JEE Main 2019 — 12 Jan Shift 1 (Previous Year Paper)

The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure 5 $\mu$m diameter of a wire is:

1. A.50
2. B.100
3. C.200✓
4. D.500

Correct answer: C

A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle $60^\circ$ with ground level. But he finds the aeroplane right vertically above his position. If $v$ is the speed of sound, speed of the plane is:

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

Correct answer: B

A passenger train of length 60 m travels at a speed of 80 km/hr. Another freight train of length 120 m travels at a speed of 30 km/hr. The ratio of times taken by the passenger train to completely cross the freight train when: (i) they are moving in the same direction, and (ii) in the opposite directions is:

1. A.$\frac{25}{11}$
2. B.$\frac{11}{5}$✓
3. C.$\frac52$
4. D.$\frac32$

Correct answer: B

A simple pendulum, made of a string of length $l$ and a bob of mass $m$, is released from a small angle $\theta_0$. It strikes a block of mass M, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle $\theta_1$. Then M is given by:

1. A.$\frac m2\left(\frac{\theta_0-\theta_1}{\theta_0+\theta_1}\right)$
2. B.$\frac m2\left(\frac{\theta_0+\theta_1}{\theta_0-\theta_1}\right)$
3. C.$m\left(\frac{\theta_0+\theta_1}{\theta_0-\theta_1}\right)$✓
4. D.$m\left(\frac{\theta_0-\theta_1}{\theta_0+\theta_1}\right)$

Correct answer: C

A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be:

1. A.in the same circular orbit of radius R
2. B.in a circular orbit of a different radius
3. C.in an elliptical orbit✓
4. D.such that it escapes to infinity

Correct answer: C

The position vector of the centre of mass $\vec r_{cm}$ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is:

1. A.$\vec r_{cm}=\frac{13}{8}L\hat x+\frac58L\hat y$✓
2. B.$\vec r_{cm}=\frac58L\hat x+\frac{13}{8}L\hat y$
3. C.$\vec r_{cm}=\frac{11}{8}L\hat x+\frac38L\hat y$
4. D.$\vec r_{cm}=\frac38L\hat x+\frac{11}{8}L\hat y$

Correct answer: A

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is:

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

Correct answer: D

A straight rod of length L extends from $x=a$ to $x=L+a$. The gravitational force it exerts on a point mass $m$ at $x=0$, if the mass per unit length of the rod is $A+Bx^2$, is given by:

1. A.$Gm\left[A\left(\frac{1}{a+L}-\frac1a\right)+BL\right]$
2. B.$Gm\left[A\left(\frac1a-\frac{1}{a+L}\right)+BL\right]$✓
3. C.$Gm\left[A\left(\frac{1}{a+L}-\frac1a\right)-BL\right]$
4. D.$Gm\left[A\left(\frac1a-\frac{1}{a+L}\right)-BL\right]$

Correct answer: B

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

1. A.$K_1+K_2$
2. B.$\frac{K_1+K_2}{2}$
3. C.$\frac{K_1+3K_2}{4}$✓
4. D.$\frac{2K_1+3K_2}{5}$

Correct answer: C

For the given cyclic process CAB as shown for a gas, the work done is:

1. A.30 J
2. B.10 J✓
3. C.5 J
4. D.1 J

Correct answer: B

An ideal gas occupies a volume of 2 m$^3$ at a pressure of $3\times10^{6}$ Pa. The energy of the gas is:

1. A.$10^{8}$ J
2. B.$3\times10^{2}$ J
3. C.$9\times10^{6}$ J✓
4. D.$6\times10^{4}$ J

Correct answer: C

Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform horizontal rod AB of length $l$ and mass $m$. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

1. A.$\frac{1}{2\pi}\sqrt{\frac{2k}{m}}$
2. B.$\frac{1}{2\pi}\sqrt{\frac km}$
3. C.$\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$
4. D.$\frac{1}{2\pi}\sqrt{\frac{6k}{m}}$✓

Correct answer: D

A travelling harmonic wave is represented by the equation $y(x,t)=10^{-3}\sin(50t+2x)$, where $x$ and $y$ are in meter and $t$ is in seconds. Which of the following is a correct statement about the wave?

