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*Multiple options can be correct

QUESTION: 1

A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure P_{0}, volume V_{0} and temperature T_{0}. If the gas mixture is adiabatically compressed to a volume V_{0}/4, then the correct statement(s) is/are,

(Give 2^{1.2} = 2.3 ; 2^{3.2} = 9.2; R is gas constant)

Solution:

∴ Option 4 is correct

which is between 9P_{0} and 10P_{0}

= 10RT

To calculate T

so

Now average

*Multiple options can be correct

QUESTION: 2

An electric dipole with dipole moment is held fixed at the origin O in the presence of an uniform electric field of magnitude E_{0}. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are:

(ε_{0} is permittivity of free space, R >> dipole size)

Solution:

E.F. at B along tangent should be zero since circle is equipotential.

So, (1) is correct

(2) Because E_{0} is uniform & due to dipole E.F. is different at different points, so magnitude of total E.F. will also be different at different points.

So, (2) is incorrect

So, (3) is wrong

(4) E_{B} = 0

so, (4) is correct

*Multiple options can be correct

QUESTION: 3

A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping.

Which of the following statement(s) is/are correct, when the rod makes an angle 60º with vertical? [g is the acceleration due to gravity]

Solution:

We can treat contact point as hinged.

Applying work energy theorem

W_{g} = ΔK.E.

radial acceleration of C.M. of rod

Using τ = I α about contact point

Net vertical acceleration of C.M. of rod

Applying F_{net} = ma in vertical direction on rod as system

*Multiple options can be correct

QUESTION: 4

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is L = L_{0} the particle speed is v = v_{0}. The piston is moved inward at a very low speed V such that where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

Solution:

(1) average rate of collision = 2L/v.

(2) speed of particle after collision = 2V + v_{0}

change in speed = (2V + v_{0}) – v_{0}

after each collision = 2V

no. of collision per unit time (frequency) = v/2L

change in speed in dt time = 2V × number of collision in dt time

or

vx = constant ⇒ on decreasing length to half K.E. becomes 1/4

vdx + xdv = 0

*Multiple options can be correct

QUESTION: 5

Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a horizontal surface shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same (R = 3m). If H_{1}, H_{2} and H_{3} are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are

Solution:

QUESTION: 6

In a Young's double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1m. A parallel beam of light of wavelength 600nm is incident on the slits at angle α as shown in figure. On the screen, the point O is equidistant from the slits and distance PO is 11.0 mm. Which of the following statement(s) is/are correct?

Solution:

(1) Δx = dsinα

= dα (as α is very small)

so constructive interference

= 3 × 10^{–4 }(2 × 10^{–3} + 11 × 10^{–3})

= 39 × 10^{–7}

= 33 × 10^{–7}

QUESTION: 7

A block of mass 2M is attached to a massless spring with spring constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are a_{1}, a_{2} and a_{3} as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is x_{0}. Which of the following option(s) is/are correct? [g is the acceleration due to gravity. Neglect friction]

Solution:

OR

that means, block 2m (connected with the spring) will perform SHM about therefore.

maximum elongation in the spring

on comparing equation (1) with

at block will be passing through its mean position therefore at mean position

*Multiple options can be correct

QUESTION: 8

A free hydrogen atom after absorbing a photon of wavelength λ_{a} gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength λ_{e}. Let the change in momentum of atom due to the absorption and the emission are Δp_{a} and Δp_{e}, respectively. If Which of the option(s) is/are correct?

[Use hc = 1242 eV nm; 1 nm = 10^{–9} m, h and c are Planck's constant and speed of light, respectively]

Solution:

from (ii)

we have

*Answer can only contain numeric values

QUESTION: 9

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that with h as Planck's constant. N photons of wavelength λ = 8π × 10^{–6}m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1µm. If the value of N is x × 10^{12}, then the value of x is ____.

