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This mock test of Test: Mathematical Physics - 1 for GATE helps you for every GATE entrance exam.
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QUESTION: 1

The particular integral of (4D^{2} + 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 a_{n} of a Fourier series for |x|, -π < x < π

Solution:

*Answer can only contain numeric values

QUESTION: 6

Kroncker delta S^{i}_{j} 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 z^{3} = 1. Let z_{0},z_{1} and z_{2} be the complex roots of the above equation.If z_{0} = 1, then the value of z_{1}z_{2} 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, a_{ii} are real but a_{ij} (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 = 3^{2} = 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 ds^{2} = 5(dx^{1})^{2} + 3(dx^{2})^{2} + 4(dx^{3})^{2} - 6dx^{1}dx^{2 }+ 4dx^{2}dx^{3} is

Solution:

Comparing with equation standard expression for the metric tensor

QUESTION: 13

Consider a vector v = (v_{1}, v_{2}, v_{3}) in three dimensional complex vector space c^{3}. A linear operator T is designed as follows

T( v_{1}, v_{2,} v_{3}) = ( v_{1,} v_{2} - v_{3,}iv_{2})

Find T^{+ }matrix representation using orthonormal basis

Solution:

QUESTION: 14

Given the Legendre polynomial P_{0}(x) = 1, P_{1 }(x) = x and then polynomial (3x^{2} + x -1)

Solution:

Polynomial, 3x^{2} + x - 1

QUESTION: 15

The matrix A defined by is orthogonal if

Solution:

A square matrix A is said to be orthogonal if AA^{T} = A^{T}A = 1

for orthogonal a^{2} + b^{2} = 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:

C_{n} = 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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