1. A.The wave is propagating along the positive $x$-axis with speed 25 m s$^{-1}$.
2. B.The wave is propagating along the negative $x$-axis with speed 25 m s$^{-1}$.✓
3. C.The wave is propagating along the positive $x$-axis with speed 100 m s$^{-1}$.
4. D.The wave is propagating along the negative $x$-axis with speed 100 m s$^{-1}$.

Correct answer: B

In the figure shown, after the switch 'S' is turned from position 'A' to position 'B', the energy dissipated in the circuit in terms of capacitance 'C' and total charge 'Q' is:

1. A.$\frac34\frac{Q^2}{C}$
2. B.$\frac18\frac{Q^2}{C}$
3. C.$\frac58\frac{Q^2}{C}$
4. D.$\frac38\frac{Q^2}{C}$✓

Correct answer: D

Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle, as shown in the figure:

1. A.$(ql)\frac{\hat i+\hat j}{\sqrt2}$
2. B.$-\sqrt3ql\hat j$✓
3. C.$2ql\hat j$
4. D.$\sqrt3ql\frac{\hat j-\hat i}{\sqrt2}$

Correct answer: B

There is a uniform spherically symmetric surface charge density at a distance $R_0$ from the origin. The charge distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best the speed $V(R(t))$ of the distribution as a function of its instantaneous radius $R(t)$ is:

1. A.Graph: V starts at zero at $R_0$ and increases ever more steeply (concave up, unbounded).
2. B.Graph: V rises from zero at $R_0$ to a maximum and then decreases.
3. C.Graph: V rises from zero at $R_0$ steeply at first and then levels off towards a constant value $V_0$ (concave down, saturating).✓
4. D.Graph: V rises slowly, then steeply, then levels off (S-shaped curve).

Correct answer: C

An ideal battery of 4 V and resistance R are connected in series in the primary circuit of a potentiometer of length 1 m and resistance $5\ \Omega$. The value of R, to give a potential difference of 5 mV across 10 cm of potentiometer wire, is:

1. A.$480\ \Omega$
2. B.$490\ \Omega$
3. C.$495\ \Omega$
4. D.$395\ \Omega$✓

Correct answer: D

Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers $P_1$ and $P_2$ respectively, then:

1. A.$P_1=9$ W, $P_2=16$ W
2. B.$P_1=16$ W, $P_2=9$ W
3. C.$P_1=16$ W, $P_2=4$ W✓
4. D.$P_1=4$ W, $P_2=16$ W

Correct answer: C

As shown in the figure, two infinitely long, identical wires are bent by $90^\circ$ and placed in such a way that the segments LP and QM are along the $x$-axis, while segments PS and QN are parallel to the $y$-axis. If OP = OQ = 4 cm, and the magnitude of the magnetic field at O is $10^{-4}$ T, and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at O will be ($\mu_0=4\pi\times10^{-7}$ N A$^{-2}$):

1. A.20 A, perpendicular into the page✓
2. B.20 A, perpendicular out of the page
3. C.40 A, perpendicular into the page
4. D.40 A, perpendicular out of the page

Correct answer: A

A proton and an $\alpha$-particle (with their masses in the ratio of 1 : 4 and charges in the ratio of 1 : 2) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii $r_p:r_\alpha$ of the circular paths described by them will be:

1. A.1 : 2
2. B.$1:\sqrt2$✓
3. C.1 : 3
4. D.$1:\sqrt3$

Correct answer: B

In the figure shown, a circuit contains two identical resistors with resistance $R=5\ \Omega$ and an inductance with $L=2$ mH. An ideal battery of 15 V is connected in the circuit. What will be the current through the battery long after the switch is closed?

1. A.5.5 A
2. B.7.5 A
3. C.6 A✓
4. D.3 A

Correct answer: C

A light wave is incident normally on a glass slab of refractive index 1.5. If 4% of light gets reflected and the amplitude of the electric field of the incident light is 30 V/m, then the amplitude of the electric field for the wave propagating in the glass medium will be:

1. A.10 V/m
2. B.24 V/m✓
3. C.6 V/m
4. D.30 V/m

Correct answer: B

What is the position and nature of image formed by lens combination shown in figure? ($f_1,f_2$ are focal lengths)

1. A.40 cm from point B at right; real
2. B.70 cm from point B at left; virtual
3. C.70 cm from point B at right; real✓
4. D.$\frac{20}{3}$ cm from point B at right, real

