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Category: IIT-JEE 2017 Physics Paper 1

Question 1:

A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true?

A,B and C

A,B, and D

B, C and D

B, D and A

Category: IIT-JEE 2017 Physics Paper 1

Question 2:

A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?

A	The position of the point mass m is: $x = - \sqrt{2} \frac{m R}{M + m}$	B	The velocity of the point mass m is: $v = \sqrt{\frac{2 g R}{1 + \frac{m}{M}}}$
C	The x component of displacement of the center of mass of the block M is: $- \frac{m R}{M + m}$	D	The velocity of the block M is: $V = - \frac{m}{M} \sqrt{2 g R}$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2017 Physics Paper 1

Question 3:

A block M hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength

λ_{0}

is produced at point O on the rope. The pulse takes time

T_{O A}

to reach point A. If the wave pulse of wavelength

λ_{0}

is produced at point A (Pulse 2) without disturbing the position of M it takes time

T_{A O}

to reach point O. Which of the following options is/are correct?

A	The time $T_{A O} = T_{O A}$	B	The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope
C	The wavelength of Pulse 1 becomes longer when it reaches point A	D	The velocity of any pulse along the rope is independent of its frequency and wavelength

A and D

B and D

C and A

D and C

Category: IIT-JEE 2017 Physics Paper 1

Question 4:

A human body has a surface area of approximately

1 m^{2}

. The normal body temperature is 10 K above the surrounding room temperature

T_{0}

. Take the room temperature to be

T_{0}

=300 K. For

T_{0}

= 300 K, the value of

σ T_{0}^{4}

= 460

W m^{- 2}

(where

σ

is the Stefan-Boltzmann constant). Which of the following options is/are correct?

A	The amount of energy radiated by the body in 1 second is close to 60 Joules	B	If the surrounding temperature reduces by a small amount $∆ T_{0} < < T_{0}$ , then to maintain the same body temperature the same (living) human being needs to radiate $∆ W = 4 σ T_{0}^{3} ∆ T_{0}$ more energy per unit time
C	Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation	D	If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

Category: IIT-JEE 2017 Physics Paper 1

Question 5:

A circular insulated copper wire loop is twisted to form two loops of area A and 2A as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field

\vec{B}

points into the plane of the paper. At t = 0, the loop starts rotating about the common diameter as axis with a constant angular velocity

ω

in the magnetic field. Which of the following options is/are correct?

A	The emf induced in the loop is proportional to the sum of the areas of the two loops	B	The amplitude of the maximum net emf induced due to both the loops is equal to the amplitude of maximum emf induced in the smaller loop alone
C	The net emf induced due to both the loops is proportional to cos $ω t$	D	The rate of change of the flux is maximum when the plane of the loops is perpendicular to plane of the paper

A and B

A and C

B and C

B and D

Category: IIT-JEE 2017 Physics Paper 1

Question 6:

In the circuit shown, L = 1

μ

H, C = 1

μ

F and R = 1 kΩ. They are connected in series with an a.c. source V =

V_{0}

sin

ω t

as shown. Which of the following options is/are correct?

A	The current will be in phase with the voltage if $ω$ = $10^{4}$ rad. $s^{- 1}$	B	The frequency at which the current will be in phase with the voltage is independent of R
C	At $ω$ ~0 the current flowing through the circuit becomes nearly zero	D	At $ω$ ≫ $10^{6}$ rad. $s^{- 1}$ , the circuit behaves like a capacitor

A and B

B and C

C and D

D and A

Category: IIT-JEE 2017 Physics Paper 1

Question 7:

For an isosceles prism of angle A and refractive index µ, it is found that the angle of minimum deviation

δ_{m}

= A . Which of the following options is/are correct?

A	For the angle of incidence $i_{1}$ = A, the ray inside the prism is parallel to the base of the prism	B	For this prism, the refractive index µ and the angle of prism A are related as A = $\frac{1}{2} \cos^{- 1} (\frac{μ}{2})$
C	At minimum deviation, the incident angle $i_{1}$ and the refracting angle $r_{1}$ at the first refracting surface are related by $r_{i}$ = ( $i_{1}$ /2)	D	For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is $i_{1} = \sin^{- 1} [\sin A \sqrt{4 \cos^{2} \frac{A}{2} - 1 - \cos A}]$

A,B and C

A,B and D

A,C and D

None of these

Category: IIT-JEE 2017 Physics Paper 1

Question 8:

A drop of liquid of radius R =

10^{2}

m having surface tension S =

\frac{0.1}{4 π} N m^{- 1}

divides itself into k identical drops. In this process the total change in the surface energy ∆U =

10^{- 3}

J. If K =

10^{a}

then the value of α is

Category: IIT-JEE 2017 Physics Paper 1

Question 9:

An electron in a hydrogen atom undergoes a transition from an orbit with quantum number

n_{i}

to another with quantum number

n_{f} . V_{i}

and

V_{f}

are respectively the initial and final potential energies of the electron. If

\frac{v_{i}}{v_{f}}

= 6.25, then the smallest possible

n_{f}

Category: IIT-JEE 2017 Physics Paper 1

Question 10:

A monochromatic light is travelling in a medium of refractive index n = 1.6. It enters a stack of glass layers from the bottom side at an angle

θ

30^{0}

. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as

n_{m}

= n – mΔn, where

n_{m}

is the refractive index of the

m^{t h}

slab and Δn = 0.1 (see the figure). The ray is refracted out parallel to the interface between the

{(m - 1)}^{t h}

and

m^{t h}

slabs from the right side of the stack. What is the value of m?

