JEE Main 2019 — 9 Apr Shift 1 (Previous Year Paper)

In the density measurement of a cube, the mass and edge length are measured as $(10.00 \pm 0.10)$ kg and $(0.10 \pm 0.01)$ m, respectively. The error in the measurement of density is:

1. A.0.10 kg/m$^3$
2. B.0.07 kg/m$^3$
3. C.0.31 kg/m$^3$✓
4. D.0.01 kg/m$^3$

Correct answer: C

The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?

1. A.150°
2. B.60°
3. C.90°
4. D.120°✓

Correct answer: D

A ball is thrown vertically up (taken as +z-axis) from the ground. The correct momentum-height (p-h) diagram is:

1. A.Curved (parabolic) p-h graph from +p on the p-axis to the h-axis and on to -p, with arrows running from -p up to +p (reverse sense)
2. B.Straight-line (triangular) p-h graph from +p to the h-axis and to -p
3. C.Curved (parabolic) p-h graph starting at +p on the p-axis, reaching p = 0 at maximum height on the h-axis and continuing to -p, arrows running from +p to -p✓
4. D.Straight-line (triangular) p-h graph with arrows from +p to -p

Correct answer: C

A body of mass 2 kg makes an elastic collision with a second body at rest and continues to move in the original direction but with one fourth of its original speed. What is the mass of the second body?

1. A.1.5 kg
2. B.1.2 kg✓
3. C.1.0 kg
4. D.1.8 kg

Correct answer: B

A uniform cable of mass 'M' and length 'L' is placed on a horizontal surface such that its $\left(\frac{1}{n}\right)^{th}$ part is hanging below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be:

1. A.$\dfrac{MgL}{n^2}$
2. B.$\dfrac{2MgL}{n^2}$
3. C.$\dfrac{MgL}{2n^2}$✓
4. D.$nMgL$

Correct answer: C

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$, where $\theta$ is the angle by which it has rotated, is given as $k\theta^2$. If its moment of inertia is I then the angular acceleration of the disc is:

1. A.$\dfrac{k}{I}\theta$
2. B.$\dfrac{2k}{I}\theta$✓
3. C.$\dfrac{k}{2I}\theta$
4. D.[option 4 not visible in the NTA portal image]

Correct answer: B

A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be:

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

Correct answer: C

For a given gas at 1 atm pressure, rms speed of the molecules is 200 m/s at 127°C. At 2 atm pressure and at 227°C, the rms speed of the molecules will be:

1. A.$100\sqrt{5}$ m/s✓
2. B.$80\sqrt{5}$ m/s
3. C.100 m/s
4. D.80 m/s

Correct answer: A

A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}$th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

1. A.$2T\sqrt{\dfrac{1}{10}}$
2. B.$4T\sqrt{\dfrac{1}{15}}$✓
3. C.$4T\sqrt{\dfrac{1}{14}}$
4. D.$2T\sqrt{\dfrac{1}{14}}$

Correct answer: B

Following figure shows two processes A and B for a gas. If $\Delta Q_A$ and $\Delta Q_B$ are the amount of heat absorbed by the system in two cases, and $\Delta U_A$ and $\Delta U_B$ are changes in internal energies, respectively, then:

1. A.$\Delta Q_A = \Delta Q_B$; $\Delta U_A = \Delta U_B$
2. B.$\Delta Q_A > \Delta Q_B$, $\Delta U_A > \Delta U_B$
3. C.$\Delta Q_A > \Delta Q_B$, $\Delta U_A = \Delta U_B$✓
4. D.$\Delta Q_A < \Delta Q_B$, $\Delta U_A < \Delta U_B$

Correct answer: C

An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is $\bar{v}$, m is its mass and $k_B$ is Boltzmann constant, then its temperature will be:

1. A.$\dfrac{m\bar{v}^2}{7k_B}$
2. B.$\dfrac{m\bar{v}^2}{5k_B}$
3. C.$\dfrac{m\bar{v}^2}{3k_B}$✓
4. D.$\dfrac{m\bar{v}^2}{6k_B}$

Correct answer: C

A string is clamped at both the ends and it is vibrating in its 4$^{th}$ harmonic. The equation of the stationary wave is $Y = 0.3\sin(0.157x)\cos(200\pi t)$. The length of the string is: (All quantities are in SI units.)

