JEE Main 2018 — 16 Apr Morning (online CBT) (Previous Year Paper)

The percentage errors in quantities P, Q, R and S are 0.5%, 1%, 3% and 1.5% respectively in the measurement of a physical quantity $A = \frac{P^3Q^2}{\sqrt{R}\,S}$. The maximum percentage error in the value of A will be :

1. A.6.0%
2. B.7.5%
3. C.8.5%
4. D.6.5%✓

Correct answer: D

Let $\vec{A} = (\hat{i} + \hat{j})$ and, $\vec{B} = (2\hat{i} - \hat{j})$. The magnitude of a coplanar vector $\vec{C}$ such that $\vec{A}\cdot\vec{C} = \vec{B}\cdot\vec{C} = \vec{A}\cdot\vec{B}$, is given by :

1. A.$\sqrt{\frac{10}{9}}$
2. B.$\sqrt{\frac{5}{9}}$✓
3. C.$\sqrt{\frac{20}{9}}$
4. D.$\sqrt{\frac{9}{12}}$

Correct answer: B

A body of mass m starts moving from rest along x-axis so that its velocity varies as $v = a\sqrt{s}$ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

1. A.$\frac{1}{8}ma^4t^2$✓
2. B.$8ma^4t^2$
3. C.$4ma^4t^2$
4. D.$\frac{1}{4}ma^4t^2$

Correct answer: A

Two particles of the same mass m are moving in circular orbits because of force, given by $F(r) = \frac{-16}{r} - r^3$. The first particle is at a distance r = 1, and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to :

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

Correct answer: A

An oscillator of mass M is at rest in its equilibrium position in a potential $V = \frac{1}{2}k(x - X)^2$. A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)

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

Correct answer: A

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

1. A.Weight of the object, everywhere on the earth, will increase.
2. B.Weight of the object, everywhere on the earth, will decrease.
3. C.There will be no change in weight anywhere on the earth.
4. D.Except at poles, weight of the object on the earth will decrease.✓

Correct answer: D

A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and z' axes will be :

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

Correct answer: A

The relative uncertainty in the period of a satellite orbiting around the earth is $10^{-2}$. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

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

Correct answer: B

A small soap bubble of radius 4 cm is trapped inside another bubble of radius 6 cm without any contact. Let $P_2$ be the pressure inside the inner bubble and $P_0$, the pressure outside the outer bubble. Radius of another bubble with pressure difference $P_2 - P_0$ between its inside and outside would be :

1. A.12 cm
2. B.2.4 cm✓
3. C.6 cm
4. D.4.8 cm

Correct answer: B

One mole of an ideal monoatomic gas is taken along the path ABCA as shown in the PV diagram. The maximum temperature attained by the gas along the path BC is given by :

1. A.$\frac{25}{16}\frac{P_0V_0}{R}$
2. B.$\frac{25}{8}\frac{P_0V_0}{R}$✓
3. C.$\frac{25}{4}\frac{P_0V_0}{R}$
4. D.$\frac{5}{8}\frac{P_0V_0}{R}$

Correct answer: B

Two moles of helium are mixed with n moles of hydrogen. If $\frac{C_P}{C_V} = \frac{3}{2}$ for the mixture, then the value of n is :

1. A.1
2. B.3
3. C.2✓
4. D.3/2

Correct answer: C

A particle executes simple harmonic motion and is located at x = a, b and c at times $t_0$, $2t_0$ and $3t_0$ respectively. The frequency of the oscillation is :

1. A.$\frac{1}{2\pi t_0}\cos^{-1}\left(\frac{a+c}{2b}\right)$✓
2. B.$\frac{1}{2\pi t_0}\cos^{-1}\left(\frac{a+b}{2c}\right)$
3. C.$\frac{1}{2\pi t_0}\cos^{-1}\left(\frac{2a+3c}{b}\right)$
4. D.$\frac{1}{2\pi t_0}\cos^{-1}\left(\frac{a+2b}{3c}\right)$

