Test: Combined (CM+MP) - Physics MCQ

# Test: Combined (CM+MP) - Physics MCQ

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## 30 Questions MCQ Test GATE Physics Mock Test Series 2025 - Test: Combined (CM+MP)

Test: Combined (CM+MP) for Physics 2024 is part of GATE Physics Mock Test Series 2025 preparation. The Test: Combined (CM+MP) questions and answers have been prepared according to the Physics exam syllabus.The Test: Combined (CM+MP) MCQs are made for Physics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Combined (CM+MP) below.
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Test: Combined (CM+MP) - Question 1

### Consider a particle of mass m moving in a one dimension under a force with the potential V ( x ) = a |x3 - 3xl). where the constant k > 0. The frequency of a small amplitude oscillation of the panicle about the equilibrium position

Detailed Solution for Test: Combined (CM+MP) - Question 1

Given: V ( x ) = a( x3 - 3xl), k > 0
At equilibrium
Stability analysis:

∴

Test: Combined (CM+MP) - Question 2

### If the functions G and F depends on the position co-ordinates qi momenta pi and time t, then poisson bracket of G and F is defined as Then [G,pr] (Here r is arbitrary number)

Detailed Solution for Test: Combined (CM+MP) - Question 2

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Test: Combined (CM+MP) - Question 3

### Two forces are given below in spherical polar and cartersian coordiantes Where A, B and R are constants. Then

Detailed Solution for Test: Combined (CM+MP) - Question 3

Test: Combined (CM+MP) - Question 4

The Lagrangian of a particle is

where k is a positive constant, q1 and q2 in generalised coordinates. The equation of motion are

Detailed Solution for Test: Combined (CM+MP) - Question 4

Using Euler-Lagrange equations for q1

Using Euler-largrange equations for q2

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 5

Let (p.q) and (P, Q) be two pairs of canonical variables. The transformation P -q cot βp and  is canonical for β = nα. The value of n is_______

Detailed Solution for Test: Combined (CM+MP) - Question 5

For canonical transformation, {Q. P}qp= 1

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 6

The mass of a particle is measured to be √2 times of its rest mass. The ratio of kinetic energy and total energy is ____________ (up to two decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 6

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Test: Combined (CM+MP) - Question 7

The percentage contraction of a rod moving with a velocity 0.8 c in a direction inclined at 60° to its own length ______% (up to one decimal place)

Detailed Solution for Test: Combined (CM+MP) - Question 7

Let the proper length be L0. The proper length component of rod perpendicular to the velocity v0 = Lcos 60°.
There is no length contrction along the direction perpendicular to velocity.
The length of the perpendicular component to the velocity in lab frame is

Total lensth of rod in Lab frame.

Percentage contraction =

Test: Combined (CM+MP) - Question 8

The Hamiltonian of a particle with mass m =1/2  units moving along x-axis inside a potential V(x) = eis given by H = p2 + ex Assume p> 0. The phase space trajectory of the particle will be

Detailed Solution for Test: Combined (CM+MP) - Question 8

Test: Combined (CM+MP) - Question 9

The Lagrangian for the case when the Hamiltonian is

Detailed Solution for Test: Combined (CM+MP) - Question 9

Given:

Test: Combined (CM+MP) - Question 10

In a central force field, the trajecting (in plane polar co-oridantes) of a particle is given by  where m is the mass of the particle. L is angular momentiun and e is the eccentricity of the particle's motion. Whcih one of the following conditions given rise to circular trajectory?

Detailed Solution for Test: Combined (CM+MP) - Question 10

For circular orbit, eccentricity e — 0,

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 11

The minimum value of coefficient of friction for an inclined plane of angle θ = 450 in order that a hoop will roll down without slipping is______(Upto one decimal place)

Detailed Solution for Test: Combined (CM+MP) - Question 11

Let the radius ot hoop be R and mass be m. Equation ot motion

Test: Combined (CM+MP) - Question 12

A distant galaxy is observed to have its hydrogen- β line shifted to a wavelength of 580 nm, away from the laboratory value of 434 mn. The approximate velocity of recession of the distant galaxy is

Detailed Solution for Test: Combined (CM+MP) - Question 12

From the relativistic dropper sliift we have

(source moving away from observer)
In terms of wavelength.

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 13

Two balls, of mass m and mass 2m. approach from perpendicular directions with identical speeds and collide. After the collision, the more massive ball moves with the same speed v but downward, perpendicular to its original direction. The less massive ball moves with speed U at an angle θ with respect to the horizontal. If no external forces act during the collision then the final speed of less massive ball is αv. The value of α is_______ (up to 1 decimal place)

Detailed Solution for Test: Combined (CM+MP) - Question 13

Applying the law of conservation
of momentum along x-axis,

Applying the law of conservation of momentum along y-axis

Dividing (ii) by (i).

⇒ α = 2.2

Test: Combined (CM+MP) - Question 14

The orthogonal trajectories to the curve x2 + (y-1)2 = 1 are a family of

Detailed Solution for Test: Combined (CM+MP) - Question 14

Given curve is x+ ( y - 1)2 = 1
As shown in figure, the orthogonal trajectories are straight lines passing through (0, 1).

