Mathematical Physics Quiz for JAM, JEST, CUET PG, CPET & other MSc Physics Entrance Exams. Please wait... 1. The function ecosx is Taylor expanded about x=0. The coefficient of x2 is−12−e2e20 2. Let M be a 2×2 matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are2 and 43 and 32 and 6-2 and -3 3. The solution y(x) of the differential equation ydydx+3x=0, y(1)=0 is described byan ellipsea circlea parabolaa straight line 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 to4π08π43πR3 5. If P and Q are ermitian matrices, which of the following is/are true?A. PQ+QP is always Hermitian.B. i(PQ−QP) is always Hermitian.C. PQ is always Hermitian.D. PQ−QP is always Hermitian.Choose the correct answer from the options given below:A, BC, DA, DB, C 6. The line integral of the vector function A→(x,y)=2yi^+xj^ along the straight line from (0, 0) to (2, 4) is12642 7. The function f(x)=8xx2+9 is continuous everywhere except atx=0x=±9x=±9ix=±3i 8. If ϕ(x,y,z) is a scalar function which satisfies the Laplace equation, then the gradient of ϕ issolenoidal and irrotationalsolenoidal but not irrotationalirrotational but not solenoidalneither solenoidal nor irrotational 9. The eigenvalues of (3i0−i30006) are2, 4 and 62i, 4i and 62i, 4 and 80, 4 and 8 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:A, B, CC, DA, DB, D 11. Let f(x,y)=x3−2y3. The curve along which ∇→2f=0 isx=2yx=2yx=6yx=−y/2 12. A curve is given by r→(t)=ti^+t2j^+t3k^. The unit vector of the tangent to the curve at t=1 isi^+j^+k^3i^+j^+2k^6i^+2j^+2k^3i^+2j^+3k^14 13. The function f(x)={x,−π<x<0−x,0<x<π is expanded as a Fourier series of the form a0+∑n=1∞ancos(nx)+∑n=1∞bnsin(nx). Which of the following is true?a0≠0,bn=0a0≠0,bn≠0a0=0,bn=0a0=0,bn≠0 14. Let f(x)=3x6−2x2−8. 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.A, B, DA, C, DC, DB, C 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:1236 16. If λ is an eigen value of a non-singular matrix A, then the eigen value of A−1 is−1λ−λλ1λ 17. A unit vector perpendicular to the plane containing A→=i^+j^−2k^ and B→=2i^−j^+k^ is135(−i^−5j^−3k^)135(−i^+5j^−3k^)119(−i^+3j^−3k^)126(−i^+3j^−4k^) 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 is2ππ4π0 19. Which of the following is not true for Hermite polynomials?H3(x)=8x3−12x+2H2(x)=4x2−2H1(x)=2xH0(x)=1 20. The modulus and phase of the complex number (1+i)i in polar representation are2 and 3π43 and 3π42 and π43 and π4 21. Consider the differential equation y′′+2y′+y=0. If y(0)=0 and y′(0)=1, then the value of y(2) is2e22e4e24e 22. Consider a 2×2 matrix M=(0aab), where a,b>0. ThenA. 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:A, B, CC, DB, DB, C ,D 23. The equation z2+z¯2=4 in the complex plane (where z¯ is the complex conjugate of z) representshyperbolaellipsecircle of radius 2circle of radius 4 24. Consider a unit circle C in the xy plane, centered at the origin. The value of the integral ∮[(sinx−y)dx−(siny−x)dy] over the circle C, traversed anticlockwise is02π3π4π 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 ∮F→⋅dr→ is28π4π−4π−28π 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 byx′=2x+3yy′=−3x+4y.The area element in C′ is117dx′dy′12dx′dy′dx′dy′9dx′dy′ 27. The unit vector perpendicular to the surface x2+y2+z2=3 at the point (1,1,1) isi^−j^+k^3i^+j^+k^3i^−j^−k^3i^+j^−k^3 28. What is the equation of the plane which is tangent to the surface xyz=4 at the point (1,2,2)?x+2y+4z=124x+2y+z=12x+4y+z=02x+y+z=6 29. If r→ is a position vector, rnr→ is solenoidal forn=−3n=3n=−2n=−1 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-112-3 31. If F→ is a constant vector and r→ is the position vector then ∇→(F→⋅r→) would be(∇→⋅r→)F→F→(∇→⋅F→)r→|r→|F→ 32. Let r→ be the position vector of a point on a closed contour C. What is the value of the line integral ∮r→⋅dr→ ?10-13 33. The value of the integral ∫−∞∞x2e−ax2dx, where a>0 is12πa312πaπ2a3π2a 34. One of the solutions of the equation (1−x2)d2ydx2−2xdydx+12y=0 isH4(x)P3(x)P4(x)H3(x) 35. The directional derivative of ϕ=x2y+xz at (1,2,-1) in the direction A→=2i^−2j^+k^ is53231352 36. If δ(x) is Delta function thenxδ(x)=0xδ(x)=1xδ(x)=∞xδ(x)=x 37. The Fourier transform of e−|x| is11+k221+k221+k11+k 38. The residue of the complex function f(z)=e1/z at z=0 is10-12 39. If J1/2(x) are J−1/2(x) are Bessel's functions, the value of [J1/2(x)]2+[J−1/2(x)]2 is1πx2πx12πx2xπ 40. For the function f(z)=zsinz(z−π)3, the residue at the pole z=π is12-1-2 41. The value of Γ(−3/2) is∞34π43πZero 42. If r→=xi^+yj^+zk^ is the position vector, then the value of ∇→(logr) isr→rr→r2−r→r3−r→r 43. The value of α so that eαy2 is an integrating factor of the differential equation (e−y2/2−xy)dy−dx=0 is−1112−12 44. The value of i+−i, where i=−1 is2120−2 45. The value of the integral ∫Cz2+1(z+1)(z+2)dz, where C is |z|=32 isπi02πi4πi 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 to01/(2n+1)2/(2n+1)n/(2n+1) 47. Laplace transform of e−2tsin4t is4s2+4s+20s−4s2+4s+20s−2s2+4s+202s2+4s+20 48. The value of the integral ∫−π/2π/2sin2θδ(3θ+π)dθ isπ/2π/41/41/2 49. Out of the given equations, the only equation which is an exact differential is(4x3y3−2xy)dx+(3x4y2−x2)dy=0(x2+y2+x)dx+xydy=0(3e3x−4x)dx+6e3xdy=0cosydx+(sinx−siny)dy=0 50. The Laplace transform of t3δ(t−4) ise−4s64e−4s4e−3s4e3s3 Loading...