Category: IIT-JEE 2017 Physics Paper 1

Question 11:

A stationary source emits sound of frequency

f_{0}

= 492 Hz. The sound is reflected by a large car approaching the source with a speed of 2

m s^{- 1}

. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330

m s^{- 1}

and the car reflects the sound at the frequency it has received).

Category: IIT-JEE 2017 Physics Paper 1

Question 12:

^{131} I

is an isotope of Iodine that β decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with

^{131} I

is injected into the blood of a person. The activity of the amount of

^{131} I

injected was 2.4 ×

10^{5}

Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person’s body, and gives an activity of 115 Bq. The total volume of blood in the person’s body, in liters is approximately (you may use ≈ 1 + x for

|x|

≪ 1 and ln 2 ≈ 0.7).

Category: IIT-JEE 2017 Physics Paper 1

Question 13:

In which case will the particle move in a straight line with constant velocity?

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(III) (ii) (R)	[B]	(IV) (i) (S)
[C]	(III) (iii)(P)	[D]	(II) (iii) (S)

Category: IIT-JEE 2017 Physics Paper 1

Question 14:

In which case will the particle describe a helical path with axis along the positive z direction?

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(IV) (i) (S)	[B]	(II) (ii) (R)
[C]	(III) (iii) (P)	[D]	(IV) (ii) (R)

Category: IIT-JEE 2017 Physics Paper 1

Question 15:

In which case would the particle move in a straight line along the negative direction of �- axis (i.e., move along –

\hat{y}

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(II) (iii) (Q)	[B]	(III) (ii) (R)
[C]	(IV) (ii) (S)	[D]	(III) (ii) (P)

Category: IIT-JEE 2017 Physics Paper 1

Question 16:

Which of the following options is the only correct representation of a process in which ΔU = ΔQ – PΔV ?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(II) (iv) (R)	[B]	(III) (iii) (P)
[C]	(II) (iii) (S)	[D]	(II) (iii) (P)

Category: IIT-JEE 2017 Physics Paper 1

Question 17:

Which one of the following options is the correct combination?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(IV) (ii) (S)	[B]	(III) (ii) (S)
[C]	(II) (iv) (P)	[D]	(II) (iv) (R)

Category: IIT-JEE 2017 Physics Paper 1

Question 18:

Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(I) (ii) (Q)	[B]	(IV) (ii) (R)
[C]	(III) (iv) (R)	[D]	(I) (iv) (Q)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 19:

An ideal gas is expanded from (

p_{1}, V_{1}, T_{1}

) to (

p_{2}, V_{2}, T_{2}

) under different conditions. The correct statement(s) among the following is(are)

[A]	The work done on the gas is maximum when it is compressed irreversibly from ( $p_{2}, V_{2}$ ) to ( $p_{1}, V_{1}$ ) against constant pressure $p_{1}$
[B]	If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic
[C]	The work done by the gas is less when it is expanded reversibly from $V_{1}$ to $V_{2}$ under adiabatic conditions as compared to that when expanded reversibly from $V_{1}$ to $V_{2}$ under isothermal conditions
[D}	The change in internal energy of the gas is (i) zero, if it is expanded reversibly with $T_{1} = T_{2}$ , and (ii) positive, if it is expanded reversibly under adiabatic conditions with $T_{1}$ ≠ $T_{2}$

A,B and C

B,C and D

C, D and A

None of these

Category: IIT-JEE 2017 Chemistry Paper 1

Question 20:

For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here

x_{1}

and

x_{M}

represent mole fractions of L and M, respectively, in the solution. The correct statement(s) applicable to this system is(are)

[A]	The point Z represents vapour pressure of pure liquid M and Raoult’s law is obeyed from $x_{L}$ = 0 to $x_{L}$ = 1
[B]	The point Z represents vapour pressure of pure liquid L and Raoult’s law is obeyed when $x_{L} \to$ 1
[C]	The point Z represents vapour pressure of pure liquid M and Raoult’s law is obeyed when $x_{L} \to 0$
[D]	Attractive intermolecular interactions between L-L in pure liquid L and M-M in pure liquid M are stronger than those between L-M when mixed in solution

A and D

B and D

C and D

Category: IIT-JEE 2017 Chemistry Paper 1

Question 21:

The correct statement(s) about the oxoacids,

H C I O_{4}

and HClO, is(are)

[A]	The central atom in both $H C I O_{4}$ and HClO is $s p^{3}$ hybridized
[B]	$H C I O_{4}$ is more acidic than HClO because of the resonance stabilization of its anion
[C]	$H C I O_{4}$ is formed in the reaction between $C I_{2}$ and $H_{2} O$
[D]	The conjugate base of $H C I O_{4}$ is weaker base than $H_{2} O$

A,B and C

A,B and D

A,D and C

None of these

Category: IIT-JEE 2017 Chemistry Paper 1

Question 22:

The colour of the

X_{2}

molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to

[A]	The physical state of $X_{2}$ at room temperature changes from gas to solid down the group
[B]	Decrease in ionization energy down the group
[C]	Decrease in $π^{} - 0^{}$ gap down the group
[D]	decrease in HOMO-LUMO gap down the group

A and B

B and C

C and D

D and A

Category: IIT-JEE 2017 Chemistry Paper 1

Question 23:

The IUPAC name(s) of the following compound is(are)

[A]	1-chloro-4-methylbenzene
[B]	4-chlorotoluene
[C]	4-methylchlorobenzene
[D]	1-methyl-4-chlorobenzene

A and B

B and C

C and D

D and D

Category: IIT-JEE 2017 Chemistry Paper 1

Question 24:

The correct statement(s) for the following addition reactions is(are)

[A]	O and P are identical molecules
[B]	(M and O) and (N and P) are two pairs of diastereomers
[C]	(M and O) and (N and P) are two pairs of enantiomers
[D]	Bromination proceeds through trans-addition in both the reactions

A and D

B and C

B and A

B and D

Category: IIT-JEE 2017 Chemistry Paper 1

Question 25:

A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is 8 g

c m^{- 3}

, then the number of atoms present in 256 g of the crystal is

N \times 10^{24}

. The value of N is

Category: IIT-JEE 2017 Chemistry Paper 1

Question 26:

The conductance of a 0.0015 M aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is 120 cm with an area of cross section of 1

c m^{2}

. The conductance of this solution was found to be

5 \times 10^{- 7}

S. The pH of the solution is 4. The value of limiting molar conductivity

(\land_{m}^{o})

of this weak monobasic acid in aqueous solution is

Z \times 10^{2}

c m^{- 1}

m o l^{- 1}

. The value of Z is

Category: IIT-JEE 2017 Chemistry Paper 1

Question 27:

The sum of the number of lone pairs of electrons on each central atom in the following species is

{[T e B r_{6}]}^{2 -}, {[B r F_{2}]}^{+}

S N F_{3}, a n d {[X e F_{3}]}^{-}

(Atomic numbers: N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 28:

Among

H_{2}, H w_{2}^{+}, B e_{2,} B_{2}, C_{2}, N_{2}, O_{2}

, and

F_{2}

, the number of diamagnetic species is (Atomic numbers: H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 29:

Among the following, the number of aromatic compound(s) is

Category: IIT-JEE 2017 Chemistry Paper 1

Question 30:

For the given orbital in Column 1, the only CORRECT combination for any hydrogen-like species is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (ii) (S)	[B]	(IV) (iv) (R)
[C]	(II) (ii) (P)	[D]	(III) (iii) (P)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 31:

For hydrogen atom, the only CORRECT combination is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (i) (S)	[B]	(II) (i) (Q)
[C]	(I) (i) (P)	[D]	(I) (iv) (R)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 32:

For

H e^{+}

ion, the only INCORRECT combination is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (i) (R)	[B]	(II) (ii) (Q)
[C]	(I) (iii) (R)	[D]	(I) (i) (S)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 33:

For the synthesis of benzoic acid, the only CORRECT combination is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(II) (i) (S)	[B]	(IV) (ii) (P)
[C]	(I) (iv) (Q)	[D]	(III) (iv) (R)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 34:

The only CORRECT combination that gives two different carboxylic acids is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(II) (iv) (R)	[B]	(IV) (iii) (Q)
[C]	(III) (iii) (P)	[D]	(I) (i) (S)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 35:

The only CORRECT combination in which the reaction proceeds through radical mechanism is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(III) (ii) (P)	[B]	(IV) (i) (Q)
[C]	(II) (iii) (R)	[D]	(I) (ii) (R)

Category: IIT-JEE 2017 Chemistry Paper 1

Question 36:

Addition of excess aqueous ammonia to a pink coloured aqueous solution of

M C {I_{2}}^{.} 6 H_{2} O

(X) and

N H_{4} C I

gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1:3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y. Among the following options, which statement(s) is(are) correct?

[A]	Addition of silver nitrate to Y gives only two equivalents of silver chloride
[B]	The hybridization of the central metal ion in Y is $d^{2} s p^{3}$
[C]	Z is a tetrahedral complex
[D]	When X and Z are in equilibrium at 0°C, the colour of the solution is pink

A,B and C

B, C, and D

C,D and A

None of these

Category: IIT-JEE 2017 Math Paper 1

Question 37:

If 2x − y + 1 = 0 is a tangent to the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1

, then which of the following CANNOT be sides of a right angled triangle?

[A]	a, 4, 1	[B]	a, 4, 2
[C]	2a, 8, 1	[D]	2a, 4, 1

A,B and C

A,C and D

B,C and D

B,D and A

Category: IIT-JEE 2017 Math Paper 1

Question 38:

If a chord, which is not a tangent, of the parabola

y^{2}

= 16x has the equation 2x + y = p, and midpoint (ℎ, k), then which of the following is(are) possible value(s) of p, ℎ and k?

[A]	P = −2, ℎ = 2, k = −4	[B]	p = −1, ℎ = 1, k = −3
[C]	p=2, h=3, k=-4	[D]	p = 5, ℎ = 4, k = −3

Category: IIT-JEE 2017 Math Paper 1

Question 39:

Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f (x) = x cos(

π

(x + [x])) is discontinuous?

[A]	x = −1	[B]	x = 0
[C]	x = 1	[D]	x = 2

Category: IIT-JEE 2017 Math Paper 1

Question 40:

Let f: ℝ → (0, 1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

[A]	$x^{9} - f (x)$	[B]	$x - \int_{0}^{\frac{π}{2} - x} f (t) \cos t d t$
[C]	$e^{x} - f_{0}^{x} f (t) \sin t d t$	[D]	$e^{x} + f_{0}^{π} f (t) \sin t d t$

A and B

B and D

C and D

D and A

Category: IIT-JEE 2017 Math Paper 1

Question 41:

Let a, b, x and y be real numbers such that a − b = 1 and y ≠ 0. If the complex number z = x + iy satisfies Im

(\frac{a z + b}{z + 1})

= y, then which of the following is(are) possible value(s) of x?

