Test: Mathematical Physics - 2


20 Questions MCQ Test GATE Physics Mock Test Series | Test: Mathematical Physics - 2


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QUESTION: 1

For the function  at the point (1, 2, -1), find its rate of change with distance in the direction 

Solution:



Therefore, 
Therefore, rate of change with distance in the direction of 

QUESTION: 2

Evaluate the line integral  (exy + cos x sin y) dx + (ex + sin x cos y)dy, around the curve 'C' 

Solution:


Note:  . then vector function is conservative and hence line integral (work clone) along a closed path is zero)  

QUESTION: 3

If a unitary matrix U is written as A + iB, where A and B are Hermitian matrix with non degenerate eigenval­ ues. then

Solution:

Since. U is unitary matrix.


or (A + iB) (A - iB) = I  (since, A and B are Hermitian = B)
A2 + B2 + i(BA - AB) = I
Now. equating real and imaginary
Part. A2 + B2 = I and BA - AB = 0

QUESTION: 4

If (A)nxn is a square matrix, and elements of matrix A(aij) are such that

then value of det (eA) is equal to

Solution:


We know that, det (eA) = etrace A

*Answer can only contain numeric values
QUESTION: 5

If solution y(x), of the differential equation subjected to the boundary conditions y(0) = 2, y'(0) = 1 has the form y(x) = Ae-x + Bex + Ce2x, then the value of A + B - C is
 (upto two decimal places)


Solution:

D2 - 3D + 2 = 0
(D - 2) (D - 1) = 0 
Therefore, complementary function Bex + Ce2x and



*Answer can only contain numeric values
QUESTION: 6

The range of values ofz, for which the following complex power series converges, is Izl < A, then A is .................. (answer should be an integer)


Solution:


QUESTION: 7

If the matrix  can be diagonalised by a transformation of the form S* AS = A', where S has the normalized eigenvectors of A as its columns, then A' is

Solution:

A' will be a diagonal matrix, whose diagonal elements are the eigenvalues of A.
Therefore, to find eigenvalues of A.

Therefore, eigenvalues are λ = - 2, - 2, 4

QUESTION: 8

If  is the fourier transformation of  then the fourier trans form of f ' (t) is 

Solution:

Fourier transfonn of f' (t)

QUESTION: 9

If a complex function f (z) lias a pole of order m at z = z0. then f'(z) has a pole of order

Solution:

 where g is analytic at z0 and g(z0) ≠ 0
 where h(z) = (z - z0) g'(z) - mg(z) is analytic at z = z0 and h(z0) = - mg (z0) ≠ 0
∴ f' (z) has a pole of order (m + 1) at z = z0.

QUESTION: 10

The inverse laplace transfonn of  

Solution:


Using partial fraction.


*Answer can only contain numeric values
QUESTION: 11

If the fourier series for Isin θ| in the range -π < θ < π is given by  then the value of A is .........


Solution:



*Answer can only contain numeric values
QUESTION: 12

Value of the integral 


Solution:


Poles are at z = + 2
Therefore, residue at 2 = 2




*Answer can only contain numeric values
QUESTION: 13

If the fourier expansion of   then the value of ar if 


Solution:


QUESTION: 14

If z = xy, where y = tan-1 t and x = sin t, then the value of dz/dt is equal to

Solution:


QUESTION: 15

If y (x) is the solution of the differential equation  with y(1) = -1, then

Solution:




Now, y (1) = -1, put x = 1, y = -1


∴ for y to be defined log x must be defined i.e.. x > 0
Also when log x - 1 = 0, log x = 1 
So in the range 0 < x < 3 (option (b)). there will be a point x = 2.73 at whichy is not defined.
Also at x = e : y → ∞ (blow up).

QUESTION: 16

 then the value of integral  where ‘S’ is the surface given by z = 12, x2 - y2 < 25 (taken anticlockwise), is 

Solution:


∴ 

QUESTION: 17

The fourier complex transfoem of f(x), where

Solution:

Fourier transform of f (x)

 (integrating by parts)

QUESTION: 18

 then which of the following statements is true about A and B

Solution:

Since A is upper trinagular and B is lower triangular matrix. Hence, their eigenvalues are the principle diagonal elements.
Therefore, eigenvalues of A are 1, 0, 2 and eigenvalues of B are -1,1, 3
Since eigenvalues of A and B are all distinct, hence their corresponding eigenvectors will be linearly independent.
Hence, both A and B are digonalizable.

QUESTION: 19

The Laplace transform of f (t) = t2 cos at is

Solution:



*Answer can only contain numeric values
QUESTION: 20

The value of the integral  where C is closecl contour defined by the equation 2 |z - 1| - 3 = 0, traversed in the clockwise direction, is _______________________ (answer should be an integers)


Solution:



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