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Category: IIT-JEE 2018 Physics Paper 1

Question 1:

The potential energy of a particle of mass 𝑚 at a distance 𝑟 from a fixed point 𝑂 is given by

v (r) = k r^{2} / 2,

where 𝑘 is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius 𝑅 about the point 𝑂. If 𝑣 is the speed of the particle and 𝐿 is the magnitude of its angular momentum about 𝑂, which of the following statements is (are) true?

A	$v = \sqrt{\frac{k}{2 m}} R$	B	$V = \sqrt{\frac{K}{m}} R$
C	$L = \sqrt{m k} R^{2}$	D	$L = \sqrt{\frac{m k}{2}} R^{2}$

A and B

B and C

C and D

Category: IIT-JEE 2018 Physics Paper 1

Question 2:

Consider a body of mass 1.0 𝑘𝑔 at rest at the origin at time 𝑡=0. A force

\vec{F}

= (α t \hat{I} + \hat{j})

is applied on the body, where

α = 1.0 N s^{- 1} a n d β

=1.0 𝑁. The torque acting on the body about the origin at time 𝑡=

= 1, 0 s i s \vec{τ}

. Which of the following statements is (are) true?

A	$\|\vec{τ}\| = \frac{1}{3} N m$	B	$T h e t o r q u e \vec{τ} i s i n t h e d i r e c t i o n o f t h e u n i t v e c t o r + \hat{k}$
C	The Velocity of the body at t =1 s is $\vec{v} = 12 (\hat{i} + 2 \hat{j}) m s^{- 1}$	D	The magnitude of displacement of the body at 𝑡=1 𝑠 is $\frac{1}{6} m$

A and B

B and C

C and A

D and C

Category: IIT-JEE 2018 Physics Paper 1

Question 3:

A uniform capillary tube of inner radius 𝑟 is dipped vertically into a beaker filled with water. The water rises to a height ℎ in the capillary tube above the water surface in the beaker. The surface tension of water is 𝜎. The angle of contact between water and the wall of the capillary tube is 𝜃. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

A	For a given material of the capillary tube, ℎ decreases with increase in 𝑟	B	For a given material of the capillary tube, ℎ is independent of 𝜎
C	If this experiment is performed in a lift going up with a constant acceleration, then ℎ decreases	D	ℎ is proportional to contact angle 𝜃

A and B

A and C

B and C

D and B

Category: IIT-JEE 2018 Physics Paper 1

Question 4:

In the figure below, the switches

s_{1}

and

s_{2}

are closed simultaneously at 𝑡=0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current 𝐼 in the middle wire reaches its maximum magnitude 𝐼𝑚𝑎𝑥 at time 𝑡=𝜏. Which of the following statements is (are) true?

A	$I_{m a x} = \frac{v}{2 R}$	B	$I_{m a x} = \frac{v}{4 R}$
C	$τ = \frac{L}{R} I n 2$	D	$τ = \frac{2 L}{R} I n 2$

A and B

B and C

C and A

D and B

Category: IIT-JEE 2018 Physics Paper 1

Question 5:

Two infinitely long straight wires lie in the 𝑥𝑦-plane along the lines 𝑥=±𝑅. The wire located at 𝑥=+𝑅 carries a constant current

I_{1}

and the wire located at 𝑥=−𝑅 carries a constant current

I_{2}

. A circular loop of radius 𝑅 is suspended with its centre at

(0, 0, \sqrt{3 R})

and in a plane parallel to the 𝑥𝑦-plane. This loop carries a constant current 𝐼 in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the +𝑗̂ direction. Which of the following statements regarding the magnetic field 𝐵⃗ is (are) true?

A	If cannot be equal to zero at the origin (0,0,0)	B	If $I_{1} > 0 a n d I_{2} < 0, \vec{B}$ can be equal to zero at the origin (0,0,0)
C	If $I_{1} < 0 a n d I_{2} > 0, \vec{B}$ can be equal to zero at the origin (0,0,0)	D	If $I_{1} = I_{2}$ then the 𝑧-component of the magnetic field at the centre of the loop is $(- \frac{μ_{o} 1}{2 R})$

A,B and C

B,C and D

C,D and A

A,B and D

Category: IIT-JEE 2018 Physics Paper 1

Question 6:

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?

A	Process I is an isochoric process	B	In process II, gas absorbs heat
C	In process IV, gas releases heat	D	Processes I and III are not isobaric

A,B and C

B,C and D

C,D and A

Non of these

Category: IIT-JEE 2018 Physics Paper 1

Question 7:

Two vectors

\vec{A}

and

\vec{B}

are defined as

\vec{A}

=𝑎 𝑖̂ and

\vec{B}

=𝑎 (cos𝜔𝑡𝑖̂+sin𝜔𝑡𝑗̂), where 𝑎 is a constant and 𝜔=𝜋/6 𝑟𝑎𝑑

S^{- 1}

. If |

|\vec{A} + \vec{B}|

= \sqrt{3} |\vec{A} + \vec{B}|

time 𝑡=𝜏 for the first time, the value of 𝜏, in 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, is __________.

2.25

2.00

2.50

1.75

Category: IIT-JEE 2018 Physics Paper 1

Question 8:

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 𝑚

s^{- 1}

and the man behind walks at a speed 2.0 𝑚

s^{- 1}

. A third man is standing at a height 12 𝑚 above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 𝐻𝑧. The speed of sound in air is 330 𝑚

s^{- 1}

. At the instant, when the moving men are 10 𝑚 apart, the stationary man is equidistant from them. The frequency of beats in 𝐻𝑧, heard by the stationary man at this instant, is __________.

5.00

4.50

5.50

6.00

Category: IIT-JEE 2018 Physics Paper 1

Question 9:

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is

(2 - \sqrt{3}) / \sqrt{10} s,

, then the height of the top of the inclined plane, in 𝑚𝑒𝑡𝑟𝑒𝑠, is __________. Take

g = 10 m s^{- 2}

0.25

0.75

0.50

1.00

Category: IIT-JEE 2018 Physics Paper 1

Question 10:

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 𝑁

m^{- 1}

and the mass of the block is 2.0 𝑘𝑔. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 𝑘𝑔 moving with a speed of 2.0

m s^{- 1}

collides elastically with the first block. The collision is such that the 2.0 𝑘𝑔 block does not hit the wall. The distance, in 𝑚𝑒𝑡𝑟𝑒𝑠, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.

2.07

2.09

2.06

2.08

Category: IIT-JEE 2018 Physics Paper 1

Question 11:

Three identical capacitors

c_{1}

c_{2}

and

c_{3}

have a capacitance of 1.0 𝜇𝐹 each and they are uncharged initially. They are connected in a circuit as shown in the figure and 𝐶1 is then filled completely with a dielectric material of relative permittivity 𝜖𝑟. The cell electromotive force (emf) 𝑉0=8 𝑉. First the switch

s_{1}

is closed while the switch

s_{2}

is kept open. When the capacitor

c_{3}

is fully charged,

s_{1}

is opened and

s_{2}

is closed simultaneously. When all the capacitors reach equilibrium, the charge on

c_{3}

is found to be 5 𝜇𝐶. The value of

ε_{r}

=____________.

