - 8y^2 + 8y = 16$
2. B.$9x^2 + 8y^2 - 8y = 16$✓
3. C.$8x^2 - 9y^2 + 9y = 18$
4. D.$8x^2 + 9y^2 - 9y = 18$

Correct answer: B

A point on the straight line, $3x + 5y = 15$ which is equidistant from the coordinate axes will lie only in:

1. A.1st, 2nd and 4th quadrants
2. B.1st quadrant
3. C.1st and 2nd quadrants✓
4. D.4th quadrant

Correct answer: C

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n$, $n \in N$, where N is the set of all natural numbers, is:

1. A.105
2. B.210✓
3. C.320
4. D.160

Correct answer: B

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is:

1. A.$\dfrac{11}{4\sqrt{2}}$
2. B.$\dfrac{7}{4\sqrt{2}}$✓
3. C.$\dfrac{7}{8}$
4. D.2

Correct answer: B

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points (1, 2) and (a, b) are perpendicular to each other, then $a^2$ is equal to:

1. A.$\dfrac{2}{17}$✓
2. B.$\dfrac{128}{17}$
3. C.$\dfrac{4}{17}$
4. D.$\dfrac{64}{17}$

Correct answer: A

The length of the perpendicular from the point (2, −1, 4) on the straight line, $\dfrac{x+3}{10} = \dfrac{y-2}{-7} = \dfrac{z}{1}$ is:

1. A.less than 2
2. B.greater than 2 but less than 3
3. C.greater than 3 but less than 4✓
4. D.greater than 4

Correct answer: C

The equation of a plane containing the line of intersection of the planes $2x - y - 4 = 0$ and $y + 2z - 4 = 0$ and passing through the point (1, 1, 0) is:

1. A.$x - y - z = 0$✓
2. B.$x + 3y + z = 4$
3. C.$x - 3y - 2z = -2$
4. D.$2x - z = 2$

Correct answer: A

The magnitude of the projection of the vector $2\hat{i} + 3\hat{j} + \hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + 2\hat{j} + 3\hat{k}$, is:

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

Correct answer: D

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is:

1. A.40
2. B.49
3. C.45
4. D.48✓

Correct answer: D

Let A and B be two non-null events such that $A \subset B$. Then, which of the following statements is always correct?

1. A.$P(A|B) = 1$
2. B.$P(A|B) \le P(A)$
3. C.$P(A|B) \ge P(A)$✓
4. D.$P(A|B) = P(B) - P(A)$

Correct answer: C

If $\cos(\alpha + \beta) = \dfrac{3}{5}$, $\sin(\alpha - \beta) = \dfrac{5}{13}$ and $0 < \alpha, \beta < \dfrac{\pi}{4}$, then $\tan(2\alpha)$ is equal to:

1. A.$\dfrac{21}{16}$
2. B.$\dfrac{63}{16}$✓
3. C.$\dfrac{63}{52}$
4. D.$\dfrac{33}{52}$

Correct answer: B

If $\alpha = \cos^{-1}\left(\dfrac{3}{5}\right)$, $\beta = \tan^{-1}\left(\dfrac{1}{3}\right)$, where $0 < \alpha, \beta < \dfrac{\pi}{2}$, then $\alpha - \beta$ is equal to:

1. A.$\tan^{-1}\left(\dfrac{9}{14}\right)$
2. B.$\sin^{-1}\left(\dfrac{9}{5\sqrt{10}}\right)$✓
3. C.$\cos^{-1}\left(\dfrac{9}{5\sqrt{10}}\right)$
4. D.$\tan^{-1}\left(\dfrac{9}{5\sqrt{10}}\right)$