Correct answer: C

A point source of light, S is placed at a distance L in front of the centre of plane mirror of width d which is hanging vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror, at a distance 2L as shown below. The distance over which the man can see the image of the light source in the mirror is:

1. A.$\frac d2$
2. B.3d✓
3. C.2d
4. D.d

Correct answer: B

A particle A of mass $m$ and charge $q$ is accelerated by a potential difference of 50 V. Another particle B of mass $4m$ and charge $q$ is accelerated by a potential difference of 2500 V. The ratio of de-Broglie wavelengths $\frac{\lambda_A}{\lambda_B}$ is close to:

1. A.4.47
2. B.10.00
3. C.0.07
4. D.14.14✓

Correct answer: D

A particle of mass $m$ moves in a circular orbit in a central potential field $U(r)=\frac12kr^2$. If Bohr's quantization conditions are applied, radii of possible orbits and energy levels vary with quantum number $n$ as:

1. A.$r_n\propto n^2$, $E_n\propto\frac{1}{n^2}$
2. B.$r_n\propto n$, $E_n\propto n$
3. C.$r_n\propto\sqrt n$, $E_n\propto\frac1n$
4. D.$r_n\propto\sqrt n$, $E_n\propto n$✓

Correct answer: D

The output of the given logic circuit is:

1. A.$A\bar B+\bar AB$
2. B.$A\bar B$✓
3. C.$\bar AB$
4. D.$AB+\overline{AB}$

Correct answer: B

A 100 V carrier wave is made to vary between 160 V and 40 V by a modulating signal. What is the modulation index?

1. A.0.3
2. B.0.6✓
3. C.0.5
4. D.0.4

Correct answer: B

The galvanometer deflection, when key $K_1$ is closed but $K_2$ is open, equals $\theta_0$ (see figure). On closing $K_2$ also and adjusting $R_2$ to $5\ \Omega$, the deflection in galvanometer becomes $\frac{\theta_0}{5}$. The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery]:

1. A.$22\ \Omega$✓
2. B.$12\ \Omega$
3. C.$5\ \Omega$
4. D.$25\ \Omega$

Correct answer: A

In a meter bridge, the wire of length 1 m has a non-uniform cross-section such that, the variation $\frac{dR}{dl}$ of its resistance R with length $l$ is $\frac{dR}{dl}\propto\frac{1}{\sqrt l}$. Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point P. What is the length AP?

1. A.0.2 m
2. B.0.25 m✓
3. C.0.3 m
4. D.0.35 m

Correct answer: B

Poly-$\beta$-hydroxybutyrate-co-$\beta$-hydroxyvalerate (PHBV) is a copolymer of ______.

1. A.2-hydroxybutanoic acid and 3-hydroxypentanoic acid
2. B.3-hydroxybutanoic acid and 3-hydroxypentanoic acid✓
3. C.3-hydroxybutanoic acid and 2-hydroxypentanoic acid
4. D.3-hydroxybutanoic acid and 4-hydroxypentanoic acid

Correct answer: B

In the following reaction: Aldehyde + Alcohol $\xrightarrow{\text{HCl}}$ Acetal. Aldehydes: HCHO, CH$_3$CHO; Alcohols: $^t$BuOH, MeOH. The best combination is:

1. A.HCHO and MeOH✓
2. B.HCHO and $^t$BuOH
3. C.CH$_3$CHO and $^t$BuOH
4. D.CH$_3$CHO and MeOH

Correct answer: A

The increasing order of reactivity of the following compounds towards reaction with alkyl halides directly is: (A) benzamide (B) phthalimide (C) 2-aminobenzonitrile (D) aniline

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

Correct answer: D

$\text{CH}_3\text{CH}_2-\text{C(OH)(Ph)}-\text{CH}_3$ cannot be prepared by:

1. A.PhCOCH$_3$ + CH$_3$CH$_2$MgX
2. B.CH$_3$CH$_2$COCH$_3$ + PhMgX
3. C.PhCOCH$_2$CH$_3$ + CH$_3$MgX
4. D.HCHO + PhCH(CH$_3$)CH$_2$MgX✓

Correct answer: D

In the following reactions, products A and B are: $\text{CH}_3\text{CO}-\text{C(CH}_3)_2-\text{CH}_2\text{CH}_2\text{CHO}\xrightarrow{\text{dil NaOH}}[A]$; $[A]\xrightarrow[\Delta]{\text{H}_3\text{O}^+}[B]$

