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

QUESTION: 1

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

A.
x²e^-x/2
B.
x²/ 2
C.
e^-x/2
D.
x³e^-x/2

Solution:

QUESTION: 2

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

A.
linearly dependent
B.
linearly independent
C.
orthogonal
D.
none of these

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

A.
B.
1/S
C.
D.

Solution:

QUESTION: 4

The projection of vector on vector

A.
5/8
B.
8/3
C.
8/7
D.
8/9

Solution:

Projection of vactor

QUESTION: 5

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

A.
B.
0
C.
D.

Solution:

*Answer can only contain numeric values

QUESTION: 6

Kroncker delta Sⁱ_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 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, 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 = 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, v_2, v₃) = ( v_1, v₂ - v_3,iv₂)
Find T⁺matrix representation using orthonormal basis

Solution:

QUESTION: 14

Given the Legendre polynomial P₀(x) = 1, P₁(x) = x and then polynomial (3x² + x -1)

A.
P₂(x) - P₁(x)
B.
2P₂(x) +P₁(x)
C.
P₂(x) +P₁(x)
D.
2P₂(x) +P₁(x)+P₀(x)

Solution:

Polynomial, 3x² + x - 1

QUESTION: 15

The matrix A defined by is orthogonal if

A.
a = 1, b = -1
B.
C.
D.
a = 1, b = 1

Solution:

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

for orthogonal a² + b² = 1
Therefore,

QUESTION: 16

Find the inverse Laplace transform of f(s) =

A.
1 - x + sinx + cos x
B.
1 + x - cosx - sin x
C.
x + sinx - cos x
D.
1 + x - 2cosx - sin x

Solution:

QUESTION: 17

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

A.
C_n = 0
B.
C_n = 1/nπ
C.
C_n = 1/inπ
D.
C_n = 1/in

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

A.
x + 2y + 4z = 12
B.
4x + 2y + z = 12
C.
x + 4y + 2z = 12
D.
x + y + z = 7

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).