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QUESTION: 1

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 TOA 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_{AO} to reach point O. Which of the following options is/are correct ?.

Solution:

(A) Speed of wave is property of medium so time taken to cross the string will be equal

(B) Speeds are same but velocity is vector, has opposite directions

(D) Velocity of any pulse is and it is property of medium.

*Multiple options can be correct

QUESTION: 2

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

Solution:

Assumption : e = 1 (Black body radiation)

(C) Surface area decrease ⇒ Energy radiation decreases

QUESTION: 3

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 ?

Solution:

*Multiple options can be correct

QUESTION: 4

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 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?

Solution:

Net emf will be difference of emfs in both loops because their polarities are opposite.

*Multiple options can be correct

QUESTION: 5

For an isosceles prism of angle A and refractive index µ, it is found that the angle of minimum deviation . Which of the following options is/are correct ?

Solution:

*Multiple options can be correct

QUESTION: 6

In the circuit shown, L = 1 µH, C = 1 µF and R = 1 kΩ. They are connected in series with an a.c. source V = V_{0} sin ωt as shown. Which of the following options is/are correct ?

Solution:

*Multiple options can be correct

QUESTION: 7

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 ?

Solution:

QUESTION: 8

A drop of liquid of radius R = 10^{–2} m having surface tension divides itself into K identical drops. In this process the total change in the surface energy then the value of α is

Solution:

QUESTION: 9

^{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

Solution:

120 Bq is the activity of 2.5 ml

∴ 2.4 * 10^{5 }Bq is the activity of

∴ Total volume of blood = 5 litres

*Answer can only contain numeric values

QUESTION: 10

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. then the smallest possible n_{f }is.

Solution:

*Answer can only contain numeric values

QUESTION: 11

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°. 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^{th} slab and Δn = 0.1 (see the figure). The ray is refracted out parallel to the interface between the (m - 1)^{th} and m^{th} slabs from the right side of the stack. What is the value of m ?

Solution:

Applying snell's law between entry & exit surfaces, n sin θ = µ sin (π/2)

*Answer can only contain numeric values

QUESTION: 12

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 ms^{–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 ms^{–1} and the car reflects the sound at the frequency it has received).

Solution:

Frequency of sound as received by large car approaching the source.

This car now acts as source for reflected sound wave

frequency of sound received by source,

= 6 Hz

QUESTION: 13

(Direction) Q.13, Q.14 and Q.15 by appropriately matching the information given in the three columns of the following table.

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity . A uniform electric field and a uniform magnetic field exist everywhere. The velocity , electric field and magnetic field are given in column 1, 2 and 3, respectively. The quantities E_{0}, B_{0} are positive in magnitude.

**Column-1 ** **Column-2 ** **Column-3**

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

Solution:

For particle to move in straight line with constant velocity,

QUESTION: 14

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity . A uniform electric field and a uniform magnetic field exist everywhere. The velocity , electric field and magnetic field are given in column 1, 2 and 3, respectively. The quantities E_{0}, B_{0} are positive in magnitude.

**Column-1 ** **Column-2 ** **Column-3**

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

Solution:

For path to be helix with axis along +ve z-direction, particle should experience a centripetal acceleration in x-y plane.

For the given set of options only option (C) satisfy the condition. Path is helical with increasing pitch.

QUESTION: 15

A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity . A uniform electric field and a uniform magnetic field exist everywhere. The velocity , electric field and magnetic field are given in column 1, 2 and 3, respectively. The quantities E_{0}, B_{0} are positive in magnitude.

**Column-1 ** **Column-2 ** **Column-3**

**Q. In which case would the particle move in a straight line along the negative direction of y-axis (i.e., move along - ) ?**

Solution:

For particle to move in -ve y-direction, either its velocity must be in –ve y-direction (if initial velocity 0) & force should be parallel to velocity or it must experience a net force in –ve y-direction only (if initial velocity = 0)

QUESTION: 16

(Direction) Q.16, Q.17 and Q.18 by appropriately matching the information given in the three columns of the following table.

