JEE Main 2019 — 11 Jan Shift 2 (Previous Year Paper)

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be:

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

Correct answer: A

A particle moves from the point $(2.0\hat i+4.0\hat j)$ m, at $t=0$, with an initial velocity $(5.0\hat i+4.0\hat j)\ \text{m s}^{-1}$. It is acted upon by a constant force which produces a constant acceleration $(4.0\hat i+4.0\hat j)\ \text{m s}^{-2}$. What is the distance of the particle from the origin at time 2 s?

1. A.$20\sqrt2$ m✓
2. B.15 m
3. C.$10\sqrt2$ m
4. D.5 m

Correct answer: A

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians):

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

Correct answer: B

A particle of mass $m$ is moving in a straight line with momentum $p$. Starting at time $t=0$, a force $F=kt$ acts in the same direction on the moving particle during time interval $T$ so that its momentum changes from $p$ to $3p$. Here $k$ is a constant. The value of $T$ is:

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

Correct answer: B

A particle of mass $m$ and charge $q$ is in an electric and magnetic field given by $\vec E=2\hat i+3\hat j$; $\vec B=4\hat j+6\hat k$. The charged particle is shifted from the origin to the point $P(x=1;\ y=1)$ along a straight path. The magnitude of the total work done is:

1. A.$5q$✓
2. B.$(2.5)q$
3. C.$(0.15)q$
4. D.$(0.35)q$

Correct answer: A

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string):

1. A.10 rad/s$^2$
2. B.12 rad/s$^2$
3. C.16 rad/s$^2$✓
4. D.20 rad/s$^2$

Correct answer: C

A circular disc $D_1$ of mass $M$ and radius $R$ has two identical discs $D_2$ and $D_3$ of the same mass $M$ and radius $R$ attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis $OO'$, passing through the centre of $D_1$, as shown in the figure, will be:

1. A.$MR^2$
2. B.$\frac23MR^2$
3. C.$3MR^2$✓
4. D.$\frac45MR^2$

Correct answer: C

The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is 2 s. The period of oscillation of the same pendulum on the planet would be:

1. A.$\frac32$ s
2. B.$\frac{\sqrt3}{2}$ s
3. C.$\frac{2}{\sqrt3}$ s
4. D.$2\sqrt3$ s✓

Correct answer: D

When 100 g of a liquid A at $100^\circ$C is added to 50 g of a liquid B at temperature $75^\circ$C, the temperature of the mixture becomes $90^\circ$C. The temperature of the mixture, if 100 g of liquid A at $100^\circ$C is added to 50 g of liquid B at $50^\circ$C, will be:

1. A.$60^\circ$C
2. B.$70^\circ$C
3. C.$80^\circ$C✓
4. D.$85^\circ$C

Correct answer: C

In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation $VT=K$, where $K$ is a constant. In this process the temperature of the gas is increased by $\Delta T$. The amount of heat absorbed by gas is ($R$ is gas constant):

1. A.$\frac12R\Delta T$✓
2. B.$\frac12KR\Delta T$
3. C.$\frac{2K}{3}\Delta T$
4. D.$\frac32R\Delta T$

Correct answer: A

A metal ball of mass 0.1 kg is heated upto $500^\circ$C and dropped into a vessel of heat capacity 800 J K$^{-1}$ and containing 0.5 kg water. The initial temperature of water and vessel is $30^\circ$C. What is the approximate percentage increment in the temperature of the water? [Specific Heat Capacities of water and metal are, respectively, 4200 J kg$^{-1}$K$^{-1}$ and 400 J kg$^{-1}$K$^{-1}$]

1. A.30%
2. B.25%
3. C.15%
4. D.20%✓

Correct answer: D

A pendulum is executing simple harmonic motion and its maximum kinetic energy is $K_1$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is $K_2$. Then:

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

Correct answer: C

A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of $10^{-2}$ m. The relative change in the angular frequency of the pendulum is best given by:

1. A.$10^{-1}$ rad/s
2. B.$10^{-3}$ rad/s✓
3. C.$10^{-5}$ rad/s
4. D.1 rad/s

Correct answer: B

Seven capacitors, each of capacitance $2\ \mu$F, are to be connected in a configuration to obtain an effective capacitance of $\left(\frac{6}{13}\right)\ \mu$F. Which of the combinations, shown in figures below, will achieve the desired value?