1.20

2.10

2.00

1.50

Category: IIT-JEE 2018 Physics Paper 1

Question 12:

In the 𝑥𝑦-plane, the region 𝑦>0 has a uniform magnetic field

B_{1} \hat{K}

and the region 𝑦<0 has another uniform magnetic field

B_{2} \hat{K}

. A positively charged particle is projected from the origin along the positive 𝑦-axis with speed 𝑣0=𝜋 𝑚

S^{- 1}

at 𝑡=0, as shown in the figure. Neglect gravity in this problem. Let 𝑡=𝑇 be the time when the particle crosses the 𝑥-axis from below for the first time. If

B_{2}

B_{1}

, the average speed of the particle, in 𝑚

S^{- 1}

, along the 𝑥-axis in the time interval 𝑇 is __________.

2.00

4.00

3.00

1.00

Category: IIT-JEE 2018 Physics Paper 1

Question 13:

Sunlight of intensity 1.3 𝑘𝑊

m^{- 2}

is incident normally on a thin convex lens of focal length 20 𝑐𝑚. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in 𝑘𝑊

m^{- 2}

, at a distance 22 𝑐𝑚 from the lens on the other side is __________.

120.00

125.00

130.00

140.00

Category: IIT-JEE 2018 Physics Paper 1

Question 14:

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures

T_{1}

=300 𝐾 and

T_{2}

=100 𝐾, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are

K_{1}

and

K_{2}

respectively. If the temperature at the junction of the two cylinders in the steady state is 200 𝐾, then

K_{1} / K_{2}

=__________.

4.00

3.50

4.80

2.00

Category: IIT-JEE 2018 Physics Paper 1

Question 15:

[\in_{0}]

and

[μ_{0}]

stand for dimensions of the permittivity and permeability of free space respectively. [𝐿] and [𝑇] are dimensions of length and time respectively. All the quantities are given in SI units. The relation between [E] and [B] is

A	[E] =[B] [L][T]	B	[E]= $[B] {[L]}^{- 1} [T]$
C	$[E] = [B] [L] {[T]}^{- 1}$	D	$[E] = [B] {[L]}^{- 1} {[T]}^{- 1}$

Category: IIT-JEE 2018 Physics Paper 1

Question 16:

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation 𝑧= 𝑥/𝑦. If the errors in 𝑥, 𝑦 and 𝑧 are Δ𝑥,Δ𝑦 and Δ𝑧, respectively, then

z \pm Δ z = \frac{x + Δ x}{y + Δ y} = \frac{x}{y} (1 \pm \frac{Δ x}{x}) {(1 \pm \frac{Δ y}{y})}^{- 1}

The series expansion for

{(1 \pm \frac{Δ y}{y})}^{- 1}

to first power inΔ𝑦/𝑦, is 1∓(Δ𝑦/𝑦). The relative errors in independent variables are always added. So the error in 𝑧 will be

Δ z = z (\frac{Δ x}{x}) + (\frac{Δ y}{y}) .

The above derivation makes the assumption that Δ𝑥/𝑥≪1, Δ𝑦/𝑦≪1. Therefore, the higher powers of these quantities are neglected. Consider the ratio 𝑟=

\frac{(1 - a)}{(1 + a)}

to be determined by measuring a dimensionless quantity 𝑎. If the error in the measurement of 𝑎 is Δ𝑎 (Δ𝑎/𝑎≪1) , then what is the error Δ𝑟 in determining 𝑟?

A	$\frac{∆ a}{{(1 + a)}^{2}}$	B	$\frac{2 ∆ a}{{(1 + a)}^{2}}$
C	$\frac{∆ a}{(1 - a^{2})}$	D	$\frac{2 a ∆ a}{(1 - a^{2})}$

Category: IIT-JEE 2018 Physics Paper 1

Question 17:

z \pm Δ z = \frac{x + Δ x}{y + Δ y} = \frac{x}{y} (1 \pm \frac{Δ x}{x}) {(1 \pm \frac{Δ y}{y})}^{- 1}

The series expansion for

{(1 \pm \frac{Δ y}{y})}^{- 1}

to first power inΔ𝑦/𝑦, is 1∓(Δ𝑦/𝑦). The relative errors in independent variables are always added. So the error in 𝑧 will be

Δ z = z (\frac{Δ x}{x}) + (\frac{Δ y}{y}) .

The above derivation makes the assumption that Δ𝑥/𝑥≪1, Δ𝑦/𝑦≪1. Therefore, the higher powers of these quantities are neglected. Consider the ratio 𝑟=

r = \frac{(1 - a)}{(1 + a)}

In an experiment, the initial number of radioactive nuclei is 3000. It is found that 1000±40 nuclei decayed in the first 1.0 𝑠. For |𝑥|≪1, ln(1+𝑥)=𝑥 up to first power in 𝑥. The error Δ𝜆, in the determination of the decay constant 𝜆, in 𝑠−1, is

A	0.04	B	0.03
C	0.02	D	0.01

Category: IIT-JEE 2018 Physics Paper 1

Question 18:

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [𝐸] and [𝐵] stand for dimensions of electric and magnetic fields respectively, while and stand for dimensions of the permittivity and permeability of free space respectively. [𝐿] and [𝑇] are dimensions of length and time respectively. All the quantities are given in SI units. The relation between

[\in_{0}]

and

[μ_{0}]

A	$[μ_{0}] = [\in_{0}] {[L]}^{2} {[T]}^{- 2}$	B	$[μ_{0}] = [\in_{0}] {[L]}^{- 2} {[T]}^{2}$
C	$[μ_{0}] = {[\in_{0}]}^{- 1} {[L]}^{2} {[T]}^{- 2}$	D	$[μ_{0}] = {[\in_{0}]}^{- 1} {[L]}^{- 2} {[T]}^{2}$

Category: IIT-JEE 2018 Chemistry Paper 1

Question 19:

The compound(s) which generate(s)

N_{2}

gas upon thermal decomposition below 300C is (are)

A	$N H_{4} N O_{3}$	B	${(N H_{4})}_{2} C r_{2} O_{7}$
C	$B a {(N_{3})}^{2}$	D	$M g_{3} N_{2}$

A and B

B and C

C and D

A and C

Category: IIT-JEE 2018 Chemistry Paper 1

Question 20:

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28)

A	Total number of valence shell electrons at metal centre in Fe(CO)5 or $N i {(C O)}_{4}$ is 16	B	These are predominantly low spin in nature
C	Metal–carbon bond strengthens when the oxidation state of the metal is lowered	D	The carbonyl C−O bond weakens when the oxidation state of the metal is increased

A and D

A and C

B and C

B and D

Category: IIT-JEE 2018 Chemistry Paper 1

Question 21:

Based on the compounds of group 15 elements, the correct statement(s) is (are)

A	$B i_{2} O_{5} i s m o r e b a s i c t h a n N_{2} O_{5}$	B	$N F_{3} i s m o r e c o v a l e n t t h a n B i F_{3}$
C	$P H_{3} b o i l s a t l o w e r t e m p e r a t u r e t h a n N H_{3}$	D	The N−N single bond is stronger than the P−P single bond

A,B and C

B,C and D

C, D and A

None of these

Category: IIT-JEE 2018 Chemistry Paper 1

Question 22:

I n the following reaction sequence, the correct structure(s) of X is (are)

\to_{\frac{2) N a l, M e_{2} C O}{3) N a N_{3} H C O M e_{2}}}^{1) P B r_{3}, E t_{2} O}

Category: IIT-JEE 2018 Chemistry Paper 1

Question 23:

The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

A,B and C

B,D and C

A,B and D

none of these

Category: IIT-JEE 2018 Chemistry Paper 1

Question 24:

A reversible cyclic process for an ideal gas is shown below. Here, P, V, and T are pressure, volume and temperature, respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and internal energy, respectively.