1. A.A = 5-hydroxy-2,2-dimethylcyclohexan-1-one; B = 6,6-dimethylcyclohex-2-en-1-one (C=C conjugated with C=O)✓
2. B.A = 5-hydroxy-2,2-dimethylcyclohexan-1-one; B = 6,6-dimethylcyclohex-3-en-1-one (C=C not conjugated)
3. C.A = cyclobutane bearing OH, two CH$_3$ groups on one carbon, a CH$_3$ and a CHO; B = the corresponding cyclobutene-carbaldehyde
4. D.A = same cyclobutanol-aldehyde as option 3; B = the exocyclic methylene (C=CH$_2$) cyclobutane-carbaldehyde

Correct answer: A

The major product of the following reaction is: [benzofuranone-type lactone: benzene ring fused to a five-membered lactone (ring O attached to the benzene, C=O on the CH$_2$ side) and bearing a CN group ortho to the ring oxygen] $\xrightarrow[\text{(ii) H}_3\text{O}^+]{\text{(i) DIBAL-H}}$

1. A.Lactone intact, CN converted to CHO
2. B.Benzene ring bearing CHO, OH (phenol) and CH$_2$CHO on three adjacent carbons✓
3. C.Benzene ring bearing CH=NH, OH and CH$_2$CH$_2$OH on adjacent carbons
4. D.Dihydrobenzofuran (cyclic ether) bearing CHO

Correct answer: B

The major product of the following reaction is: 1-methoxy-3-(but-3-en-1-yl)benzene $\xrightarrow[\text{(2) AlCl}_3\ (\text{anhyd.})]{\text{(1) Cl}_2/\text{CCl}_4}$

1. A.Indane bearing a CH$_2$Cl group at C-1, with OCH$_3$ on the benzene carbon adjacent to the ring fusion
2. B.Indane bearing a CH$_2$Cl group at C-1, with OCH$_3$ para to the C-1 ring-fusion carbon✓
3. C.Tetralin (tetrahydronaphthalene) bearing Cl at C-2, with OCH$_3$ adjacent to the ring fusion
4. D.Tetralin bearing Cl at C-2, with OCH$_3$ para to the ring fusion

Correct answer: B

Among the following compounds most basic amino acid is:

1. A.Histidine
2. B.Serine
3. C.Lysine✓
4. D.Asparagine

Correct answer: C

The correct order for acid strength of compounds CH$\equiv$CH, CH$_3$-C$\equiv$CH and CH$_2$=CH$_2$ is as follows:

1. A.HC$\equiv$CH > CH$_3$-C$\equiv$CH > CH$_2$=CH$_2$✓
2. B.CH$_3$-C$\equiv$CH > CH$\equiv$CH > CH$_2$=CH$_2$
3. C.CH$_3$-C$\equiv$CH > CH$_2$=CH$_2$ > HC$\equiv$CH
4. D.CH$\equiv$CH > CH$_2$=CH$_2$ > CH$_3$-C$\equiv$CH

Correct answer: A

Among the following four aromatic compounds, which one will have the lowest melting point?

1. A.Naphthalene✓
2. B.2-Naphthol
3. C.Phthalic acid (benzene-1,2-dicarboxylic acid)
4. D.2,3-Diacetylnaphthalene (drawn with the two acetyl groups on adjacent carbons)

Correct answer: A

The element with Z = 120 (not yet discovered) will be an/a:

1. A.alkali metal
2. B.alkaline earth metal✓
3. C.transition metal
4. D.inner-transition metal

Correct answer: B

In the Hall-Heroult process, aluminium is formed at the cathode. The cathode is made out of:

1. A.Platinum
2. B.Pure aluminium
3. C.Copper
4. D.Carbon✓

Correct answer: D

The hardness of a water sample (in terms of equivalents of CaCO$_3$) containing $10^{-3}$ M CaSO$_4$ is: (molar mass of CaSO$_4$ = 136 g mol$^{-1}$)

1. A.90 ppm
2. B.100 ppm✓
3. C.10 ppm
4. D.50 ppm

Correct answer: B

A metal on combustion in excess air forms X. X upon hydrolysis with water yields H$_2$O$_2$ and O$_2$ along with another product. The metal is:

1. A.Li
2. B.Na
3. C.Rb✓
4. D.Mg

Correct answer: C

Iodine reacts with concentrated HNO$_3$ to yield Y along with other products. The oxidation state of iodine in Y, is:

1. A.1
2. B.3
3. C.5✓
4. D.7

Correct answer: C

The pair of metal ions that can give a spin-only magnetic moment of 3.9 BM for the complex [M(H$_2$O)$_6$]Cl$_2$, is:

1. A.Cr$^{2+}$ and Mn$^{2+}$
2. B.V$^{2+}$ and Fe$^{2+}$
3. C.V$^{2+}$ and Co$^{2+}$✓
4. D.Co$^{2+}$ and Fe$^{2+}$

Correct answer: C

The metal d-orbitals that are directly facing the ligands in K$_3$[Co(CN)$_6$] are:

1. A.$d_{xy}$ and $d_{x^2-y^2}$
2. B.$d_{xy}$, $d_{xz}$ and $d_{yz}$
3. C.$d_{xz}$, $d_{yz}$ and $d_{z^2}$
4. D.$d_{x^2-y^2}$ and $d_{z^2}$✓

Correct answer: D

Mn$_2$(CO)$_{10}$ is an organometallic compound due to the presence of:

1. A.Mn-C bond✓
2. B.Mn-Mn bond
3. C.C-O bond
4. D.Mn-O bond

Correct answer: A

Water samples with BOD values of 4 ppm and 18 ppm, respectively, are:

1. A.Clean and Clean
2. B.Highly polluted and Clean
3. C.Clean and Highly polluted✓
4. D.Highly polluted and Highly polluted

Correct answer: C

The molecule that has minimum/no role in the formation of photochemical smog, is:

1. A.O$_3$
2. B.N$_2$✓
3. C.NO
4. D.CH$_2$=O

Correct answer: B

The volume of gas A is twice than that of gas B. The compressibility factor of gas A is thrice than that of gas B at same temperature. The pressures of the gases for equal number of moles are:

1. A.$P_A=2P_B$
2. B.$P_A=3P_B$
3. C.$2P_A=3P_B$✓
4. D.$3P_A=2P_B$

Correct answer: C

What is the work function of the metal if the light of wavelength 4000 Å generates photoelectrons of velocity $6\times10^{5}$ m s$^{-1}$ from it? (Mass of electron $=9\times10^{-31}$ kg, Velocity of light $=3\times10^{8}$ m s$^{-1}$, Planck's constant $=6.626\times10^{-34}$ J s, Charge of electron $=1.6\times10^{-19}$ J eV$^{-1}$)

1. A.4.0 eV
2. B.2.1 eV✓
3. C.0.9 eV
4. D.3.1 eV

Correct answer: B

Two solids dissociate as follows: $\text{A(s)}\rightleftharpoons\text{B(g)}+\text{C(g)}$; $K_{P_1}=x$ atm$^2$ and $\text{D(s)}\rightleftharpoons\text{C(g)}+\text{E(g)}$; $K_{P_2}=y$ atm$^2$. The total pressure when both the solids dissociate simultaneously is:

1. A.$\sqrt{x+y}$ atm
2. B.$(x+y)$ atm
3. C.$2\left(\sqrt{x+y}\right)$ atm✓
4. D.$x^2+y^2$ atm

Correct answer: C

For a diatomic ideal gas in a closed system, which of the following plots does not correctly describe the relation between various thermodynamic quantities?

1. A.Graph: U versus T - straight line with a steeper segment in the middle (slope increases stepwise).
2. B.Graph: $C_v$ versus T - constant, then rising to a higher constant plateau (step).
3. C.Graph: $C_p$ versus P - straight line increasing with P.✓
4. D.Graph: $C_v$ versus V - horizontal line (constant).