1. A.Figure: 5 capacitors in parallel, followed in series by 2 capacitors.
2. B.Figure: 4 capacitors in parallel, followed in series by 3 capacitors.
3. C.Figure: 3 capacitors in parallel, followed in series by 4 capacitors.✓
4. D.Figure: 2 capacitors in parallel, followed in series by 5 capacitors.

Correct answer: C

An electric field of 1000 V/m is applied to an electric dipole at angle of $45^\circ$. The value of electric dipole moment is $10^{-29}$ C m. What is the potential energy of the electric dipole?

1. A.$-7\times10^{-27}$ J✓
2. B.$-10\times10^{-29}$ J
3. C.$-9\times10^{-20}$ J
4. D.$-20\times10^{-18}$ J

Correct answer: A

Two rods A and B of identical dimensions are at temperature $30^\circ$C. If A is heated upto $180^\circ$C and B upto $T^\circ$C, then the new lengths are the same. If the ratio of the coefficients of linear expansion of A and B is 4 : 3, then the value of $T$ is:

1. A.$200^\circ$C
2. B.$230^\circ$C✓
3. C.$250^\circ$C
4. D.$270^\circ$C

Correct answer: B

In the experimental set up of metre bridge shown in the figure, the null point is obtained at a distance of 40 cm from A. If a $10\ \Omega$ resistor is connected in series with $R_1$, the null point shifts by 10 cm. The resistance that should be connected in parallel with $(R_1+10)\ \Omega$ such that the null point shifts back to its initial position is:

1. A.$60\ \Omega$✓
2. B.$40\ \Omega$
3. C.$30\ \Omega$
4. D.$20\ \Omega$

Correct answer: A

In the circuit shown, the potential difference between A and B is:

1. A.2 V✓
2. B.3 V
3. C.6 V
4. D.1 V

Correct answer: A

A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of $20\times10^{-6}$ J/T when a magnetic intensity of $60\times10^{3}$ A/m is applied. Its magnetic susceptibility is:

1. A.$3.3\times10^{-2}$
2. B.$4.3\times10^{-2}$
3. C.$2.3\times10^{-2}$
4. D.$3.3\times10^{-4}$✓

Correct answer: D

The region between $y=0$ and $y=d$ contains a magnetic field $\vec B=B\hat z$. A particle of mass $m$ and charge $q$ enters the region with a velocity $\vec v=v\hat i$. If $d=\frac{mv}{2qB}$, the acceleration of the charged particle at the point of its emergence at the other side is:

1. A.$\frac{qvB}{m}\left(\frac{\sqrt3}{2}\hat i+\frac12\hat j\right)$✓
2. B.$\frac{qvB}{m}\left(\frac12\hat i-\frac{\sqrt3}{2}\hat j\right)$
3. C.$\frac{qvB}{m}\left(\frac{\hat i+\hat j}{\sqrt2}\right)$
4. D.$\frac{qvB}{m}\left(\frac{-\hat j+\hat i}{\sqrt2}\right)$

Correct answer: A

A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3, keeping the number of turns of the coil per unit length of the frame the same, then the self inductance of the coil:

1. A.increases by a factor of 3
2. B.decreases by a factor of 9
3. C.increases by a factor of 27✓
4. D.decreases by a factor of $9\sqrt3$

Correct answer: C

A 27 mW laser beam has a cross-sectional area of 10 mm$^2$. The magnitude of the maximum electric field in this electromagnetic wave is given by: [Given permittivity of space $\epsilon_0=9\times10^{-12}$ SI units, Speed of light $c=3\times10^{8}$ m/s]

1. A.1.4 kV/m✓
2. B.1 kV/m
3. C.2 kV/m
4. D.0.7 kV/m

Correct answer: A

A monochromatic light is incident at a certain angle on an equilateral triangular prism and suffers minimum deviation. If the refractive index of the material of the prism is $\sqrt3$, then the angle of incidence is:

1. A.$45^\circ$
2. B.$30^\circ$
3. C.$60^\circ$✓
4. D.$90^\circ$

Correct answer: C

In a photoelectric experiment, the wavelength of the light incident on a metal is changed from 300 nm to 400 nm. The decrease in the stopping potential is close to: ($\frac{hc}{e}=1240$ nm-V)

1. A.1.0 V✓
2. B.2.0 V
3. C.0.5 V
4. D.1.5 V

Correct answer: A

In a hydrogen like atom, when an electron jumps from the M-shell to the L-shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from N-shell to the L-shell, the wavelength of emitted radiation will be:

1. A.$\frac{16}{25}\lambda$
2. B.$\frac{25}{16}\lambda$
3. C.$\frac{20}{27}\lambda$✓
4. D.$\frac{27}{20}\lambda$

Correct answer: C

The circuit shown below contains two ideal diodes, each with a forward resistance of $50\ \Omega$. If the battery voltage is 6 V, the current through the $100\ \Omega$ resistance (in Amperes) is:

1. A.0.020✓
2. B.0.027
3. C.0.030
4. D.0.036

Correct answer: A

An amplitude modulated signal is plotted below: [graph of $V(t)$ with maximum amplitude 10 V, minimum amplitude 8 V, envelope period 100 $\mu$s and carrier period 8 $\mu$s]. Which one of the following best describes the above signal?

1. A.$(9+\sin(2.5\pi\times10^{5}t))\sin(2\pi\times10^{4}t)$ V
2. B.$(1+9\sin(2\pi\times10^{4}t))\sin(2.5\pi\times10^{5}t)$ V
3. C.$(9+\sin(4\pi\times10^{4}t))\sin(5\pi\times10^{5}t)$ V
4. D.$(9+\sin(2\pi\times10^{4}t))\sin(2.5\pi\times10^{5}t)$ V✓

Correct answer: D

A galvanometer having a resistance of $20\ \Omega$ and 30 divisions on both sides has figure of merit 0.005 ampere/division. The resistance that should be connected in series such that it can be used as a voltmeter upto 15 volt, is:

1. A.$80\ \Omega$✓
2. B.$100\ \Omega$
3. C.$120\ \Omega$
4. D.$125\ \Omega$

Correct answer: A

A thermometer graduated according to a linear scale reads a value $x_0$ when in contact with boiling water, and $x_0/3$ when in contact with ice. What is the temperature of an object in $^\circ$C, if this thermometer in the contact with the object reads $x_0/2$?

1. A.60
2. B.25✓
3. C.35
4. D.40

Correct answer: B

Which of the following compounds reacts with ethylmagnesium bromide and also decolourizes bromine water solution:

1. A.2-Vinylphenol (benzene ring with OH and an adjacent CH=CH$_2$)✓
2. B.Cyclohexene ring bearing CN and CH$_2$CO$_2$CH$_3$ on adjacent carbons
3. C.2-Acetylbenzonitrile (benzene ring with CN and an adjacent COCH$_3$)
4. D.2-Methoxystyrene (benzene ring with OCH$_3$ and an adjacent CH=CH$_2$)

Correct answer: A

The correct match between Item I and Item II is:<br>Item I: (A) Ester test (B) Carbylamine test (C) Phthalein dye test<br>Item II: (P) Tyr (Q) Asp (R) Ser (S) Lys

1. A.(A)$\to$(Q); (B)$\to$(S); (C)$\to$(R)
2. B.(A)$\to$(Q); (B)$\to$(S); (C)$\to$(P)✓
3. C.(A)$\to$(R); (B)$\to$(Q); (C)$\to$(P)
4. D.(A)$\to$(R); (B)$\to$(S); (C)$\to$(Q)

Correct answer: B

The homopolymer formed from 4-hydroxybutanoic acid is:

1. A.$\left[-\text{O-C(=O)(CH}_2)_3-\text{O}-\right]_n$
2. B.$\left[-\text{C(=O)(CH}_2)_2\text{C(=O)}-\right]_n$
3. C.$\left[-\text{C(=O)(CH}_2)_2\text{C(=O)-O}-\right]_n$
4. D.$\left[-\text{C(=O)(CH}_2)_3-\text{O}-\right]_n$✓

Correct answer: D

In the following compound (adenine, with sites labelled: (a) the exocyclic NH$_2$, (b) and (c) the two pyrimidine-ring N atoms, (d) the imidazole-ring N=, (e) the imidazole N-H), the favourable site/s for protonation is/are:

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

Correct answer: C

A compound 'X' on treatment with Br$_2$/NaOH, provided C$_3$H$_9$N, which gives positive carbylamine test. Compound 'X' is:

1. A.CH$_3$CH$_2$COCH$_2$NH$_2$
2. B.CH$_3$CH$_2$CH$_2$CONH$_2$✓
3. C.CH$_3$CON(CH$_3$)$_2$
4. D.CH$_3$COCH$_2$NHCH$_3$

Correct answer: B

The major product obtained in the following reaction is: [cyclohexane ring bearing COOH, CH=CH$_2$, COCH$_3$ and NO$_2$ groups] $\xrightarrow{\text{LiAlH}_4\ (\text{excess})}$

1. A.COOH reduced to CH$_2$OH and COCH$_3$ reduced to CH(OH)CH$_3$; NO$_2$ and CH=CH$_2$ unchanged
2. B.COOH unchanged, COCH$_3$ reduced to CH(OH)CH$_3$; NO$_2$ and CH=CH$_2$ unchanged
3. C.CH$_2$OH, CH(OH)CH$_3$, NH$_2$, and the vinyl group reduced to ethyl
4. D.CH$_2$OH, CH(OH)CH$_3$ and NH$_2$ (NO$_2$ reduced); CH=CH$_2$ unchanged✓

Correct answer: D

The major product of the following reaction is: 3-(but-3-en-1-yl)phenol $\xrightarrow[\text{(2) AlCl}_3\ (\text{anhyd.})]{\text{(1) HCl}}$

1. A.Indane with a methyl group at C-1 and OH on the benzene ring adjacent (ortho) to the ring fusion near C-1
2. B.Same indane skeleton as option 1 but with Cl in place of OH
3. C.Indane with a methyl group at C-1 and OH on the benzene ring para to the C-1 ring-fusion carbon (1-methylindan-5-ol type)✓
4. D.Same indane skeleton as option 3 but with Cl in place of OH

Correct answer: C

The major product obtained in the following conversion is: 2-(but-2-enoyl)phenyl acetate $\xrightarrow[\text{MeOH}]{\text{Br}_2\ (1\ \text{eqv.})}$

1. A.Ring-brominated: Br ortho to the acetoxy group, side chain unchanged
2. B.Ring-brominated: Br para to the acetoxy group, side chain unchanged
3. C.Ar-CO-CH(Br)-CH(OMe)-CH$_3$ (Br on the carbon next to C=O, OMe on the next carbon)✓
4. D.Ar-CO-CH(OMe)-CH(Br)-CH$_3$ (OMe on the carbon next to C=O, Br on the next carbon)

Correct answer: C

Which of the following compounds will produce a precipitate with AgNO$_3$?

1. A.Bromobenzene
2. B.Bromocyclohexane
3. C.3-Bromopyridine
4. D.7-Bromocyclohepta-1,3,5-triene (tropylium bromide)✓

Correct answer: D

The correct match between Item I and Item II is:<br>Item I: (A) Allosteric effect (B) Competitive inhibitor (C) Receptor (D) Poison<br>Item II: (P) Molecule binding to the active site of enzyme (Q) Molecule crucial for communication in the body (R) Molecule binding to a site other than the active site of enzyme (S) Molecule binding to the enzyme covalently

1. A.(A)$\to$(P); (B)$\to$(R); (C)$\to$(Q); (D)$\to$(S)
2. B.(A)$\to$(P); (B)$\to$(R); (C)$\to$(S); (D)$\to$(Q)
3. C.(A)$\to$(R); (B)$\to$(P); (C)$\to$(Q); (D)$\to$(S)✓
4. D.(A)$\to$(R); (B)$\to$(P); (C)$\to$(S); (D)$\to$(Q)

Correct answer: C

The correct option with respect to the Pauling electronegativity values of the elements is:

1. A.Te > Se
2. B.P > S
3. C.Si < Al
4. D.Ga < Ge✓

Correct answer: D

The reaction that does NOT define calcination is:

1. A.$\text{Fe}_2\text{O}_3\cdot x\text{H}_2\text{O}\xrightarrow{\Delta}\text{Fe}_2\text{O}_3+x\text{H}_2\text{O}$
2. B.$\text{ZnCO}_3\xrightarrow{\Delta}\text{ZnO}+\text{CO}_2$
3. C.$\text{CaCO}_3\cdot\text{MgCO}_3\xrightarrow{\Delta}\text{CaO}+\text{MgO}+2\text{CO}_2$
4. D.$2\text{Cu}_2\text{S}+3\text{O}_2\xrightarrow{\Delta}2\text{Cu}_2\text{O}+2\text{SO}_2$✓

Correct answer: D

The hydride that is NOT electron deficient is:

1. A.SiH$_4$✓
2. B.AlH$_3$
3. C.B$_2$H$_6$
4. D.GaH$_3$

Correct answer: A

Match the following items in column I with the corresponding items in column II.<br>Column I: (i) Na$_2$CO$_3\cdot$10H$_2$O (ii) Mg(HCO$_3$)$_2$ (iii) NaOH (iv) Ca$_3$Al$_2$O$_6$<br>Column II: (A) Portland cement ingredient (B) Castner-Kellner process (C) Solvay process (D) Temporary hardness

1. A.(i)$\to$(B); (ii)$\to$(C); (iii)$\to$(A); (iv)$\to$(D)
2. B.(i)$\to$(C); (ii)$\to$(B); (iii)$\to$(D); (iv)$\to$(A)
3. C.(i)$\to$(C); (ii)$\to$(D); (iii)$\to$(B); (iv)$\to$(A)✓
4. D.(i)$\to$(D); (ii)$\to$(A); (iii)$\to$(B); (iv)$\to$(C)

Correct answer: C

$\underline{A}\xrightarrow{4\text{KOH},\ \text{O}_2}2\underline{B}\ (\text{Green})+2\text{H}_2\text{O}$<br>$3\underline{B}\xrightarrow{4\text{HCl}}2\underline{C}\ (\text{Purple})+\text{MnO}_2+2\text{H}_2\text{O}$<br>$2\underline{C}\xrightarrow{\text{H}_2\text{O},\ \text{KI}}2\underline{A}+2\text{KOH}+\underline{D}$<br>In the above sequence of reactions, $\underline{A}$ and $\underline{D}$, respectively, are:

1. A.KI and K$_2$MnO$_4$
2. B.MnO$_2$ and KIO$_3$✓
3. C.KIO$_3$ and MnO$_2$
4. D.KI and KMnO$_4$

Correct answer: B

The number of bridging CO ligand(s) and Co-Co bond(s) in Co$_2$(CO)$_8$, respectively are:

1. A.0 and 2
2. B.2 and 0
3. C.2 and 1✓
4. D.4 and 0

Correct answer: C

The coordination number of Th in K$_4$[Th(C$_2$O$_4$)$_4$(OH$_2$)$_2$] is: (C$_2$O$_4^{2-}$ = Oxalato)

1. A.6
2. B.8
3. C.10✓
4. D.14

Correct answer: C

Taj Mahal is being slowly disfigured and discoloured. This is primarily due to:

1. A.soil pollution
2. B.global warming
3. C.acid rain✓
4. D.water pollution

Correct answer: C

The higher concentration of which gas in air can cause stiffness of flower buds?

1. A.SO$_2$✓
2. B.CO$_2$
3. C.NO$_2$
4. D.CO

Correct answer: A

25 ml of the given HCl solution requires 30 mL of 0.1 M sodium carbonate solution. What is the volume of this HCl solution required to titrate 30 mL of 0.2 M aqueous NaOH solution?

1. A.25 mL✓
2. B.50 mL
3. C.75 mL
4. D.12.5 mL

Correct answer: A

The radius of the largest sphere which fits properly at the centre of the edge of a body centred cubic unit cell is: (Edge length is represented by $a$)

1. A.0.134 a
2. B.0.067 a✓
3. C.0.027 a
4. D.0.047 a

Correct answer: B

The de Broglie wavelength ($\lambda$) associated with a photoelectron varies with the frequency ($\nu$) of the incident radiation as, [$\nu_0$ is threshold frequency]:

1. A.$\lambda\propto\frac{1}{(\nu-\nu_0)^{1/2}}$✓
2. B.$\lambda\propto\frac{1}{(\nu-\nu_0)}$
3. C.$\lambda\propto\frac{1}{(\nu-\nu_0)^{1/4}}$
4. D.$\lambda\propto\frac{1}{(\nu-\nu_0)^{3/2}}$

Correct answer: A

The standard reaction Gibbs energy for a chemical reaction at an absolute temperature $T$ is given by $\Delta_rG^\circ=A-BT$, where A and B are non-zero constants. Which of the following is TRUE about this reaction?