[A]	$- 1 + \sqrt{1 - y^{2}}$	[B]	$- 1 - \sqrt{1 - y^{2}}$
[C]	$1 + \sqrt{1 - y^{2}}$	[D]	$1 + \sqrt{1 - y^{2}}$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2017 Math Paper 1

Question 42:

Let X and Y be two events such that P(X) =

\frac{1}{3}

,P(X| Y)=

\frac{1}{2}

and P(Y| X) =

\frac{2}{5}

, Then

[A]	$P (Y) = \frac{4}{15}$	[B]	$P (X' \| Y) = \frac{1}{2}$
[C]	$P (X \cap Y) = \frac{1}{5}$	[D]	$P (X \cup Y) = \frac{2}{5}$

A and B

A and C

C and D

C and A

Category: IIT-JEE 2017 Math Paper 1

Question 43:

For how many values of P, the circle

x^{2} + y^{2} + 2 x + 4 y - p = 0

and the coordinate axes have exactly three common points?

Category: IIT-JEE 2017 Math Paper 1

Question 44:

Let

f; ℝ \to ℝ

be a differentiable Function such that

f (0) = 0, f (\frac{π}{2}) = 3

and

f' (0) = 1 .

g (x) = \int_{x}^{\frac{π}{2}} [f' (t) \cos e c t - c o t t \cos e c t f (t)] d t

for x

\in (0, \frac{π}{2}],

then

\lim_{x \to \infty} g (x) =

Category: IIT-JEE 2017 Math Paper 1

Question 45:

For a real number a, if the system

[\begin{matrix} 1 & a & a^{2} \\ a & 1 & a \\ a^{2} & a & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]

of linear equations, has infinitely many solutions, then 1+a+

a^{2}

Category: IIT-JEE 2017 Math Paper 1

Question 46:

Words of length 10 are formed using the letters A, B, C, D, E, F, G,H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then,

\frac{y}{9 x} =

Category: IIT-JEE 2017 Math Paper 1

Question 47:

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Category: IIT-JEE 2017 Math Paper 1

Question 48:

For a =

\sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(I) (i) (P)		(I) (ii) (Q)
[C]	(II) (ii) (Q)		(III) (i) (P)

Category: IIT-JEE 2017 Math Paper 1

Question 49:

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(I) (ii) (Q)	[B]	(II) (iv) (R)
[C]	(III) (i) (P)	[D]	(III) (ii) (Q)

Category: IIT-JEE 2017 Math Paper 1

Question 50:

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3 x} + 2 y = 4

, then which of the following options is the only CORRECT combination?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(IV) (iii) (S)	[B]	(IV) (iv) (S)
[C]	(II) (iii) (R)	[D]	(II) (iv) (R)

Category: IIT-JEE 2017 Math Paper 1

Question 51:

Which of the following options is the only CORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (i) (P)	[B]	(II) (ii) (Q)
[C]	(III) (iii) (R)	[D]	(IV) (iv) (S)

Category: IIT-JEE 2017 Math Paper 1

Question 52:

Which of the following options is the only CORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (ii) (R)	[B]	(II) (iii) (S)
[C]	(III) (iv) (P)	[D]	(IV) (i) (S)

Category: IIT-JEE 2017 Math Paper 1

Question 53:

Which of the following options is the only INCORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (iii) (P)	[B]	(II) (iv) (Q)
[C]	(III) (i) (R)	[D]	(II) (iii) (P)

Category: IIT-JEE 2017 Math Paper 1

Question 54:

Which of the following is(are) NOT the square of a 3×3 matrix with real entries?

(A)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$	(B)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$
(C)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$	(D)	$[\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$

A and C

A and D

B and C

B and D

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Question 1:

Correct Answer : A,B, and D

Question 2:

A	The position of the point mass m is: $x = - \sqrt{2} \frac{m R}{M + m}$	B	The velocity of the point mass m is: $v = \sqrt{\frac{2 g R}{1 + \frac{m}{M}}}$
C	The x component of displacement of the center of mass of the block M is: $- \frac{m R}{M + m}$	D	The velocity of the block M is: $V = - \frac{m}{M} \sqrt{2 g R}$

Correct Answer : B and C

Question 3:

λ_{0}

is produced at point O on the rope. The pulse takes time

T_{O A}

to reach point A. If the wave pulse of wavelength

λ_{0}

is produced at point A (Pulse 2) without disturbing the position of M it takes time

T_{A O}

to reach point O. Which of the following options is/are correct?

A	The time $T_{A O} = T_{O A}$	B	The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope
C	The wavelength of Pulse 1 becomes longer when it reaches point A	D	The velocity of any pulse along the rope is independent of its frequency and wavelength

Correct Answer : A and D

Question 4:

A human body has a surface area of approximately

1 m^{2}

. The normal body temperature is 10 K above the surrounding room temperature

T_{0}

. Take the room temperature to be

T_{0}

=300 K. For

T_{0}

= 300 K, the value of

σ T_{0}^{4}

= 460

W m^{- 2}

(where

σ

is the Stefan-Boltzmann constant). Which of the following options is/are correct?