Correct answer: A

Two sitar strings, A and B, playing the note 'Dha' are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease by 3 Hz. If the frequency of A is 425 Hz, the original frequency of B is :

1. A.430 Hz
2. B.420 Hz✓
3. C.428 Hz
4. D.422 Hz

Correct answer: B

Two identical conducting spheres A and B, carry equal charge. They are separated by a distance much larger than their diameters, and the force between them is F. A third identical conducting sphere, C, is uncharged. Sphere C is first touched to A, then to B, and then removed. As a result, the force between A and B would be equal to :

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

Correct answer: C

A heating element has a resistance of 100 $\Omega$ at room temperature. When it is connected to a supply of 220 V, a steady current of 2 A passes in it and temperature is 500°C more than room temperature. What is the temperature coefficient of resistance of the heating element ?

1. A.$0.5\times10^{-4}$ °C$^{-1}$
2. B.$5\times10^{-4}$ °C$^{-1}$
3. C.$1\times10^{-4}$ °C$^{-1}$
4. D.$2\times10^{-4}$ °C$^{-1}$✓

Correct answer: D

A galvanometer with its coil resistance 25 $\Omega$ requires a current of 1 mA for its full deflection. In order to construct an ammeter to read up to a current of 2 A, the approximate value of the shunt resistance should be :

1. A.$2.5\times10^{-3}\,\Omega$
2. B.$1.25\times10^{-2}\,\Omega$✓
3. C.$1.25\times10^{-3}\,\Omega$
4. D.$2.5\times10^{-2}\,\Omega$

Correct answer: B

In the following circuit, the switch S is closed at t = 0. The charge on the capacitor $C_1$ as a function of time will be given by $\left(C_{eq} = \frac{C_1C_2}{C_1+C_2}\right)$

1. A.$C_1E\left[1 - \exp(-tR/C_1)\right]$
2. B.$C_2E\left[1 - \exp(-t/RC_2)\right]$
3. C.$C_{eq}E\left[1 - \exp(-t/RC_{eq})\right]$✓
4. D.$C_{eq}E\exp(-t/RC_{eq})$

Correct answer: C

A coil of cross-sectional area A having n turns is placed in a uniform magnetic field B. When it is rotated with an angular velocity $\omega$, the maximum e.m.f. induced in the coil will be :

1. A.$3nBA\omega$
2. B.$\frac{3}{2}nBA\omega$
3. C.$nBA\omega$✓
4. D.$\frac{1}{2}nBA\omega$

Correct answer: C

A charge q is spread uniformly over an insulated loop of radius r. If it is rotated with an angular velocity $\omega$ with respect to normal axis then the magnetic moment of the loop is :

1. A.$q\omega r^2$
2. B.$\frac{4}{3}q\omega r^2$
3. C.$\frac{3}{2}q\omega r^2$
4. D.$\frac{1}{2}q\omega r^2$✓

Correct answer: D

A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns, giving the output power at 230 V. If the current in the primary of the transformer is 5 A, and its efficiency is 90%, the output current would be :

1. A.50 A
2. B.45 A✓
3. C.25 A
4. D.20 A

Correct answer: B

A plane electromagnetic wave of wavelength $\lambda$ has an intensity I. It is propagating along the positive Y-direction. The allowed expressions for the electric and magnetic fields are given by :

1. A.$\vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}}\cos\left[\frac{2\pi}{\lambda}(y - ct)\right]\hat{k}$ ; $\vec{B} = +\frac{1}{c}E\,\hat{i}$✓
2. B.$\vec{E} = \sqrt{\frac{2I}{\epsilon_0 c}}\cos\left[\frac{2\pi}{\lambda}(y + ct)\right]\hat{k}$ ; $\vec{B} = \frac{1}{c}E\,\hat{i}$
3. C.$\vec{E} = \sqrt{\frac{I}{\epsilon_0 c}}\cos\left[\frac{2\pi}{\lambda}(y - ct)\right]\hat{k}$ ; $\vec{B} = \frac{1}{c}E\,\hat{i}$
4. D.$\vec{E} = \sqrt{\frac{I}{\epsilon_0 c}}\cos\left[\frac{2\pi}{\lambda}(y - ct)\right]\hat{i}$ ; $\vec{B} = \frac{1}{c}E\,\hat{k}$