1. A.80 m✓
2. B.60 m
3. C.40 m
4. D.20 m

Correct answer: A

The pressure wave, $P = 0.01\sin[1000t - 3x]$ N m$^{-2}$, corresponds to the sound produced by a vibrating blade on a day when atmospheric temperature is 0°C. On some other day when temperature is T, the speed of sound produced by the same blade and at the same frequency is found to be 336 m s$^{-1}$. Approximate value of T is:

1. A.12°C
2. B.11°C
3. C.15°C
4. D.4°C✓

Correct answer: D

A capacitor with capacitance 5 μF is charged to 5 μC. If the plates are pulled apart to reduce the capacitance to 2 μF, how much work is done?

1. A.$6.25\times10^{-6}$ J
2. B.$2.16\times10^{-6}$ J
3. C.$2.55\times10^{-6}$ J
4. D.$3.75\times10^{-6}$ J✓

Correct answer: D

A system of three charges are placed as shown in the figure: If $D \gg d$, the potential energy of the system is best given by:

1. A.$\dfrac{1}{4\pi\epsilon_0}\left[+\dfrac{q^2}{d} + \dfrac{qQd}{D^2}\right]$
2. B.$\dfrac{1}{4\pi\epsilon_0}\left[-\dfrac{q^2}{d} - \dfrac{qQd}{2D^2}\right]$
3. C.$\dfrac{1}{4\pi\epsilon_0}\left[-\dfrac{q^2}{d} + \dfrac{2qQd}{D^2}\right]$
4. D.$\dfrac{1}{4\pi\epsilon_0}\left[-\dfrac{q^2}{d} - \dfrac{qQd}{D^2}\right]$✓

Correct answer: D

A wire of resistance R is bent to form a square ABCD as shown in the figure. The effective resistance between E and C is: (E is mid-point of arm CD)

1. A.$\dfrac{3}{4}R$
2. B.$\dfrac{1}{16}R$
3. C.$\dfrac{7}{64}R$✓
4. D.R

Correct answer: C

Determine the charge on the capacitor in the following circuit:

1. A.2 μC
2. B.200 μC✓
3. C.60 μC
4. D.10 μC

Correct answer: B

A rectangular coil (Dimension 5 cm × 2.5 cm) with 100 turns, carrying a current of 3 A in the clock-wise direction, is kept centered at the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through 45° about Z-axis, then the torque on the coil is:

1. A.0.27 Nm✓
2. B.0.55 Nm
3. C.0.38 Nm
4. D.0.42 Nm

Correct answer: A

A rigid square loop of side 'a' and carrying current $I_2$ is lying on a horizontal surface near a long current $I_1$ carrying wire in the same plane as shown in figure. The net force on the loop due to the wire will be:

1. A.Zero
2. B.Attractive and equal to $\dfrac{\mu_0 I_1 I_2}{3\pi}$
3. C.Repulsive and equal to $\dfrac{\mu_0 I_1 I_2}{4\pi}$✓
4. D.Repulsive and equal to $\dfrac{\mu_0 I_1 I_2}{2\pi}$

Correct answer: C

The total number of turns and cross-section area in a solenoid is fixed. However, its length L is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to:

1. A.L
2. B.$L^2$
3. C.$1/L^2$
4. D.$1/L$✓

Correct answer: D

The magnetic field of a plane electromagnetic wave is given by: $\vec{B} = B_0\hat{i}[\cos(kz - \omega t)] + B_1\hat{j}\cos(kz + \omega t)$ where $B_0 = 3\times10^{-5}$ T and $B_1 = 2\times10^{-6}$ T. The rms value of the force experienced by a stationary charge $Q = 10^{-4}$ C at z = 0 is closest to:

1. A.0.6 N✓
2. B.$3\times10^{-2}$ N
3. C.0.9 N
4. D.0.1 N

Correct answer: A

A concave mirror for face viewing has focal length of 0.4 m. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of 5 is:

1. A.0.16 m
2. B.1.60 m
3. C.0.32 m✓
4. D.0.24 m

Correct answer: C

Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2$^{nd}$ Balmer line (n = 4 to n = 2) will be:

1. A.488.9 nm✓
2. B.889.2 nm
3. C.388.9 nm
4. D.642.7 nm

Correct answer: A

The electric field of light wave is given as $\vec{E} = 10^{-3}\cos\left(\dfrac{2\pi x}{5\times10^{-7}} - 2\pi\times6\times10^{14}t\right)\hat{x}\ \frac{\text{N}}{\text{C}}$. This light falls on a metal plate of work function 2 eV. The stopping potential of the photo-electrons is: Given, E (in eV) $= \dfrac{12375}{\lambda(\text{in Å})}$