A	The amount of energy radiated by the body in 1 second is close to 60 Joules	B	If the surrounding temperature reduces by a small amount $∆ T_{0} < < T_{0}$ , then to maintain the same body temperature the same (living) human being needs to radiate $∆ W = 4 σ T_{0}^{3} ∆ T_{0}$ more energy per unit time
C	Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation	D	If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

Correct Answer : C

Question 5:

\vec{B}

points into the plane of the paper. At t = 0, the loop starts rotating about the common diameter as axis with a constant angular velocity

ω

in the magnetic field. Which of the following options is/are correct?

A	The emf induced in the loop is proportional to the sum of the areas of the two loops	B	The amplitude of the maximum net emf induced due to both the loops is equal to the amplitude of maximum emf induced in the smaller loop alone
C	The net emf induced due to both the loops is proportional to cos $ω t$	D	The rate of change of the flux is maximum when the plane of the loops is perpendicular to plane of the paper

Correct Answer : B and D

Question 6:

In the circuit shown, L = 1

μ

H, C = 1

μ

F and R = 1 kΩ. They are connected in series with an a.c. source V =

V_{0}

sin

ω t

as shown. Which of the following options is/are correct?

A	The current will be in phase with the voltage if $ω$ = $10^{4}$ rad. $s^{- 1}$	B	The frequency at which the current will be in phase with the voltage is independent of R
C	At $ω$ ~0 the current flowing through the circuit becomes nearly zero	D	At $ω$ ≫ $10^{6}$ rad. $s^{- 1}$ , the circuit behaves like a capacitor

Correct Answer : B and C

Question 7:

For an isosceles prism of angle A and refractive index µ, it is found that the angle of minimum deviation

δ_{m}

= A . Which of the following options is/are correct?

A	For the angle of incidence $i_{1}$ = A, the ray inside the prism is parallel to the base of the prism	B	For this prism, the refractive index µ and the angle of prism A are related as A = $\frac{1}{2} \cos^{- 1} (\frac{μ}{2})$
C	At minimum deviation, the incident angle $i_{1}$ and the refracting angle $r_{1}$ at the first refracting surface are related by $r_{i}$ = ( $i_{1}$ /2)	D	For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is $i_{1} = \sin^{- 1} [\sin A \sqrt{4 \cos^{2} \frac{A}{2} - 1 - \cos A}]$

Correct Answer : A,C and D

Question 8:

A drop of liquid of radius R =

10^{2}

m having surface tension S =

\frac{0.1}{4 π} N m^{- 1}

divides itself into k identical drops. In this process the total change in the surface energy ∆U =

10^{- 3}

J. If K =

10^{a}

then the value of α is

Correct Answer : 6

Question 9:

An electron in a hydrogen atom undergoes a transition from an orbit with quantum number

n_{i}

to another with quantum number

n_{f} . V_{i}

and

V_{f}

are respectively the initial and final potential energies of the electron. If

\frac{v_{i}}{v_{f}}

= 6.25, then the smallest possible

n_{f}

Correct Answer : 5

Question 10:

A monochromatic light is travelling in a medium of refractive index n = 1.6. It enters a stack of glass layers from the bottom side at an angle

θ

30^{0}

. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as

n_{m}

= n – mΔn, where

n_{m}

is the refractive index of the

m^{t h}

slab and Δn = 0.1 (see the figure). The ray is refracted out parallel to the interface between the

{(m - 1)}^{t h}

and

m^{t h}

slabs from the right side of the stack. What is the value of m?

Correct Answer : 8

Question 11:

A stationary source emits sound of frequency

f_{0}

= 492 Hz. The sound is reflected by a large car approaching the source with a speed of 2

m s^{- 1}

. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330

m s^{- 1}

and the car reflects the sound at the frequency it has received).

Correct Answer : 6

Question 12:

^{131} I

is an isotope of Iodine that β decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with

^{131} I

is injected into the blood of a person. The activity of the amount of

^{131} I

injected was 2.4 ×

10^{5}

|x|

≪ 1 and ln 2 ≈ 0.7).

Correct Answer : 5

Question 13:

In which case will the particle move in a straight line with constant velocity?

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(III) (ii) (R)	[B]	(IV) (i) (S)
[C]	(III) (iii)(P)	[D]	(II) (iii) (S)

Correct Answer : D

Question 14:

In which case will the particle describe a helical path with axis along the positive z direction?

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(IV) (i) (S)	[B]	(II) (ii) (R)
[C]	(III) (iii) (P)	[D]	(IV) (ii) (R)

Correct Answer : A

Question 15:

In which case would the particle move in a straight line along the negative direction of �- axis (i.e., move along –

\hat{y}

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity $\vec{v}$ . A uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ exist everywhere. The velocity $\vec{v}$ , electric field $\vec{E}$ and magnetic field $\vec{B}$ are given in columns 1, 2 and 3, respectively. The quantities $E_{0} . B_{0}$ are positive in magnitude.
	Column 1		Column 2		Column 3
(I)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i)	$\vec{E} = E_{0} \hat{z}$	(P)	$\vec{B} = - B_{0} \hat{x}$
(II)	Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii)	$\vec{E} = - E_{0} \hat{y}$	(Q)	$\vec{B} = B_{0} \hat{x}$
(III)	Proton with $\vec{v} = 0$	(iii)	$\vec{E} = - E_{0} \hat{x}$	(R)	$\vec{B} = B_{0} \hat{y}$
(IV)	Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv)	$\vec{E} = E_{0} \hat{x}$	(S)	$\vec{B} = B_{0} \hat{z}$