An ideal gas is undergoing a cyclic thermodynamics 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 thermodynamics processes. Here g is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

**Q. Which of the following options is the only correct representation of a process in which ΔU = ΔQ - PΔV?**

Solution:

Work (Column-I), process (Column-II) & corresponding graph (Column-III) are in this sequence.

Only "B" option follow the sequence.

QUESTION: 17

An ideal gas is undergoing a cyclic thermodynamics 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 thermodynamics processes. Here g is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

**Q. Which one of the following options is the correct combination ?**

Solution:

Only option "A" follow the sequence.

QUESTION: 18

An ideal gas is undergoing a cyclic thermodynamics 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 thermodynamics processes. Here g is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

**Q. Which one of the following options correctly represents a thermodynamics process that is used as a correction in the determination of the speed of sound in an ideals gas ?**

Solution:

It is for an adiabatic process.

*Multiple options can be correct

QUESTION: 19

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

Solution:

IUPAC Name- "Toluene" is accepted by IUPAC as a name of parent carbon chain.

So it can also be named as 4-chlorotoluene.

*Multiple options can be correct

QUESTION: 20

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

Solution:

(ii)

(O) and (P) are enantiomers

Explanation of 4 options : (A) (M) and (O) are distereomers of each other.

(N) and (P) are distereomers of each other.

(B) Addition of Br_{2} on alkene follows non-classical carbocation mechanism. It is anti or trans addition.

(C) (O) and (P) are enantiomers

(D) (M) and (N) are identical and (O) and (P) are enantiomers.

(M and O) are distereomers and (N and P) are distereomers.

*Multiple options can be correct

QUESTION: 21

Addition of excess aqueous ammonia to a pink coloured aqueous solution of MCl_{2} . 6H_{2}O (X) and NH_{4}Cl 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.

**Q. Among the following options, which statment is (are) ?**

Solution:

(A) Hyridisation of (Y) is d^{2}sp^{3} as NH_{3} is strong field ligand

(B) [COCl_{4}]^{2-} have sp^{3} hybridisation as Cl- is weak field ligand

When ice is added to the solution the equilibrium shifts right hence pink colour will remain predominant So, correct answer is (A,B& D)

*Multiple options can be correct

QUESTION: 22

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

Solution:

*Multiple options can be correct

QUESTION: 23

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)

Solution:

*Multiple options can be correct

QUESTION: 24

The correct statements(s) about the oxoacids, HClO_{4} and HClO, is (are) -

Solution:

*Multiple options can be correct

QUESTION: 25

The colour of the X_{2} molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to -

Solution:

Halogens are coloured due to HOMO-LUMO transition of electrons.

On moving down the group HOMO-LUMO energy gap decreases so transition of electrons become easier therefore colour intensify.

QUESTION: 26

Among H_{2}, He^{+}_{2}, Li_{2}, Be_{2}, B_{2}, C_{2}, N_{2}, O^{-}_{2}, the number of diamagnetic species is -( Atomic number) : H =1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7,I = 8, F = 9)

Solution:

If existence of Be_{2} is considered in atomic form or very weak bonded higher energetic species having zero bond order then it is diamagnetic , then answer will be 6. But if existence of molecular form of Be_{2} is not considered then magnetic property can't be predicted then answer will be 5

QUESTION: 27

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

Solution:

Cyclooctatetraene ; non aromatic

Due to nonplanarity of ring the π-electrons are not delocalised.

Cyclopropcnyl anion : Anti aromatic 4π-electrons delocalised.

Cyclopropenyl cation ; Aromatic 2π-clcctrons delocalised.

Cyclohexadiene : Non-aromatic.

Tropylium ion : Arom atic 6π-electrons delocalised.

Cyclo pentadienyl cation . Anti-aromatic 4π-electrons delocalised.

Cyclo pentadienyl anion ; Aromatic 6π-electrons delocalised.

