JEE Main 2019 — 8 Apr Shift 2 (Previous Year Paper)

If Surface tension (S), Moment of Inertia (I) and Planck's constant (h), were to be taken as the fundamental units, the dimensional formula for linear momentum would be:

1. A.$S^{1/2}I^{1/2}h^0$✓
2. B.$S^{3/2}I^{1/2}h^0$
3. C.$S^{1/2}I^{3/2}h^{-1}$
4. D.$S^{1/2}I^{1/2}h^{-1}$

Correct answer: A

Let $|\vec{A}_1| = 3$, $|\vec{A}_2| = 5$ and $|\vec{A}_1 + \vec{A}_2| = 5$. The value of $(2\vec{A}_1 + 3\vec{A}_2)\cdot(3\vec{A}_1 - 2\vec{A}_2)$ is:

1. A.−99.5
2. B.−106.5
3. C.−112.5
4. D.−118.5✓

Correct answer: D

A body of mass $m_1$ moving with an unknown velocity of $v_1\hat{i}$, undergoes a collinear collision with a body of mass $m_2$ moving with a velocity $v_2\hat{i}$. After collision, $m_1$ and $m_2$ move with velocities of $v_3\hat{i}$ and $v_4\hat{i}$, respectively. If $m_2 = 0.5m_1$ and $v_3 = 0.5v_1$, then $v_1$ is:

1. A.$v_4 - v_2$✓
2. B.$v_4 - \dfrac{v_2}{2}$
3. C.$v_4 + v_2$
4. D.$v_4 - \dfrac{v_2}{4}$

Correct answer: A

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{sph}$ and $h_{cyl}$ on the incline. The ratio $\dfrac{h_{sph}}{h_{cyl}}$ is given by:

1. A.1
2. B.$\dfrac{2}{\sqrt{5}}$
3. C.$\dfrac{14}{15}$✓
4. D.$\dfrac{4}{5}$

Correct answer: C

A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion HBGO is cut-off, the coordinates of the centre of mass of the remaining portion will be:

1. A.$\left(\dfrac{5a}{12}, \dfrac{5b}{12}\right)$✓
2. B.$\left(\dfrac{5a}{3}, \dfrac{5b}{3}\right)$
3. C.$\left(\dfrac{3a}{4}, \dfrac{3b}{4}\right)$
4. D.$\left(\dfrac{2a}{3}, \dfrac{2b}{3}\right)$

Correct answer: A

A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon.

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

Correct answer: C

Young's moduli of two wires A and B are in the ratio 7 : 4. Wire A is 2 m long and has radius R. Wire B is 1.5 m long and has radius 2 mm. If the two wires stretch by the same length for a given load, then the value of R is close to:

1. A.1.3 mm
2. B.1.5 mm
3. C.1.7 mm✓
4. D.1.9 mm

Correct answer: C

The given diagram shows four processes i.e., isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order is given by:

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

Correct answer: D

The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, is closest to: [Boltzmann Constant $k_B = 1.38\times10^{-23}$ J/K, Avogadro Number $N_A = 6.02\times10^{26}$ /kg, Radius of Earth: $6.4\times10^6$ m, Gravitational acceleration on Earth = 10 m s$^{-2}$]

1. A.800 K
2. B.650 K
3. C.$10^4$ K✓
4. D.$3\times10^5$ K

Correct answer: C

A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\dfrac{1}{1000}$ of the original amplitude is close to:

1. A.100 s
2. B.50 s
3. C.20 s✓
4. D.10 s

Correct answer: C

An electric dipole is formed by two equal and opposite charges q with separation d. The charges have same mass m. It is kept in a uniform electric field E. If it is slightly rotated from its equilibrium orientation, then its angular frequency $\omega$ is:

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

Correct answer: B

A parallel plate capacitor has 1 μF capacitance. One of its two plates is given +2 μC charge and the other plate, +4 μC charge. The potential difference developed across the capacitor is:

1. A.1 V✓
2. B.5 V
3. C.2 V
4. D.3 V

Correct answer: A

A positive point charge is released from rest at a distance $r_0$ from a positive line charge with uniform density. The speed ($v$) of the point charge, as a function of instantaneous distance r from line charge, is proportional to:

1. A.$v \propto \ln\left(\dfrac{r}{r_0}\right)$
2. B.$v \propto \left(\dfrac{r}{r_0}\right)$
3. C.$v \propto \sqrt{\ln\left(\dfrac{r}{r_0}\right)}$✓
4. D.$v \propto e^{+r/r_0}$

Correct answer: C

The electric field in a region is given by $\vec{E} = (Ax + B)\hat{i}$, where E is in N C$^{-1}$ and x is in metres. The values of constants are A = 20 SI unit and B = 10 SI unit. If the potential at x = 1 is $V_1$ and that at x = −5 is $V_2$, then $V_1 - V_2$ is:

1. A.−520 V
2. B.320 V
3. C.180 V✓
4. D.−48 V

Correct answer: C

A cell of internal resistance r drives current through an external resistance R. The power delivered by the cell to the external resistance will be maximum when:

1. A.R = 1000 r
2. B.R = 0.001 r
3. C.R = r✓
4. D.R = 2r

Correct answer: C

In the figure shown, what is the current (in Ampere) drawn from the battery? You are given: $R_1 = 15\ \Omega$, $R_2 = 10\ \Omega$, $R_3 = 20\ \Omega$, $R_4 = 5\ \Omega$, $R_5 = 25\ \Omega$, $R_6 = 30\ \Omega$, E = 15 V

1. A.9/32✓
2. B.13/24
3. C.20/3
4. D.7/18

Correct answer: A

Two very long, straight, and insulated wires are kept at 90° angle from each other in xy-plane as shown in the figure. These wires carry currents of equal magnitude I, whose directions are shown in the figure. The net magnetic field at point P will be:

1. A.$\dfrac{\mu_0 I}{2\pi d}(\hat{x} + \hat{y})$
2. B.$-\dfrac{\mu_0 I}{2\pi d}(\hat{x} + \hat{y})$
3. C.$+\dfrac{\mu_0 I}{\pi d}(\hat{z})$
4. D.Zero✓

Correct answer: D

Two magnetic dipoles X and Y are placed at a separation d, with their axes perpendicular to each other. The dipole moment of Y is twice that of X. A particle of charge q is passing through their mid-point P, at angle $\theta = 45°$ with the horizontal line, as shown in figure. What would be the magnitude of force on the particle at that instant? (d is much larger than the dimensions of the dipole)

1. A.$\left(\dfrac{\mu_0}{4\pi}\right)\dfrac{2M}{(d/2)^3}\times qv$
2. B.$\left(\dfrac{\mu_0}{4\pi}\right)\dfrac{M}{(d/2)^3}\times qv$
3. C.0✓
4. D.$\sqrt{2}\left(\dfrac{\mu_0}{4\pi}\right)\dfrac{M}{(d/2)^3}\times qv$

Correct answer: C

A circuit connected to an ac source of emf $e = e_0\sin(100t)$ with t in seconds, gives a phase difference of $\dfrac{\pi}{4}$ between the emf e and current i. Which of the following circuits will exhibit this?

