JEE Main 2019 — 10 Apr Shift 2 (Previous Year Paper)

In the formula $X=5YZ^2$, $X$ and $Z$ have dimensions of capacitance and magnetic field, respectively. What are the dimensions of $Y$ in SI units?

1. A.$[M^{-1}L^{-2}T^4A^2]$
2. B.$[M^{-3}L^{-2}T^8A^4]$✓
3. C.$[M^{-2}L^{-2}T^6A^3]$
4. D.$[M^{-2}L^0T^{-4}A^{-2}]$

Correct answer: B

A plane is inclined at an angle $\alpha=30°$ with respect to the horizontal. A particle is projected with a speed $u=2\ \text{ms}^{-1}$, from the base of the plane, making an angle $\theta=15°$ with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to: (Take $g=10\ \text{ms}^{-2}$)

1. A.14 cm
2. B.18 cm
3. C.20 cm✓
4. D.26 cm

Correct answer: C

The time dependence of the position of a particle of mass $m=2$ is given by $\vec{r}(t)=2t\hat{i}-3t^2\hat{j}$. Its angular momentum, with respect to the origin, at time $t=2$ is:

1. A.$36\hat{k}$
2. B.$-48\hat{k}$✓
3. C.$-34(\hat{k}-\hat{i})$
4. D.$48(\hat{i}+\hat{j})$

Correct answer: B

Two blocks A and B of masses $m_A=1$ kg and $m_B=3$ kg are kept on the table as shown in figure. The coefficient of friction between A and B is 0.2 and between B and the surface of the table is also 0.2. The maximum force F that can be applied on B horizontally, so that the block A does not slide over the block B is: [Take $g=10\ \text{m/s}^2$]

1. A.8 N
2. B.12 N
3. C.16 N✓
4. D.40 N

Correct answer: C

A bullet of mass 20 g has an initial speed of $1\ \text{ms}^{-1}$, just before it starts penetrating a mud wall of thickness 20 cm. If the wall offers a mean resistance of $2.5\times10^{-2}$ N, the speed of the bullet after emerging from the other side of the wall is close to:

1. A.0.7 $\text{ms}^{-1}$✓
2. B.0.3 $\text{ms}^{-1}$
3. C.0.4 $\text{ms}^{-1}$
4. D.0.1 $\text{ms}^{-1}$

Correct answer: A

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The first part has a mass of $\frac{7M}{8}$ and is converted into a uniform disc of radius $2R$. The second part is converted into a uniform solid sphere. Let $I_1$ be the moment of inertia of the disc about its axis and $I_2$ be the moment of inertia of the new sphere about its axis. The ratio $I_1/I_2$ is given by:

1. A.140✓
2. B.65
3. C.285
4. D.185

Correct answer: A

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s, is close to:

1. A.$4.0\times10^{-6}$ Nm
2. B.$7.9\times10^{-6}$ Nm
3. C.$1.6\times10^{-5}$ Nm
4. D.$2.0\times10^{-5}$ Nm✓

Correct answer: D

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet? [Given: Mass of planet $=8\times10^{22}$ kg, Radius of planet $=2\times10^6$ m, Gravitational constant $G=6.67\times10^{-11}\ \text{Nm}^2/\text{kg}^2$]

1. A.13
2. B.17
3. C.9
4. D.11✓

Correct answer: D

The elastic limit of brass is 379 MPa. What should be the minimum diameter of a brass rod if it is to support a 400 N load without exceeding its elastic limit?

1. A.1.00 mm
2. B.0.90 mm
3. C.1.16 mm✓
4. D.1.36 mm

Correct answer: C

Water from a tap emerges vertically downwards with an initial speed of $1.0\ \text{ms}^{-1}$. The cross-sectional area of the tap is $10^{-4}\ \text{m}^2$. Assume that the pressure is constant throughout the stream of water and that the flow is streamlined. The cross-sectional area of the stream, 0.15 m below the tap would be: (Take $g=10\ \text{ms}^{-2}$)

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

Correct answer: D

One mole of an ideal gas passes through a process where pressure and volume obey the relation $P=P_0\left[1-\frac{1}{2}\left(\frac{V_0}{V}\right)^2\right]$. Here $P_0$ and $V_0$ are constants. Calculate the change in the temperature of the gas if its volume changes from $V_0$ to $2V_0$.