The correct option(s) is (are)

A	$q_{A C} = Δ U_{B C} a n d w_{A B} = P_{2} (V_{2} - V_{1})$	B	$w_{B C} = P_{2} (V_{2} - V_{1}) a n d q_{B C} = Δ H_{A C}$
C	$Δ H_{C A} < Δ U_{C A} a n d q_{A C} = Δ U_{B C}$	D	$q_{B C} = Δ {��}_{A C} a n d Δ H_{C A} > Δ U_{C A}$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2018 Chemistry Paper 1

Question 25:

Among the species given below, the total number of diamagnetic species is ___. H atom,

N O_{2}

monomer,

O_{2}^{-}

(superoxide), dimeric sulphur in vapour phase,

M n_{3} O_{4}, {(N H_{4})}_{2} [F e C I]

₄,(NH4)2NiCI₄,K2MnO4K2CrO4

0.01

0.11

0.1

Category: IIT-JEE 2018 Chemistry Paper 1

Question 26:

The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by

N i C I_{2} 6 H_{2} O

to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl2.6H2O are used in the preparation, the combined weight (in grams) of gypsum and the nickel-ammonia coordination compound thus produced is ____. (Atomic weights in g

m o l^{- 1}

H = 1, N = 14, O = 16, S = 32, C I = 35.5

C a = 40, N i = 59)

2992

2998

2995

2996

Category: IIT-JEE 2018 Chemistry Paper 1

Question 27:

Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance. (i) Remove all the anions (X) except the central one (ii) Replace all the face centered cations (M) by anions (X) (iii) Remove all the corner cations (M) (iv) Replace the central anion (X) with cation (M) The value of

(\frac{n u m b e r s o f a n i o n s}{n u m b e r o f c a t i o n s}) i n Z i s

____.

Category: IIT-JEE 2018 Chemistry Paper 1

Question 28:

F or the electrochemical cell,

M g (s) M g^{2 +} (a q, 1 M) ∥ C u^{2 +} (a q, 1 M) C u (s)

the standard emf of the cell is 2.70 V at 300 K. When the concentration of

M g^{2 +} i s

is changed to 𝒙 M, the cell potential changes to 2.67 V at 300 K. The value of 𝒙 is ____. (given

\frac{F}{R}

11500 K

V^{- 1}

, where 𝐹 is the Faraday constant and 𝑅 is the gas constant, ln(10) = 2.30)

Category: IIT-JEE 2018 Chemistry Paper 1

Question 29:

A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to leak across (Figure 2), the volume

(i n m^{3})

of the compartment A after the system attains equilibrium is ____.

22.22

2.2

22.2

2.22

Category: IIT-JEE 2018 Chemistry Paper 1

Question 30:

Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions

x_{A} a n d x_{B}

, respectively, has vapour pressure of 22.5 Torr. The value of

x_{A} / x_{B}

in the new solution is ____. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

None of these

Category: IIT-JEE 2018 Chemistry Paper 1

Question 31:

The solubility of a salt of weak acid (AB) at pH 3 is

Y x 10^{- 3}

m o l^{- 1}

.The value of Y is ____. (Given that the value of solubility product of AB

(k_{s p})

2 x 10^{- 10}

and the value of ionization constant of HB

(k_{a}) = 1 x 10^{- 8})

4.45

4.46

4.47

none of these

Category: IIT-JEE 2018 Chemistry Paper 1

Question 32:

The plot given below shows 𝑃−𝑇 curves (where P is the pressure and T is the temperature) for two solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the solvents.

On addition of equal number of moles of a non-volatile solute S in equal amount (in kg) of these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent Y, the degree of dimerization in solvent X is ____.

0.10

0.05

0.07

0.01

Category: IIT-JEE 2018 Chemistry Paper 1

Question 33:

Treatment of benzene with CO/HCl in the presence of anhydrous

A I C I_{3} / C U C I

followed by reaction with

A c_{2} O

NaOAc gives compound X as the major product.Compound X upon reaction with

B r_{2} / N a_{2} / C O_{3}

followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by

H_{3} P O_{4}

treatment gives Z as the major product. T he compound Y is

Category: IIT-JEE 2018 Chemistry Paper 1

Question 34:

Treatment of benzene with CO/HCl in the presence of anhydrous

A I C I_{3} / C U C I

followed by reaction with

A c_{2} O

NaOAc gives compound X as the major product.Compound X upon reaction with

B r_{2} / N a_{2} / C O_{3}

followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by

H_{3} P O_{4}

treatment gives Z as the major product. T he compound Z is

Category: IIT-JEE 2018 Chemistry Paper 1

Question 35:

An organic acid P

(C_{11} H_{12} O_{2})

can easily be oxidized to a dibasic acid which reacts with ethyleneglycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.

S \leftarrow_{\frac{C H C I_{3} K O H, ∆}{H_{2} / P d - c}}^{\frac{\frac{1) H_{2} / P d - c}{2) N H_{3} / ∆}}{3) B r_{2} N a O H}}

P \to_{\frac{3) M e M g B r, C d C I_{2}}{N a B H_{2}}}^{\frac{1) H_{2} / P d - c}{2) S O C I_{2}}}

\to_{\frac{3) C O_{2} (d e y i c e)}{H_{3} O^{+}}}^{\frac{1) H C I}{2) M g / E t_{2 O}}}

T he compound R is

Category: IIT-JEE 2018 Chemistry Paper 1

Question 36:

An organic acid P

(C_{11} H_{12} O_{2})

S \leftarrow_{\frac{C H C I_{3} K O H, ∆}{H_{2} / P d - c}}^{\frac{\frac{1) H_{2} / P d - c}{2) N H_{3} / ∆}}{3) B r_{2} N a O H}}

P \to_{\frac{3) M e M g B r, C d C I_{2}}{N a B H_{2}}}^{\frac{1) H_{2} / P d - c}{2) S O C I_{2}}}

\to_{\frac{3) C O_{2} (d e y i c e)}{H_{3} O^{+}}}^{\frac{1) H C I}{2) M g / E t_{2 O}}}

T he compound S is

Category: IIT-JEE 2018 Math Paper 1

Question 37:

For a non-zero complex number 𝑧, let arg(𝑧) denote the principal argument with

- π < \arg (z) \leq π

Then, which of the following statement(s) is (are) FALSE?