1. A.RC circuit with R = 1 kΩ and C = 1 μF
2. B.RC circuit with R = 1 kΩ and C = 10 μF✓
3. C.RL circuit with R = 1 kΩ and L = 10 mH
4. D.RL circuit with R = 1 kΩ and L = 1 mH

Correct answer: B

The magnetic field of an electromagnetic wave is given by: $\vec{B} = 1.6\times10^{-6}\cos(2\times10^7 z + 6\times10^{15} t)(2\hat{i} + \hat{j})\ \dfrac{\text{Wb}}{\text{m}^2}$. The associated electric field will be:

1. A.$\vec{E} = 4.8\times10^2\cos(2\times10^7 z + 6\times10^{15} t)(-\hat{i} + 2\hat{j})\ \frac{\text{V}}{\text{m}}$✓
2. B.$\vec{E} = 4.8\times10^2\cos(2\times10^7 z - 6\times10^{15} t)(-2\hat{j} + \hat{i})\ \frac{\text{V}}{\text{m}}$
3. C.$\vec{E} = 4.8\times10^2\cos(2\times10^7 z + 6\times10^{15} t)(\hat{i} - 2\hat{j})\ \frac{\text{V}}{\text{m}}$
4. D.$\vec{E} = 4.8\times10^2\cos(2\times10^7 z - 6\times10^{15} t)(2\hat{i} + \hat{j})\ \frac{\text{V}}{\text{m}}$

Correct answer: A

A convex lens (of focal length 20 cm) and a concave mirror, having their principal axes along the same lines, are kept 80 cm apart from each other. The concave mirror is to the right of the convex lens. When an object is kept at a distance of 30 cm to the left of the convex lens, its image remains at the same position even if the concave mirror is removed. The maximum distance of the object for which this concave mirror, by itself would produce a virtual image would be:

1. A.30 cm
2. B.25 cm
3. C.20 cm
4. D.10 cm✓

Correct answer: D

Calculate the limit of resolution of a telescope objective having a diameter of 200 cm, if it has to detect light of wavelength 500 nm coming from a star.

1. A.$152.5\times10^{-9}$ radian
2. B.$457.5\times10^{-9}$ radian
3. C.$610\times10^{-9}$ radian
4. D.$305\times10^{-9}$ radian✓

Correct answer: D

A nucleus A, with a finite de-broglie wavelength $\lambda_A$, undergoes spontaneous fission into two nuclei B and C of equal mass. B flies in the same direction as that of A, while C flies in the opposite direction with a velocity equal to half of that of B. The de-Broglie wavelengths $\lambda_B$ and $\lambda_C$ of B and C are respectively:

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

Correct answer: C

The ratio of mass densities of nuclei of $^{40}$Ca and $^{16}$O is close to:

1. A.0.1
2. B.5
3. C.2
4. D.1✓

Correct answer: D

A common emitter amplifier circuit, built using an npn transistor, is shown in the figure. Its dc current gain is 250, $R_C = 1$ kΩ and $V_{CC} = 10$ V. What is the minimum base current for $V_{CE}$ to reach saturation?

1. A.10 μA
2. B.40 μA✓
3. C.7 μA
4. D.100 μA

Correct answer: B

In a line of sight radio communication, a distance of about 50 km is kept between the transmitting and receiving antennas. If the height of the receiving antenna is 70 m, then the minimum height of the transmitting antenna should be: (Radius of the Earth = $6.4\times10^6$ m).

1. A.40 m
2. B.32 m✓
3. C.51 m
4. D.20 m

Correct answer: B

In the circuit shown, a four-wire potentiometer is made of a 400 cm long wire, which extends between A and B. The resistance per unit length of the potentiometer wire is r = 0.01 Ω/cm. If an ideal voltmeter is connected as shown with jockey J at 50 cm from end A, the expected reading of the voltmeter will be:

1. A.0.25 V✓
2. B.0.20 V
3. C.0.50 V
4. D.0.75 V

Correct answer: A

In a simple pendulum experiment for determination of acceleration due to gravity (g), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken comes out to be 30 s. The length of pendulum is measured by using a meter scale of least count 1 mm and the value obtained is 55.0 cm. The percentage error in the determination of g is close to:

1. A.0.2 %
2. B.0.7 %
3. C.3.5 %
4. D.6.8 %✓

Correct answer: D

The structure of Nylon-6 is:

1. A.$\left[-\overset{O}{\overset{\|}{C}}-(CH_2)_5-\overset{H}{\overset{|}{N}}-\right]_n$✓
2. B.$\left[-\overset{O}{\overset{\|}{C}}-(CH_2)_6-\overset{H}{\overset{|}{N}}-\right]_n$
3. C.$\left[-(CH_2)_4-\overset{O}{\overset{\|}{C}}-\overset{H}{\overset{|}{N}}-\right]_n$
4. D.$\left[-(CH_2)_6-\overset{O}{\overset{\|}{C}}-\overset{H}{\overset{|}{N}}-\right]_n$

Correct answer: A

The major product obtained in the following reaction is: a benzene ring bearing $-NH_2$, an ortho $-COCH_3$ and a $-CN$ group (ortho to the acetyl) $\xrightarrow[\text{(ii) Pd/C/H}_2]{\text{(i) CHCl}_3\text{/KOH}}$

1. A.Ring bearing $-NHCHCl_2$, $-CH(OH)CH_3$ and $-CN$
2. B.Ring bearing $-NHCH_3$, $-COCH_3$ and $-CN$
3. C.Ring bearing $-NHCH_3$, $-CH(OH)CH_3$ and $-CH_2NH_2$✓
4. D.Ring bearing $-NHCH_3$, $-CH(OH)CH_3$ and $-CN$

Correct answer: C

Polysubstitution is a major drawback in:

1. A.Friedel Craft's alkylation✓
2. B.Friedel Craft's acylation
3. C.Acetylation of aniline
4. D.Reimer Tiemann reaction

Correct answer: A

Fructose and glucose can be distinguished by:

1. A.Fehling's test
2. B.Benedict's test
3. C.Barfoed's test
4. D.Seliwanoff's test✓

Correct answer: D

The major product obtained in the following reaction is: $OHC-CH(CH_3)-CH_2-CH_2-CH_2-CO-CH_3$ (2-methyl-6-oxoheptanal) $\xrightarrow{NaOH,\ \Delta}$

1. A.1-(3-Methylcyclopent-2-en-1-yl)ethan-1-one: cyclopentene ring with the acetyl group on an sp3 ring carbon and CH3 on a ring alkene carbon
2. B.1-(3-Methylcyclopent-1-en-1-yl)ethan-1-one: cyclopentene ring with the acetyl group on a ring alkene carbon (C1=C2) and CH3 on the sp3 carbon C3 next to C2✓
3. C.1-Methyl-2-methylenecyclopentane-1-carbaldehyde: cyclopentane bearing CH3 and CHO on the same carbon, with an exocyclic =CH2 on the adjacent carbon
4. D.1,2-Dimethylcyclopent-2-ene-1-carbaldehyde: cyclopentene bearing CH3 and CHO on the same sp3 carbon, and CH3 on the adjacent alkene carbon

Correct answer: B

The major product of the following reaction is: 4-chlorotoluene $\xrightarrow[\text{(2) H}_2\text{O, }\Delta]{\text{(1) Cl}_2/h\nu}$

1. A.$4-ClC_6H_4CH_2OH$
2. B.$4-ClC_6H_4CHO$✓
3. C.$4-ClC_6H_4CHCl_2$
4. D.$4-ClC_6H_4CO_2H$

Correct answer: B

The major product of the following reaction is: $3-(CH_3)_2CH-C_6H_4-CH_2-CO-CH_2-CH_2Cl$ (a meta-isopropylbenzyl ketone bearing a 2-chloroethyl group) $\xrightarrow[\text{(2) Conc. H}_2\text{SO}_4/\Delta]{\text{(1) }^t\text{BuOK}}$

1. A.An indan-2-one (five-membered ketone ring fused to benzene) bearing a methyl on the ring carbon next to the benzene and the isopropyl group on the benzene ring next to the ring fusion
2. B.A 2-tetralone (six-membered ketone ring fused to benzene) with the isopropyl group on the benzene ring next to the ring fusion
3. C.An indan-2-one bearing a methyl on the ring carbon next to the benzene and the isopropyl group on the benzene ring away from the ring fusion
4. D.A 2-tetralone with the isopropyl group on the benzene ring away from the ring fusion (para to the new C-C bond)✓

Correct answer: D

Which of the following compounds will show the maximum 'enol' content?