Correct answer: C

Freezing point of a 4% aqueous solution of X is equal to freezing point of 12% aqueous solution of Y. If molecular weight of X is A, then molecular weight of Y is:

1. A.A
2. B.2A
3. C.3A✓
4. D.4A

Correct answer: C

In a chemical reaction, $\text{A}+2\text{B}\overset{K}{\rightleftharpoons}2\text{C}+\text{D}$, the initial concentration of B was 1.5 times of the concentration of A, but the equilibrium concentrations of A and B were found to be equal. The equilibrium constant (K) for the aforesaid chemical reaction is:

1. A.$\frac14$
2. B.4✓
3. C.1
4. D.16

Correct answer: B

The standard electrode potential $E^\ominus$ and its temperature coefficient $\left(\frac{dE^\ominus}{dT}\right)$ for a cell are 2 V and $-5\times10^{-4}$ V K$^{-1}$ at 300 K respectively. The cell reaction is $\text{Zn(s)}+\text{Cu}^{2+}(\text{aq})\to\text{Zn}^{2+}(\text{aq})+\text{Cu(s)}$. The standard reaction enthalpy ($\Delta_rH^\ominus$) at 300 K in kJ mol$^{-1}$ is, [Use $R=8$ J K$^{-1}$ mol$^{-1}$ and $F=96{,}000$ C mol$^{-1}$]

1. A.$-412.8$✓
2. B.206.4
3. C.$-384.0$
4. D.192.0

Correct answer: A

Decomposition of X exhibits a rate constant of 0.05 $\mu$g/year. How many years are required for the decomposition of 5 $\mu$g of X into 2.5 $\mu$g?

1. A.20
2. B.50✓
3. C.25
4. D.40

Correct answer: B

Given: critical temperatures (K) - H$_2$: 33, CH$_4$: 190, CO$_2$: 304, SO$_2$: 630. On the basis of data given above, predict which of the following gases shows least adsorption on a definite amount of charcoal?

1. A.SO$_2$
2. B.CO$_2$
3. C.CH$_4$
4. D.H$_2$✓

Correct answer: D

Let $S=\{1,2,3,\ldots,100\}$. The number of non-empty subsets A of S such that the product of elements in A is even is:

1. A.$2^{100}-1$
2. B.$2^{50}-1$
3. C.$2^{50}(2^{50}-1)$✓
4. D.$2^{50}+1$

Correct answer: C

If $\frac{z-\alpha}{z+\alpha}$ $(\alpha\in\mathbb{R})$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is:

1. A.$\frac12$
2. B.2✓
3. C.$\sqrt2$
4. D.1

Correct answer: B

If $\lambda$ be the ratio of the roots of the quadratic equation in $x$, $3m^2x^2+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda+\frac1\lambda=1$, is:

1. A.$4-3\sqrt2$✓
2. B.$4-2\sqrt3$
3. C.$2-\sqrt3$
4. D.$-2+\sqrt2$

Correct answer: A

Let $P=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}$ and $Q=[q_{ij}]$ be two $3\times3$ matrices such that $Q-P^5=I_3$. Then $\frac{q_{21}+q_{31}}{q_{32}}$ is equal to:

1. A.9
2. B.10✓
3. C.15
4. D.135

Correct answer: B

An ordered pair $(\alpha,\beta)$ for which the system of linear equations<br>$(1+\alpha)x+\beta y+z=2$<br>$\alpha x+(1+\beta)y+z=3$<br>$\alpha x+\beta y+2z=2$<br>has a unique solution, is:

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

Correct answer: D

Consider three boxes, each containing 10 balls labelled $1,2,\ldots,10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i^{\text{th}}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $n_1<n_2<n_3$ is:

1. A.120✓
2. B.164
3. C.82
4. D.240

Correct answer: A

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $\left(2^{1/3}+\frac{1}{2(3)^{1/3}}\right)^{10}$ is:

1. A.$1:2(6)^{1/3}$
2. B.$2(36)^{1/3}:1$
3. C.$4(36)^{1/3}:1$✓
4. D.$1:4(16)^{1/3}$

Correct answer: C

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is:

1. A.36
2. B.32
3. C.28✓
4. D.24

Correct answer: C

Let $S_k=\frac{1+2+3+\ldots+k}{k}$. If $S_1^2+S_2^2+\cdots+S_{10}^2=\frac{5}{12}A$, then A is equal to:

1. A.156
2. B.283
3. C.301
4. D.303✓

Correct answer: D

$\lim\limits_{x\to\pi/4}\frac{\cot^3x-\tan x}{\cos\left(x+\frac{\pi}{4}\right)}$ is:

1. A.$4\sqrt2$
2. B.4
3. C.$8\sqrt2$
4. D.8✓

Correct answer: D

For $x>1$, if $(2x)^{2y}=4e^{2x-2y}$, then $(1+\log_e2x)^2\frac{dy}{dx}$ is equal to:

1. A.$\frac{x\log_e2x+\log_e2}{x}$
2. B.$\log_e2x$
3. C.$\frac{x\log_e2x-\log_e2}{x}$✓
4. D.$x\log_e2x$

Correct answer: C

Let S be the set of all points in $(-\pi,\pi)$ at which the function, $f(x)=\min\{\sin x,\cos x\}$ is not differentiable. Then S is a subset of which of the following?