A	$a r g (- 1 - i) = \frac{π}{4} =, w h e r e i = \sqrt{1}$	B	The function 𝑓: ℝ $(- π, π] defined by f (t) = \arg (- 1 + it)$ for all 𝑡∈ℝ, is continuous at all points of ℝ, where $i = \sqrt{- 1}$
C	For any two non-zero complex numbers $Z_{1} a n d Z_{2}$ $a r g (\frac{Z_{1}}{Z_{2}}) - a r g (Z_{1}) + a r g (Z_{2})$ is an integer multiple of 2𝜋	D	For any three given distinct complex numbers $Z_{1}, Z_{2} a n d Z_{3}$ the locus of the point 𝑧 satisfying the condition $a r g (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})})$ = 𝜋, lies on a straight line

A,D and C

A,B and D

A,B and C

None of these

Category: IIT-JEE 2018 Math Paper 1

Question 38:

In a triangle 𝑃𝑄𝑅, let

< P Q R = 30 °

and the sides 𝑃𝑄 and 𝑄𝑅 have lengths and 10

\sqrt{3}

, respectively. Then, which of the following statement(s) is (are) TRUE?

A	$< Q P R = 45 °$	B	The area of the triangle 𝑃𝑄𝑅 is 25 $\sqrt{3}$ and $< Q R P = 120 °$
C	The radius of the incircle of the triangle 𝑃𝑄𝑅 is $10 \sqrt{3} - 15$	D	The area of the circumcircle of the triangle $P Q R i s 100 π$

A,B and C

B,C and D

C,D and A

Category: IIT-JEE 2018 Math Paper 1

Question 39:

Let

P_{1}

: 2𝑥+𝑦−𝑧=3 and

P_{2}

: 𝑥+2𝑦+𝑧=2 be two planes. Then, which of the following statement(s) is (are) TRUE?A

A	The line of intersection of $p_{1} a n d p_{2}$ has direction ratios 1,2,−1.	B	The line $\frac{3 x - 4}{9} = \frac{1 - 3 y}{9} = \frac{z}{3}$
C	The acute angle between $p_{1} a n d p_{2} i s 60 °$	D	If $p_{3}$ is the plane passing through the point (4,2,−2) and perpendicular to the line of intersection of $p_{1} a n d p_{2}$ , then the distance of the point (2,1,1) from the plane $p_{3} i s \frac{2}{\sqrt{3}}$

A and B

B and C

C and D

D and A

Category: IIT-JEE 2018 Math Paper 1

Question 40:

For every twice differentiable function 𝑓: ℝ→[−2,2] with

{(f (0))}^{2} + {(f' (0))}^{2}

=85, which of the following statement(s) is (are) TRUE?

A	There exist 𝑟,𝑠∈ℝ, where 𝑟<𝑠, such that 𝑓 is one-one on the open interval (𝑟,𝑠)	B	There exists 𝑥0∈ $x_{0} \in (- 4, 0)$ such that $f' (o x) \leq 1$
C	$\lim_{x \to \infty} f (x) = 1$	D	There exists 𝛼∈(−4,4) such that 𝑓(𝛼)+𝑓′′(𝛼) = 0 and 𝑓′(𝛼)≠0

A,B and D

B, C and D

C,B and D

None of these

Category: IIT-JEE 2018 Math Paper 1

Question 41:

Let 𝑓:ℝ→ℝ and 𝑔:ℝ→ℝ be two non-constant differentiable functions. If

f' (x) = (e^{(f (x) - g (x))}) g' (x)

for all 𝑥∈ℝ, and 𝑓(1)=𝑔(2)=1, then which of the following statement(s) is (are) TRUE?

A	$f (2) < 1 - \log_{e} 2$	B	$f (2) > 1 - \log_{e} 2$
C	$g (1) > 1 - \log_{e} 2$	D	$g (1) < 1 - \log_{e} 2$

A and B

B and C

C and A

D and A

Category: IIT-JEE 2018 Math Paper 1

Question 42:

Let 𝑓:[0,∞)→ℝ be a continuous function such that

f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) d t

for all 𝑥∈[0,∞). Then, which of the following statement(s) is (are) TRUE?

A	for all 𝑥∈[0,∞). Then, which of the following statement(s) is (are) TRUE?	B	The curve 𝑦=𝑓(𝑥) passes through the point (2,−1)
C	The area of the region ${(x, y) \in [0, 1] x ℝ : f (x) \leq y \leq \sqrt{1 - x^{2}}} i s \frac{π - 2}{4}$	D	The area of the region ${(x, y) \in [0, 1] x ℝ : f (x) \leq y \leq \sqrt{1 - x^{2}}} i s \frac{π - 1}{4}$

A and B

B and A

C and B

D and A

Category: IIT-JEE 2018 Math Paper 1

Question 43:

The value of

({(\log_{2} 9)}^{2}) \frac{1}{\log_{2} (\log_{2} 9) x (\sqrt{7})} \frac{1}{\log_{4} 7}

is ______ .

Category: IIT-JEE 2018 Math Paper 1

Question 44:

The number of 5 digit numbers which are divisible by 4, with digits from the set {1,2,3,4,5} and the repetition of digits is allowed, is _____ .

625

650

600

675

Category: IIT-JEE 2018 Math Paper 1

Question 45:

Let 𝑋 be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11,… , and 𝑌 be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,… . Then, the number of elements in the set 𝑋∪𝑌 is _____.

3746

3748

3747

None of these

Category: IIT-JEE 2018 Math Paper 1

Question 46:

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} i_{x} + 1_{- x} \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

(- \frac{1}{2}, \frac{1}{2})

is _____ . (Here, the inverse trigonometric functions

\sin^{- 1}

𝑥 and

\cos^{- 1}

𝑥 assume values in

[- \frac{π}{r}, \frac{π}{r}]

and [0,𝜋], respectively.)

2.1

2.5

2.2

Category: IIT-JEE 2018 Math Paper 1

Question 47:

For each positive integer 𝑛, let

y_{n} = \frac{1}{n} {((n + 1) (n + 2) . . . (n + n))}^{\frac{1}{2}}

For 𝑥∈ℝ, let [𝑥] be the greatest integer less than or equal to 𝑥.if

\lim_{n \to \infty} y_{n} =

𝐿, then the value of [𝐿] is _____ .

0.1

Category: IIT-JEE 2018 Math Paper 1

Question 48:

let

\vec{a} a n d \vec{b}

be two unit vectors such that

\vec{a} . \vec{b}

For some 𝑥,𝑦∈ℝ, let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} + \vec{b)}

.If

|\vec{c}|

⃗|=2 and the vector

\vec{c}

is inclined at the same angle 𝛼 to both

\vec{a} a n d \vec{b}

then the value of

\cos^{2}

𝛼 is _____.

Category: IIT-JEE 2018 Math Paper 1

Question 49:

Let 𝑎,𝑏,𝑐 be three non-zero real numbers such that the equation

\sqrt{3}

𝑎cos𝑥+2 𝑏sin𝑥=𝑐, 𝑥

\in [- \frac{π}{2}, \frac{π}{2}],

has two distinct real roots

α a n d β

with 𝛼+𝛽=

\frac{π}{3}

Then, the value of

\frac{b}{a} i s .