[A]	(II) (iii) (Q)	[B]	(III) (ii) (R)
[C]	(IV) (ii) (S)	[D]	(III) (ii) (P)

Correct Answer : B

Question 16:

Which of the following options is the only correct representation of a process in which ΔU = ΔQ – PΔV ?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(II) (iv) (R)	[B]	(III) (iii) (P)
[C]	(II) (iii) (S)	[D]	(II) (iii) (P)

Correct Answer : D

Question 17:

Which one of the following options is the correct combination?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(IV) (ii) (S)	[B]	(III) (ii) (S)
[C]	(II) (iv) (P)	[D]	(II) (iv) (R)

Correct Answer : B

Question 18:

Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P − V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.
	Column 1		Column 2		Column 3
(I)	$W_{1 \to 2} = \frac{1}{y - 1} (P_{2} V_{2} - P_{1} V_{1})$	(i)	Isothermal	(P)
(II)	$W_{1 \to 2} = (P V_{2} P V_{1})$	(ii)	Isochoric	(Q)
(III)	$W_{1 \to 2} = 0$	(iii)	Isobaric	(R)
(IV)	$W_{1 \to 2} = \frac{1}{y - 1} n R T \ln (\frac{V_{2}}{V_{1}})$	(iv)	Adiabatic	(S)

[A]	(I) (ii) (Q)	[B]	(IV) (ii) (R)
[C]	(III) (iv) (R)	[D]	(I) (iv) (Q)

Correct Answer : D

Question 19:

An ideal gas is expanded from (

p_{1}, V_{1}, T_{1}

) to (

p_{2}, V_{2}, T_{2}

) under different conditions. The correct statement(s) among the following is(are)

[A]	The work done on the gas is maximum when it is compressed irreversibly from ( $p_{2}, V_{2}$ ) to ( $p_{1}, V_{1}$ ) against constant pressure $p_{1}$
[B]	If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic
[C]	The work done by the gas is less when it is expanded reversibly from $V_{1}$ to $V_{2}$ under adiabatic conditions as compared to that when expanded reversibly from $V_{1}$ to $V_{2}$ under isothermal conditions
[D}	The change in internal energy of the gas is (i) zero, if it is expanded reversibly with $T_{1} = T_{2}$ , and (ii) positive, if it is expanded reversibly under adiabatic conditions with $T_{1}$ ≠ $T_{2}$

Correct Answer : A,B and C

Question 20:

For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here

x_{1}

and

x_{M}

represent mole fractions of L and M, respectively, in the solution. The correct statement(s) applicable to this system is(are)

[A]	The point Z represents vapour pressure of pure liquid M and Raoult’s law is obeyed from $x_{L}$ = 0 to $x_{L}$ = 1
[B]	The point Z represents vapour pressure of pure liquid L and Raoult’s law is obeyed when $x_{L} \to$ 1
[C]	The point Z represents vapour pressure of pure liquid M and Raoult’s law is obeyed when $x_{L} \to 0$
[D]	Attractive intermolecular interactions between L-L in pure liquid L and M-M in pure liquid M are stronger than those between L-M when mixed in solution

Correct Answer : B and D

Question 21:

The correct statement(s) about the oxoacids,

H C I O_{4}

and HClO, is(are)

[A]	The central atom in both $H C I O_{4}$ and HClO is $s p^{3}$ hybridized
[B]	$H C I O_{4}$ is more acidic than HClO because of the resonance stabilization of its anion
[C]	$H C I O_{4}$ is formed in the reaction between $C I_{2}$ and $H_{2} O$
[D]	The conjugate base of $H C I O_{4}$ is weaker base than $H_{2} O$

Correct Answer : A,B and D

Question 22:

The colour of the

X_{2}

molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to

[A]	The physical state of $X_{2}$ at room temperature changes from gas to solid down the group
[B]	Decrease in ionization energy down the group
[C]	Decrease in $π^{} - 0^{}$ gap down the group
[D]	decrease in HOMO-LUMO gap down the group

Correct Answer : C and D

Question 23:

The IUPAC name(s) of the following compound is(are)

[A]	1-chloro-4-methylbenzene
[B]	4-chlorotoluene
[C]	4-methylchlorobenzene
[D]	1-methyl-4-chlorobenzene

Correct Answer : A and B

Question 24:

The correct statement(s) for the following addition reactions is(are)

[A]	O and P are identical molecules
[B]	(M and O) and (N and P) are two pairs of diastereomers
[C]	(M and O) and (N and P) are two pairs of enantiomers
[D]	Bromination proceeds through trans-addition in both the reactions

Correct Answer : B and D

Question 25:

A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is 8 g

c m^{- 3}

, then the number of atoms present in 256 g of the crystal is

N \times 10^{24}

. The value of N is

Correct Answer : 2

Question 26:

c m^{2}

. The conductance of this solution was found to be

5 \times 10^{- 7}

S. The pH of the solution is 4. The value of limiting molar conductivity

(\land_{m}^{o})

of this weak monobasic acid in aqueous solution is

Z \times 10^{2}

c m^{- 1}

m o l^{- 1}

. The value of Z is

Correct Answer : 6

Question 27:

The sum of the number of lone pairs of electrons on each central atom in the following species is

{[T e B r_{6}]}^{2 -}, {[B r F_{2}]}^{+}

S N F_{3}, a n d {[X e F_{3}]}^{-}

(Atomic numbers: N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

Correct Answer : 6

Question 28:

Among

H_{2}, H w_{2}^{+}, B e_{2,} B_{2}, C_{2}, N_{2}, O_{2}

, and

F_{2}

, the number of diamagnetic species is (Atomic numbers: H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)

Correct Answer : 0

Question 29:

Among the following, the number of aromatic compound(s) is

Correct Answer : 5

Question 30:

For the given orbital in Column 1, the only CORRECT combination for any hydrogen-like species is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (ii) (S)	[B]	(IV) (iv) (R)
[C]	(II) (ii) (P)	[D]	(III) (iii) (P)

Correct Answer : C

Question 31:

For hydrogen atom, the only CORRECT combination is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (i) (S)	[B]	(II) (i) (Q)
[C]	(I) (i) (P)	[D]	(I) (iv) (R)

Correct Answer : A

Question 32:

For

H e^{+}

ion, the only INCORRECT combination is

The wave function, $ψ_{n, I, m_{I}}$ is a mathematical function whose value depends upon spherical polar coordinates (r,θ, $ϕ$ ) of the electron and characterized by the quantum numbers n, l and $m_{I}$ . Here r is distance from nucleus, θ is colatitude and $ϕ$ is azimuth. In the mathematical functions given in the Table, Z is atomic number and $a_{o}$ is Bohr radius.
	Column 1		Column 2		Column 3
(I)	1s orbital	(i)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{3}{2}} e^{- (\frac{Z r}{a_{o}})}$	(P)
(II)	2s orbital	(ii)	One radial node	(Q)	Probability density at nucleus $\infty \frac{1}{a_{o}^{3}}$
(III)	2 $p_{z}$ orbital	(iii)	$ψ_{n, I, m_{I}} \infty {(\frac{Z}{a_{o}})}^{\frac{5}{2}} r e^{- (\frac{Z r}{a_{o}})} \cos θ$	(R)	Probability density at nucleus
(IV)	3 $d_{z}^{2}$ orbital	(iv)	xy-plane is a nodal plane	(S)	Energy needed to excite electron from n = 2 state to n = 4 state is $\frac{27}{32}$ times the energy needed to excite electron from n = 2 state to n = 6 state

[A]	(I) (i) (R)	[B]	(II) (ii) (Q)
[C]	(I) (iii) (R)	[D]	(I) (i) (S)

Correct Answer : C

Question 33:

For the synthesis of benzoic acid, the only CORRECT combination is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(II) (i) (S)	[B]	(IV) (ii) (P)
[C]	(I) (iv) (Q)	[D]	(III) (iv) (R)

Correct Answer : A

Question 34:

The only CORRECT combination that gives two different carboxylic acids is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(II) (iv) (R)	[B]	(IV) (iii) (Q)
[C]	(III) (iii) (P)	[D]	(I) (i) (S)

Correct Answer : C

Question 35:

The only CORRECT combination in which the reaction proceeds through radical mechanism is

Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively.
	Column 1		Column 2		Column 3
(I)	Toluene	(i)	NaOH/ $B r_{2}$	(P)	Condensation
(II)	Acetophenone	(ii)	$B r_{2} / h v$	(Q)	Carboxylation
(III)	Benzaldehyde	(iii)	${(C H_{3} C O)}_{2} O / C H_{3} C O O K$	(R)	Substitution
(IV)	Phenol	(iv)	$N a O H / C O_{2}$	(S)	Haloform

[A]	(III) (ii) (P)	[B]	(IV) (i) (Q)
[C]	(II) (iii) (R)	[D]	(I) (ii) (R)

Correct Answer : D

Question 36:

Addition of excess aqueous ammonia to a pink coloured aqueous solution of

M C {I_{2}}^{.} 6 H_{2} O

(X) and

N H_{4} C I

[A]	Addition of silver nitrate to Y gives only two equivalents of silver chloride
[B]	The hybridization of the central metal ion in Y is $d^{2} s p^{3}$
[C]	Z is a tetrahedral complex
[D]	When X and Z are in equilibrium at 0°C, the colour of the solution is pink

Correct Answer : B, C, and D

Question 37:

If 2x − y + 1 = 0 is a tangent to the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1

, then which of the following CANNOT be sides of a right angled triangle?

[A]	a, 4, 1	[B]	a, 4, 2
[C]	2a, 8, 1	[D]	2a, 4, 1

Correct Answer : A,B and C

Question 38:

If a chord, which is not a tangent, of the parabola

y^{2}

= 16x has the equation 2x + y = p, and midpoint (ℎ, k), then which of the following is(are) possible value(s) of p, ℎ and k?

[A]	P = −2, ℎ = 2, k = −4	[B]	p = −1, ℎ = 1, k = −3
[C]	p=2, h=3, k=-4	[D]	p = 5, ℎ = 4, k = −3

Correct Answer : C

Question 39:

Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f (x) = x cos(

π

(x + [x])) is discontinuous?