1. A.0.72 V
2. B.0.48 V✓
3. C.2.48 V
4. D.2.0 V

Correct answer: B

An NPN transistor is used in common emitter configuration as an amplifier with 1 kΩ load resistance. Signal voltage of 10 mV is applied across the base-emitter. This produces a 3 mA change in the collector current and 15 μA change in the base current of the amplifier. The input resistance and voltage gain are:

1. A.0.33 kΩ, 300
2. B.0.33 kΩ, 1.5
3. C.0.67 kΩ, 300✓
4. D.0.67 kΩ, 200

Correct answer: C

A signal $A\cos\omega t$ is transmitted using $v_0\sin\omega_0 t$ as carrier wave. The correct amplitude modulated (AM) signal is:

1. A.$v_0\sin\omega_0 t + A\cos\omega t$
2. B.$(v_0 + A)\cos\omega t\sin\omega_0 t$
3. C.$v_0\sin\omega_0 t + \dfrac{A}{2}\sin(\omega_0 - \omega)t + \dfrac{A}{2}\sin(\omega_0 + \omega)t$✓
4. D.$v_0\sin[\omega_0(1 + 0.01A\sin\omega t)t]$

Correct answer: C

If 'M' is the mass of water that rises in a capillary tube of radius 'r', then mass of water which will rise in a capillary tube of radius '2r' is:

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

Correct answer: B

A moving coil galvanometer has resistance 50 Ω and it indicates full deflection at 4 mA current. A voltmeter is made using this galvanometer and a 5 kΩ resistance. The maximum voltage, that can be measured using this voltmeter, will be close to:

1. A.10 V
2. B.15 V
3. C.20 V✓
4. D.40 V

Correct answer: C

The major product of the following reaction is: $CH_3CH=CHCO_2CH_3 \xrightarrow{LiAlH_4}$

1. A.$CH_3CH_2CH_2CO_2CH_3$
2. B.$CH_3CH=CHCH_2OH$✓
3. C.$CH_3CH_2CH_2CH_2OH$
4. D.$CH_3CH_2CH_2CHO$

Correct answer: B

The increasing order of reactivity of the following compounds towards aromatic electrophilic substitution reaction is: chlorobenzene (A), anisole (B, $C_6H_5OMe$), toluene (C, $C_6H_5Me$), benzonitrile (D, $C_6H_5CN$)

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

Correct answer: B

The major product of the following reaction is: $CH_3C\equiv CH \xrightarrow[\text{(ii) DI}]{\text{(i) DCl (1 equiv.)}}$

1. A.$CH_3CD_2CH(Cl)(I)$
2. B.$CH_3CD(I)CHD(Cl)$
3. C.$CH_3CD(Cl)CHD(I)$
4. D.$CH_3C(I)(Cl)CHD_2$✓

Correct answer: D

Which of the following statements is not true about sucrose?

1. A.On hydrolysis, it produces glucose and fructose
2. B.It is a non reducing sugar
3. C.The glycosidic linkage is present between $C_1$ of α-glucose and $C_1$ of β-fructose✓
4. D.It is also named as invert sugar

Correct answer: C

The major product of the following reaction is: 4-hydroxycyclohexan-1-one $\xrightarrow[\text{2. KOH (alc.)}]{\text{1. PBr}_3}$

1. A.Phenol
2. B.Cyclohex-2-en-1-one (C=C conjugated with C=O)✓
3. C.5-Hydroxycyclohex-2-en-1-one
4. D.Cyclohex-3-en-1-one (C=C not conjugated with C=O)

Correct answer: B

The major product of the following reaction is: $4-ClC_6H_4-CH_2-CH(Cl)-CH_3 \xrightarrow[\text{(2) Free radical polymerisation}]{\text{(1) KOH (alc.)}}$

1. A.$-[CH_2-CH(CH_2C_6H_4Cl)]_n-$ (polymer of 3-(4-chlorophenyl)propene)
2. B.$-[CH(CH_3)-CH(C_6H_4Cl)]_n-$ (polymer of 1-(4-chlorophenyl)propene)✓
3. C.$-[CH_2-CH(CH_2C_6H_4OH)]_n-$
4. D.$-[CH(CH_3)-CH(C_6H_4OH)]_n-$

Correct answer: B

The major product of the following reaction is: ethylbenzene $\xrightarrow[\text{(ii) H}_3\text{O}^+]{\text{(i) alkaline KMnO}_4}$

1. A.$C_6H_5COCH_3$
2. B.$C_6H_5COOH$✓
3. C.$C_6H_5CH_2CHO$
4. D.$C_6H_5CH_2COOH$

Correct answer: B

The organic compound that gives following qualitative analysis is: Test / Inference - (a) Dil. HCl: Insoluble; (b) NaOH solution: soluble; (c) Br$_2$/water: Decolourization

1. A.Cyclohexanol
2. B.Aniline
3. C.Phenol✓
4. D.Cyclohexylamine

Correct answer: C

The correct IUPAC name of the following compound is: benzene ring with $-NO_2$ at the top, $-CH_3$ at the bottom (para to $NO_2$) and $-Cl$ adjacent to the $CH_3$

1. A.5-chloro-4-methyl-1-nitrobenzene
2. B.2-chloro-1-methyl-4-nitrobenzene✓
3. C.2-methyl-5-nitro-1-chlorobenzene
4. D.3-chloro-4-methyl-1-nitrobenzene

Correct answer: B

Aniline dissolved in dilute HCl is reacted with sodium nitrite at 0°C. This solution was added dropwise to a solution containing equimolar mixture of aniline and phenol in dil. HCl. The structure of the major product is:

1. A.$C_6H_5-N=N-NH-C_6H_5$
2. B.$C_6H_5-N=N-O-C_6H_5$
3. C.$C_6H_5-N=N-C_6H_4-NH_2$ (para)✓
4. D.$C_6H_5-N=N-C_6H_4-OH$ (para)

Correct answer: C

The element having greatest difference between its first and second ionization energies, is:

1. A.K✓
2. B.Ca
3. C.Ba
4. D.Sc

Correct answer: A

The ore that contains the metal in the form of fluoride is:

1. A.sphalerite
2. B.cryolite✓
3. C.magnetite
4. D.malachite

Correct answer: B

The number of water molecule(s) not coordinated to copper ion directly in $CuSO_4\cdot5H_2O$, is:

1. A.1✓
2. B.2
3. C.3
4. D.4

Correct answer: A

Magnesium powder burns in air to give:

1. A.MgO only
2. B.$Mg(NO_3)_2$ and $Mg_3N_2$
3. C.MgO and $Mg_3N_2$✓
4. D.MgO and $Mg(NO_3)_2$

Correct answer: C

The correct order of the oxidation states of nitrogen in NO, N$_2$O, NO$_2$ and N$_2$O$_3$ is:

1. A.$NO_2 < NO < N_2O_3 < N_2O$
2. B.$NO_2 < N_2O_3 < NO < N_2O$
3. C.$N_2O < NO < N_2O_3 < NO_2$✓
4. D.$N_2O < N_2O_3 < NO < NO_2$

Correct answer: C

$C_{60}$, an allotrope of carbon contains:

1. A.20 hexagons and 12 pentagons.✓
2. B.12 hexagons and 20 pentagons.
3. C.16 hexagons and 16 pentagons.
4. D.18 hexagons and 14 pentagons.

Correct answer: A

Match the catalysts (Column I) with products (Column II). Column I (Catalyst): (A) $V_2O_5$; (B) $TiCl_4/Al(Me)_3$; (C) $PdCl_2$; (D) Iron Oxide. Column II (Product): (i) Polyethylene; (ii) ethanal; (iii) $H_2SO_4$; (iv) $NH_3$

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

Correct answer: D

The one that will show optical activity is: (en = ethane-1,2-diamine)

1. A.[M(A)2(B)2(en)] with the two A ligands cis, the two B ligands cis and en spanning two cis positions✓
2. B.[M(A)2(B)4] with the two A ligands trans (axial)
3. C.[M(en)2(A)2] with the two A ligands trans (axial)
4. D.[M(A)3(B)3]

Correct answer: A

The degenerate orbitals of $[Cr(H_2O)_6]^{3+}$ are:

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

Correct answer: B

Excessive release of CO$_2$ into the atmosphere results in:

1. A.depletion of ozone
2. B.formation of smog
3. C.global warming✓
4. D.polar vortex

Correct answer: C

For a reaction, $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$; identify dihydrogen (H$_2$) as a limiting reagent in the following reaction mixtures.