_____ .

0.0

0.5

0.3

0.1

Category: IIT-JEE 2018 Math Paper 1

Question 50:

A farmer

F_{1}

has a land in the shape of a triangle with vertices at 𝑃(0,0),𝑄(1,1) and 𝑅(2,0).From this land, a neighbouring farmer

F_{2}

takes away the region which lies between the side 𝑃𝑄 and a curve of the form

y = x^{n} (n > 1)

If the area of the region taken away by the farmer

F_{2}

is exactly 30% of the area of Δ𝑃𝑄𝑅, then the value of 𝑛 is _____ .

Category: IIT-JEE 2018 Math Paper 1

Question 51:

Let 𝑆 be the circle in the 𝑥𝑦-plane defined by the equation

x^{2} + y^{2} = 4

.Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of 𝑆 passing through the point

P_{0}

(1,1) and parallel to the x-axis and the y-axis, respectively. Let

G_{1} G_{2}

be the chord of S passing through

P_{0}

and having slope −1.Let the tangents to 𝑆 at

E_{1} a n d E_{2}

meet at

E_{3}

the tangents to 𝑆 at

F_{1} a n d F_{2}

meet at

F_{3}

and the tangents to 𝑆 at

G_{1} a n d G_{2} m e e t a t G_{3}

Then, the points

E_{3} F_{3}

and

G_{3}

.lie on the curve

A	𝑥+𝑦=4	B	( $�� - 4)^{2} + {(�� - 4)}^{2} = 16$
C	(𝑥−4)(𝑦−4)=4	D	𝑥𝑦=4

Category: IIT-JEE 2018 Math Paper 1

Question 52:

Let 𝑆 be the circle in the 𝑥𝑦-plane defined by the equation

x^{2} + y^{2} = 4

Let 𝑃 be a point on the circle 𝑆 with both coordinates being positive. Let the tangent to 𝑆 at 𝑃 intersect the coordinate axes at the points 𝑀 and 𝑁. Then, the mid-point of the line segment 𝑀𝑁 must lie on the curve

A	${(�� + ��)}^{2} = 3 �� $	B	$x^{2 / 3} + y^{2 / 3} = 2^{4 / 3}$
C	$x^{2} + y^{2} = 2 �� $	D	$x^{2} + y^{2} = x^{2} y^{2}$

Category: IIT-JEE 2018 Math Paper 1

Question 53:

There are five students

S_{1} S_{2} S_{3} S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1} R_{2} R_{3} R_{4} a n d R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

𝑖=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student

s_{1}

gets the previously allotted seat

R_{1}

, and NONE of the remaining students gets the seat previously allotted to him/her is.

A	$\frac{3}{40}$	B	$\frac{1}{8}$
C	$\frac{7}{40}$	D	$\frac{1}{5}$

Category: IIT-JEE 2018 Math Paper 1

Question 54:

There are five students

S_{1} S_{2} S_{3} S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1} R_{2} R_{3} R_{4} a n d R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

𝑖=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. For 𝑖=1,2,3,4, let 𝑇𝑖 denote the event that the students

S_{1} a n d s_{i + 1}

do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event

T_{1} \cap T_{2} \cap T_{3} \cap T_{4} i s

A	$\frac{1}{15}$	B	$\frac{1}{10}$
C	$\frac{7}{60}$	D	$\frac{1}{5}$

⮜ Back Next ⮞

Not Visited	0	Not Answered
Answered	0	Marked for Review
Answered & Marked for Review (will be considered for evaluation)

Question 1:

The potential energy of a particle of mass 𝑚 at a distance 𝑟 from a fixed point 𝑂 is given by

v (r) = k r^{2} / 2,

A	$v = \sqrt{\frac{k}{2 m}} R$	B	$V = \sqrt{\frac{K}{m}} R$
C	$L = \sqrt{m k} R^{2}$	D	$L = \sqrt{\frac{m k}{2}} R^{2}$

Correct Answer : B and C

Question 2:

Consider a body of mass 1.0 𝑘𝑔 at rest at the origin at time 𝑡=0. A force

\vec{F}

= (α t \hat{I} + \hat{j})

is applied on the body, where

α = 1.0 N s^{- 1} a n d β

=1.0 𝑁. The torque acting on the body about the origin at time 𝑡=

= 1, 0 s i s \vec{τ}

. Which of the following statements is (are) true?

A	$\|\vec{τ}\| = \frac{1}{3} N m$	B	$T h e t o r q u e \vec{τ} i s i n t h e d i r e c t i o n o f t h e u n i t v e c t o r + \hat{k}$
C	The Velocity of the body at t =1 s is $\vec{v} = 12 (\hat{i} + 2 \hat{j}) m s^{- 1}$	D	The magnitude of displacement of the body at 𝑡=1 𝑠 is $\frac{1}{6} m$

Correct Answer : C and A

Question 3:

A	For a given material of the capillary tube, ℎ decreases with increase in 𝑟	B	For a given material of the capillary tube, ℎ is independent of 𝜎
C	If this experiment is performed in a lift going up with a constant acceleration, then ℎ decreases	D	ℎ is proportional to contact angle 𝜃

Correct Answer : A and C

Question 4:

In the figure below, the switches

s_{1}

and

s_{2}

A	$I_{m a x} = \frac{v}{2 R}$	B	$I_{m a x} = \frac{v}{4 R}$
C	$τ = \frac{L}{R} I n 2$	D	$τ = \frac{2 L}{R} I n 2$

Correct Answer : D and B

Question 5:

Two infinitely long straight wires lie in the 𝑥𝑦-plane along the lines 𝑥=±𝑅. The wire located at 𝑥=+𝑅 carries a constant current

I_{1}

and the wire located at 𝑥=−𝑅 carries a constant current

I_{2}

. A circular loop of radius 𝑅 is suspended with its centre at

(0, 0, \sqrt{3 R})

A	If cannot be equal to zero at the origin (0,0,0)	B	If $I_{1} > 0 a n d I_{2} < 0, \vec{B}$ can be equal to zero at the origin (0,0,0)
C	If $I_{1} < 0 a n d I_{2} > 0, \vec{B}$ can be equal to zero at the origin (0,0,0)	D	If $I_{1} = I_{2}$ then the 𝑧-component of the magnetic field at the centre of the loop is $(- \frac{μ_{o} 1}{2 R})$

Correct Answer : A,B and D

Question 6:

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?

A	Process I is an isochoric process	B	In process II, gas absorbs heat
C	In process IV, gas releases heat	D	Processes I and III are not isobaric

Correct Answer : B,C and D

Question 7:

Two vectors

\vec{A}

and

\vec{B}

are defined as

\vec{A}

=𝑎 𝑖̂ and

\vec{B}

=𝑎 (cos𝜔𝑡𝑖̂+sin𝜔𝑡𝑗̂), where 𝑎 is a constant and 𝜔=𝜋/6 𝑟𝑎𝑑

S^{- 1}

. If |

|\vec{A} + \vec{B}|

= \sqrt{3} |\vec{A} + \vec{B}|

time 𝑡=𝜏 for the first time, the value of 𝜏, in 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, is __________.