1. A.Endothermic if $A>0$✓
2. B.Exothermic if $B<0$
3. C.Endothermic if $A<0$ and $B>0$
4. D.Exothermic if $A>0$ and $B<0$

Correct answer: A

The reaction, $\text{MgO(s)}+\text{C(s)}\to\text{Mg(s)}+\text{CO(g)}$, for which $\Delta_rH^\circ=+491.1$ kJ mol$^{-1}$ and $\Delta_rS^\circ=198.0$ J K$^{-1}$ mol$^{-1}$, is not feasible at 298 K. Temperature above which reaction will be feasible is:

1. A.2040.5 K
2. B.2480.3 K✓
3. C.2380.5 K
4. D.1890.0 K

Correct answer: B

K$_2$HgI$_4$ is 40% ionised in aqueous solution. The value of its van't Hoff factor ($i$) is:

1. A.1.6
2. B.1.8✓
3. C.2.0
4. D.2.2

Correct answer: B

For the equilibrium, $2\text{H}_2\text{O}\rightleftharpoons\text{H}_3\text{O}^++\text{OH}^-$, the value of $\Delta G^\circ$ at 298 K is approximately:

1. A.$-100$ kJ mol$^{-1}$
2. B.$-80$ kJ mol$^{-1}$
3. C.80 kJ mol$^{-1}$✓
4. D.100 kJ mol$^{-1}$

Correct answer: C

Given the equilibrium constant $K_C$ of the reaction: $\text{Cu(s)}+2\text{Ag}^+(\text{aq})\to\text{Cu}^{2+}(\text{aq})+2\text{Ag(s)}$ is $10\times10^{15}$, calculate the $E^\circ_{\text{cell}}$ of this reaction at 298 K. [$2.303\frac{RT}{F}$ at 298 K $=0.059$ V]

1. A.0.04736 V
2. B.0.4736 V✓
3. C.0.04736 mV
4. D.0.4736 mV

Correct answer: B

The reaction $2X\to B$ is a zeroth order reaction. If the initial concentration of X is 0.2 M, the half-life is 6 h. When the initial concentration of X is 0.5 M, the time required to reach its final concentration of 0.2 M will be:

1. A.7.2 h
2. B.18.0 h✓
3. C.9.0 h
4. D.12.0 h

Correct answer: B

Among the colloids cheese (C), milk (M) and smoke (S), the correct combination of the dispersed phase and dispersion medium, respectively is:

1. A.C: solid in liquid; M: liquid in liquid; S: gas in solid
2. B.C: solid in liquid; M: solid in liquid; S: solid in gas
3. C.C: liquid in solid; M: liquid in solid; S: solid in gas
4. D.C: liquid in solid; M: liquid in liquid; S: solid in gas✓

Correct answer: D

Let a function $f:(0,\infty)\to(0,\infty)$ be defined by $f(x)=\left|1-\frac1x\right|$. Then $f$ is:

1. A.injective only
2. B.not injective but it is surjective✓
3. C.neither injective nor surjective
4. D.both injective as well as surjective

Correct answer: B

Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to:

1. A.$\frac53$✓
2. B.$\frac54$
3. C.$\frac{\sqrt{34}}{3}$
4. D.$\frac{\sqrt{41}}{4}$

Correct answer: A

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin\theta-x(\sin\theta\cos\theta+1)+\cos\theta=0$ $(0<\theta<45^\circ)$, and $\alpha<\beta$. Then $\sum_{n=0}^{\infty}\left(\alpha^n+\frac{(-1)^n}{\beta^n}\right)$ is equal to:

1. A.$\frac{1}{1+\cos\theta}-\frac{1}{1-\sin\theta}$
2. B.$\frac{1}{1-\cos\theta}-\frac{1}{1+\sin\theta}$
3. C.$\frac{1}{1+\cos\theta}+\frac{1}{1-\sin\theta}$
4. D.$\frac{1}{1-\cos\theta}+\frac{1}{1+\sin\theta}$✓

Correct answer: D

Let A and B be two invertible matrices of order $3\times3$. If $\det(ABA^T)=8$ and $\det(AB^{-1})=8$, then $\det(BA^{-1}B^T)$ is equal to:

1. A.1
2. B.$\frac14$
3. C.16
4. D.$\frac{1}{16}$✓

Correct answer: D

If $\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}=(a+b+c)(x+a+b+c)^2$, $x\ne0$ and $a+b+c\ne0$, then $x$ is equal to:

1. A.$abc$
2. B.$2(a+b+c)$
3. C.$-(a+b+c)$
4. D.$-2(a+b+c)$✓

Correct answer: D

The number of functions $f$ from $\{1,2,3,\ldots,20\}$ onto $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of 3, whenever $k$ is a multiple of 4, is:

1. A.$5!\times6!$
2. B.$5^6\times15$
3. C.$6^5\times(15)!$
4. D.$(15)!\times6!$✓

Correct answer: D

Let $(x+10)^{50}+(x-10)^{50}=a_0+a_1x+a_2x^2+\ldots+a_{50}x^{50}$, for all $x\in\mathbb{R}$; then $\frac{a_2}{a_0}$ is equal to:

1. A.12.75
2. B.12.50
3. C.12.25✓
4. D.12.00

Correct answer: C

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is:

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

Correct answer: D

Let $S_n=1+q+q^2+\ldots+q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^2+\ldots+\left(\frac{q+1}{2}\right)^n$ where $q$ is a real number and $q\ne1$. If ${}^{101}C_1+{}^{101}C_2\cdot S_1+\ldots+{}^{101}C_{101}\cdot S_{100}=\alpha T_{100}$, then $\alpha$ is equal to:

1. A.202
2. B.200
3. C.$2^{100}$✓
4. D.$2^{99}$

Correct answer: C

$\lim\limits_{x\to0}\frac{x\cot(4x)}{\sin^2x\cot^2(2x)}$ is equal to:

1. A.2
2. B.0
3. C.1✓
4. D.4

Correct answer: C

Let K be the set of all real values of $x$ where the function $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable. Then the set K is equal to:

1. A.$\{0\}$
2. B.$\{\pi\}$
3. C.$\{0,\pi\}$
4. D.$\phi$ (an empty set)✓

Correct answer: D

Let $f(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x)^2}}$, $x\in\mathbb{R}$, where $a$, $b$ and $d$ are non-zero real constants. Then:

1. A.$f$ is an increasing function of $x$✓
2. B.$f$ is neither increasing nor decreasing function of $x$
3. C.$f$ is a decreasing function of $x$
4. D.$f'$ is not a continuous function of $x$

Correct answer: A

Let $x,y$ be positive real numbers and $m,n$ positive integers. The maximum value of the expression $\frac{x^my^n}{(1+x^{2m})(1+y^{2n})}$ is:

1. A.$\frac14$✓
2. B.1
3. C.$\frac{m+n}{6mn}$
4. D.$\frac12$

Correct answer: A

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

1. A.$\frac13(x+1)$
2. B.$\frac23(x+2)$
3. C.$\frac23(x-4)$
4. D.$\frac13(x+4)$✓

Correct answer: D

The integral $\int_{\pi/6}^{\pi/4}\frac{dx}{\sin2x\,(\tan^5x+\cot^5x)}$ equals:

1. A.$\frac{1}{10}\left(\frac{\pi}{4}-\tan^{-1}\left(\frac{1}{9\sqrt3}\right)\right)$✓
2. B.$\frac15\left(\frac{\pi}{4}-\tan^{-1}\left(\frac{1}{3\sqrt3}\right)\right)$
3. C.$\frac{\pi}{40}$
4. D.$\frac{1}{20}\tan^{-1}\left(\frac{1}{9\sqrt3}\right)$

Correct answer: A

The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^2+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is:

1. A.$\frac{37}{24}$✓
2. B.$\frac83$
3. C.$\frac{187}{24}$
4. D.$\frac{14}{3}$

Correct answer: A

The solution of the differential equation, $\frac{dy}{dx}=(x-y)^2$, when $y(1)=1$, is:

1. A.$-\log_e\left|\frac{1-x+y}{1+x-y}\right|=2(x-1)$✓
2. B.$\log_e\left|\frac{2-y}{2-x}\right|=2(y-1)$
3. C.$\log_e\left|\frac{2-x}{2-y}\right|=x-y$
4. D.$-\log_e\left|\frac{1+x-y}{1-x+y}\right|=x+y-2$

Correct answer: A

If in a parallelogram ABDC, the coordinates of A, B and C are respectively $(1,2)$, $(3,4)$ and $(2,5)$, then the equation of the diagonal AD is:

1. A.$5x+3y-11=0$
2. B.$5x-3y+1=0$✓
3. C.$3x-5y+7=0$
4. D.$3x+5y-13=0$

Correct answer: B

Let the length of the latus rectum of an ellipse with its major axis along $x$-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

1. A.$(4\sqrt2,2\sqrt2)$
2. B.$(4\sqrt2,2\sqrt3)$
3. C.$(4\sqrt3,2\sqrt3)$
4. D.$(4\sqrt3,2\sqrt2)$✓