QUESTION: 28

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 cm^{2}. The conductance of this solution was found to be 5 × 10^{–7}S. The pH of the solution is 4. The value of limiting molar conductivity of this weak monobasic acid in aquence solution is

Solution:

For weak acid [H^{+}]

QUESTION: 29

The metal used to recover copper from a solution of copper sulphate is

Solution:

QUESTION: 30

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 8g cm^{–3}, then the number of atoms present in 256g of the crystal is N × 10^{24}. The value of N is

Solution:

Formula of density =

For FCC unit cell Z = 4

Edge length a = 4 * 10^{-8} cm

QUESTION: 31

(Direction) Q.31, Q.32 and Q.33 by appropriately matching the information given in the three columns of the following table.

The wave function 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_{1}. 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 a_{0} is Bohr radius.

**Q. For the given orbital in column 1, the only CORRECT combination for any hydrogen - like species is :**

Solution:

(A) (IV) (iv) (R) ⇒ incorrect, because, ^{d}z^{2} has no nodal plane.

(B) (II) (ii) (P) ⇒ correct, because 2s orbtial has 1 radial node.

(C) (III) (iii) (P) ⇒ incorrect, because probability density for 2p at nucleus is zero.

(D) (I) (ii) (S) ⇒ incorrect, because 1s orbital has no radial node.

QUESTION: 32

The wave function 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_{1}. 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 a_{0} is Bohr radius.

**Q. For He+ ion, the only INCORRECT combination is **

Solution:

The option (D) is incorrect because in the wave function of 1s orbital , no angular function should be present.

QUESTION: 33

The wave function 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_{1}. 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 a_{0} is Bohr radius.

**Q. For hydrogen atom, the only CORRECT combination is**

Solution:

We have to select only correct combination hence, the option (D) is correct.

QUESTION: 34

**Q.**

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

Solution:

(D). (II)(i)(S)

(A)

(B)

(C)

(D)

QUESTION: 35

**Q.**

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

Solution:

Ans. [A](I)(ii)(R)

mechanism involved is free radical substitution

(B)

(C)

(D)

QUESTION: 36

**Q.**

The only CORRECT combination that gives two different carboxylic acids is

Solution:

**Ans. (B**)

*Multiple options can be correct

QUESTION: 37

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

Solution:

Ans. (A,B)

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 ?

Solution:

Ans. (D)

Equation of chord with mid point (h, k) :

⇒ 8x – ky + k^{2} – 8h = 0

Comparing with 2x + y – p = 0, we get k = –4; 2h – p = 4 only (D) satisfies above relation.

*Multiple options can be correct

QUESTION: 39

Let a, b, x and y be r eal n um ber s su ch th at a – b = 1 and y ¹ 0 . If t he co mp lex nu mb er z = x + iy satisfies

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

Solution:

**Ans. (A,D)**

*Multiple options can be correct

QUESTION: 40

Let X and Y be two events such that and Then

Solution:

**Ans. (A,D)**

**from this information, we get**

*Multiple options can be correct

QUESTION: 41

Let [x] be the greatest integer less than or equal to x. Then, at which of the following point(s) the function ƒ(x) = xcos(π(x + [x])) is discontinuous ?

Solution:

**Ans. (A,C,D)**

Discontinuous at all integers except zero.

*Multiple options can be correct

QUESTION: 42

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

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

Solution:

**Ans. (B,C,D)**

The line y= mx + c is tangent to the hyperbola

c^{2 }= a^{2}m^{2 }- b^{2}

*Multiple options can be correct

QUESTION: 43

Let ƒ : 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)?

Solution:

**Ans. (B,D)**

For option (A),

⇒ g(x) is strictly incrasing function.

Also, g(0) = 1

∴ option (A) is not possible.

For option (B), let

⇒ k(0). k(1) < 0

So, option(B) is correct.

For option (C), let

so, option(C) is not possible.

For option (D),

⇒ M(0). M(1) < 0

∴ option (D) is correct.

