Mathematical Physics Quiz for JAM, JEST, CUET PG, CPET & other MSc Physics Entrance Exams.

1. The function ecosx is Taylor expanded about x=0. The coefficient of x2 is
2. Let M be a 2×2 matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are
3. The solution y(x) of the differential equation ydydx+3x=0, y(1)=0 is described by
4. The volume integral Ve(r/R)2(r^r2)d3r, where V is the volume of a sphere of radius R  centered at the origin, is equal to
5. If P and Q are ermitian matrices, which of the following is/are true?

A. PQ+QP is always Hermitian.

B. i(PQQP) is always Hermitian.

C. PQ is always Hermitian.

D. PQQP is always Hermitian.

Choose the correct answer from the options given below:
6. The line integral of the vector function A(x,y)=2yi^+xj^ along the straight line from (0, 0) to (2, 4) is
7. The function f(x)=8xx2+9 is continuous everywhere except at
8. If ϕ(x,y,z) is a scalar function which satisfies the Laplace equation, then the gradient of ϕ is
9. The eigenvalues of (3i0i30006) are
10. The gradient of a scalar field S(x,y,z) has the following characteristic(s).

A. Line integral of the gradient is path independent.

B. Closed line integral of the gradient is zero.

C. Gradient is a measure of the maximum rate of change.

D. Gradiet is scalar quantity.

Choose the most appropriate answer from the options given below:
11. Let f(x,y)=x32y3. The curve along which 2f=0 is
12. A curve is given by r(t)=ti^+t2j^+t3k^. The unit vector of the tangent to the curve at t=1 is
13. The function f(x)={x,π<x<0x,0<x<π is expanded as a Fourier series of the form a0+n=1ancos(nx)+n=1bnsin(nx). Which of the following is true?
14. Let f(x)=3x62x28. Which of the following statements are true?

A. The sum of all its roots is zero.

B. The product of its roots is 83.

C. The sum of all its roots is 23.

D. Complex roots are conjugates of each other.
15. For the Fourier series of the following function of period 2π,

f(x)={0,π<x<0,1,0<x<π, the ratio of the Fourier coefficients of the first and the third harmonic is:
16. If λ is an eigen value of a non-singular matrix A, then the eigen value of A1 is
17. A unit vector perpendicular to the plane containing A=i^+j^2k^ and B=2i^j^+k^ is
18. A hemispherical shell is placed on the x-y plane centered at the origin. For a vector field E=yi^+xj^x2+y2, the value of the integral S(×E)ds over the hemispherical surface is
19. Which of the following is not true for Hermite polynomials?
20. The modulus and phase of the complex number (1+i)i in polar representation are
21. Consider the differential equation y+2y+y=0. If y(0)=0 and y(0)=1, then the value of y(2) is
22. Consider a 2×2 matrix M=(0aab), where a,b>0. Then

A. M is real symmetric matrix.

B. One of the eigenvalues of M is greater than b.

C. One of the eigenvalues of M is negative.

D. Product of eigenvalues of M is b.

Choose the most appropriate answer from the options given below:
23. The equation z2+z¯2=4 in the complex plane (where z¯ is the complex conjugate of z) represents
24. Consider a unit circle C in the xy plane, centered at the origin. The value of the integral [(sinxy)dx(sinyx)dy] over the circle C, traversed anticlockwise is
25. Consider a vector field F=yi^+xz3j^zyk^. Let C be the circle x2+y2=4 on the plane z=2, oriented counter-clockwise. The value of the contour integral Fdr is
26. Let (x,y) denote the coordinates in a rectangular Cartesian coordinate system C. Let (x,y) denote the coordinates in another coordinate system C, defined by

x=2x+3y

y=3x+4y.

The area element in C is
27. The unit vector perpendicular to the surface x2+y2+z2=3 at the point (1,1,1) is
28. What is the equation of the plane which is tangent to the surface xyz=4 at the point (1,2,2)?
29. If r is a position vector, rnr is solenoidal for
30. Three vectors A, B and C are given by A=αi^2j^+2k^, B=6i^+4j^2k^ and C=3i^2j^4k^. The value of α for which the vectors will be coplanar is
31. If F is a constant vector and r is the position vector then (Fr) would be
32. Let r be the position vector of a point on a closed contour C. What is the value of the line integral rdr ?
33. The value of the integral x2eax2dx, where a>0 is
34. One of the solutions of the equation (1x2)d2ydx22xdydx+12y=0 is
35. The directional derivative of ϕ=x2y+xz at (1,2,-1) in the direction A=2i^2j^+k^ is
36. If δ(x) is Delta function then
37. The Fourier transform of e|x| is
38. The residue of the complex function  f(z)=e1/z  at z=0 is
39. If  J1/2(x) are J1/2(x) are Bessel's functions, the value of [J1/2(x)]2+[J1/2(x)]2 is
40. For the function f(z)=zsinz(zπ)3, the residue at the pole z=π is
41. The value of Γ(3/2) is
42. If r=xi^+yj^+zk^ is the position vector, then the value of (logr) is
43. The value of α so that eαy2 is an integrating factor of the differential equation (ey2/2xy)dydx=0 is
44. The value of i+i, where i=1 is
45. The value of the integral Cz2+1(z+1)(z+2)dz, where C is |z|=32 is
46. Let Pn(x) be the Legendre polynomial of degree n>1, then the value of the integral 11(1+x)Pn(x)dx is equal to
47. Laplace transform of e2tsin4t is
48. The value of the integral π/2π/2sin2θδ(3θ+π)dθ is
49. Out of the given equations, the only equation which is an exact differential is
50. The Laplace transform of t3δ(t4) is