1. A.$\left\{-\frac\pi2,-\frac\pi4,\frac\pi4,\frac\pi2\right\}$
2. B.$\left\{-\frac{3\pi}{4},-\frac\pi4,\frac{3\pi}{4},\frac\pi4\right\}$✓
3. C.$\left\{-\frac{3\pi}{4},-\frac\pi2,\frac\pi2,\frac{3\pi}{4}\right\}$
4. D.$\left\{-\frac\pi4,0,\frac\pi4\right\}$

Correct answer: B

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

1. A.36
2. B.32✓
3. C.$20\sqrt2$
4. D.$18\sqrt3$

Correct answer: B

The integral $\int\cos(\log_ex)\,dx$ is equal to: (where C is a constant of integration)

1. A.$x\left[\cos(\log_ex)+\sin(\log_ex)\right]+C$
2. B.$x\left[\cos(\log_ex)-\sin(\log_ex)\right]+C$
3. C.$\frac x2\left[\sin(\log_ex)-\cos(\log_ex)\right]+C$
4. D.$\frac x2\left[\cos(\log_ex)+\sin(\log_ex)\right]+C$✓

Correct answer: D

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $\int_0^af(x)g(x)\,dx$ is equal to:

1. A.$\int_0^af(x)\,dx$
2. B.$2\int_0^af(x)\,dx$✓
3. C.$4\int_0^af(x)\,dx$
4. D.$-3\int_0^af(x)\,dx$

Correct answer: B

The area (in sq. units) of the region bounded by the parabola, $y=x^2+2$ and the lines, $y=x+1$, $x=0$ and $x=3$, is:

1. A.$\frac{15}{2}$✓
2. B.$\frac{21}{2}$
3. C.$\frac{17}{4}$
4. D.$\frac{15}{4}$

Correct answer: A

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x\log_ex$, $(x>1)$. If $2y(2)=\log_e4-1$, then $y(e)$ is equal to:

1. A.$\frac e4$✓
2. B.$\frac{e^2}{4}$
3. C.$-\frac e2$
4. D.$-\frac{e^2}{2}$

Correct answer: A

If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta)$, then $\beta$ equals:

1. A.$-5$
2. B.$\frac{35}{3}$
3. C.5✓
4. D.$-\frac{35}{3}$

Correct answer: C

If a variable line, $3x+4y-\lambda=0$ is such that the two circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval:

1. A.$(23,31)$
2. B.$(2,17)$
3. C.$[13,23]$
4. D.$[12,21]$✓

Correct answer: D

Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $PC_1QC_2$ is:

1. A.4✓
2. B.6
3. C.8
4. D.9

Correct answer: A

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, $y^2=4x$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $\Delta PXQ$ is maximum. Then this maximum area (in sq. units) is:

1. A.$\frac{125}{2}$
2. B.$\frac{625}{4}$
3. C.$\frac{125}{4}$✓
4. D.$\frac{75}{2}$

Correct answer: C

If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

1. A.$(4,\sqrt{15})$
2. B.$(2\sqrt6,5)$
3. C.$(6,5\sqrt2)$✓
4. D.$(-6,2\sqrt{10})$

Correct answer: C

The perpendicular distance from the origin to the plane containing the two lines, $\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$, is:

1. A.11
2. B.$11\sqrt6$
3. C.$\frac{11}{\sqrt6}$✓
4. D.$6\sqrt{11}$

Correct answer: C

A tetrahedron has vertices $P(1,2,1)$, $Q(2,1,3)$, $R(-1,1,2)$ and $O(0,0,0)$. The angle between the faces OPQ and PQR is:

1. A.$\cos^{-1}\left(\frac{19}{35}\right)$✓
2. B.$\cos^{-1}\left(\frac{17}{31}\right)$
3. C.$\cos^{-1}\left(\frac{9}{35}\right)$
4. D.$\cos^{-1}\left(\frac{7}{31}\right)$