1. A.$CH_3COCH_2COCH_3$✓
2. B.$CH_3COCH_2COOC_2H_5$
3. C.$CH_3COCH_3$
4. D.$CH_3COCH_2CONH_2$

Correct answer: A

Which one of the following alkenes when treated with HCl yields majorly an anti Markovnikov product?

1. A.$CH_3O-CH=CH_2$
2. B.$F_3C-CH=CH_2$✓
3. C.$Cl-CH=CH_2$
4. D.$H_2N-CH=CH_2$

Correct answer: B

The IUPAC symbol for the element with atomic number 119 would be:

1. A.une
2. B.uun
3. C.unh
4. D.uue✓

Correct answer: D

The Mond process is used for the:

1. A.extraction of Zn
2. B.extraction of Mo
3. C.purification of Ni✓
4. D.purification of Zr and Ti

Correct answer: C

The strength of 11.2 volume solution of H$_2$O$_2$ is: [Given that molar mass of H = 1 g mol$^{-1}$ and O = 16 g mol$^{-1}$]

1. A.13.6%
2. B.3.4%✓
3. C.1.7%
4. D.34%

Correct answer: B

The covalent alkaline earth metal halide (X = Cl, Br, I) is:

1. A.$BeX_2$✓
2. B.$MgX_2$
3. C.$CaX_2$
4. D.$SrX_2$

Correct answer: A

The correct statement about ICl$_5$ and ICl$_4^-$ is:

1. A.both are isostructural.
2. B.$ICl_5$ is trigonal bipyramidal and $ICl_4^-$ is tetrahedral.
3. C.$ICl_5$ is square pyramidal and $ICl_4^-$ is square planar.✓
4. D.$ICl_5$ is square pyramidal and $ICl_4^-$ is tetrahedral.

Correct answer: C

The ion that has sp$^3$d$^2$ hybridization for the central atom, is:

1. A.$[BrF_2]^-$
2. B.$[ICl_4]^-$✓
3. C.$[IF_6]^-$
4. D.$[ICl_2]^-$

Correct answer: B

The statement that is INCORRECT about the interstitial compounds is:

1. A.they have high melting points.
2. B.they are very hard.
3. C.they have metallic conductivity.
4. D.they are chemically reactive.✓

Correct answer: D

The compound that inhibits the growth of tumors is:

1. A.cis-$[Pt(Cl)_2(NH_3)_2]$✓
2. B.trans-$[Pt(Cl)_2(NH_3)_2]$
3. C.cis-$[Pd(Cl)_2(NH_3)_2]$
4. D.trans-$[Pd(Cl)_2(NH_3)_2]$

Correct answer: A

The maximum prescribed concentration of copper in drinking water is:

1. A.0.5 ppm
2. B.0.05 ppm
3. C.3 ppm✓
4. D.5 ppm

Correct answer: C

The percentage composition of carbon by mole in methane is:

1. A.25%
2. B.75%
3. C.20%✓
4. D.80%

Correct answer: C

Consider the bcc unit cells of the solids 1 and 2 with the position of atoms as shown below. The radius of atom B is twice that of atom A. The unit cell edge length is 50% more in solid 2 than in 1. What is the approximate packing efficiency in solid 2?

1. A.90%✓
2. B.75%
3. C.65%
4. D.45%

Correct answer: A

If p is the momentum of the fastest electron ejected from a metal surface after the irradiation of light having wavelength $\lambda$, then for 1.5 p momentum of the photoelectron, the wavelength of the light should be: (Assume kinetic energy of ejected photoelectron to be very high in comparison to work function):

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

Correct answer: D

Among the following molecules/ions, $C_2^{2-}$, $N_2^{2-}$, $O_2^{2-}$, $O_2$ which one is diamagnetic and has the shortest bond length?