Correct answer: A

A ray of light is incident at an angle of $60^\circ$ on one face of a prism of angle $30^\circ$. The emergent ray of light makes an angle of $30^\circ$ with incident ray. The angle made by the emergent ray with second face of prism will be :

1. A.$0^\circ$
2. B.$90^\circ$✓
3. C.$45^\circ$
4. D.$30^\circ$

Correct answer: B

Unpolarized light of intensity I is incident on a system of two polarizers, A followed by B. The intensity of emergent light is I/2. If a third polarizer C is placed between A and B, the intensity of emergent light is reduced to I/3. The angle between the polarizers A and C is $\theta$. Then :

1. A.$\cos\theta = \left(\frac{2}{3}\right)^{1/2}$
2. B.$\cos\theta = \left(\frac{2}{3}\right)^{1/4}$✓
3. C.$\cos\theta = \left(\frac{1}{3}\right)^{1/2}$
4. D.$\cos\theta = \left(\frac{1}{3}\right)^{1/4}$

Correct answer: B

The de-Broglie wavelength $(\lambda_B)$ associated with the electron orbiting in the second excited state of hydrogen atom is related to that in the ground state $(\lambda_G)$ by :

1. A.$\lambda_B = 2\lambda_G$
2. B.$\lambda_B = 3\lambda_G$✓
3. C.$\lambda_B = \lambda_G/2$
4. D.$\lambda_B = \lambda_G/3$

Correct answer: B

Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths $\lambda_N$, $\lambda_A$ respectively. The ratio $\frac{\lambda_N}{\lambda_A}$ is closest to :

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

Correct answer: A

At some instant, a radioactive sample $S_1$ having an activity 5 $\mu$Ci has twice the number of nuclei as another sample $S_2$ which has an activity of 10 $\mu$Ci. The half lives of $S_1$ and $S_2$ are :

1. A.20 years and 5 years, respectively✓
2. B.20 years and 10 years, respectively
3. C.5 years and 20 years, respectively
4. D.10 years and 20 years, respectively

Correct answer: A

In the given circuit, the current through zener diode is :

1. A.5.5 mA
2. B.6.7 mA
3. C.2.5 mA
4. D.3.3 mA✓

Correct answer: D

A carrier wave of peak voltage 14 V is used for transmitting a message signal. The peak voltage of modulating signal given to achieve a modulation index of 80% will be :

1. A.7 V
2. B.28 V
3. C.11.2 V✓
4. D.22.4 V

Correct answer: C

In a circuit for finding the resistance of a galvanometer by half deflection method, a 6 V battery and a high resistance of 11 k$\Omega$ are used. The figure of merit of the galvanometer is 60 $\mu$A/division. In the absence of shunt resistance, the galvanometer produces a deflection of $\theta = 9$ divisions when current flows in the circuit. The value of the shunt resistance that can cause the deflection of $\theta/2$, is closest to :

1. A.550 $\Omega$
2. B.220 $\Omega$
3. C.55 $\Omega$
4. D.110 $\Omega$✓

Correct answer: D

The end correction of a resonance column is 1 cm. If the shortest length resonating with the tuning fork is 10 cm, the next resonating length should be :

1. A.28 cm
2. B.32 cm✓
3. C.36 cm
4. D.40 cm

Correct answer: B

An unknown chlorohydrocarbon has 3.55% of chlorine. If each molecule of the hydrocarbon has one chlorine atom only; chlorine atoms present in 1 g of chlorohydrocarbon are : (Atomic wt. of Cl = 35.5 u; Avogadro constant = $6.023\times10^{23}$ mol$^{-1}$)