Test: Combined (CM+MP) - Question 15

Consider a charged sphere of radius R with charge density f(r ) = e-r. The gradient vector to the family of equipotential surfaces of this charged surface points

Detailed Solution for Test: Combined (CM+MP) - Question 15

Given, charge density f(r ) = e-r
This density depends only on the radius r and not (θ, φ) of spherical polar coordinates.
Therefore, the equipotential surfaces for this volume charge are spheres of different radii with potential decreasing radially outwards  is radially outwards). This means that V (r, θ, φ) decreases with radius. The gradient of a surface is in the direction along the maximum rate o f increase o f the function (here V (r, θ, φ)).

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 16

The value of the integral  over a contour of unit circle centered at origin. (Taken in the anticlockwise sense) is_______(Answer should be an integer)

Detailed Solution for Test: Combined (CM+MP) - Question 16

Requited intergral =

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 17

Given the value of H5(0 ) is  _________ (Answer should be an integer)

Detailed Solution for Test: Combined (CM+MP) - Question 17

Comparing the coefficient of t2n+1 on both sides, we get

Test: Combined (CM+MP) - Question 18

The Gaimna function is defined by the integral  will be

Detailed Solution for Test: Combined (CM+MP) - Question 18

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Test: Combined (CM+MP) - Question 19

Consider a metric tensor   Then det (gij) = _____

Detailed Solution for Test: Combined (CM+MP) - Question 19

Test: Combined (CM+MP) - Question 20

Which one of the following combinations can be a possible set of eigenvalues for a 4 x 4 unitary matrix?

Detailed Solution for Test: Combined (CM+MP) - Question 20

A real unitary matrix will have complex eigenvalues are in conjugate pairs and the modulus of each eigenvalue is 1.
Therefore,  is the possible set of eigenvalues of a (4 x 4) unitary matrix

Test: Combined (CM+MP) - Question 21

Consider a complex function f(x. y) = eax + i In by. If the function is analytic at (0. 1) then the possible values of (a, b) is

Detailed Solution for Test: Combined (CM+MP) - Question 21

f ( x, y ) = eax + In by
According to Cauchy-Riemann equations,

The other equation is trivially satisfied

∴ b can have any positive value for which In (by) is real.
∴ Out of the given options (1, 1) satisfies the condition

Test: Combined (CM+MP) - Question 22

If δ(x) is the Dirac-Delta function. and x has the dimension of angular momentum, then δ(x) has a dimension which can be written as [LαMβTy]. The value of (α,β,γ) is

Detailed Solution for Test: Combined (CM+MP) - Question 22

Dirac-Delta function has the property  has the dimension of [x]
δ(x) must have the dimension of [1/x]
If [x] = [Angular momentum] = [ML2T-1]]. then [δ (x )] = M-1L-2 T1
Therefore, α = - 2 β = - 1 γ = 1

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 23

The value of Trace  ______ where  is the pauli matrix  (upto 2 decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 23

The eigenvalues of the Pauli matrix
Therefore, eigenvalues of the matrix

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 24

Consider a vector   The line integral  over the contour as shown is equal to _____ (upto 2 decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 24

The semi-major (a) and semi-minor (b) axis lengths are

= 6.28

Test: Combined (CM+MP) - Question 25

Given below are forces in Cartesian and spherical coordinates. Which of the following are conservative?

Detailed Solution for Test: Combined (CM+MP) - Question 25

only are conservative

Test: Combined (CM+MP) - Question 26

A particle of m moves with speed v. Ail explosion divides the particle into two half, giving each half a speed V in the centre of mass frame. Assume all motion is confined to one dimension. The increase in kinetic energy in the Lab frame is

Detailed Solution for Test: Combined (CM+MP) - Question 26

The final velocities of each half in the lab frame are y+V and y - y.
The total kinetic energy after explosion

The initial K.E. = 1/2 mV2
The increase in K.E.= 1/2 mV2

Test: Combined (CM+MP) - Question 27

A particle is thrown from earth's surface with speed  where M and R are mass and radius of the earth respectively. The trajectory of the particle will be a /an

Detailed Solution for Test: Combined (CM+MP) - Question 27

Given :
Force on the particle is F =
Potential energy of the particle is.

Potential energy near earth’s surface

Total energy of the particle is

Since E > 0, therefore trajecting of the particle will be a hyperbola.

Test: Combined (CM+MP) - Question 28

A cricular disc is rotaing about an axis, shown in figure, makes an angle θ = 300 with the vertical axis. The moment of inertia of the disc about that axis is

Detailed Solution for Test: Combined (CM+MP) - Question 28

The projection of circular disc on the plane parallel to AB. as shown in figure, is an ellipse with AB = 2 R cosθ and CD = 2R

Test: Combined (CM+MP) - Question 29

A block of mass M with the shape of an inverted trapezoid is pushed downwards minimum force required to tip the block is

Detailed Solution for Test: Combined (CM+MP) - Question 29

The centre of mass of right angle triangle
By Symmetry. Mass of square ABCD = 2M/3

Mass of triangle CBE = M/3
Taking point C as origin andy-axis along CB.
When the block is just about to tip, net torque about origin is zero.

*Answer can only contain numeric values
Test: Combined (CM+MP) - Question 30

An electron moves in the lab with a speed of 0.6c. An observer moves with a velocity of 0.8 c along the direction of the motion of the electron. The kinetic energy of the electron as determined by the observer is  _____  MeV (Upto three decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 30

Using Lorentz velocity transformations, velocity of election in the frame of the observer (S') is

Kinetic energy of the electron as observed by the observer

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