1. A.$O_2^{2-}$
2. B.$C_2^{2-}$✓
3. C.$N_2^{2-}$
4. D.$O_2$

Correct answer: B

5 moles of an ideal gas at 100 K are allowed to undergo reversible compression till its temperature becomes 200 K. If $C_V = 28$ J K$^{-1}$ mol$^{-1}$, calculate $\Delta U$ and $\Delta pV$ for this process. (R = 8.0 J K$^{-1}$ mol$^{-1}$)

1. A.$\Delta U = 14$ kJ; $\Delta(pV) = 18$ kJ
2. B.$\Delta U = 14$ J; $\Delta(pV) = 0.8$ J
3. C.$\Delta U = 14$ kJ; $\Delta(pV) = 4$ kJ✓
4. D.$\Delta U = 2.8$ kJ; $\Delta(pV) = 0.8$ kJ

Correct answer: C

For the solution of the gases w, x, y and z in water at 298 K, the Henry's law constants ($K_H$) are 0.5, 2, 35 and 40 kbar, respectively. The correct plot for the given data is:

1. A.Plot of partial pressure vs mole fraction of water: four nearly horizontal lines starting at different heights on the pressure axis, rising slightly, in order w (lowest), x, y, z (highest)
2. B.Plot of partial pressure vs mole fraction of water: four parallel rising straight lines, labelled w, x, y, z
3. C.Plot of partial pressure vs mole fraction of water: four straight lines falling from different intercepts on the pressure axis towards the x-axis, steepest for z, then y, x, w✓
4. D.Plot of partial pressure vs mole fraction of water: lines starting on the pressure axis, w and x falling, y and z nearly horizontal

Correct answer: C

For the following reactions, equilibrium constants are given: $S(s) + O_2(g) \rightleftharpoons SO_2(g)$; $K_1 = 10^{52}$; $2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g)$; $K_2 = 10^{129}$. The equilibrium constant for the reaction, $2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$ is:

1. A.$10^{77}$
2. B.$10^{181}$
3. C.$10^{154}$
4. D.$10^{25}$✓

Correct answer: D

Calculate the standard cell potential (in V) of the cell in which following reaction takes place: $Fe^{2+}(aq) + Ag^+(aq) \rightarrow Fe^{3+}(aq) + Ag(s)$. Given that $E^o_{Ag^+/Ag} = x$ V, $E^o_{Fe^{2+}/Fe} = y$ V, $E^o_{Fe^{3+}/Fe} = z$ V

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

Correct answer: D

For a reaction scheme $A \xrightarrow{k_1} B \xrightarrow{k_2} C$, if the rate of formation of B is set to be zero then the concentration of B is given by:

1. A.$\left(\dfrac{k_1}{k_2}\right)[A]$✓
2. B.$(k_1 + k_2)[A]$
3. C.$(k_1 - k_2)[A]$
4. D.$k_1k_2[A]$

Correct answer: A

0.27 g of a long chain fatty acid was dissolved in 100 cm$^3$ of hexane. 10 mL of this solution was added dropwise to the surface of water in a round watch glass. Hexane evaporates and a monolayer is formed. The distance from edge to centre of the watch glass is 10 cm. What is the height of the monolayer? [Density of fatty acid = 0.9 g cm$^{-3}$; $\pi = 3$]

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

Correct answer: B

Let $f(x) = a^x$ $(a > 0)$ be written as $f(x) = f_1(x) + f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x + y) + f_1(x - y)$ equals:

1. A.$2f_1(x)f_1(y)$✓
2. B.$2f_1(x+y)f_1(x-y)$
3. C.$2f_1(x)f_2(y)$
4. D.$2f_1(x+y)f_2(x-y)$

Correct answer: A

If $z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2}$ $(i = \sqrt{-1})$, then $(1 + iz + z^5 + iz^8)^9$ is equal to:

1. A.1
2. B.−1✓
3. C.$(-1 + 2i)^9$
4. D.0

Correct answer: B

The number of integral values of m for which the equation $(1 + m^2)x^2 - 2(1 + 3m)x + (1 + 8m) = 0$ has no real root is:

1. A.2
2. B.3
3. C.infinitely many✓
4. D.1

Correct answer: C

Let the numbers 2, b, c be in an A.P. and $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^2 & c^2 \end{bmatrix}$. If det(A) $\in [2, 16]$, then c lies in the interval:

1. A.$[3, 2 + 2^{3/4}]$
2. B.[4, 6]✓
3. C.[2, 3)
4. D.$(2 + 2^{3/4}, 4)$

Correct answer: B

If the system of linear equations $x - 2y + kz = 1$, $2x + y + z = 2$, $3x - y - kz = 3$ has a solution (x, y, z), $z \ne 0$, then (x, y) lies on the straight line whose equation is:

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

Correct answer: D

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is:

1. A.310✓
2. B.360
3. C.288
4. D.306

Correct answer: A

If three distinct numbers a, b, c are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?

1. A.d, e, f are in G.P.
2. B.d, e, f are in A.P.
3. C.$\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c}$ are in G.P.
4. D.$\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c}$ are in A.P.✓

Correct answer: D

The sum $\displaystyle\sum_{k=1}^{20} k\frac{1}{2^k}$ is equal to:

1. A.$2 - \dfrac{11}{2^{19}}$✓
2. B.$2 - \dfrac{21}{2^{20}}$
3. C.$1 - \dfrac{11}{2^{20}}$
4. D.$2 - \dfrac{3}{2^{17}}$

Correct answer: A

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function satisfying $f'(3) + f'(2) = 0$. Then $\displaystyle\lim_{x\to 0}\left(\frac{1 + f(3 + x) - f(3)}{1 + f(2 - x) - f(2)}\right)^{\frac{1}{x}}$ is equal to:

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

Correct answer: B

If $f(1) = 1$, $f'(1) = 3$, then the derivative of $f(f(f(x))) + (f(x))^2$ at $x = 1$ is:

1. A.9
2. B.12
3. C.15
4. D.33✓

Correct answer: D

Let $f: [-1, 3] \rightarrow \mathbb{R}$ be defined as $f(x) = \begin{cases} |x| + [x], & -1 \le x < 1 \\ x + |x|, & 1 \le x < 2 \\ x + [x], & 2 \le x \le 3, \end{cases}$ where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at:

1. A.only one point
2. B.only two points
3. C.only three points✓
4. D.four or more points

Correct answer: C

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:

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

Correct answer: B

If $\displaystyle\int \frac{dx}{x^3(1 + x^6)^{2/3}} = xf(x)(1 + x^6)^{\frac{1}{3}} + C$ where C is a constant of integration, then the function f(x) is equal to:

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

Correct answer: C

Let $f(x) = \displaystyle\int_0^x g(t)\,dt$, where g is a non-zero even function. If $f(x + 5) = g(x)$, then $\displaystyle\int_0^x f(t)\,dt$ equals:

1. A.$\displaystyle\int_{x+5}^{5} g(t)\,dt$✓
2. B.$\displaystyle\int_{5}^{x+5} g(t)\,dt$
3. C.$2\displaystyle\int_{5}^{x+5} g(t)\,dt$
4. D.$5\displaystyle\int_{x+5}^{5} g(t)\,dt$

Correct answer: A

Let $S(\alpha) = \{(x, y) : y^2 \le x,\ 0 \le x \le \alpha\}$ and $A(\alpha)$ is area of the region $S(\alpha)$. If for a $\lambda$, $0 < \lambda < 4$, $A(\lambda) : A(4) = 2 : 5$, then $\lambda$ equals:

1. A.$4\left(\dfrac{4}{25}\right)^{\frac{1}{3}}$✓
2. B.$4\left(\dfrac{2}{5}\right)^{\frac{1}{3}}$
3. C.$2\left(\dfrac{2}{5}\right)^{\frac{1}{3}}$
4. D.$2\left(\dfrac{4}{25}\right)^{\frac{1}{3}}$

Correct answer: A

Given that the slope of the tangent to a curve $y = y(x)$ at any point (x, y) is $\dfrac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2 + y^2 - 2x - 2y = 0$, then its equation is:

1. A.$x\log_e|y| = 2(x - 1)$✓
2. B.$x\log_e|y| = x - 1$
3. C.$x\log_e|y| = -2(x - 1)$
4. D.$x^2\log_e|y| = -2(x - 1)$

Correct answer: A

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^2 + y^2 = 4$ and the x-axis form a triangle. The area of this triangle (in square units) is:

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

Correct answer: D

Suppose that the points (h, k), (1, 2) and (−3, 4) lie on the line $L_1$. If a line $L_2$ passing through the points (h, k) and (4, 3) is perpendicular to $L_1$, then $\dfrac{k}{h}$ equals:

1. A.$\dfrac{1}{3}$✓
2. B.3
3. C.0
4. D.$-\dfrac{1}{7}$

Correct answer: A

The tangent to the parabola $y^2 = 4x$ at the point where it intersects the circle $x^2 + y^2 = 5$ in the first quadrant, passes through the point:

1. A.$\left(-\dfrac{1}{3}, \dfrac{4}{3}\right)$
2. B.$\left(-\dfrac{1}{4}, \dfrac{1}{2}\right)$
3. C.$\left(\dfrac{3}{4}, \dfrac{7}{4}\right)$✓
4. D.$\left(\dfrac{1}{4}, \dfrac{3}{4}\right)$

Correct answer: C

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0, 5\sqrt{3})$, then the length of its latus rectum is:

1. A.8
2. B.10
3. C.5✓
4. D.6

Correct answer: C

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:

1. A.$3x - 2y = 0$
2. B.$2x - y - 2 = 0$✓
3. C.$2x - 3y + 10 = 0$
4. D.$x - 2y + 8 = 0$

Correct answer: B

The vector equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0$ is:

1. A.$\vec{r}\times(\hat{i} + \hat{k}) + 2 = 0$
2. B.$\vec{r}\times(\hat{i} - \hat{k}) + 2 = 0$
3. C.$\vec{r}\cdot(\hat{i} - \hat{k}) - 2 = 0$
4. D.$\vec{r}\cdot(\hat{i} - \hat{k}) + 2 = 0$✓

Correct answer: D

If a point R(4, y, z) lies on the line segment joining the points P(2, −3, 4) and Q(8, 0, 10), then the distance of R from the origin is:

1. A.$\sqrt{53}$
2. B.$2\sqrt{14}$✓
3. C.$2\sqrt{21}$
4. D.6

Correct answer: B

Let $\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$, for some real x. Then $|\vec{a}\times\vec{b}| = r$ is possible if:

1. A.$0 < r \le \sqrt{\dfrac{3}{2}}$
2. B.$\sqrt{\dfrac{3}{2}} < r \le 3\sqrt{\dfrac{3}{2}}$
3. C.$3\sqrt{\dfrac{3}{2}} < r < 5\sqrt{\dfrac{3}{2}}$
4. D.$r \ge 5\sqrt{\dfrac{3}{2}}$✓

Correct answer: D

A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:

1. A.$\dfrac{100}{3}$
2. B.$\dfrac{10}{3}$
3. C.$\dfrac{10}{\sqrt{3}}$✓
4. D.$\dfrac{100}{\sqrt{3}}$

Correct answer: C

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is:

1. A.2
2. B.3
3. C.4✓
4. D.5