1. A.$6.023\times10^{20}$✓
2. B.$6.023\times10^{9}$
3. C.$6.023\times10^{21}$
4. D.$6.023\times10^{23}$

Correct answer: A

The gas phase reaction $2NO_2(g) \rightarrow N_2O_4(g)$ is an exothermic reaction. The decomposition of $N_2O_4$, in equilibrium mixture of $NO_2(g)$ and $N_2O_4(g)$, can be increased by :

1. A.lowering the temperature.
2. B.increasing the pressure.
3. C.addition of an inert gas at constant volume.
4. D.addition of an inert gas at constant pressure.✓

Correct answer: D

Assuming ideal gas behaviour, the ratio of density of ammonia to that of hydrogen chloride at same temperature and pressure is : (Atomic wt. of Cl = 35.5 u)

1. A.1.46
2. B.0.46✓
3. C.1.64
4. D.0.64

Correct answer: B

When 9.65 ampere current was passed for 1.0 hour into nitrobenzene in acidic medium, the amount of p-aminophenol produced is :

1. A.9.81 g✓
2. B.10.9 g
3. C.98.1 g
4. D.109.0 g

Correct answer: A

For which of the following processes, $\Delta S$ is negative ?

1. A.$H_2(g) \rightarrow 2H(g)$
2. B.$N_2(g, 1\text{ atm}) \rightarrow N_2(g, 5\text{ atm})$✓
3. C.C(diamond) $\rightarrow$ C(graphite)
4. D.$N_2(g, 273\text{ K}) \rightarrow N_2(g, 300\text{ K})$

Correct answer: B

Which one of the following is not a property of physical adsorption ?

1. A.Higher the pressure, more the adsorption
2. B.Lower the temperature, more the adsorption
3. C.Greater the surface area, more the adsorption
4. D.Unilayer adsorption occurs✓

Correct answer: D

If 50% of a reaction occurs in 100 second and 75% of the reaction occurs in 200 second, the order of this reaction is :

1. A.Zero
2. B.1✓
3. C.2
4. D.3

Correct answer: B

Which of the following statements is false ?

1. A.Photon has momentum as well as wavelength.
2. B.Splitting of spectral lines in electrical field is called Stark effect.
3. C.Rydberg constant has unit of energy.
4. D.Frequency of emitted radiation from a black body goes from a lower wavelength to higher wavelength as the temperature increases.✓

Correct answer: D

At 320 K, a gas $A_2$ is 20% dissociated to A(g). The standard free energy change at 320 K and 1 atm in J mol$^{-1}$ is approximately : (R = 8.314 J K$^{-1}$ mol$^{-1}$; ln 2 = 0.693; ln 3 = 1.098)

1. A.4763✓
2. B.2068
3. C.1844
4. D.4281

Correct answer: A

The incorrect statement is :

1. A.$Cu^{2+}$ salts give red coloured borax bead test in reducing flame.
2. B.$Cu^{2+}$ and $Ni^{2+}$ ions give black precipitate with $H_2S$ in presence of HCl solution.✓
3. C.Ferric ion gives blood red colour with potassium thiocyanate.
4. D.$Cu^{2+}$ ion gives chocolate coloured precipitate with potassium ferrocyanide solution.

Correct answer: B

The incorrect geometry is represented by :

1. A.$BF_3$ - trigonal planar
2. B.$H_2O$ - bent
3. C.$NF_3$ - trigonal planar✓
4. D.$AsF_5$ - trigonal bipyramidal

Correct answer: C

In Wilkinson's catalyst, the hybridization of central metal ion and its shape are respectively :

1. A.$sp^3d$, trigonal bipyramidal
2. B.$sp^3$, tetrahedral
3. C.$dsp^2$, square planar✓
4. D.$d^2sp^3$, octahedral

Correct answer: C

Among the oxides of nitrogen : $N_2O_3$, $N_2O_4$ and $N_2O_5$ ; the molecule(s) having nitrogen-nitrogen bond is/are :

1. A.Only $N_2O_5$
2. B.$N_2O_3$ and $N_2O_5$
3. C.$N_2O_4$ and $N_2O_5$
4. D.$N_2O_3$ and $N_2O_4$✓

Correct answer: D

Which of the following complexes will show geometrical isomerism ?

1. A.aquachlorobis(ethylenediamine)cobalt(II) chloride✓
2. B.pentaaquachlorochromium(III) chloride
3. C.potassium amminetrichloroplatinate(II)
4. D.potassium tris(oxalato)chromate(III)

Correct answer: A

In a complexometric titration of metal ion with ligand M(Metal ion) + L(Ligand) $\rightarrow$ C(Complex) end point is estimated spectrophotometrically (through light absorption). If 'M' and 'C' do not absorb light and only 'L' absorbs, then the titration plot between absorbed light (A) versus volume of ligand 'L' (V) would look like :

1. A.Graph (1): A decreases linearly with V to a minimum at the end point, then increases linearly (V-shaped)
2. B.Graph (2): A stays constant (flat, near zero) up to the end point, then increases linearly✓
3. C.Graph (3): A increases linearly to a maximum at the end point, then decreases linearly (inverted V)
4. D.Graph (4): A decreases linearly up to the end point, then stays constant

Correct answer: B

In the extraction of copper from its sulphide ore, metal is finally obtained by the oxidation of cuprous sulphide with :

1. A.$Fe_2O_3$
2. B.$Cu_2O$✓
3. C.$SO_2$
4. D.CO

Correct answer: B

Which of the following conversions involves change in both shape and hybridisation ?

1. A.$NH_3 \rightarrow NH_4^+$
2. B.$CH_4 \rightarrow C_2H_6$
3. C.$H_2O \rightarrow H_3O^+$
4. D.$BF_3 \rightarrow BF_4^-$✓

Correct answer: D

A group 13 element 'X' reacts with chlorine gas to produce a compound $XCl_3$. $XCl_3$ is electron deficient and easily reacts with $NH_3$ to form $Cl_3X \leftarrow NH_3$ adduct; however, $XCl_3$ does not dimerize. X is :

1. A.B✓
2. B.Al
3. C.Ga
4. D.In

Correct answer: A

When $XO_2$ is fused with an alkali metal hydroxide in presence of an oxidizing agent such as $KNO_3$; a dark green product is formed which disproportionates in acidic solution to afford a dark purple solution. X is :

1. A.Ti
2. B.V
3. C.Cr
4. D.Mn✓

Correct answer: D

The major product of the following reaction is : $C_6H_5-CH=CHCH_3 \xrightarrow{HBr}$

1. A.$C_6H_5-CH(Br)CH_2CH_3$✓
2. B.$C_6H_5-CH_2CH(Br)CH_3$
3. C.$C_6H_5-CH_2CH_2CH_2Br$
4. D.$p$-$Br-C_6H_4-CH=CHCH_3$ (ring-brominated)

Correct answer: A

For standardizing NaOH solution, which of the following is used as a primary standard ?

1. A.Ferrous Ammonium Sulfate
2. B.dil. HCl
3. C.Oxalic acid✓
4. D.Sodium tetraborate

Correct answer: C

The most polar compound among the following is :

1. A.1-Fluoro-3-(1-fluoroethylidene)cyclohex-1-ene, geometric isomer drawn in option (1) (exocyclic F drawn to the right, pointing up)✓
2. B.1,4-Difluoro-3-(propan-2-ylidene)cyclohex-1-ene (ring vinylic F plus an sp$^3$ C-F, exocyclic $=C(CH_3)_2$)
3. C.1-Fluoro-3-(1-fluoroethylidene)cyclohex-1-ene, the other geometric isomer (exocyclic F drawn pointing down)
4. D.1-Methyl-3-(propan-2-ylidene)cyclohex-1-ene (no fluorine)

Correct answer: A

The correct match between items of List - I and List - II is : List - I: (A) Phenelzine (B) Chloroxylenol (C) Uracil (D) Ranitidine. List - II: (P) Pyrimidine (Q) Furan (R) Hydrazine (S) Phenol

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

Correct answer: A

Among the following, the incorrect statement is :

1. A.Maltose and lactose has 1,4-glycosidic linkage.
2. B.Sucrose and amylose has 1,2-glycosidic linkage.✓
3. C.Cellulose and amylose has 1,4-glycosidic linkage.
4. D.Lactose contains $\beta$-D-galactose and $\beta$-D-glucose.

Correct answer: B

Which of the following compounds will most readily be dehydrated to give alkene under acidic condition ?

1. A.1-Pentanol
2. B.4-Hydroxypentan-2-one✓
3. C.3-Hydroxypentan-2-one
4. D.2-Hydroxycyclopentanone

Correct answer: B

Products A and B formed in the following reactions are respectively : $p$-$HO_3S-C_6H_4-\overset{+}{N}H_3\ CH_3CO\overset{-}{O} + HNO_2 \rightarrow A \xrightarrow{C_6H_5NH_2} B$

1. A.A = $p$-$HO_3S-C_6H_4-N=N-COCH_3$ ; B = $p$-$HO_3S-C_6H_4-N=N-C_6H_5$
2. B.A = $p$-$HO_3S-C_6H_4-N=N-COCH_3$ ; B = $HO_3S-C_6H_4-NH-C_6H_4-NH_2$
3. C.A = $p$-$HO_3S-C_6H_4-N=N-OCOCH_3$ ; B = $p$-$HO_3S-C_6H_4-N=N-NH-C_6H_5$
4. D.A = $p$-$HO_3S-C_6H_4-N=N-OCOCH_3$ ; B = $p$-$HO_3S-C_6H_4-N=N-C_6H_4-NH_2$ ($p$-amino azo compound)✓

Correct answer: D

The major product B formed in the following reaction sequence is : $p$-$MeO-C_6H_4-CHO \xrightarrow[\text{(ii) } H_2O]{\text{(i) } C_2H_5MgBr} A \xrightarrow{HCl} B$

1. A.$p$-$Cl-C_6H_4-CH_2CH(Cl)CH_3$
2. B.$p$-$MeO-C_6H_4-CH(Cl)CH_2CH_3$✓
3. C.$p$-$MeO-C_6H_4-CH=CHCH_3$
4. D.$p$-$MeO-C_6H_4-CH_2CH=CH_2$

Correct answer: B

The major product of the following reaction is : [(E)-hex-3-ene chain with Br on C-2 (wedge) and $NH_2$ on C-5 (hashed), treated with KOH ($S_N2$)]

1. A.Hex-3-ene chain drawn with $H_2N$ on a bold wedge at the left end carbon and OH on a bold wedge at the right end carbon
2. B.Chain drawn with OH on a bold wedge at the left end carbon and $NH_2$ on a hashed bond at the right end carbon (same drawing as the substrate with Br replaced by OH)
3. C.Chain drawn with $H_2N$ on a bold wedge at the left end carbon and OH on a hashed bond at the right end carbon✓
4. D.Chain drawn with OH on a hashed bond at the left end carbon and $NH_2$ on a bold wedge at the right end carbon

Correct answer: C

The major product of the following reaction is : 3-methoxyphenol $\xrightarrow[\text{(ii) } H_2SO_4,\ \text{heat}]{\text{(i) } OHC\,CH_2COCl}$

1. A.7-Methoxy-2H-chromen-2-one (7-methoxycoumarin)✓
2. B.7-Methoxy-4H-chromen-4-one (7-methoxychromone)
3. C.5-Methoxy-2H-chromen-2-one (5-methoxycoumarin)
4. D.5-Methoxy-4H-chromen-4-one (5-methoxychromone)

Correct answer: A

Let $\mathbb{N}$ denote the set of all natural numbers. Define two binary relations on $\mathbb{N}$ as $R_1 = \{(x, y) \in \mathbb{N}\times\mathbb{N} : 2x + y = 10\}$ and $R_2 = \{(x, y) \in \mathbb{N}\times\mathbb{N} : x + 2y = 10\}$. Then :

1. A.Range of $R_1$ is $\{2, 4, 8\}$.
2. B.Range of $R_2$ is $\{1, 2, 3, 4\}$.✓
3. C.Both $R_1$ and $R_2$ are symmetric relations.
4. D.Both $R_1$ and $R_2$ are transitive relations.

Correct answer: B

Let p, q and r be real numbers $(p \ne q, r \ne 0)$, such that the roots of the equation $\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

1. A.$\frac{p^2+q^2}{2}$
2. B.$p^2 + q^2$✓
3. C.$2(p^2 + q^2)$
4. D.$p^2 + q^2 + r^2$

Correct answer: B

The least positive integer n for which $\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)^n = 1$, is :

1. A.2
2. B.3✓
3. C.5
4. D.6

Correct answer: B

Let $A = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1\end{bmatrix}$ and $B = A^{20}$. Then the sum of the elements of the first column of B is :

1. A.210
2. B.211
3. C.231✓
4. D.251

Correct answer: C

The number of values of k for which the system of linear equations, $(k + 2)x + 10y = k$, $kx + (k + 3)y = k - 1$ has no solution, is :

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

Correct answer: A

The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

1. A.24
2. B.30✓
3. C.36
4. D.48

Correct answer: B

The coefficient of $x^2$ in the expansion of the product $(2 - x^2)\cdot\left((1 + 2x + 3x^2)^6 + (1 - 4x^2)^6\right)$ is :

1. A.107
2. B.106✓
3. C.108
4. D.155

Correct answer: B

Let $\frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_n}$ $(x_i \ne 0$ for $i = 1, 2, \ldots, n)$ be in A.P. such that $x_1 = 4$ and $x_{21} = 20$. If n is the least positive integer for which $x_n > 50$, then $\sum_{i=1}^{n}\left(\frac{1}{x_i}\right)$ is equal to :

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

Correct answer: D

The sum of the first 20 terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \ldots$, is :

1. A.$38 + \frac{1}{2^{19}}$✓
2. B.$38 + \frac{1}{2^{20}}$
3. C.$39 + \frac{1}{2^{20}}$
4. D.$39 + \frac{1}{2^{19}}$

Correct answer: A

$\lim_{x\to0}\frac{(27+x)^{1/3} - 3}{9 - (27+x)^{2/3}}$ equals :

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

Correct answer: C

If the function f defined as $f(x) = \frac{1}{x} - \frac{k-1}{e^{2x}-1}$, $x \ne 0$, is continuous at $x = 0$, then the ordered pair $(k, f(0))$ is equal to :

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

Correct answer: B

If $x = \sqrt{2^{\text{cosec}^{-1}t}}$ and $y = \sqrt{2^{\sec^{-1}t}}$ $(|t| \ge 1)$, then $\frac{dy}{dx}$ is equal to :

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

Correct answer: C

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, $f(x) = 2x^3 - 9x^2 + 12x + 5$ in the interval [0, 3]. Then $M - m$ is equal to :

1. A.5
2. B.9✓
3. C.4
4. D.1

Correct answer: B

If $\int\frac{\tan x}{1 + \tan x + \tan^2 x}dx = x - \frac{K}{\sqrt{A}}\tan^{-1}\left(\frac{K\tan x + 1}{\sqrt{A}}\right) + C$, (C is a constant of integration), then the ordered pair (K, A) is equal to :

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

Correct answer: C

If $f(x) = \int_0^x t(\sin x - \sin t)\,dt$ then :

1. A.$f'''(x) + f''(x) = \sin x$
2. B.$f'''(x) + f''(x) - f'(x) = \cos x$
3. C.$f'''(x) + f'(x) = \cos x - 2x\sin x$✓
4. D.$f'''(x) - f''(x) = \cos x - 2x\sin x$

Correct answer: C

If the area of the region bounded by the curves, $y = x^2$, $y = \frac{1}{x}$ and the lines $y = 0$ and $x = t$ $(t > 1)$ is 1 sq. unit, then t is equal to :

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

Correct answer: D

The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is :

1. A.$xyy'' + x(y')^2 - yy' = 0$
2. B.$x + yy'' = 0$
3. C.$xyy' + y^2 - 9 = 0$
4. D.$xyy' - y^2 + 9 = 0$✓

Correct answer: D

The locus of the point of intersection of the lines, $\sqrt{2}x - y + 4\sqrt{2}k = 0$ and $\sqrt{2}kx + ky - 4\sqrt{2} = 0$ (k is any non-zero real parameter), is :

1. A.an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$.
2. B.an ellipse with length of its major axis $8\sqrt{2}$.
3. C.a hyperbola whose eccentricity is $\sqrt{3}$.
4. D.a hyperbola with length of its transverse axis $8\sqrt{2}$.✓

Correct answer: D

If a circle C, whose radius is 3, touches externally the circle, $x^2 + y^2 + 2x - 4y - 4 = 0$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

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

Correct answer: A

Let P be a point on the parabola, $x^2 = 4y$. If the distance of P from the centre of the circle, $x^2 + y^2 + 6x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at P, is :

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

Correct answer: C

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is :

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

Correct answer: B

The sum of the intercepts on the coordinate axes of the plane passing through the point $(-2, -2, 2)$ and containing the line joining the points $(1, -1, 2)$ and $(1, 1, 1)$, is :

1. A.4
2. B.$-4$✓
3. C.$-8$
4. D.12

Correct answer: B

If the angle between the lines, $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{5-x}{-2} = \frac{7y-14}{p} = \frac{z-3}{4}$ is $\cos^{-1}\left(\frac{2}{3}\right)$, then p is equal to :

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

Correct answer: A

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{c} = \hat{j} - \hat{k}$ and a vector $\vec{b}$ be such that $\vec{a}\times\vec{b} = \vec{c}$ and $\vec{a}\cdot\vec{b} = 3$. Then $|\vec{b}|$ equals :

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

Correct answer: C

The mean and the standard deviation (s.d.) of five observations are 9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is :

1. A.0
2. B.1
3. C.2✓
4. D.4

Correct answer: C

Let A, B and C be three events, which are pair-wise independent and $\overline{E}$ denotes the complement of an event E. If $P(A\cap B\cap C) = 0$ and $P(C) > 0$, then $P\left[(\overline{A}\cap\overline{B})\,|\,C\right]$ is equal to :

1. A.$P(\overline{A}) - P(B)$✓
2. B.$P(A) + P(\overline{B})$
3. C.$P(\overline{A}) - P(\overline{B})$
4. D.$P(\overline{A}) + P(\overline{B})$

Correct answer: A

Two different families A and B are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is $\frac{1}{12}$, then the number of children in each family is :

1. A.3
2. B.4
3. C.5✓
4. D.6

Correct answer: C

If an angle A of a $\Delta ABC$ satisfies $5\cos A + 3 = 0$, then the roots of the quadratic equation, $9x^2 + 27x + 20 = 0$ are :

1. A.$\sec A$, $\cot A$
2. B.$\sin A$, $\sec A$
3. C.$\sec A$, $\tan A$✓
4. D.$\tan A$, $\cos A$