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

Correct answer: D

When heat $Q$ is supplied to a diatomic gas of rigid molecules, at constant volume its temperature increases by $\Delta T$. The heat required to produce the same change in temperature, at a constant pressure is:

1. A.$\frac{5}{3}Q$
2. B.$\frac{7}{5}Q$✓
3. C.$\frac{2}{3}Q$
4. D.$\frac{3}{2}Q$

Correct answer: B

A submarine experiences a pressure of $5.05\times10^6$ Pa at a depth of $d_1$ in a sea. When it goes further to a depth of $d_2$, it experiences a pressure of $8.08\times10^6$ Pa. Then $d_2-d_1$ is approximately: (density of water $=10^3\ \text{kg/m}^3$ and acceleration due to gravity $=10\ \text{ms}^{-2}$)

1. A.300 m✓
2. B.600 m
3. C.500 m
4. D.400 m

Correct answer: A

A source of sound S is moving with a velocity of 50 m/s towards a stationary observer. The observer measures the frequency of the source as 1000 Hz. What will be the apparent frequency of the source when it is moving away from the observer after crossing him? (Take velocity of sound in air is 350 m/s)

1. A.857 Hz
2. B.1143 Hz
3. C.750 Hz✓
4. D.807 Hz

Correct answer: C

A simple pendulum of length $L$ is placed between the plates of a parallel plate capacitor having electric field $E$, as shown in figure. Its bob has mass $m$ and charge $q$. The time period of the pendulum is given by:

1. A.$2\pi\sqrt{\frac{L}{\left(g+\frac{qE}{m}\right)}}$
2. B.$2\pi\sqrt{\frac{L}{\left(g-\frac{qE}{m}\right)}}$
3. C.$2\pi\sqrt{\frac{L}{\sqrt{g^2+\left(\frac{qE}{m}\right)^2}}}$✓
4. D.$2\pi\sqrt{\frac{L}{\sqrt{g^2-\frac{q^2E^2}{m^2}}}}$

Correct answer: C

The space between two concentric spheres of radii $a$ and $b$ $(b>a)$ is filled with a medium of resistivity $\rho$. The resistance between the two spheres will be:

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

Correct answer: B

The magnitude of the magnetic field at the center of an equilateral triangular loop of side 1 m which is carrying a current of 10 A is: [Take $\mu_0=4\pi\times10^{-7}\ \text{NA}^{-2}$]

1. A.1 $\mu$T
2. B.3 $\mu$T
3. C.9 $\mu$T
4. D.18 $\mu$T✓

Correct answer: D

A square loop is carrying a steady current $I$ and the magnitude of its magnetic dipole moment is $m$. If this square loop is changed to a circular loop and it carries the same current, the magnitude of the magnetic dipole moment of circular loop will be:

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

Correct answer: D

A coil of self inductance 10 mH and resistance 0.1 $\Omega$ is connected through a switch to a battery of internal resistance 0.9 $\Omega$. After the switch is closed, the time taken for the current to attain 80% of the saturation value is: [take $\ln5=1.6$]

1. A.0.016 s✓
2. B.0.002 s
3. C.0.103 s
4. D.0.324 s

Correct answer: A

Light is incident normally on a completely absorbing surface with an energy flux of $25\ \text{Wcm}^{-2}$. If the surface has an area of $25\ \text{cm}^2$, the momentum transferred to the surface in 40 min time duration will be:

1. A.$1.4\times10^{-6}$ Ns
2. B.$3.5\times10^{-6}$ Ns
3. C.$5.0\times10^{-3}$ Ns✓
4. D.$6.3\times10^{-4}$ Ns

Correct answer: C

The graph shows how the magnification $m$ produced by a thin lens varies with image distance $v$. What is the focal length of the lens used?