Correct Answer : 2.00

Question 8:

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 𝑚

s^{- 1}

and the man behind walks at a speed 2.0 𝑚

s^{- 1}

s^{- 1}

. At the instant, when the moving men are 10 𝑚 apart, the stationary man is equidistant from them. The frequency of beats in 𝐻𝑧, heard by the stationary man at this instant, is __________.

Correct Answer : 5.00

Question 9:

(2 - \sqrt{3}) / \sqrt{10} s,

, then the height of the top of the inclined plane, in 𝑚𝑒𝑡𝑟𝑒𝑠, is __________. Take

g = 10 m s^{- 2}

Correct Answer : 0.75

Question 10:

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 𝑁

m^{- 1}

and the mass of the block is 2.0 𝑘𝑔. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 𝑘𝑔 moving with a speed of 2.0

m s^{- 1}

Correct Answer : 2.09

Question 11:

Three identical capacitors

c_{1}

c_{2}

and

c_{3}

s_{1}

is closed while the switch

s_{2}

is kept open. When the capacitor

c_{3}

is fully charged,

s_{1}

is opened and

s_{2}

is closed simultaneously. When all the capacitors reach equilibrium, the charge on

c_{3}

is found to be 5 𝜇𝐶. The value of

ε_{r}

=____________.

Correct Answer : 1.50

Question 12:

In the 𝑥𝑦-plane, the region 𝑦>0 has a uniform magnetic field

B_{1} \hat{K}

and the region 𝑦<0 has another uniform magnetic field

B_{2} \hat{K}

. A positively charged particle is projected from the origin along the positive 𝑦-axis with speed 𝑣0=𝜋 𝑚

S^{- 1}

at 𝑡=0, as shown in the figure. Neglect gravity in this problem. Let 𝑡=𝑇 be the time when the particle crosses the 𝑥-axis from below for the first time. If

B_{2}

B_{1}

, the average speed of the particle, in 𝑚

S^{- 1}

, along the 𝑥-axis in the time interval 𝑇 is __________.

Correct Answer : 2.00

Question 13:

Sunlight of intensity 1.3 𝑘𝑊

m^{- 2}

m^{- 2}

, at a distance 22 𝑐𝑚 from the lens on the other side is __________.

Correct Answer : 130.00

Question 14:

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures

T_{1}

=300 𝐾 and

T_{2}

=100 𝐾, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are

K_{1}

and

K_{2}

respectively. If the temperature at the junction of the two cylinders in the steady state is 200 𝐾, then

K_{1} / K_{2}

=__________.

Correct Answer : 4.00

Question 15:

[\in_{0}]

and

[μ_{0}]

A	[E] =[B] [L][T]	B	[E]= $[B] {[L]}^{- 1} [T]$
C	$[E] = [B] [L] {[T]}^{- 1}$	D	$[E] = [B] {[L]}^{- 1} {[T]}^{- 1}$

Correct Answer : C

Question 16:

z \pm Δ z = \frac{x + Δ x}{y + Δ y} = \frac{x}{y} (1 \pm \frac{Δ x}{x}) {(1 \pm \frac{Δ y}{y})}^{- 1}

The series expansion for

{(1 \pm \frac{Δ y}{y})}^{- 1}

to first power inΔ𝑦/𝑦, is 1∓(Δ𝑦/𝑦). The relative errors in independent variables are always added. So the error in 𝑧 will be

Δ z = z (\frac{Δ x}{x}) + (\frac{Δ y}{y}) .

The above derivation makes the assumption that Δ𝑥/𝑥≪1, Δ𝑦/𝑦≪1. Therefore, the higher powers of these quantities are neglected. Consider the ratio 𝑟=

\frac{(1 - a)}{(1 + a)}

to be determined by measuring a dimensionless quantity 𝑎. If the error in the measurement of 𝑎 is Δ𝑎 (Δ𝑎/𝑎≪1) , then what is the error Δ𝑟 in determining 𝑟?

A	$\frac{∆ a}{{(1 + a)}^{2}}$	B	$\frac{2 ∆ a}{{(1 + a)}^{2}}$
C	$\frac{∆ a}{(1 - a^{2})}$	D	$\frac{2 a ∆ a}{(1 - a^{2})}$

Correct Answer : B

Question 17:

z \pm Δ z = \frac{x + Δ x}{y + Δ y} = \frac{x}{y} (1 \pm \frac{Δ x}{x}) {(1 \pm \frac{Δ y}{y})}^{- 1}

The series expansion for

{(1 \pm \frac{Δ y}{y})}^{- 1}

to first power inΔ𝑦/𝑦, is 1∓(Δ𝑦/𝑦). The relative errors in independent variables are always added. So the error in 𝑧 will be

Δ z = z (\frac{Δ x}{x}) + (\frac{Δ y}{y}) .

The above derivation makes the assumption that Δ𝑥/𝑥≪1, Δ𝑦/𝑦≪1. Therefore, the higher powers of these quantities are neglected. Consider the ratio 𝑟=

r = \frac{(1 - a)}{(1 + a)}

A	0.04	B	0.03
C	0.02	D	0.01

Correct Answer : C

Question 18:

[\in_{0}]

and

[μ_{0}]

A	$[μ_{0}] = [\in_{0}] {[L]}^{2} {[T]}^{- 2}$	B	$[μ_{0}] = [\in_{0}] {[L]}^{- 2} {[T]}^{2}$
C	$[μ_{0}] = {[\in_{0}]}^{- 1} {[L]}^{2} {[T]}^{- 2}$	D	$[μ_{0}] = {[\in_{0}]}^{- 1} {[L]}^{- 2} {[T]}^{2}$

Correct Answer : D

Question 19:

The compound(s) which generate(s)

N_{2}

gas upon thermal decomposition below 300C is (are)

A	$N H_{4} N O_{3}$	B	${(N H_{4})}_{2} C r_{2} O_{7}$
C	$B a {(N_{3})}^{2}$	D	$M g_{3} N_{2}$

Correct Answer : B and C

Question 20:

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28)

A	Total number of valence shell electrons at metal centre in Fe(CO)5 or $N i {(C O)}_{4}$ is 16	B	These are predominantly low spin in nature
C	Metal–carbon bond strengthens when the oxidation state of the metal is lowered	D	The carbonyl C−O bond weakens when the oxidation state of the metal is increased

Correct Answer : B and C

Question 21:

Based on the compounds of group 15 elements, the correct statement(s) is (are)

A	$B i_{2} O_{5} i s m o r e b a s i c t h a n N_{2} O_{5}$	B	$N F_{3} i s m o r e c o v a l e n t t h a n B i F_{3}$
C	$P H_{3} b o i l s a t l o w e r t e m p e r a t u r e t h a n N H_{3}$	D	The N−N single bond is stronger than the P−P single bond

Correct Answer : A,B and C

Question 22:

I n the following reaction sequence, the correct structure(s) of X is (are)

\to_{\frac{2) N a l, M e_{2} C O}{3) N a N_{3} H C O M e_{2}}}^{1) P B r_{3}, E t_{2} O}

Correct Answer : B

Question 23:

The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

Correct Answer : A,B and D

Question 24:

The correct option(s) is (are)

A	$q_{A C} = Δ U_{B C} a n d w_{A B} = P_{2} (V_{2} - V_{1})$	B	$w_{B C} = P_{2} (V_{2} - V_{1}) a n d q_{B C} = Δ H_{A C}$
C	$Δ H_{C A} < Δ U_{C A} a n d q_{A C} = Δ U_{B C}$	D	$q_{B C} = Δ {��}_{A C} a n d Δ H_{C A} > Δ U_{C A}$

Correct Answer : B and C

Question 25:

Among the species given below, the total number of diamagnetic species is ___. H atom,

N O_{2}

monomer,

O_{2}^{-}

(superoxide), dimeric sulphur in vapour phase,

M n_{3} O_{4}, {(N H_{4})}_{2} [F e C I]

₄,(NH4)2NiCI₄,K2MnO4K2CrO4

Correct Answer : 1

Question 26:

The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by

N i C I_{2} 6 H_{2} O

m o l^{- 1}

H = 1, N = 14, O = 16, S = 32, C I = 35.5

C a = 40, N i = 59)

Correct Answer : 2992

Question 27:

(\frac{n u m b e r s o f a n i o n s}{n u m b e r o f c a t i o n s}) i n Z i s

____.

Correct Answer : 3

Question 28:

F or the electrochemical cell,

M g (s) M g^{2 +} (a q, 1 M) ∥ C u^{2 +} (a q, 1 M) C u (s)

the standard emf of the cell is 2.70 V at 300 K. When the concentration of

M g^{2 +} i s

is changed to 𝒙 M, the cell potential changes to 2.67 V at 300 K. The value of 𝒙 is ____. (given

\frac{F}{R}

11500 K

V^{- 1}

, where 𝐹 is the Faraday constant and 𝑅 is the gas constant, ln(10) = 2.30)

Correct Answer : 10

Question 29:

(i n m^{3})

of the compartment A after the system attains equilibrium is ____.

Correct Answer : 2.22

Question 30:

x_{A} a n d x_{B}

, respectively, has vapour pressure of 22.5 Torr. The value of

x_{A} / x_{B}

in the new solution is ____. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

Correct Answer : 19

Question 31:

The solubility of a salt of weak acid (AB) at pH 3 is

Y x 10^{- 3}

m o l^{- 1}

.The value of Y is ____. (Given that the value of solubility product of AB

(k_{s p})

2 x 10^{- 10}

and the value of ionization constant of HB

(k_{a}) = 1 x 10^{- 8})

Correct Answer : 4.47

Question 32:

Correct Answer : 0.05

Question 33:

Treatment of benzene with CO/HCl in the presence of anhydrous

A I C I_{3} / C U C I

followed by reaction with

A c_{2} O

NaOAc gives compound X as the major product.Compound X upon reaction with

B r_{2} / N a_{2} / C O_{3}

followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by

H_{3} P O_{4}

treatment gives Z as the major product. T he compound Y is

Correct Answer : C

Question 34:

Treatment of benzene with CO/HCl in the presence of anhydrous

A I C I_{3} / C U C I

followed by reaction with

A c_{2} O

NaOAc gives compound X as the major product.Compound X upon reaction with

B r_{2} / N a_{2} / C O_{3}

followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H2/Pd-C, followed by

H_{3} P O_{4}

treatment gives Z as the major product. T he compound Z is

Correct Answer : A

Question 35:

An organic acid P

(C_{11} H_{12} O_{2})

S \leftarrow_{\frac{C H C I_{3} K O H, ∆}{H_{2} / P d - c}}^{\frac{\frac{1) H_{2} / P d - c}{2) N H_{3} / ∆}}{3) B r_{2} N a O H}}

P \to_{\frac{3) M e M g B r, C d C I_{2}}{N a B H_{2}}}^{\frac{1) H_{2} / P d - c}{2) S O C I_{2}}}

\to_{\frac{3) C O_{2} (d e y i c e)}{H_{3} O^{+}}}^{\frac{1) H C I}{2) M g / E t_{2 O}}}

T he compound R is

Correct Answer : A

Question 36:

An organic acid P

(C_{11} H_{12} O_{2})

S \leftarrow_{\frac{C H C I_{3} K O H, ∆}{H_{2} / P d - c}}^{\frac{\frac{1) H_{2} / P d - c}{2) N H_{3} / ∆}}{3) B r_{2} N a O H}}

P \to_{\frac{3) M e M g B r, C d C I_{2}}{N a B H_{2}}}^{\frac{1) H_{2} / P d - c}{2) S O C I_{2}}}

\to_{\frac{3) C O_{2} (d e y i c e)}{H_{3} O^{+}}}^{\frac{1) H C I}{2) M g / E t_{2 O}}}

T he compound S is

Correct Answer : B

Question 37:

For a non-zero complex number 𝑧, let arg(𝑧) denote the principal argument with

- π < \arg (z) \leq π

Then, which of the following statement(s) is (are) FALSE?

A	$a r g (- 1 - i) = \frac{π}{4} =, w h e r e i = \sqrt{1}$	B	The function 𝑓: ℝ $(- π, π] defined by f (t) = \arg (- 1 + it)$ for all 𝑡∈ℝ, is continuous at all points of ℝ, where $i = \sqrt{- 1}$
C	For any two non-zero complex numbers $Z_{1} a n d Z_{2}$ $a r g (\frac{Z_{1}}{Z_{2}}) - a r g (Z_{1}) + a r g (Z_{2})$ is an integer multiple of 2𝜋	D	For any three given distinct complex numbers $Z_{1}, Z_{2} a n d Z_{3}$ the locus of the point 𝑧 satisfying the condition $a r g (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})})$ = 𝜋, lies on a straight line

Correct Answer : A,B and D

Question 38:

In a triangle 𝑃𝑄𝑅, let

< P Q R = 30 °

and the sides 𝑃𝑄 and 𝑄𝑅 have lengths and 10

\sqrt{3}

, respectively. Then, which of the following statement(s) is (are) TRUE?

A	$< Q P R = 45 °$	B	The area of the triangle 𝑃𝑄𝑅 is 25 $\sqrt{3}$ and $< Q R P = 120 °$
C	The radius of the incircle of the triangle 𝑃𝑄𝑅 is $10 \sqrt{3} - 15$	D	The area of the circumcircle of the triangle $P Q R i s 100 π$

Correct Answer : B,C and D

Question 39:

Let

P_{1}

: 2𝑥+𝑦−𝑧=3 and

P_{2}

: 𝑥+2𝑦+𝑧=2 be two planes. Then, which of the following statement(s) is (are) TRUE?A

A	The line of intersection of $p_{1} a n d p_{2}$ has direction ratios 1,2,−1.	B	The line $\frac{3 x - 4}{9} = \frac{1 - 3 y}{9} = \frac{z}{3}$
C	The acute angle between $p_{1} a n d p_{2} i s 60 °$	D	If $p_{3}$ is the plane passing through the point (4,2,−2) and perpendicular to the line of intersection of $p_{1} a n d p_{2}$ , then the distance of the point (2,1,1) from the plane $p_{3} i s \frac{2}{\sqrt{3}}$

Correct Answer : C and D

Question 40:

For every twice differentiable function 𝑓: ℝ→[−2,2] with

{(f (0))}^{2} + {(f' (0))}^{2}

=85, which of the following statement(s) is (are) TRUE?

