Test: Mathematical Physics - 1


20 Questions MCQ Test GATE Physics Mock Test Series | Test: Mathematical Physics - 1


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QUESTION: 1

The particular integral of (4D2 + 4D + 1) y = 8e-x/2 is

Solution:


QUESTION: 2

The vector [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] are

Solution:

Let a = [1, 2, 3], b = [1, 0, 0], c = [0, 1, 0]
 d = [0, 0, 1]
a = b + 2c + 3d
Therefore, vector a, b, c, d are linearly dependent.

QUESTION: 3

Find 

Solution:

QUESTION: 4

The projection of vector   on vector 

Solution:

Projection of vactor 

QUESTION: 5

Find an of a Fourier series for |x|, -π < x < π

Solution:

*Answer can only contain numeric values
QUESTION: 6

Kroncker delta Sij is a mixed tensor of rank _____


Solution:
*Answer can only contain numeric values
QUESTION: 7

Real part of the  is ______ (upto two decimal places)


Solution:



*Answer can only contain numeric values
QUESTION: 8

Given z3 = 1. Let z0,z1 and z2 be the complex roots of the above equation.If z0 = 1, then the value of z1z2 is ____ (Answer should be an integer)


Solution:

*Answer can only contain numeric values
QUESTION: 9

Find the value of integral,  is ____ (upto one decimal place)


Solution:


*Answer can only contain numeric values
QUESTION: 10

The dimensionality  of the vector space of hermitian 3 x 3 matrices is ____ (answer should be an integer)


Solution:

For the Hermitian 
Therefore, aii are real but aij (j≠i) can be complex. 
For  n x n Hermitiain matrices,


Therefore, for 3 x 3 Hermitian matrices, number of independent entries in the matrices = 32 = 9 Dimentionality = 9

*Answer can only contain numeric values
QUESTION: 11

Given vector  the line integral  where C is a circle of radius 5 units with its center at origin is ________


Solution:



*Answer can only contain numeric values
QUESTION: 12

The determinant of the metric tensor corresponding to ds2 = 5(dx1)2 + 3(dx2)2 + 4(dx3)2 - 6dx1dx2 + 4dx2dx3 is


Solution:

Comparing with equation standard expression for the metric tensor 

QUESTION: 13

Consider a vector v = (v1, v2, v3) in three dimensional complex vector space c3. A linear operator T is designed as follows
T( v1, v2, v3) = ( v1, v2 - v3,iv2)
Find Tmatrix representation using orthonormal basis

Solution:


QUESTION: 14

Given the Legendre polynomial P0(x) = 1, P(x) = x and  then polynomial (3x2 + x -1)

Solution:


Polynomial, 3x2 + x - 1

QUESTION: 15

The matrix A defined by  is orthogonal if

Solution:

A square matrix A is said to be orthogonal if AAT = ATA = 1


for orthogonal a2 + b2 = 1
Therefore, 

QUESTION: 16

Find the inverse Laplace transform of f(s) = 

Solution:

QUESTION: 17

Find the complex coefficient Cn of the fourier series of the function  for n is odd.

Solution:


Cn = 1/ inπ

QUESTION: 18

The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) is

Solution:

Suppose T(x,y,z) be any point on tangent plane  is normal to surface  at point P(1,2,4). Therefore,  is perpendicular to vector  lying in the tangent plane of the given surface.

*Answer can only contain numeric values
QUESTION: 19

The value of the integral  is ______ (upto two decimal places)


Solution:



*Answer can only contain numeric values
QUESTION: 20

The value of the Contour integral
 
and the contour C is a circle of radius 2 centred at the origin traversed in the counterclockwise direction is ______ (answer should be an integer).


Solution:


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