1. A.$\frac{b^2}{ac}$
2. B.$\frac{a}{c}$
3. C.$\frac{b^2c}{a}$
4. D.$\frac{b}{c}$✓

Correct answer: D

In a Young's double slit experiment, the ratio of the slit's width is 4 : 1. The ratio of the intensity of maxima to minima, close to the central fringe on the screen, will be:

1. A.4 : 1
2. B.9 : 1✓
3. C.$(\sqrt3+1)^4:16$
4. D.25 : 9

Correct answer: B

A 2 mW laser operates at a wavelength of 500 nm. The number of photons that will be emitted per second is: [Given Planck's constant $h=6.6\times10^{-34}$ Js, speed of light $c=3.0\times10^8$ m/s]

1. A.$5\times10^{15}$✓
2. B.$1\times10^{16}$
3. C.$1.5\times10^{16}$
4. D.$2\times10^{16}$

Correct answer: A

In $Li^{++}$, electron in first Bohr orbit is excited to a level by a radiation of wavelength $\lambda$. When the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of $\lambda$? (Given: $h=6.63\times10^{-34}$ Js; $c=3\times10^8\ \text{ms}^{-1}$)

1. A.9.4 nm
2. B.10.8 nm✓
3. C.11.4 nm
4. D.12.3 nm

Correct answer: B

Two radioactive substances A and B have decay constants $5\lambda$ and $\lambda$ respectively. At $t=0$, a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become $\left(\frac{1}{e}\right)^2$ will be:

1. A.$1/\lambda$
2. B.$2/\lambda$
3. C.$1/2\lambda$✓
4. D.$1/4\lambda$

Correct answer: C

The figure represents a voltage regulator circuit using a Zener diode. The breakdown voltage of the Zener diode is 6 V and the load resistance is, $R_L=4$ k$\Omega$. The series resistance of the circuit is $R_i=1$ k$\Omega$. If the battery voltage $V_B$ varies from 8 V to 16 V, what are the minimum and maximum values of the current through Zener diode?

1. A.0.5 mA; 6 mA
2. B.1.5 mA; 8.5 mA
3. C.0.5 mA; 8.5 mA✓
4. D.1 mA; 8.5 mA

Correct answer: C

The correct figure that shows, schematically, the wave pattern produced by superposition of two waves of frequencies 9 Hz and 11 Hz, is:

1. A.Figure 1: displacement y vs t over 0–2 s showing an envelope with 2 beats per second (4 envelope lobes in 2 s)
2. B.Figure 2: displacement y vs t over 0–2 s showing an envelope with 4 lobes per second (8 lobes in 2 s)✓
3. C.Figure 3: displacement y vs t over 0–2 s showing an envelope with one crossing at t = 1 s (2 lobes in 2 s)
4. D.Figure 4: displacement y vs t over 0–2 s showing an envelope with lobes of alternating size

Correct answer: B

A cubical block of side 0.5 m floats on water with 30% of its volume under water. What is the maximum weight that can be put on the block without fully submerging it under water? [Take, density of water $=10^3\ \text{kg/m}^3$]

1. A.30.1 kg
2. B.87.5 kg✓
3. C.46.3 kg
4. D.65.4 kg

Correct answer: B

The major product 'Y' in the following reaction is: Acetophenone ($PhCOCH_3$) $\xrightarrow{NaOCl}$ X $\xrightarrow{\text{(i) }SOCl_2\text{; (ii) aniline}}$ Y

1. A.2-Aminobenzophenone (benzene ring bearing $NH_2$ and, ortho to it, $COPh$)
2. B.$PhN=CH-CO-Ph$ (N-phenyl imine of phenylglyoxal)
3. C.Benzanilide, $PhNH-CO-Ph$✓
4. D.4-Aminobenzophenone (benzene ring bearing $NH_2$ and, para to it, $COPh$)

Correct answer: C

Which of these factors does not govern the stability of a conformation in acyclic compounds?

1. A.Steric interactions
2. B.Angle strain✓
3. C.Torsional strain
4. D.Electrostatic forces of interaction

Correct answer: B

In chromatography, which of the following statements is INCORRECT for $R_f$?

1. A.Higher $R_f$ value means higher adsorption.✓
2. B.The value of $R_f$ depends on the type of chromatography.
3. C.The value of $R_f$ can not be more than one.
4. D.$R_f$ value is dependent on the mobile phase.

