Test: Combined (CM+MP) - Physics MCQ

Test: Combined (CM+MP) - Physics MCQ

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

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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

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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

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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,

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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.

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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)).

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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 =

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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

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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

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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.

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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

Test: Combined (CM+MP) - Question 31

The relativistic Lagrangian of a point particle moving in 1 -D is given by

Here m is mass of particle and c is speed of light. The graph of x vs t is a

Detailed Solution for Test: Combined (CM+MP) - Question 31

Test: Combined (CM+MP) - Question 32

For a dynamical system, time evolution of dynamical variable is given by

Which one of the following statement is correct?

Detailed Solution for Test: Combined (CM+MP) - Question 32

Stability analysis of fixed points.

Test: Combined (CM+MP) - Question 33

The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2. If pdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?

Detailed Solution for Test: Combined (CM+MP) - Question 33

is not a cyclic coordiante and hence px is not conserved.

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Test: Combined (CM+MP) - Question 34

The space-time coordinates of two events as measured by oserver O are and  The spatial separation of the two events as measured by O'. where the two events occur simultaneously ______ x 104m. (Upto three decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 34

In O' frame.

Spatial separation in O' frame,

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Test: Combined (CM+MP) - Question 35

A three particle system consists of masses m. and coordinates (x. y. z) as follows

If Lij represents the elements of moment of inertia tensor then value of  ( Up to two decimal places)

Detailed Solution for Test: Combined (CM+MP) - Question 35

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Test: Combined (CM+MP) - Question 36

A particle of mass m moves wider the action of a central force whose potential energy is U(r) = kr6 , k > O. If the orbit be a circle of radius a about origin, then the total energy is ηka6. The value of η is _______.

Detailed Solution for Test: Combined (CM+MP) - Question 36

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Test: Combined (CM+MP) - Question 37

A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is propertional to tβ . The value of β is ____________ .

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Test: Combined (CM+MP) - Question 38

Consider an equation  Then f(x) can possibly look like

Detailed Solution for Test: Combined (CM+MP) - Question 38

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Test: Combined (CM+MP) - Question 39

Consider the following function:

which repeats itself outside the interval [-5. -5]. In the Fourier series expansion of the fraction, the Fourier coefficient b1oo will be equal to ________ (Symbols have their usual meanings)

Detailed Solution for Test: Combined (CM+MP) - Question 39

Since. Given function f(x) is even, hence there is no 'sine' term in fourier series of f(x) and hence co-efficient 'bn’ are zero.

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Test: Combined (CM+MP) - Question 40

Given the Fourier series  The value of

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Test: Combined (CM+MP) - Question 41

If  Then S = ________

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Test: Combined (CM+MP) - Question 42

Given  The Laplace transform  is

Detailed Solution for Test: Combined (CM+MP) - Question 42

Test: Combined (CM+MP) - Question 43

Given  The Fourier transform   given by

Detailed Solution for Test: Combined (CM+MP) - Question 43

∴

Test: Combined (CM+MP) - Question 44

Consider a complex function   Which of the following is true regarding f(z)?

Detailed Solution for Test: Combined (CM+MP) - Question 44

Test: Combined (CM+MP) - Question 45

Evaluate

Detailed Solution for Test: Combined (CM+MP) - Question 45

Test: Combined (CM+MP) - Question 46

For which of the following functions, Fourier series expansion is not possible in the given interval?

Detailed Solution for Test: Combined (CM+MP) - Question 46

Test: Combined (CM+MP) - Question 47

Suppose we have differentiation equation.

with boundary condition,
Which of the following relations between y and x is correct.

Detailed Solution for Test: Combined (CM+MP) - Question 47

Given:

Multiplying I.F. and then taking integration both side, we get

∴

Test: Combined (CM+MP) - Question 48

Let  be the Pauli matrices and  Then the coordinates are related as follows:

Detailed Solution for Test: Combined (CM+MP) - Question 48

Test: Combined (CM+MP) - Question 49

The value of the following series  is

Detailed Solution for Test: Combined (CM+MP) - Question 49

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Test: Combined (CM+MP) - Question 50

Consider the two equations  How many simultaneous real solutions does this pair of equations have?

Detailed Solution for Test: Combined (CM+MP) - Question 50

We have   .....(i)
.....(ii)
(ii) is the positive (square root) branch of
......(iii)
Now plotting them on a single coordinate axes, we have

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Test: Combined (CM+MP) - Question 51

Given a function f(x,y,z) = xyz. The magnitude of rate of change of f(x,y,z) along the direction of the vactor  at a point (2, -1, -1) is ____ (upto 2 decimal places).

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Test: Combined (CM+MP) - Question 52

Consider the integral  where P is a real constant. This integral has a real, non-singular value if and only if and only if P → α. Then, α is ______.

Detailed Solution for Test: Combined (CM+MP) - Question 52

Now, any value of P such that |P| > 1 will contain the interval -1 < x <1 which would be non - real.
Now, let us look at P < 1

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Test: Combined (CM+MP) - Question 53

Consider a 2 x 2 matrix A, whose characteristic equation is given by 2A2 - 5 A + 2I = 0 . Then, the value of 2(Tr A)-3(det A) = ___________

Detailed Solution for Test: Combined (CM+MP) - Question 53

The characteristic equation o f A is
According to Cayley - Hamilton theorem, eigenvalues of the matrix will also satisfy the characteristic equation i.e.

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Test: Combined (CM+MP) - Question 54

The number of nodes in the solution of the differential equation

Detailed Solution for Test: Combined (CM+MP) - Question 54

Comparing the given differential equation with the Legendre differential equation i.e.

Therefore, the solution of the given equation will be is P2(x) which is a polynomial of degree 2.

Therefore, there are 2 nodes in the solution

Test: Combined (CM+MP) - Question 55

The matrix  can be related by a similarity transformation to the matrix

Detailed Solution for Test: Combined (CM+MP) - Question 55

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