1. A.14 g of $N_2$ + 4 g of $H_2$
2. B.28 g of $N_2$ + 6 g of $H_2$
3. C.35 g of $N_2$ + 8 g of $H_2$
4. D.56 g of $N_2$ + 10 g of $H_2$✓

Correct answer: D

Consider the van der Waals constants, a and b, for the following gases. Gas: Ar, Ne, Kr, Xe; a/(atm dm$^6$ mol$^{-2}$): 1.3, 0.2, 5.1, 4.1; b/($10^{-2}$ dm$^3$ mol$^{-1}$): 3.2, 1.7, 1.0, 5.0. Which gas is expected to have the highest critical temperature?

1. A.Ar
2. B.Ne
3. C.Kr✓
4. D.Xe

Correct answer: C

For any given series of spectral lines of atomic hydrogen, let $\Delta\bar{v} = \bar{v}_{max} - \bar{v}_{min}$ be the difference in maximum and minimum frequencies in cm$^{-1}$. The ratio $\Delta\bar{v}_{Lyman}/\Delta\bar{v}_{Balmer}$ is:

1. A.27 : 5
2. B.4 : 1
3. C.9 : 4✓
4. D.5 : 4

Correct answer: C

Among the following, the molecule expected to be stabilized by anion formation is: $C_2$, $O_2$, NO, $F_2$

1. A.$O_2$
2. B.$C_2$✓
3. C.NO
4. D.$F_2$

Correct answer: B

Among the following, the set of parameters that represents path functions, is: (A) q + w (B) q (C) w (D) H − TS

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

Correct answer: B

Liquid 'M' and liquid 'N' form an ideal solution. The vapour pressures of pure liquids 'M' and 'N' are 450 and 700 mmHg, respectively, at the same temperature. Then correct statement is: ($x_M$ = Mole fraction of 'M' in solution; $x_N$ = Mole fraction of 'N' in solution; $y_M$ = Mole fraction of 'M' in vapour phase; $y_N$ = Mole fraction of 'N' in vapour phase)

1. A.$\dfrac{x_M}{x_N} = \dfrac{y_M}{y_N}$
2. B.$\dfrac{x_M}{x_N} > \dfrac{y_M}{y_N}$✓
3. C.$\dfrac{x_M}{x_N} < \dfrac{y_M}{y_N}$
4. D.$(x_M - y_M) < (x_N - y_N)$

Correct answer: B

The osmotic pressure of a dilute solution of an ionic compound XY in water is four times that of a solution of 0.01 M BaCl$_2$ in water. Assuming complete dissociation of the given ionic compounds in water, the concentration of XY (in mol L$^{-1}$) in solution is:

1. A.$4\times10^{-2}$
2. B.$16\times10^{-4}$
3. C.$6\times10^{-2}$✓
4. D.$4\times10^{-4}$

Correct answer: C

The standard Gibbs energy for the given cell reaction in kJ mol$^{-1}$ at 298 K is: $Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)$, $E^o = 2$ V at 298 K (Faraday's constant, F = 96000 C mol$^{-1}$)

1. A.192
2. B.384
3. C.−384✓
4. D.−192

Correct answer: C

The given plots represent the variation of the concentration of a reactant R with time for two different reactions (i) and (ii). The respective orders of the reactions are:

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

Correct answer: B

The aerosol is a kind of colloid in which:

1. A.liquid is dispersed in water
2. B.gas is dispersed in liquid
3. C.gas is dispersed in solid
4. D.solid is dispersed in gas✓

Correct answer: D

If the function $f: \mathbb{R} - \{1, -1\} \rightarrow A$ defined by $f(x) = \dfrac{x^2}{1 - x^2}$, is surjective, then A is equal to:

1. A.$\mathbb{R} - [-1, 0)$✓
2. B.$\mathbb{R} - \{-1\}$
3. C.$\mathbb{R} - (-1, 0)$
4. D.$[0, \infty)$

Correct answer: A

All the points in the set $S = \left\{\dfrac{\alpha + i}{\alpha - i} : \alpha \in \mathbb{R}\right\}$ $(i = \sqrt{-1})$ lie on a:

1. A.circle whose radius is $\sqrt{2}$.
2. B.circle whose radius is 1.✓
3. C.straight line whose slope is 1.
4. D.straight line whose slope is −1.