Correct answer: A

The number of chiral carbons present in the molecule of glucose in its open chain and cyclic structures respectively are:

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

Correct answer: B

The major product 'Y' in the following reaction is: $CH_3-CH(Cl)-CH(CH_3)-CH_3$ (2-chloro-3-methylbutane) $\xrightarrow{\text{EtONa, Heat}}$ X $\xrightarrow{HBr}$ Y

1. A.$CH_3-CH(Br)-CH(CH_3)_2$ (2-bromo-3-methylbutane)
2. B.$CH_3-CH(OH)-CH(CH_3)_2$ (3-methylbutan-2-ol)
3. C.$(CH_3)_2C(Br)-CH_2CH_3$ (2-bromo-2-methylbutane)✓
4. D.$BrCH_2CH_2CH(CH_3)_2$ (1-bromo-3-methylbutane)

Correct answer: C

The major product obtained in the given reaction is: 3-Methylphenyl–O–$CH_2$–$CH_2$–CH(Cl)–$CH_3$ $\xrightarrow{AlCl_3}$ Product

1. A.Chromane with two methyl groups: one at C-4 of the pyran ring and one on the aromatic ring adjacent (peri) to it
2. B.$H_3C$–C$_6$H$_4$–O–$CH_2$–CH=CH–$CH_3$ (m-tolyl but-2-enyl ether)
3. C.4-Methylchromane with a $CH_3$ on the aromatic ring para to the ring-fusion carbon bearing C-4 (7-methyl-4-methylchromane)✓
4. D.$H_3C$–C$_6$H$_4$–O–$CH_2$–$CH_2$–CH=$CH_2$ (m-tolyl but-3-enyl ether)

Correct answer: C

Arrange the following nucleophiles in the increasing order of nucleophilicity: (a) $CH_3CO_2^{\ominus}$ (b) $H_2O$ (c) $CH_3SO_3^{\ominus}$ (d) $\overset{\ominus}{O}H$

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

Correct answer: C

Which of the following is not a correct method of the preparation of benzylamine from cyanobenzene?

1. A.(i) HCl/$H_2O$ (ii) $NaBH_4$✓
2. B.(i) $LiAlH_4$ (ii) $H_3O^+$
3. C.$H_2$/Ni
4. D.(i) $SnCl_2$ + HCl(gas) (ii) $NaBH_4$

Correct answer: A

The correct order of the first ionization enthalpies is:

1. A.Ti < Mn < Ni < Zn✓
2. B.Mn < Ti < Zn < Ni
3. C.Ti < Mn < Zn < Ni
4. D.Zn < Ni < Mn < Ti

Correct answer: A

The correct statement is:

1. A.Aniline is a froth stabilizer.✓
2. B.Sodium cyanide cannot be used in the metallurgy of silver.
3. C.Zincite is a carbonate ore.
4. D.Zone refining process is used for the refining of titanium.

Correct answer: A

The correct statements among (a) to (d) are: (a) Saline hydrides produce $H_2$ gas when reacted with $H_2O$. (b) Reaction of $LiAlH_4$ with $BF_3$ leads to $B_2H_6$. (c) $PH_3$ and $CH_4$ are electron-rich and electron-precise hydrides, respectively. (d) HF and $CH_4$ are called as molecular hydrides.

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

Correct answer: D

A hydrated solid X on heating initially gives a monohydrated compound Y. Y upon heating above 373 K leads to an anhydrous white powder Z. X and Z, respectively, are:

1. A.Baking soda and soda ash.
2. B.Washing soda and dead burnt plaster.
3. C.Washing soda and soda ash.✓
4. D.Baking soda and dead burnt plaster.

Correct answer: C

The number of pentagons in $C_{60}$ and trigons (triangles) in white phosphorus, respectively, are:

1. A.12 and 4✓
2. B.12 and 3
3. C.20 and 3
4. D.20 and 4

Correct answer: A

The highest possible oxidation states of uranium and plutonium, respectively, are:

1. A.7 and 6
2. B.6 and 7✓
3. C.6 and 4
4. D.4 and 6

Correct answer: B

The crystal field stabilization energy (CFSE) of $[Fe(H_2O)_6]Cl_2$ and $K_2[NiCl_4]$, respectively, are:

1. A.$-0.6\Delta_o$ and $-0.8\Delta_t$
2. B.$-0.4\Delta_o$ and $-0.8\Delta_t$✓
3. C.$-2.4\Delta_o$ and $-1.2\Delta_t$
4. D.$-0.4\Delta_o$ and $-1.2\Delta_t$

Correct answer: B

The INCORRECT statement is:

1. A.the spin-only magnetic moment of $[Ni(NH_3)_4(H_2O)_2]^{2+}$ is 2.83 BM.
2. B.the spin-only magnetic moments of $[Fe(H_2O)_6]^{2+}$ and $[Cr(H_2O)_6]^{2+}$ are nearly similar.
3. C.the color of $[CoCl(NH_3)_5]^{2+}$ is violet as it absorbs the yellow light.
4. D.the gemstone, ruby, has $Cr^{3+}$ ions occupying the octahedral sites of beryl.✓

Correct answer: D

Air pollution that occurs in sunlight is:

1. A.acid rain
2. B.fog
3. C.reducing smog
4. D.oxidising smog✓

Correct answer: D

The minimum amount of $O_2(g)$ consumed per gram of reactant is for the reaction: (Given atomic mass: Fe = 56, O = 16, Mg = 24, P = 31, C = 12, H = 1)

1. A.$4Fe(s)+3O_2(g)\to2Fe_2O_3(s)$✓
2. B.$2Mg(s)+O_2(g)\to2MgO(s)$
3. C.$P_4(s)+5O_2(g)\to P_4O_{10}(s)$
4. D.$C_3H_8(g)+5O_2(g)\to3CO_2(g)+4H_2O(l)$

Correct answer: A

Points I, II and III in the following plot respectively correspond to ($V_{mp}$: most probable velocity)

1. A.$V_{mp}$ of $H_2$ (300 K); $V_{mp}$ of $N_2$ (300 K); $V_{mp}$ of $O_2$ (400 K)
2. B.$V_{mp}$ of $O_2$ (400 K); $V_{mp}$ of $N_2$ (300 K); $V_{mp}$ of $H_2$ (300 K)
3. C.$V_{mp}$ of $N_2$ (300 K); $V_{mp}$ of $O_2$ (400 K); $V_{mp}$ of $H_2$ (300 K)✓
4. D.$V_{mp}$ of $N_2$ (300 K); $V_{mp}$ of $H_2$ (300 K); $V_{mp}$ of $O_2$ (400 K)

Correct answer: C

The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about 9. The spectral series are:

1. A.Balmer and Brackett
2. B.Lyman and Paschen✓
3. C.Paschen and Pfund
4. D.Brackett and Pfund

Correct answer: B

The difference between $\Delta H$ and $\Delta U$ $(\Delta H-\Delta U)$, when the combustion of one mole of heptane(l) is carried out at a temperature T, is equal to:

1. A.4 RT
2. B.3 RT
3. C.−4 RT✓
4. D.−3 RT

Correct answer: C

1 g of a non-volatile non-electrolyte solute is dissolved in 100 g of two different solvents A and B whose ebullioscopic constants are in the ratio of 1 : 5. The ratio of the elevation in their boiling points, $\frac{\Delta T_b(A)}{\Delta T_b(B)}$, is:

1. A.1 : 0.2
2. B.5 : 1
3. C.1 : 5✓
4. D.10 : 1

Correct answer: C

For the reaction, $2SO_2(g)+O_2(g)\rightleftharpoons2SO_3(g)$, $\Delta H=-57.2\ \text{kJ mol}^{-1}$ and $K_c=1.7\times10^{16}$. Which of the following statement is INCORRECT?

1. A.The equilibrium constant decreases as the temperature increases.
2. B.The equilibrium will shift in forward direction as the pressure increases.
3. C.The addition of inert gas at constant volume will not affect the equilibrium constant.
4. D.The equilibrium constant is large suggestive of reaction going to completion and so no catalyst is required.✓

Correct answer: D

The pH of a 0.02 M $NH_4Cl$ solution will be [given $K_b(NH_4OH)=10^{-5}$ and $\log2=0.301$]

1. A.4.65
2. B.2.65
3. C.5.35✓
4. D.4.35

Correct answer: C

Which one of the following graphs between molar conductivity $(\Lambda_m)$ versus $\sqrt{C}$ is correct?

