Online Multiple Choice Questions of EM-II of Unit-III

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1 Online Multiple Choice Questions of EM-II of Unit-III. The primitive period of a constant function is does not exist ANS: d. The constant function is an odd function an even function neither even nor odd none of these ANS: C 3. If f(x)is an even function then its graph is symmetric about y axis x axis opposite quadrants none of these. If the graph of f(x)is symmetric about origin then f(x)is an an even function an odd function neither even nor odd None of these ANS: b 5. The value of the fourier coefficient b n in half range cosine series of sinx in x is

2 n n 6. The value of the fourier coefficient a 5 in half range sine series of cosx in x is n ANS: b n 7. The value of the fourier coefficient a of an even function f(x)in x is ANS: c f(x)dx f(x)dx f(x)dx 8. The constant term in Fourier series is a a 3 a a 5 9. The value of b n in the Fourier series of f(x) = x in x is

3 cosn n cosn n cosn n ANS:C. The value of the constant term in the Fourier series of f(x) = x in x is ANS: b 3 6. The value of the fourier coefficient a in the fourier series of f(x) = x x in x 3 and period = 3 is 3. The value of a n in the Fourier series of f(x) = x, < x < is ANS:b n n n n

4 3.The amplitude of 5th harmonic is a 5 + b 5 a 5 b 5 a 5 + b 5 ANS: d. The percentage st harmonic is 5 does not exist 5. The value of the constant term in the Fourier series of is if < x < f(x) = { x if < x < cosx if < x < 6. The value of an in the Fourier series of f(x)= { cosx if < x < is

5 ( ) n n n ( ) n n ANS: d 7. The value of b n in the Fourier series of f(x) = x in < x < is cosn n cosn n ANS: C cosn n 8. The value of the constant term in the Fourier series of f(x) = x ; < x < is 3 6 ANS: b 9. Fourier Coefficient for Odd function f(x) defined in the interval x and f(x + ) are

6 a =, b n =, a n = a =, a n =, b n = f(x)cosnxdx a =, a n =, b n = None of these ANS: c f(x)sinnx dx. The Curve represented by the equation x(x + y ) = a(x y ) is ANS: b Symmetrical about y axis and passing through origin Symmetrical about x axis and passing through origin Symmetrical about y axis and passing through (a, ) Symmetrical about x axis and not passing through origin. If the curve passes through origin then tangents at origin to the Cartesian curve can be obtaining by equating to zero Coefficient of lowest degree term in the equation Coefficient of highest degree term in the equation Highest term in the equation Lowest degree term in the equation ANS:.The value of a in Harmonic analysis of y for the following tabulated data:.83 X Y

7 none of these ANS: c 3.The value of / sin (x)dx is 8 ANS:b 3.If In = / Cos n x dx then which is the following relation is correct In = n n In In = n n In In = n n In + None of these 5. Fourier Series representation of periodic function f(x)with period which satisfies the D F(x) = a + [a n cosnx + b n sinnx] F(x) = a + [a n cosnx + b n sinnx] F(x) = a + [(a n cosnx)(b n sinnx)]

8 ANS:a None of these 6. In Harmonic analysis the term a cosx + b sinx is called First harmonic Second harmonic Third harmonic None of these 7. In Harmonic analysis the amplitude of first Harmonic analysis a cosx + b sinx is a + b a b a + b a b 8. The value of sin 9 x dx is /35 8/35 5/35 /8 ANS:

9 9. If I n = (logx) n dx then ANS: C I n + n I n = x(logx) n I n n I n = x(logx) n I n + I n = x(logx) n None of these 3. Gamma function of n(n > ), is defined as ANS: c e x x n e x x n e x x n dx dx dx None of these 3. Γ is equal to ANS: 3. Reduction formula for Gamma function is Γ(n + ) = (n )Γ(n ) Γ(n + ) = n Γ(n) Γ(n + ) = (n ) Γ(n)

10 ANS: b None of these 33. Beta Function B(m, n) is x m x m x m ( x) n dx ( x) n dx None of these ( x) n dx b 3. If I(α) = f(x, α) a dx where a, b are functions of parameter α then by DUIS Rule di(α) dα is ANS : b a α b a α b a α None of these f(x, α)dx + f(a, α) da db f(b, α) dα dα f(x, α)dx + f(b, α) db da f(a, α) dα dα f(x, α)dx f(b, α) db da f(a, α) dα dα 35. Error function of x, erf(x) is defined as e u du x e u e u du du

11 None of these ANS: b 36. Error function of, erf() is None of these ANS: c 37. Complimentary error function of, erfc() is None of these ANS: 38. Error function is An odd function An even function A periodic function A Harmonic function 39. Which of the following is true erf(x) erfc(x) =

12 ANS: b erf(x) + erfc(x) = erf(x) + erfc(x) = None of these. The number of loops in rose curve r = acosθ ANS: c 3 5. A double point is called cusp if the tangents to the curve at the double point are Real and equal Imaginary Always perpendicular Real and distinct. If U n = sin n x dx and U n = n U n then the value of U is n 3 3 8

13 8 ANS: c 3. If U n = tan n x dx then the value of U n + U n+ is n n n ANS: d. If U n = cos n x cosnx dx and U n = U n then the value of U is ANS: c 5. If U n = sinnx dx = U sinx n, then the value of U n is n