A	There exist 𝑟,𝑠∈ℝ, where 𝑟<𝑠, such that 𝑓 is one-one on the open interval (𝑟,𝑠)	B	There exists 𝑥0∈ $x_{0} \in (- 4, 0)$ such that $f' (o x) \leq 1$
C	$\lim_{x \to \infty} f (x) = 1$	D	There exists 𝛼∈(−4,4) such that 𝑓(𝛼)+𝑓′′(𝛼) = 0 and 𝑓′(𝛼)≠0

Correct Answer : A,B and D

Question 41:

Let 𝑓:ℝ→ℝ and 𝑔:ℝ→ℝ be two non-constant differentiable functions. If

f' (x) = (e^{(f (x) - g (x))}) g' (x)

for all 𝑥∈ℝ, and 𝑓(1)=𝑔(2)=1, then which of the following statement(s) is (are) TRUE?

A	$f (2) < 1 - \log_{e} 2$	B	$f (2) > 1 - \log_{e} 2$
C	$g (1) > 1 - \log_{e} 2$	D	$g (1) < 1 - \log_{e} 2$

Correct Answer : B and C

Question 42:

Let 𝑓:[0,∞)→ℝ be a continuous function such that

f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) d t

for all 𝑥∈[0,∞). Then, which of the following statement(s) is (are) TRUE?

A	for all 𝑥∈[0,∞). Then, which of the following statement(s) is (are) TRUE?	B	The curve 𝑦=𝑓(𝑥) passes through the point (2,−1)
C	The area of the region ${(x, y) \in [0, 1] x ℝ : f (x) \leq y \leq \sqrt{1 - x^{2}}} i s \frac{π - 2}{4}$	D	The area of the region ${(x, y) \in [0, 1] x ℝ : f (x) \leq y \leq \sqrt{1 - x^{2}}} i s \frac{π - 1}{4}$

Correct Answer : C and B

Question 43:

The value of

({(\log_{2} 9)}^{2}) \frac{1}{\log_{2} (\log_{2} 9) x (\sqrt{7})} \frac{1}{\log_{4} 7}

is ______ .

Correct Answer : 8

Question 44:

The number of 5 digit numbers which are divisible by 4, with digits from the set {1,2,3,4,5} and the repetition of digits is allowed, is _____ .

Correct Answer : 625

Question 45:

Correct Answer : 3748

Question 46:

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} i_{x} + 1_{- x} \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

(- \frac{1}{2}, \frac{1}{2})

is _____ . (Here, the inverse trigonometric functions

\sin^{- 1}

𝑥 and

\cos^{- 1}

𝑥 assume values in

[- \frac{π}{r}, \frac{π}{r}]

and [0,𝜋], respectively.)

Correct Answer : 2

Question 47:

For each positive integer 𝑛, let

y_{n} = \frac{1}{n} {((n + 1) (n + 2) . . . (n + n))}^{\frac{1}{2}}

For 𝑥∈ℝ, let [𝑥] be the greatest integer less than or equal to 𝑥.if

\lim_{n \to \infty} y_{n} =

𝐿, then the value of [𝐿] is _____ .

Correct Answer : 1

Question 48:

let

\vec{a} a n d \vec{b}

be two unit vectors such that

\vec{a} . \vec{b}

For some 𝑥,𝑦∈ℝ, let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} + \vec{b)}

.If

|\vec{c}|

⃗|=2 and the vector

\vec{c}

is inclined at the same angle 𝛼 to both

\vec{a} a n d \vec{b}

then the value of

\cos^{2}

𝛼 is _____.

Correct Answer : 3

Question 49:

Let 𝑎,𝑏,𝑐 be three non-zero real numbers such that the equation

\sqrt{3}

𝑎cos𝑥+2 𝑏sin𝑥=𝑐, 𝑥

\in [- \frac{π}{2}, \frac{π}{2}],

has two distinct real roots

α a n d β

with 𝛼+𝛽=

\frac{π}{3}

Then, the value of

\frac{b}{a} i s .

_____ .

Correct Answer : 0.5

Question 50:

A farmer

F_{1}

has a land in the shape of a triangle with vertices at 𝑃(0,0),𝑄(1,1) and 𝑅(2,0).From this land, a neighbouring farmer

F_{2}

takes away the region which lies between the side 𝑃𝑄 and a curve of the form

y = x^{n} (n > 1)

If the area of the region taken away by the farmer

F_{2}

is exactly 30% of the area of Δ𝑃𝑄𝑅, then the value of 𝑛 is _____ .

Correct Answer : 4

Question 51:

Let 𝑆 be the circle in the 𝑥𝑦-plane defined by the equation

x^{2} + y^{2} = 4

.Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of 𝑆 passing through the point

P_{0}

(1,1) and parallel to the x-axis and the y-axis, respectively. Let

G_{1} G_{2}

be the chord of S passing through

P_{0}

and having slope −1.Let the tangents to 𝑆 at

E_{1} a n d E_{2}

meet at

E_{3}

the tangents to 𝑆 at

F_{1} a n d F_{2}

meet at

F_{3}

and the tangents to 𝑆 at

G_{1} a n d G_{2} m e e t a t G_{3}

Then, the points

E_{3} F_{3}

and

G_{3}

.lie on the curve

A	𝑥+𝑦=4	B	( $�� - 4)^{2} + {(�� - 4)}^{2} = 16$
C	(𝑥−4)(𝑦−4)=4	D	𝑥𝑦=4

Correct Answer : A

Question 52:

Let 𝑆 be the circle in the 𝑥𝑦-plane defined by the equation

x^{2} + y^{2} = 4

A	${(�� + ��)}^{2} = 3 �� $	B	$x^{2 / 3} + y^{2 / 3} = 2^{4 / 3}$
C	$x^{2} + y^{2} = 2 �� $	D	$x^{2} + y^{2} = x^{2} y^{2}$

Correct Answer : D

Question 53:

There are five students

S_{1} S_{2} S_{3} S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1} R_{2} R_{3} R_{4} a n d R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

𝑖=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student

s_{1}

gets the previously allotted seat

R_{1}

, and NONE of the remaining students gets the seat previously allotted to him/her is.

A	$\frac{3}{40}$	B	$\frac{1}{8}$
C	$\frac{7}{40}$	D	$\frac{1}{5}$

Correct Answer : A

Question 54:

There are five students

S_{1} S_{2} S_{3} S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1} R_{2} R_{3} R_{4} a n d R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

𝑖=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. For 𝑖=1,2,3,4, let 𝑇𝑖 denote the event that the students

S_{1} a n d s_{i + 1}

do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event

T_{1} \cap T_{2} \cap T_{3} \cap T_{4} i s

A	$\frac{1}{15}$	B	$\frac{1}{10}$
C	$\frac{7}{60}$	D	$\frac{1}{5}$

Correct Answer : C

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