1. A.Graph 1: two parallel straight lines falling with $\sqrt{C}$, KCl above NaCl✓
2. B.Graph 2: two parallel straight lines falling with $\sqrt{C}$, NaCl above KCl
3. C.Graph 3: two falling straight lines that cross, NaCl starting above KCl
4. D.Graph 4: two falling straight lines with different slopes, KCl steeper and starting higher, NaCl less steep

Correct answer: A

For the reaction of $H_2$ with $I_2$, the rate constant is $2.5\times10^{-4}\ \text{dm}^3\text{mol}^{-1}\text{s}^{-1}$ at 327 °C and $1.0\ \text{dm}^3\text{mol}^{-1}\text{s}^{-1}$ at 527 °C. The activation energy for the reaction, in kJ mol$^{-1}$ is: ($R=8.314\ \text{J K}^{-1}\text{mol}^{-1}$)

1. A.59
2. B.72
3. C.150
4. D.166✓

Correct answer: D

The correct option among the following is:

1. A.Addition of alum to water makes it unfit for drinking.
2. B.Colloidal medicines are more effective because they have small surface area.
3. C.Colloidal particles in lyophobic sols can be precipitated by electrophoresis.✓
4. D.Brownian motion in colloidal solution is faster if the viscosity of the solution is very high.

Correct answer: C

Let $f(x)=\log_e(\sin x)$, $(0<x<\pi)$ and $g(x)=\sin^{-1}(e^{-x})$, $(x\ge0)$. If $\alpha$ is a positive real number such that $a=(f\circ g)'(\alpha)$ and $b=(f\circ g)(\alpha)$, then:

1. A.$a\alpha^2-b\alpha-a=0$
2. B.$a\alpha^2+b\alpha+a=0$
3. C.$a\alpha^2+b\alpha-a=-2\alpha^2$
4. D.$a\alpha^2-b\alpha-a=1$✓

Correct answer: D

If $z$ and $w$ are two complex numbers such that $|zw|=1$ and $\arg(z)-\arg(w)=\frac{\pi}{2}$, then:

1. A.$\bar{z}w=-i$✓
2. B.$z\bar{w}=\frac{-1+i}{\sqrt2}$
3. C.$\bar{z}w=i$
4. D.$z\bar{w}=\frac{1-i}{\sqrt2}$

Correct answer: A

The number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$ is:

1. A.4
2. B.1✓
3. C.2
4. D.3

Correct answer: B

The sum of the real roots of the equation $\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0$, is equal to:

1. A.−4
2. B.0✓
3. C.1
4. D.6

Correct answer: B

Let $\lambda$ be a real number for which the system of linear equations $x+y+z=6$ $4x+\lambda y-\lambda z=\lambda-2$ $3x+2y-4z=-5$ has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

1. A.$\lambda^2+3\lambda-4=0$
2. B.$\lambda^2-3\lambda-4=0$
3. C.$\lambda^2+\lambda-6=0$
4. D.$\lambda^2-\lambda-6=0$✓

Correct answer: D

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

1. A.170✓
2. B.190
3. C.180
4. D.210

Correct answer: A

The smallest natural number $n$, such that the coefficient of $x$ in the expansion of $\left(x^2+\frac{1}{x^3}\right)^n$ is ${}^nC_{23}$, is:

1. A.23
2. B.38✓
3. C.35
4. D.58

Correct answer: B

Let $a$, $b$ and $c$ be in G.P. with common ratio $r$, where $a\neq0$ and $0<r\le\frac{1}{2}$. If $3a$, $7b$ and $15c$ are the first three terms of an A.P., then the $4^{\text{th}}$ term of this A.P. is:

1. A.$\frac{2}{3}a$
2. B.$a$✓
3. C.$\frac{7}{3}a$
4. D.$5a$

Correct answer: B

The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\ldots+\frac{1^3+2^3+3^3+\ldots+15^3}{1+2+3+\ldots+15}-\frac{1}{2}(1+2+3+\ldots+15)$ is equal to:

1. A.620✓
2. B.660
3. C.1240
4. D.1860

Correct answer: A

If $\lim_{x\to1}\frac{x^2-ax+b}{x-1}=5$, then $a+b$ is equal to:

1. A.1
2. B.5
3. C.−4
4. D.−7✓

Correct answer: D

Let $a_1, a_2, a_3, \ldots$ be an A.P. with $a_6=2$. Then the common difference of this A.P., which maximises the product $a_1a_4a_5$, is:

1. A.$\frac{2}{3}$
2. B.$\frac{8}{5}$✓
3. C.$\frac{6}{5}$
4. D.$\frac{3}{2}$

Correct answer: B

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of $50\ \text{cm}^3/\text{min}$. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is:

1. A.$\frac{1}{36\pi}$
2. B.$\frac{5}{6\pi}$
3. C.$\frac{1}{18\pi}$✓
4. D.$\frac{1}{9\pi}$

Correct answer: C

If the tangent to the curve $y=\frac{x}{x^2-3}$, $x\in\mathbf{R}$, $(x\neq\pm\sqrt3)$, at a point $(\alpha,\beta)\neq(0,0)$ on it is parallel to the line $2x+6y-11=0$, then:

1. A.$|6\alpha+2\beta|=19$✓
2. B.$|2\alpha+6\beta|=11$
3. C.$|6\alpha+2\beta|=9$
4. D.$|2\alpha+6\beta|=19$

Correct answer: A

If $\int x^5e^{-x^2}dx=g(x)e^{-x^2}+c$, where $c$ is a constant of integration, then $g(-1)$ is equal to:

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

Correct answer: A

The integral $\int_{\pi/6}^{\pi/3}\sec^{2/3}x\,\text{cosec}^{4/3}x\,dx$ is equal to:

1. A.$3^{5/3}-3^{1/3}$
2. B.$3^{4/3}-3^{1/3}$
3. C.$3^{5/6}-3^{2/3}$
4. D.$3^{7/6}-3^{5/6}$✓

Correct answer: D

The area (in sq. units) of the region bounded by the curves $y=2^x$ and $y=|x+1|$, in the first quadrant is:

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

Correct answer: B

Let $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}+y\tan x=2x+x^2\tan x$, $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, such that $y(0)=1$. Then:

1. A.$y\left(\frac{\pi}{4}\right)-y\left(-\frac{\pi}{4}\right)=\sqrt2$
2. B.$y\left(\frac{\pi}{4}\right)+y\left(-\frac{\pi}{4}\right)=\frac{\pi^2}{2}+2$
3. C.$y'\left(\frac{\pi}{4}\right)-y'\left(-\frac{\pi}{4}\right)=\pi-\sqrt2$✓
4. D.$y'\left(\frac{\pi}{4}\right)+y'\left(-\frac{\pi}{4}\right)=-\sqrt2$

Correct answer: C

Lines are drawn parallel to the line $4x-3y+2=0$, at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

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

Correct answer: D

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y$-axis and lie in the first quadrant, is:

1. A.$y=\sqrt{1+2x}$, $x\ge0$✓
2. B.$y=\sqrt{1+4x}$, $x\ge0$
3. C.$x=\sqrt{1+4y}$, $y\ge0$
4. D.$x=\sqrt{1+2y}$, $y\ge0$

Correct answer: A

If the line $ax+y=c$, touches both the curves $x^2+y^2=1$ and $y^2=4\sqrt2x$, then $|c|$ is equal to:

1. A.$\sqrt2$✓
2. B.$\frac{1}{\sqrt2}$
3. C.$\frac{1}{2}$
4. D.2

Correct answer: A

The tangent and normal to the ellipse $3x^2+5y^2=32$ at the point P(2, 2) meet the $x$-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is:

1. A.$\frac{34}{15}$
2. B.$\frac{16}{3}$
3. C.$\frac{14}{3}$
4. D.$\frac{68}{15}$✓

Correct answer: D

If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144$, then its corresponding focus is:

1. A.(−5, 0)✓
2. B.(5, 0)
3. C.$\left(-\frac{5}{3},0\right)$
4. D.$\left(\frac{5}{3},0\right)$

Correct answer: A

A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular Q also lies on the plane $x-y+z=3$. Then the co-ordinates of Q are:

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

Correct answer: A

If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda=0$ and $2x-y+2z+\mu=0$, respectively, then the maximum value of $\lambda+\mu$ is equal to:

1. A.5
2. B.9
3. C.13✓
4. D.15

Correct answer: C

The distance of the point having position vector $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line passing through the point (2, 3, −4) and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is:

1. A.6
2. B.7✓
3. C.$2\sqrt{13}$
4. D.$4\sqrt3$

Correct answer: B

If both the mean and the standard deviation of 50 observations $x_1, x_2, \ldots, x_{50}$ are equal to 16, then the mean of $(x_1-4)^2, (x_2-4)^2, \ldots, (x_{50}-4)^2$ is:

1. A.380
2. B.400✓
3. C.480
4. D.525

Correct answer: B

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is:

1. A.5
2. B.6
3. C.7✓
4. D.8