MATHEMATICAL PHYSICS (PREVIOUS SET QUESTIONS)

 Module 1. Mathematical physics 

 Dimensional analysis, Vector algebra and vector calculus, Linear algebra, matrices, Cayley- Hamilton Theorem, Eigen values and eigen vectors. Linear differential equations of first and second order. Fourierseries, Fourier and Laplace transforms. Elementary complex analysis, analytic functions; Taylor & Laurent series; poles,residues and evaluation of integrals. Special functions (Hermite, Bessel, Laguerre and Legendre). Elementary probability theory, random variables, binomial, Poisson and normal distributions. Central limit theorem.

PREVIOUS YEAR QUESTION PAPER

SET 2019/7

2. If   A = [√3/2]i+[1/2]j  & B = [1/2]i+[√3/2]j   then the angle between the two vectors is:

A) 60°      B) 45°    C)[pi/6] radians     D)[ pi/3] radians

3. Which among the following is TRUE?

A) ∇2φ = ∇. (∇φ)    B) ∇2φ = ∇X(∇φ)    C) ∇2φ = ∇X(∇Xφ)   D) None of these

4. Which among the following quantities is not a vector?

A) moment of inertia   B) linear momentum   C) angular momentum   D) moment of a force

5. Every square matrix A can be uniquely expressed as the sum of ------.

A) a Unitary matrix and a non-unitary matrix

B) a Hermition matrix and a Skew-Hermition matrix

C) two Hermition matrices       D) a Hermition matrix and a unitary matrix

6. Which among the following statement is TRUE for a Skew-Hermition Matrix?

A) diagonal elements must be real numbers

B) diagonal elements should be complex numbers with nonzero real and imaginary

parts

C) diagonal elements can only be pure imaginary numbers

D) diagonal elements can be only pure imaginary numbers or zero

7. For a singular matrix A, which among the following statements is TRUE?

A) |A| = 0      B) |A| < 0     C) |A| > 0     D) |A| ≠ 0

8. Newton’s equation of motion is a -------differential equation of ------order

A) non-linear, second   B) linear, second    C) nonlinear, first D) none of these

9. Find the Fourier sine transform of  exp(-x)

A) [n^2]/[1-n^2]      B)[1-n^2]  /[n^2]    C) n/[1+n^2]    D)[1+n^2]/[1-n^2]

10. The residue of      Z/{[Z-a]*[Z-b]}   at infinity is

A) a/b   B) − b/a    C) 1         D) −1

11. If f(z) is a regular function of z and if f(z) is continuous at each point within and on a

closed contour C, then ∫c f(z)dz = 0 . This is ---------.

A) Cauchy’s theorem B) Rieman equality   C) Residue theorem D) Drichlett condition

12. For the function f(z)= [z^4]/{([z-1]^4)*[z-2]*[z-3]}; the pole at z =1 is of -------order.

A) 1        B) 2       C) 3        D) none of these

13Give          d2y/dx2  -2x dy/dx+2vy = 0

, where v is a parameter . This differential equation is   called -----.

A) Legendere’s equation                       B) Laguerre’s equation                                                      C) Hermite equation                         D) Associated Lauguerre’s equation

14. If the probability that a horse A winning a race is 1/3 and the probability that another

horse B winning the race is 1/5 , the probability that one of the horses wins is --------

A) 1/15        B) 1/3          C) 2/5           D) None of these

SET 2019/2

81. If the matrixand A^2 = 49 , then the value of α is

A) 0 B) 1 C) 2 D) +-3

82. The eigen values of a skew – Hermitian matrix are

A) Zero B) Imaginary

C) Real D) Both A and B

83. For Laguerre polynomials, ∫ 0to∞f(t)Ln(t)Lm(t)dt=delta nm , where f(t)=

A) 1 B) exp(-t) C) t D) exp(-t2/2)

84. Value of ∫ 0to pi {d (theta)/(2- cos theta) is

 B) pi/√3 

85. Laplace transform of sinh(at)    for    s > 0 is

A)a/[s^2-a^2]

86. Let P be a (n x n) diagonalizable matrix. Given P is idempotent with Trace (P) = n-1.

Then det(P)=

A) 1 B) 0 C) n D) n2

88. As sample size increases, the sampling distribution must approaches to normal

distribution is termed as

A) Limited approximation theorem      B) Secondary limit theorem

C) Primary limit theorem                       D) Central limit theorem

89. A possible unit tangent vector to the plane x2+y2+z2 = 4 at (3,2,1) is

 D) [-2i/√13]+[3j/√13 ]

90. Bessel function J1/2(x) varies as

 C) cos (x)/√x