Correct answer: B

Let p, q $\in \mathbb{R}$. If $2 - \sqrt{3}$ is a root of the quadratic equation, $x^2 + px + q = 0$, then:

1. A.$p^2 - 4q + 12 = 0$
2. B.$q^2 - 4p - 16 = 0$
3. C.$p^2 - 4q - 12 = 0$
4. D.$q^2 + 4p + 14 = 0$✓

Correct answer: D

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + x + 1 = 0$. Then for $y \ne 0$ in $\mathbb{R}$, $\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{vmatrix}$ is equal to:

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

Correct answer: D

If $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\cdot\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\cdot\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\cdots\begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix}$, then the inverse of $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is:

1. A.$\begin{bmatrix} 1 & 0 \\ 12 & 1 \end{bmatrix}$
2. B.$\begin{bmatrix} 1 & 0 \\ 13 & 1 \end{bmatrix}$
3. C.$\begin{bmatrix} 1 & -13 \\ 0 & 1 \end{bmatrix}$✓
4. D.$\begin{bmatrix} 1 & -12 \\ 0 & 1 \end{bmatrix}$

Correct answer: C

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then:

1. A.n = m − 8
2. B.m = n = 78✓
3. C.m = n = 68
4. D.m + n = 68

Correct answer: B

Let the sum of the first n terms of a non-constant A.P., $a_1, a_2, a_3, \ldots$ be $50n + \dfrac{n(n-7)}{2}A$, where A is a constant. If d is the common difference of this A.P., then the ordered pair $(d, a_{50})$ is equal to:

1. A.(50, 50 + 46A)
2. B.(A, 50 + 46A)✓
3. C.(A, 50 + 45A)
4. D.(50, 50 + 45A)

Correct answer: B

Let $\displaystyle\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)$, where the function f satisfies $f(x + y) = f(x)f(y)$ for all natural numbers x, y and $f(1) = 2$. Then the natural number 'a' is:

1. A.2
2. B.3✓
3. C.4
4. D.16

Correct answer: B

If the fourth term in the Binomial expansion of $\left(\dfrac{2}{x} + x^{\log_8 x}\right)^6$ $(x > 0)$ is $20\times8^7$, then a value of x is:

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

Correct answer: A

If the function f defined on $\left(\dfrac{\pi}{6}, \dfrac{\pi}{3}\right)$ by $f(x) = \begin{cases} \dfrac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \ne \dfrac{\pi}{4} \\ k, & x = \dfrac{\pi}{4} \end{cases}$ is continuous, then k is equal to:

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

Correct answer: B

Let $f(x) = 15 - |x - 10|$; $x \in \mathbb{R}$. Then the set of all values of x, at which the function, $g(x) = f(f(x))$ is not differentiable, is:

1. A.$\{10\}$
2. B.$\{10, 15\}$
3. C.$\{5, 10, 15\}$✓
4. D.$\{5, 10, 15, 20\}$

Correct answer: C

Let S be the set of all values of x for which the tangent to the curve $y = f(x) = x^3 - x^2 - 2x$ at (x, y) is parallel to the line segment joining the points (1, f(1)) and (−1, f(−1)), then S is equal to:

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

Correct answer: D

If f(x) is a non-zero polynomial of degree four, having local extreme points at x = −1, 0, 1; then the set $S = \{x \in \mathbb{R} : f(x) = f(0)\}$ contains exactly:

1. A.two irrational and two rational numbers.
2. B.two irrational and one rational number.✓
3. C.four rational numbers.
4. D.four irrational numbers.

Correct answer: B

The integral $\displaystyle\int \sec^{2/3}x\,\text{cosec}^{4/3}x\,dx$ is equal to: (Here C is a constant of integration)

1. A.$-3\cot^{-1/3}x + C$
2. B.$-3\tan^{-1/3}x + C$✓
3. C.$-\dfrac{3}{4}\tan^{-4/3}x + C$
4. D.$3\tan^{-1/3}x + C$

Correct answer: B

The value of $\displaystyle\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x}\,dx$ is:

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

Correct answer: B

The area (in sq. units) of the region $A = \{(x, y) : x^2 \le y \le x + 2\}$ is:

1. A.$\dfrac{31}{6}$
2. B.$\dfrac{10}{3}$
3. C.$\dfrac{13}{6}$
4. D.$\dfrac{9}{2}$✓

Correct answer: D

The solution of the differential equation $x\dfrac{dy}{dx} + 2y = x^2$ $(x \ne 0)$ with $y(1) = 1$, is:

1. A.$y = \dfrac{3}{4}x^2 + \dfrac{1}{4x^2}$
2. B.$y = \dfrac{4}{5}x^3 + \dfrac{1}{5x^2}$
3. C.$y = \dfrac{x^2}{4} + \dfrac{3}{4x^2}$✓
4. D.$y = \dfrac{x^3}{5} + \dfrac{1}{5x^2}$

Correct answer: C

If the tangent to the curve, $y = x^3 + ax - b$ at the point (1, −5) is perpendicular to the line, $-x + y + 4 = 0$, then which one of the following points lies on the curve?

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

Correct answer: B

Slope of a line passing through P(2, 3) and intersecting the line, $x + y = 7$ at a distance of 4 units from P, is:

1. A.$\dfrac{\sqrt{5} - 1}{\sqrt{5} + 1}$
2. B.$\dfrac{1 - \sqrt{5}}{1 + \sqrt{5}}$
3. C.$\dfrac{\sqrt{7} - 1}{\sqrt{7} + 1}$
4. D.$\dfrac{1 - \sqrt{7}}{1 + \sqrt{7}}$✓

Correct answer: D

If a tangent to the circle $x^2 + y^2 = 1$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

1. A.$x^2 + y^2 - 2xy = 0$
2. B.$x^2 + y^2 - 2x^2y^2 = 0$
3. C.$x^2 + y^2 - 4x^2y^2 = 0$✓
4. D.$x^2 + y^2 - 16x^2y^2 = 0$

Correct answer: C

If one end of a focal chord of the parabola, $y^2 = 16x$ is at (1, 4), then the length of this focal chord is:

1. A.25✓
2. B.24
3. C.22
4. D.20

Correct answer: A

If the line $y = mx + 7\sqrt{3}$ is normal to the hyperbola $\dfrac{x^2}{24} - \dfrac{y^2}{18} = 1$, then a value of m is:

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

Correct answer: A

If the line, $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-2}{4}$ meets the plane, $x + 2y + 3z = 15$ at a point P, then the distance of P from the origin is:

1. A.7/2
2. B.9/2✓
3. C.$2\sqrt{5}$
4. D.$\sqrt{5}/2$

Correct answer: B

A plane passing through the points (0, −1, 0) and (0, 0, 1) and making an angle $\dfrac{\pi}{4}$ with the plane $y - z + 5 = 0$, also passes through the point:

1. A.$(\sqrt{2}, -1, 4)$
2. B.$(-\sqrt{2}, -1, -4)$
3. C.$(-\sqrt{2}, 1, -4)$
4. D.$(\sqrt{2}, 1, 4)$✓

Correct answer: D

Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$, where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}_1 \times \vec{\beta}_2$ is equal to:

1. A.$\dfrac{1}{2}(-3\hat{i} + 9\hat{j} + 5\hat{k})$✓
2. B.$\dfrac{1}{2}(3\hat{i} - 9\hat{j} + 5\hat{k})$
3. C.$-3\hat{i} + 9\hat{j} + 5\hat{k}$
4. D.$3\hat{i} - 9\hat{j} - 5\hat{k}$

Correct answer: A

Four persons can hit a target correctly with probabilities $\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}$ and $\dfrac{1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:

1. A.$\dfrac{25}{32}$✓
2. B.$\dfrac{1}{192}$
3. C.$\dfrac{7}{32}$
4. D.$\dfrac{25}{192}$

Correct answer: A

If the standard deviation of the numbers −1, 0, 1, k is $\sqrt{5}$ where k > 0, then k is equal to:

1. A.$\sqrt{6}$
2. B.$2\sqrt{\dfrac{10}{3}}$
3. C.$2\sqrt{6}$✓
4. D.$4\sqrt{\dfrac{5}{3}}$

Correct answer: C

The value of $\cos^2 10° - \cos 10°\cos 50° + \cos^2 50°$ is:

1. A.$\dfrac{3}{4} + \cos 20°$
2. B.3/4✓
3. C.$\dfrac{3}{2}(1 + \cos 20°)$
4. D.3/2

Correct answer: B

Let $S = \{\theta \in [-2\pi, 2\pi] : 2\cos^2\theta + 3\sin\theta = 0\}$. Then the sum of the elements of S is:

1. A.$\dfrac{13\pi}{6}$
2. B.$\dfrac{5\pi}{3}$
3. C.$2\pi$✓
4. D.$\pi$