[Consider the spring as massless]

Solution:

Let momentum of one photon is p and after reflection velocity of the mirror is v. conservation of linear momentum

mv = 2Np ...(1)

since v is velocity of mirror (spring mass system) at mean position,

v = AΩ

Where A is maximum deflection of mirror from mean position. (A = 1 µm) and Ω is angular frequency of mirror spring system, of momentum of 1 photon,

mv = 2Np ...(i)

*Answer can only contain numeric values

QUESTION: 10

A ball is thrown from ground at an angle θ with horizontal and with an initial speed u_{0}. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V_{1}. After hitting the ground, ball rebounds at the same angle θ but with a reduced speed of u_{0}/α. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V_{1}, the value of α is______

Solution:

*Answer can only contain numeric values

QUESTION: 11

A 10 cm long perfectly conducting wire PQ is moving, with a velocity 1cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor L = 1 mH and a resistance R = 1Ω as shown in figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field B = 1 T perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is x × 10^{–3}A, where the value of x is_______.

[Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: e^{–1} = 0.37, where e is base of the natural logarithm]

Solution:

Since velocity of PQ is constant. So emf developed across it remains constant.

ε = Blv where l = length of wire PQ

Current at any time t is given by

*Answer can only contain numeric values

QUESTION: 12

A monochromatic light is incident from air on a refracting surface of a prism of angle 75° and refractive index n_{0} =√3 . The other refracting surface of a prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of θ ≤ 60°. The value of n^{2} is_______.

Solution:

At θ = 60° ray incidents at critical angle at second surface

So,

√3 sin 45° = n sin 90°

*Answer can only contain numeric values

QUESTION: 13

Suppose a nucleus at rest and in ground state undergoes α-decay to a nucleus in its excited state. The kinetic energy of the emitted α particle is found to be 4.44 MeV. nucleus then goes to its ground state by γ-decay. The energy of the emitted γ-photon is _______ keV,

[Given: atomic mass of atomic mass of atomic mass of α particle = 4.000u, 1u = 931 MeV/c^{2}, c is speed of the light]

Solution:

= .135 MeV

= 135 KeV

*Answer can only contain numeric values

QUESTION: 14

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is________.

Solution:

For the given lens

u = –30cm

v = 60 cm

on differentiation

f = 20cm, du = dv = 1/4 cm

Since there are 4 divisions in 1 cm on scale

QUESTION: 15

Answer the following by appropriately matching the lists based on the information given in the paragraph.

A musical instrument is made using four different metal strings, 1,2,3 and 4 with mass per unit length µ, 2µ, 3µ and 4µ respectively. The instrument is played by vibrating the strings by varying the free length in between the range L_{0} and 2L_{0}. It is found that in string-1 (µ) at free length L_{0} and tension T_{0} the fundamental mode frequency is f_{0}.

List-I gives the above four strings while List-II lists the magnitude of some quantity.

If the tension in each string is T_{0}, the correct match for the highest fundamental frequency in f_{0} units will be,

Solution:

For fundamental mode

QUESTION: 16

Answer the following by appropriately matching the lists based on the information given in the paragraph.

A musical instrument is made using four different metal strings, 1,2,3 and 4 with mass per unit length µ, 2µ, 3µ and 4µ respectively. The instrument is played by vibrating the strings by varying the free length in between the range L_{0 }and 2L_{0}. It is found that in string-1 (µ) at free length L_{0} and tension T_{0} the fundamental mode frequency is f_{0}.

List-I gives the above four strings while List-II lists the magnitude of some quantity.

The length of the string 1,2,3 and 4 are kept fixed at respectively. Strings 1,2,3 and 4 are vibrated at their 1^{st}, 3^{rd}, 5^{th} and 14^{th} harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of T_{0} will be.

Solution:

For string (1)

Length of string = L_{0}

It is vibrating in I^{st} harmonic i.e. fundamental mode.

For string(2)

Length of string = 3L_{0}/2

It is vibrating in III^{rd} harmonic but frequency is still f_{0}.

For string (3)

Length of string = 5L_{0}/4

It is vibrating in 5^{th} harmonic but frequency is still f_{0}.

It is vibrating in 14^{th} harmonic but frequency is still f_{0}.

⇒

QUESTION: 17

Answer the following by appropriately matching the lists based on the information given in the paragraph.

In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by TΔX, where T is temperature of the system and ΔX is the infinitesimal change in a thermodynamic quantity X of the system. For a mole of monatomic ideal gas Here, R is gas constant, V is volume of gas, T_{A} and V_{A} are constants.

The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with the correct match is,

Solution:

(I) Degree of freedom f = 3

Work done in any process = Area under P–V graph

⇒ Work done in 1 → 2 → 3 = P_{0}V_{0}

(II) Change in internal energy 1 → 2 → 3

(III) Heat absorbed in 1 → 2 → 3

for any process, I^{st} law of thermodynamics

(IV) Heat absorbed in process 1 → 2

QUESTION: 18

Answer the following by appropriately matching the lists based on the information given in the paragraph.

In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by TΔX, where T is temperature of the system and ΔX is the infinitesimal change in a thermodynamic quantity X of the system. For a mole of monatomic ideal gas . Here, R is gas constant, V is volume of gas, T_{A} and V_{A} are constants.

The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

If the process on one mole of monatomic ideal gas is an shown is as shown in the TV-diagram with P_{0}V_{0} = 1/3 RT_{0}, the correct match is

Solution:

Process 1 → 2 is isothermal (temperature constant)

Process 2 → 3 is isochoric (volume constant)

*Multiple options can be correct

QUESTION: 19

The cyanide process of gold extraction involves leaching out gold from its ore with CN^{–} in the presence of Q in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the correct option(s).

Solution:

*Multiple options can be correct

QUESTION: 20

Which of the following reactions produce(s) propane as a major product?

Solution:

*Multiple options can be correct

QUESTION: 21

The ground state energy of hydrogen atom is –13.6 eV. Consider an electronic state ψ of He^{+ }whose energy, azimuthal quantum number and magnetic quantum number are –3.4 eV, 2 and 0 respectively. Which of the following statement(s) is(are) true for the state ψ?

Solution:

*Multiple options can be correct

QUESTION: 22

Choose the correct option(s) that give(s) an aromatic compound as the major product.

Solution:

*Multiple options can be correct

QUESTION: 23

Consider the following reactions (unbalanced)

Zn + hot conc. H_{2}SO_{4} → G + R + X

Zn + conc. NaOH → T + Q

G + H_{2}S + NH_{4}OH → Z (a precipitate) + X + Y

Choose the correct option(s).

Solution:

*Multiple options can be correct

QUESTION: 24

With reference to aqua regia, choose the correct option(s).

Solution:

(4) Yellow colour of aqua regia is due to it's decomposition into NOCl(orange yellow) and Cl_{2}(greenish yellow).

*Multiple options can be correct

QUESTION: 25

Choose the correct option(s) from the following

Solution:

1. Natural rubber is polyisoprene containing cis alkene units

2. Nylon-6 has amide linkage

3. Cellulose has only β-D glucose units.

*Multiple options can be correct

QUESTION: 26

Choose the correct option(s) for the following reaction sequence

Consider Q, R and S as major products

Solution:

*Answer can only contain numeric values

QUESTION: 27

The decomposition reaction is started in a closed cylinder under isothermal isochoric condition at an initial pressure of 1 atm. After Y × 10^{3} s, the pressure inside the cylinder is found to be 1.45 atm. If the rate constant of the reaction is 5 × 10^{–4} s^{–1}, assuming ideal gas behavior, the value of Y is ___

Solution:

*Answer can only contain numeric values

QUESTION: 28

Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula C_{4}H_{8}O is ___

Solution:

*Answer can only contain numeric values

QUESTION: 29

The amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc.HNO_{3} to a compound with the highest oxidation state of sulphur is __

(Given data : Molar mass of water = 18 g mol^{–1})

Solution:

S_{8} + 48 HNO_{3} → 8H_{2}SO_{4} + 48NO_{2} + 16H_{2}O

1 mole of rhombic sulphur produce 16 mole of H_{2}O i.e. 288 gm of H_{2}O

*Answer can only contain numeric values

QUESTION: 30

Total number of cis N–Mn–Cl bond angles (that is, Mn–N and Mn–Cl bonds in cis positions) present in a molecule of cis-[Mn(en)_{2}Cl_{2}] complex is ____ (en = NH_{2}CH_{2}CH_{2}NH_{2})

Solution:

Number of cis (Cl-Mn-N) = 6

*Answer can only contain numeric values

QUESTION: 31

Total number of hydroxyl groups present in a molecule of the major product P is ___

Solution:

total 6 –OH group present in a molecule of the major product.

*Answer can only contain numeric values

QUESTION: 32

The mole fraction of urea in an aqueous urea solution containing 900 g of water is 0.05. If the density of the solution is 1.2 g cm^{–3}, the molarity of urea solution is ___

(Given data : Molar masses of urea and water are 60 g mol^{–1} and 18 g mol^{–1}, respectively)

Solution:

QUESTION: 33

Answer the following by appropriately matching the lists based on the information given in the paragraph

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the n^{th} orbit of the atom and List-II contains options showing how they depend on n.

Which of the following options has the correct combination considering List-I and List-II?

Solution:

QUESTION: 34

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the n^{th} orbit of the atom and List-II contains options showing how they depend on n.

Which of the following options has the correct combination considering List-I and List-II?

Solution:

QUESTION: 35

Answer the following by appropriately matching the lists based on the information given in the paragraph

List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I

Which of the following options has correct combination considering List-I and List-II?

Solution:

IV, Q, R

QUESTION: 36

List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I

Which of the following options has correct combination considering List-I and List-II?

Solution:

I, Q, R

II, P, S, U

*Multiple options can be correct

QUESTION: 37

Let f : be given by f(x) = (x – 1) (x – 2) (x – 5). Define F(x) = Then which of the following options is/are correct?

Solution:

ƒ(x) = (x – 1) (x – 2) (x – 5)

clearly F(x) has local minimum at x = 1,5

F(x) has local maximum at x = 2

from the graph of y = F(x), clearly

*Multiple options can be correct

QUESTION: 38

For a Then the possible value(s) of a is/are:

Solution:

*Multiple options can be correct

QUESTION: 39

Three lines

are given. For which point(s) Q on L_{2} can we find a point P on L_{1 }and a point R on L_{3} so that P, Q and R are collinear?

Solution:

Hence Q cannot have coordinates (0, 1, 1) and (0, 0, 1).

*Multiple options can be correct

QUESTION: 40

Let F: be a function. We say that f has

Then which of the following options is/are correct?

Solution:

P - 1:

P-2 :

*Multiple options can be correct

QUESTION: 41

For non-negative integers n, let

Assuming cos^{–1} x takes values in [0, π], which of the following options is/are correct?

Solution:

*Multiple options can be correct

QUESTION: 42

Let

where denotes the transpose of the matrix P_{K}. Then which of the following options is/are correct?

Solution:

X is symmetric

⇒ α = 30.

Trace X =

⇒ X - 30I is non-invertible

*Multiple options can be correct

QUESTION: 43

Let and R = PQP^{–1}. Then which of the following options is/are correct?

Solution:

= det Q

= 48 – 4x^{2}

Option-1:

for x = 1 det (R) = 44 ≠ 0

∴ for equation

We will have trivial solution

Option-3:

Option-4:

*Multiple options can be correct

QUESTION: 44

Let

Let x_{1} < x_{2} < x_{3} < ... < x_{n} < ... be all the points of local maximum of f

and y_{1} < y_{2} < y_{3 }< ... < y_{n} < ... be all the points of local minimum of f.

Then which of the following options is/are correct?

Solution:

*Answer can only contain numeric values

QUESTION: 45

The value of in the interval equals

Solution:

*Answer can only contain numeric values

QUESTION: 46

Let |X| denote the number of elements in set X. Let S = {1,2,3,4,5,6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A,B) such that 1 ≤ |B| < |A|, equals

Solution:

n(A) = 2 does not satisfy the constraint (1).

*Answer can only contain numeric values

QUESTION: 47

Five person A,B,C,D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is

Solution:

*Answer can only contain numeric values

QUESTION: 48

Suppose holds for some positive integer n. Then equals

Solution:

Suppose

*Answer can only contain numeric values

QUESTION: 49

The value of the integral equals

Solution:

*Answer can only contain numeric values

QUESTION: 50

Let be two vectors. Consider a vector If the projection of on the vector then the minimum value of equals

Solution:

QUESTION: 51

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let ƒ(x) = sin(π cosx) and g(x) = cos(2π sinx) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:

List-I contains the sets X, Y, Z and W. List -II contains some information regarding these sets.

Which of the following is the only CORRECT combination?

Solution:

f(x) = 0 ⇒ sin (π cos x) = 0 ⇒ cos x = n ⇒ cos x = 1, –1, 0 ⇒ x = nπ/2

QUESTION: 52

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let ƒ(x) = sin(π cosx) and g(x) = cos(2π sinx) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:

List-I contains the sets X,Y,Z and W. List -II contains some information regarding these sets.

Which of the following is the only CORRECT combination?

Solution:

f(x) = 0 ⇒ sin (π cos x) = 0 ⇒ cos x = n ⇒ cos x = 1, –1, 0 ⇒ x = nπ/2

QUESTION: 53

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let the circles C_{1} : x^{2} + y^{2} = 9 and C^{2} : (x– 3)^{2} + (y – 4)^{2} = 16, intersect at the points X and Y. Suppose that another circle C_{3} : (x – h)^{2} + (y – k)^{2} = r^{2} satisfies the following conditions:

(i) centre of C_{3} is collinear with the centres of C_{1} and C_{2}

(ii) C_{1} and C_{2} both lie inside C_{3}, and

(iii) C_{3} touches C_{1} at M and C_{2} at N.

Let the line through X and Y intersect C_{3} at Z and W, and let a common tangent of C_{1} and C_{3} be a tangent to the parabola x^{2} = 8αy.

There are some expression given in the List-I whose values are given in List-II below:

Which of the following is the only INCORRECT combination?

Solution:

QUESTION: 54

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let the circles C_{1} : x^{2} + y^{2} = 9 and C_{2 }: (x– 3)^{2} + (y – 4)^{2} = 16, intersect at the points X and Y.

Suppose that another circle C_{3} : (x – h)^{2} + (y – k)^{2} = r^{2} satisfies the following conditions:

(i) centre of C_{3} is collinear with the centres of C_{1} and C_{2}

(ii) C_{1} and C_{2} both lie inside C_{3}, and

(iii) C_{3} touches C_{1} at M and C_{2} at N.

Let the line through X and Y intersect C_{3} at Z and W, and let a common tangent of C_{1} and C_{3 }be a tangent to the parabola x^{2} = 8αy.

There are some expression given in the List-I whose values are given in List-II below:

Which of the following is the only CORRECT combination?

Solution:

MC_{1} + C_{1}C_{2} + C_{2}N = 2r

⇒ 3 + 5 + 4 = 2r ⇒ r = 6 ⇒ Radius of C_{3} = 6

Suppose centre of C_{3} be

Equation of ZW and XY is 3x + 4y – 9 = 0

(common chord of circle C_{1} = 0 and C_{2} = 0)

Let length of perpendicular from M to ZW be

C_{1} : x^{2} + y^{2} – 9 = 0

common tangent to C_{1} and C_{3} is common chord of C_{1} and C_{3} is 3x + 4y + 15 = 0.

Now 3x + 4y + 15 = 0 is tangent to parabola x^{2} = 8αy.

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