[A]	x = −1	[B]	x = 0
[C]	x = 1	[D]	x = 2

Correct Answer : A

Question 40:

Let f: ℝ → (0, 1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

[A]	$x^{9} - f (x)$	[B]	$x - \int_{0}^{\frac{π}{2} - x} f (t) \cos t d t$
[C]	$e^{x} - f_{0}^{x} f (t) \sin t d t$	[D]	$e^{x} + f_{0}^{π} f (t) \sin t d t$

Correct Answer : A and B

Question 41:

Let a, b, x and y be real numbers such that a − b = 1 and y ≠ 0. If the complex number z = x + iy satisfies Im

(\frac{a z + b}{z + 1})

= y, then which of the following is(are) possible value(s) of x?

[A]	$- 1 + \sqrt{1 - y^{2}}$	[B]	$- 1 - \sqrt{1 - y^{2}}$
[C]	$1 + \sqrt{1 - y^{2}}$	[D]	$1 + \sqrt{1 - y^{2}}$

Correct Answer : A and B

Question 42:

Let X and Y be two events such that P(X) =

\frac{1}{3}

,P(X| Y)=

\frac{1}{2}

and P(Y| X) =

\frac{2}{5}

, Then

[A]	$P (Y) = \frac{4}{15}$	[B]	$P (X' \| Y) = \frac{1}{2}$
[C]	$P (X \cap Y) = \frac{1}{5}$	[D]	$P (X \cup Y) = \frac{2}{5}$

Correct Answer : A and B

Question 43:

For how many values of P, the circle

x^{2} + y^{2} + 2 x + 4 y - p = 0

and the coordinate axes have exactly three common points?

Correct Answer : 2

Question 44:

Let

f; ℝ \to ℝ

be a differentiable Function such that

f (0) = 0, f (\frac{π}{2}) = 3

and

f' (0) = 1 .

g (x) = \int_{x}^{\frac{π}{2}} [f' (t) \cos e c t - c o t t \cos e c t f (t)] d t

for x

\in (0, \frac{π}{2}],

then

\lim_{x \to \infty} g (x) =

Correct Answer : 2

Question 45:

For a real number a, if the system

[\begin{matrix} 1 & a & a^{2} \\ a & 1 & a \\ a^{2} & a & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]

of linear equations, has infinitely many solutions, then 1+a+

a^{2}

Correct Answer : 1

Question 46:

\frac{y}{9 x} =

Correct Answer : 5

Question 47:

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Correct Answer : 6

Question 48:

For a =

\sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(I) (i) (P)		(I) (ii) (Q)
[C]	(II) (ii) (Q)		(III) (i) (P)

Correct Answer : B

Question 49:

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(I) (ii) (Q)	[B]	(II) (iv) (R)
[C]	(III) (i) (P)	[D]	(III) (ii) (Q)

Correct Answer : C

Question 50:

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3 x} + 2 y = 4

, then which of the following options is the only CORRECT combination?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
	Column 1		Column 2		Column 3
(I)	$x^{2} + y^{2} = a^{2}$	(i)	$m y = m^{2} x + a$	(P)	$(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II)	$x^{2} + a^{2} y^{2} = a^{2}$	(ii)	$y = m x + a \sqrt{m^{2} + 1}$	(Q)	$(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III)	$y^{2} = 4 a x$	(iii)	$y = m x + \sqrt{a^{2} m^{2} - 1}$	(R)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV)	$x^{2} - a^{2} y^{2} = a^{2}$	(iv)	$y = m x + \sqrt{a^{2} m^{2} + 1}$	(S)	$(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} + 1}})$

[A]	(IV) (iii) (S)	[B]	(IV) (iv) (S)
[C]	(II) (iii) (R)	[D]	(II) (iv) (R)

Correct Answer : D

Question 51:

Which of the following options is the only CORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (i) (P)	[B]	(II) (ii) (Q)
[C]	(III) (iii) (R)	[D]	(IV) (iv) (S)

Correct Answer : B

Question 52:

Which of the following options is the only CORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (ii) (R)	[B]	(II) (iii) (S)
[C]	(III) (iv) (P)	[D]	(IV) (i) (S)

Correct Answer : B

Question 53:

Which of the following options is the only INCORRECT combination?

Let f(x) = x + $\log_{e} x - x \log_{e} x, x \in (0, \infty)$ • Column 1 contains information about zeros of f(x) , f'(x) and f"(x). • Column 2 contains information about the limiting behavior of f(x) , f'(x) and f'(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x) and f'(x) .
	Column 1		Column 2		Column 3
(I)	$f (x)$ = 0 for some x ∈ (1, $e^{2}$ )	(i)	$\lim_{x \to \infty} f (x) = 0$	(P)	f is increasing in (0, 1)
(II)	$f (x)$ = 0 for some x ∈ (1,e)	(ii)	$\lim_{x \to \infty} f (x) = - \infty$	(Q)	f is decreasing in (e, $e^{2}$ )
(III)	$f (x)$ = 0 for some x ∈ (0,1)	(iii)	$\lim_{x \to \infty} f' (x) = - \infty$	(R)	f' is increasing in (0, 1)
(IV)	$f (x)$ = 0 for some x ∈	(iv)	$\lim_{x \to \infty} f'' (x) = - \infty$	(S)	f' is decreasing in (e, $e^{2}$ )

[A]	(I) (iii) (P)	[B]	(II) (iv) (Q)
[C]	(III) (i) (R)	[D]	(II) (iii) (P)

Correct Answer : C

Question 54:

Which of the following is(are) NOT the square of a 3×3 matrix with real entries?

(A)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$	(B)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$
(C)	$[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$	(D)	$[\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$

Correct Answer : B and C

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