QUESTION: 44

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

Solution:

**Ans. 6**

**where d > 0, a > 0
⇒ length of smallest side = a - d
Now (a + d)**

**∴ a = 4d ...(1)
(As a = 0 is rejected)**

Also,

**⇒ ** a(a – d) = 48 ...(2)

**∴ **From (1) and (2),

we get a = 8, d = 2

Hence, length of smallest side

**⇒** (a - d) = (8 - 2) = 6

option d

QUESTION: 45

For how many values of p, the circle x^{2} + y^{2} + 2x + 4y – p = 0 and the coordinate axes have exactly three common points ?

Solution:

Ans. 2

We shall consider 3 cases.

Case I : When p = 0 (i.e. circle passes through origin) Now, equation of circle becomes x^{2} + y^{2} + 2x + 4y = 0

Case II : When circle intersects x-axis at 2 distinct points and touches y-axis

Now (g^{2} – c) > 0 & ƒ^{2} – c = 0

⇒ 1 - (- p) > 0 & 4 - (- p) = 0

⇒ p = - 4 ⇒ p > -1

∴ Not possible.

Case III : When circle intersects y-axis at 2 distinct points & touches x-axis.

Now, g^{2} – c = 0 & ƒ^{2 }– c > 0

⇒ 1 – (–p) = 0 & 4 – (–p) > 0

⇒ p = –1 ⇒ p > –4

∴ p = – 1 is possible.

∴ Finally we conclude that p = 0, –1

⇒ Two possible values of p.

Option 3

QUESTION: 46

For a real number a, if the system

of linear equations, has infinitely many solutions, then 1 + α + α^{2} =

Solution:

**αns. 1**

Δ = 0 ⇒ 1(1 - α^{2}) - α(α - α^{3}) + α^{2}(α^{2} - α^{2}) = 0

(1 – α^{2}) - α^{2} + α^{4} = 0

(α^{2} - 1)^{2} = 0 ⇒ α = ±1

but αt α = 1 No solution so rejected αt α = -1 αll three equαtion become x - y + z = 1 (coincident plαnes)

∴1 + α + α^{2} = 1

Hence Option. b

QUESTION: 47

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, y/9x =

Solution:

**Ans. 5**

**x = 10!**

Hence option. c

QUESTION: 48

Let f : R → R be a differentiable function such that f(0) = 0, and f'(0) = 1. If

Solution:

**Ans. 2**

Hence Option. c

QUESTION: 49

**Q.**

The tangent to a suitable conic (Column 1) at is found to be then which of the following options is the only CORRECT combination ?

Solution:

**Ans. (D)**

QUESTION: 50

**Q.**

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 ?

Solution:

**Ans. (A)**

**Sol.** y = x + 8 is tangent ⇒ m = 1; P(8, 16)

Comparing tangent with (i) of column 2, m = 1 satisfied and a = 8 obtained which matches for point of contact (P) of column 3 and (III) of column I.

QUESTION: 51

**Q.**

For a=√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 ?

Solution:

**Ans. (D)**

For a=√2 and point (-1,1) only I of column -1 satisfies. Hence equaiton of tangent is - x + y = 2 or y = x + 2

⇒ m = 1 which matches with (ii) of column 2 and also with Q of column 3

QUESTION: 52

**Q.**

Which of the following options is the only CORRECT combination ?

Solution:

**Ans. (D)**

(I) ƒ(1) ƒ(e^{2}) < 0 so true

(II) ƒ'(1) ƒ'(e) < 0 so true

(III) Graph of ƒ'(x) so (III) is false

(IV) Is false

**Alternate :**

QUESTION: 53

**Q. **Which of the following options is the only CORRECT combination ?

Solution:

**Ans. (D)**

(I) ƒ(1) ƒ(e^{2}) < 0 so true

(II) ƒ'(1) ƒ'(e) < 0 so true

(III) Graph of ƒ'(x) so (III) is false

(IV) Is false

**Alternate :**

QUESTION: 54

**Q.**

Which of the following options is the only INCORRECT combination ?

Solution:

**Ans. (D)**

(I) ƒ(1) ƒ(e^{2}) < 0 so true

(II) ƒ'(1) ƒ'(e) < 0 so true

(III) Graph of ƒ'(x) so (III) is false

(IV) Is false

**Alternate :**

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