14 6. If I n = xsin n x dx = + n I n n n, then the value of I ANS:b 7. If U n = sin(n )x sinx dx and n(u n+ U n ) = sin ( n n ), then the value of U 3 is ANS:d + 8. The value of the integral sin m x cosx dx is m m m+ m+

15 ANS: d 9. The value of the integral tan n x dx using reduction formula is ANS: c I n = I n I n = I n I n = n I n I n = I n 5. If = [logx] n dx then the reduction formula is ANS:a I n + ni n = x[logx] n I n + ni n = I n + ni n = I n + ni n = 5. If I n = xcos n x dx = n n I n n, then the value of I is + 6 ANS:a The value of I, if I n = cos n x cosnx dx is

16 ANS: b 53. If U n = sin (n )x sinx dx and n(u n+ U n ) = sin ( n n ), then the value of U is + ANS:b 5. If I n = sin n x dx = n I n then the value of I is n ANS:b 55. If U n = tan n x dx and U n+ = n U n then the value of U + U is

17 ANS:b 56. The value of the integral sin 7 x cos x dx is ANS:b 57. The value of the integral sin 8 x cos x dx is The value of the integral sin 6 x dx is 8

18 ANS: b 59. The value of the integral cos 8 x dx is ANS:a 6. The value of the integral cos n x dx is ANS: 3 cos n x dx sin n x dx sin n x dx none of these 6. The gamma function of (n + ) is defined as e x x n dx e x x n dx

19 ANS: b e x x n dx e x x n dx 6. The value of the integral e x x 5 dx is 5! 3! 6!! ANS:c 63. The value of the integral e x dx is none of these ANS:a 6. The value of the integral logx dx is ANS: b

20 65. The value of the integral ( x n ) m dx ANS: b β(n, m + ) nβ(n, m + ) β(m, n) none of these is 66. The value of the integral ( x) n dx is β(, n) β(, n) β(n, ) β(3, n) ANS:a 67. The value of β(,) is ANS:c 68. The value of β(3,3) is 3 3

21 3 none of these ANS: b 69. The value of β(5,9) β(9,5) is ANS: c 7. The value of the integral x 3 (+x) 6 dx is 3 ANS: b 7. The value of the integral tanθ dθ is 3

22 ANS: b 7. The value of the integral dθ sinθ sinθ dθ is ANS: c 73. The value of the integral tanθ dθ cotθ dθ is 7. The value of the integral dx x p+ (x ) q is β(p + q, q) β(p, q) β(p, q) β(p + q, q) ANS: d

23 75. The value of the integral sinθ dθ is ANS: b 76. The value of nβ(m +, n) is β(m, n + ) mβ(m, n + ) β(m, n) none of these ANS: 77. The value of β(n, n + ) is Ґn Ґn (Ґn) Ґn Ґn Ґn Ґn (Ґn) ANS: 78. The value of β(, ) dθ sinθ

24 β(, ) β(, ) β(, 3 ) ANS: b 79. The value of β(m +, n) is β(m, n) m m+n n m+n β(m, n) β(m, n) ANS:b none of these 8. The value of t ANS: b β(, ) β( 5, 5 ) β( 5, ) β( 5, 3 ) ( t) dt is 8.Using duplication formula, the value of Ґ Ґ 3 is

25 ANS: c 8. The value of t n ANS:c β(m, n) β ( m, n + ). β(n, m + ) none of these ( t) m dt is 83. Which of the following is relation between Bete and Gamma function? ANS.c β(m, n) = ҐmҐn Ґm β(m, n) = ҐmҐn ҐmҐn = Ґm + nβ(m, n) β(m, n) = ҐmҐn Ґm+n+ 8. Which of the following is an odd function? sinx e x + e x e x x

26 85. Which of the following is an even function? ANS:d sinx e x e x xcosx cosx 86. Which of the following function is neither even nor odd function? ANS.c xsinx x e x xcosx 87. sin x dx is equal to 8 ANS: d 88. cos x dx is equal to 8

27 89. sin 6 t dt is equal to ANS:a 9. sin 6 t dt is equal to ANS:b 9. sin 6 xcos x dx is equal to

28 ANS.b 9. sin 6 xcos x dx is equal to ANS:c x 93. If y = f(t) sin(x t) dt then by DUIS rule II, dy ANS:c x af(t)sina(x t)dt x f(t)cosa(x t)dt x af(t)cosa(x t)dt x af(t)cosa(x t)dt + f(t) dx is 9. The value of integral x(8 x 3 ) 3dx by busing substitution x 3 = 8t is 3 β( 3, 3 ) 3 β ( 3, 3 ) 3 β ( 3, 3 )

29 ANS:d 8 3 β( 3, 3 ) 95. Using duplication formula, value of β (m, ) is m β(m, m) m β(m, m) m β(m, m) none of these ANS:C 96. The value of the integral e x x dx is ANS: d 97. The value of I, if I n = cos n xcosnx dx is ANS:b

30 98. The value of a in the cosine series of f(x) = lx x ; < x < l is l 3 l 6 l l 99. The value of the Fourier coefficient a 5 in half range sine series of cosx in x is n n ANS: b. The value of the integral cos 8 x dx ****************

SOLUTIONS FOR PRACTICE FINAL EXAM

SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable