Questions & Answers

Question No: 1
The fourier transform of a convolution f*g is equal to
A)
The sum of fourier transform of f*g
B)
The difference of fourier transform of f*g
C)
The product of fourier transform of f*g
D)
The quotient of fourier transform of f*g
Correct Answer - C
Question No: 2
If f ∈ L[-π,π] then the trigonometric polynomial closest to f in the metric for If
f ∈ L2[-π,π] is
A)
The fourier series for f
B)
The nth partial sum of the fourier series for f
C)
The fourier series for f2
D)
None of these
Correct Answer - B
Question No: 3
If f ∈ L[-π,π] and x ∈ [-π, π] then the fourier series for f at x converges to
A)
F(x)
B)

C)

D)
none of these
Correct Answer - B
Question No: 4
If f is a continuous complex 2π periodic function on R[0, 2π] then the arithmetic
means of the partial sum of the fourier series of f converge uniformly to f”. It is
according to
A)
Fejer
B)
Lebesgue
C)
Fisher
D)
Bessel
Correct Answer - A
Question No: 5
If h(x) is an odd function then  =
A)

B)

C)
2π
D)
0
Correct Answer - D
Question No: 6
The Fourier transform of 
A)
√π {as .cos sa –sin sa}
B)
√π {as sin sa + sin sa}
C)
 {s cos sa +sin sa}
D)
 {s cos sa +sin sa}

Correct Answer - A
f(t) = 

F{f (t)} =  f(t) dt

= f(t) dt

= 12π 

= 

=  

= 

= 

= 12π 2iass2cossa-2is2sin(sa)

= √2is2√π as .cossa-sinsa
Question No: 7
 ds =
A)

B)
0

C)

D)

Correct Answer - C
 ds = π x<a0 x>a

Put, x=0 , a=1




Question No: 8
Find the Fourier transform of f(x) if f(x) =∈
A)

B)

C)

D)

Correct Answer - A
F(s) = dx
= 
= 
= 
= 
Question No: 9
Find the Fourier sine transform of f(x) = 
A)
√π2
B)
√2π
C)
π2
D)
2π
Correct Answer - A
 = sin sx dx
= 

Put , y = sx dy = sdx


 =  = 
=  

=
Question No: 10
Find the Fourier cosine transform of
A)

B)

C)

D)

Correct Answer - B


= 
= 
= 
Question No: 11
The fourier series of an even function f(x) with period 2π is
A)
 + a1 cosx + cos2x + . . .
B)

C)
sinx + sin2x + . . .
D)

Correct Answer - A
Question No: 12
f and g are absolutely integrable functions on (-∞, ∞) then the fourier transform
of the convolution f*g is F(f*g) is
A)
F(f) + F(g)
B)
F(f) - F(g)
C)
F(f) . F(g)
D)
F(f) / F(g)
Correct Answer - C
Question No: 13
The fourier series generated by f ∈ R(0, 2π) with in the period 2π is cesaro
summable and has (c, 1) sum s(x) which is equal to
A)

B)

C)


D)

Correct Answer - A
Question No: 14
If h(x) is an odd function then  =
A)
π
B)

C)
2π
D)
0
Correct Answer - D
Question No: 15
The primitive period of the function sin2x is
A)

B)
0
C)
2π
D)

Correct Answer - A
Question No: 16
The fourier series for the function f(x) = x() in 0≤x is f(x) =  – 4 [  + 
+ . . . ] then  -  +  . . . ∞ is
A)

B)

C)

D)

Correct Answer - B
Question No: 17
The fourier series for the function f(x) = x2 in the interval (-π, π) is
f(x) = π33 + 4n2 (-1)n cosnx, then 112 + 122 + 132 . . . ∞ is
A)

B)

C)

D)

Correct Answer - C
Question No: 18
If {θ0, θ1, … } is orthonormal on I and {Cn} is any sequence of complex number
such that Ck2 converges then f ∈ L2(I) such that (f, θk)Ck k≥0 and f2
= k=0∞Ck2 this statement is according to
A)
Fourier integral theorem
B)
Riez – fisher theorem
C)
Fejei’s theorem
D)
Convolution theorem
Correct Answer - B
Question No: 19
Fourier cosine transform of e-axx is

A)

B)

C)
-1√2π log(s2+a2)
D)
1√2π log (s2+a2 )
Correct Answer - C
Fourier cosine transform of Type equation here.
 = 
Question No: 20
Find the Fourier transform of f(x) = 
A)

B)

C)
e-s22
D)
e -2s
Correct Answer - A
The Fourier transform of
e-x22 is e-x22
Question No: 21
Find 

A)

B)

C)

D)

Correct Answer - A
Fourier sine transform of
 = 
= 
Question No: 22
If F (f (x-a)] = k F(s) then k =
A)

B)

C)

D)

Correct Answer - C
Formula :
F[F(x-a)] = 
Question No: 23
The fourier transform which is completely symmetrical was first given by
A)
Euler
B)
Cauchy
C)
Plancherel
D)
Fisher
Correct Answer - C
Question No: 24
Consider the function (32 to 35) f(x) =  = , period = 10 find 
A)

B)

C)
0
D)
None of these
Correct Answer - C
Question No: 25
Find the fourier coefficient 
A)

B)

C)

D)
None of these
Correct Answer - B
Question No: 26
the function f(z) = sinz is
A)
Odd function
B)
Even function
C)
Not at all function
D)
None of these
Correct Answer - A
Question No: 27
Find the fourier transform of F(x) defined by 
A)

B)

C)

D)
None of these
Correct Answer - A
Question No: 28
IfFs (f (ax)) = kFssa then, k=
A)
a
B)

C)

D)
1√a
Correct Answer - C
Formula :


Question No: 29
F (f(ax)) = Kf  then, k =
A)

B)

C)
a
D)
1|a2|
Correct Answer - B
Formula :
F(f(ax)) =

Question No: 30
Which of the following function is self-reciprocal under Fourier cosine transform

A)

B)

C)
√xa
D)
x2e/x
Correct Answer - B
Formula :
Fs1x = 
 is self reciprocal under fourier cosine transorm
Question No: 31
Let F(f(x)) = F(S) then,  [F(s+a) +F(s-a)]
A)
F[f(x) cos ax]
B)
F[f(x) sinax]
C)
F[f(x) tan ax]
D)
F [F(X+ 2a]
Correct Answer - A
By Modulation theorem,
F(f (x) cos ax) = [F(s+a) + F(s-a)]
Question No: 32
Find  (xf(x))
A)
 (f(x))
B)
 (f(x))
C)
-dFsds (f(x))
D)
None of thes
Correct Answer - A
Formula :
Fs =f(x)
Question No: 33
Find 
A)

B)

C)
0
D)
None of these
Correct Answer - A
Question No: 34
Using the above result find  dp
A)

B)

C)

D)
None of these
Correct Answer - B
Question No: 35
Find the fourier series if the period is not specified
A)
f(x) = 
B)
f(x) = 1
C)
can’t be determined
D)
none of these
Correct Answer - C
Question No: 36
Find the fourier transform of f(x) if 
A)

B)

C)
Sinp cosp
D)
None of these
Correct Answer - A
Question No: 37
The value of 
A)

B)
0
C)
1
D)
-1
Correct Answer - A
Question No: 38
Find finite Fourier sine transform of f(x) =  0<x<1

A)

B)


C)

D)
None of these
Correct Answer - C
By preious problem,

Put, x = 0
0 =

= 

Question No: 39
F(x) = 12 (π-x)  in the interval (0, 2π ). Then a0 ,  are respectively
A)
1, 0, 
B)
0, 1, 0
C)
0, 1, n
D)
0, 0, 1n
Correct Answer - D
F(x) = 
= 
= 
= 



 


=
= 1n

= +…

Question No: 40
F (x) =  =  Then
A)

B)
sin nx dx
C)
2π0xf(x) Cos nx dx
D)
none of these
Correct Answer - C
Half range series
Fourier cosine series
F(x) = 
Where 
an= 
Question No: 41
The Fourier transform of a convolution f * g is equal to
A)
the sum of fourier transform of f and g
B)
the difference of fourier transform of f and g
C)
the product of the fourier transform of f and g
D)
the quotient of the fourier transform of f and g
Correct Answer - C
The fourier transform of a convolution f * g is equal to the product of the fourier
transform
F and g
Question No: 42
The Fourier co efficient of a Lebsque integrable function must
A)
Approach as k
B)
Approach zero as k
C)
Approach zero as k
D)
None of these
Correct Answer - B

Question No: 43
 = is
A)
Parseval’s formula
B)
Poission’s formula
C)
Bessel’s formula
D)
Riesz – fischer’s formula
Correct Answer - A
Question No: 44
Let f be real valued and continuous on a compact interval [a, b] then for every
t>0 there is a polynomial such that  it is
A)
Parseval’s theorem
B)
Weierstrass theorem
C)
Converse of weierstrass theorem
D)
None of these
Correct Answer - B
Question No: 45
An infinite collection is called linearly independent on [a, b]
A)
Linearly dependent on [a, b]
B)
Linearly independent on[a, b]
C)
Both
D)
None of these
Correct Answer - B
Question No: 46
The infinite fourier sine transform of f(x) = 0 is also defined by [f(x)] = 
the function f(x) is called
A)
Inverse fourier sine transform of 
B)
Inverse fourier cosine transform of 
C)
Inverse function
D)
None of these
Correct Answer - A
Question No: 47
The series  converges and satisfies the inequality  is called
A)
Parseval’s formula
B)
Conjugate formula
C)
Legendre’s inequality
D)
None of these
Correct Answer - B
Question No: 48
If  then the trigonometric polynomial closet to f in the metric for  is

A)
TheFourier series for f
B)
The  partial sum of the Fourier series for f
C)
TheFourier series for 
D)
None of these
Correct Answer - B
If f , then the trigonometric polynomial closest to f (in the metric for 
sn the  partial sum of the fourier series for f
Question No: 49
he Fourier series generated by f with period 2π is cesaro summable and has
(c,1) Sum s(x) which is equal to
A)

B)

C)

D)
(x+t) – f(x-t)
Correct Answer - B
F is the sequence of partial sums then,

Question No: 50
Fourier series for f (x) is an odd function of x in the interval (- is f(x) =  +
 +  then

A)

B)


C)
an=0, bn=0
D)
a0=0,bn=0
Correct Answer - C
F(x) = + (-1)n cos nx

Put, x = 




= 
Question No: 51
If{ is ortho normal on I , and { is any sequence of complex number such that 
converges, then  (I) such that (f, ∅k)=Ck,k≥0 and  This statement is
according to
A)
Fourier internal theorem
B)
Risize–Fischer theorem
C)
Fourier theorem
D)
Convolution theorem
Correct Answer - C
Riesz - F ischer Theorem
If { is orthonormal on I, and {is any sequence of complex number such that
converges then, there is a function f in L2 Isuch that,( f , ∅k) = Ck for each
k ≥0 |f|2= k=0∞|Ck|2
Question No: 52
The theta function  given by θx=n= x > 0 can also given as the transform
equation
A)

B)

C)
θx=1xθ x, x>0
D)
θx= 1x θ1x,x>0
Correct Answer - B
Transformation formula for the theta function  Is 
Question No: 53
The converse of the bessel’s inequality is
A)
Riesz fisher theorem
B)
Convolution heorem
C)
Not true
D)
None of these
Correct Answer - A
Question No: 54
By using appropriate fourier series conjuction with parseval’s formula the value of
ε(4) =
A)
0
B)

C)

D)

Correct Answer - C
Question No: 55
If g is of bounded variation on [0, δ] then 
A)
g(0+)
B)
f(0+)
C)
g(0-)
D)
f(0-)
Correct Answer - A
Question No: 56
In the fourier integral theorem the integral must be an
A)
Proper Riemann integral
B)
Improper Riemann integral
C)
Lebesgue Riemann integral
D)
Riemann stieldges integral
Correct Answer - B
Question No: 57
The convolution will be defined at each point of an interval [a, b] if both
A)
f & g are Riemann integrable on [a, b]
B)
f & g are improper integrable on [a, b]
C)
f & g are lebegue integrable on [a, b]
D)
None of these
Correct Answer - A
Question No: 58
For each a >c,  = 
A)
Inversion formula
B)
Convolution formula
C)
Parseval’s formula
D)
None of these
Correct Answer - A
Question No: 59
The fourier coefficient  for f(x) =  in (0, 2π) is
A)

B)

C)

D)
0
Correct Answer - A
Question No: 60
If f is continuous on [] and as period 2π then {} converges uniformly to f
on [-] it is know as
A)
Fourier integral theorem
B)
Bessel’s theor
C)
Fejer theorem
D)
Wiestrass approximation theorem
Correct Answer - C
Question No: 61
Let f be continuous on (0, ∞) for z = x + iy define F(z) =  let g(x) =  for
x>c>0 if s >cand a>0 then  =
A)
 F(s+a)
B)
F(s+a)
C)

D)

Correct Answer - C
Question No: 62
Let  + k=1∞  be the fourier series of f(x) in  then for the function f(x)
= 
A)
f is even and bn=0
B)
f is even and an=0
C)
f is odd and bn=0
D)
f is odd and an=0
Correct Answer - A
Question No: 63
If F(f(x)) =  is
A)

B)

C)

D)

Correct Answer - C
Question No: 64
If a function f is not continuous at C then point C is a removable discontinuous if

A)
Either f(C+) or f(C-) does not exist
B)
Both f(C+) and f(C-) exist but have different values
C)
Both f(C+) and f(C-) exist and f(C+) = f(C-) ≠  f(C)
D)
All the above
Correct Answer - C
Question No: 65
If as m & n tend independently to ∞  then there exists a function  to which 
converges in mean with index p is a
A)
Fejer’s lebsgue theorem
B)
Riesz fisher theorem
C)
Fejer’s theorem
D)
None of these
Correct Answer - B
Question No: 66
The value of fourier coefficient a0 to the function f(x) = x in the interval 
is
A)

B)

C)
0
D)

Correct Answer - B
Question No: 67
Assume that f  then for each real β we have 

A)
0
B)
1
C)
-1
D)
None of these
Correct Answer - A
Question No: 68
In poisson summation formula each series being
A)
Absolutely convergent
B)
Conditionally convergent
C)
Convergent
D)
Oscillates
Correct Answer - A
Question No: 69
The value of  is
A)
0
B)

C)

D)

Correct Answer - C
Question No: 70
The fourier series generated by f  with period 2π is cesaro summable and has
(C, 1) sum s(x) which is equal to
A)

B)

C)

D)

Correct Answer - A
Question No: 71
Fourier series of f  at x is (C, 1) summable to f(x) if
A)
f is not continuous at x
B)
f is decreasing at x
C)
f is increasing at x
D)
f continuous at x
Correct Answer - D
Question No: 72
If f(x) has period α then f() has period
A)

B)
Kα
C)

D)

Correct Answer - B
Question No: 73
f ,  + k=1∞ ( is the fourier series for f as n→∞
A)

B)

C)

D)

Correct Answer - C
Question No: 74
the convolution of two function f & g is
A)
(f*g)x = f(x)g(t-x)
B)
(f*g)x = 
C)
(f*g)x = f(x)g(x-t)
D)
(f*g)x = 
Correct Answer - D
Question No: 75
If a function f is not continuous at C then point C is a removable discontinuous if
A)
Either f(C+) or f(C-) does not exist
B)
Both f(C+) and f(C-) exist but have different values
C)
Both f(C+) and f(C-) exist and f(C+) = f(C-) ≠  f(C)
D)
All the above
Correct Answer - C
Question No: 76
The convolution of two integrable functions f & g are defined by the equation is
A)
 
B)
 
C)
 
D)
 
Correct Answer - B
Question No: 77
For  if the fouier coefficient ak, k=0, 1, 2, … and bk, k=0, 1, 2, … are
all zero
A)

B)

C)

D)

Correct Answer - A
Question No: 78
Let f(x) =  then the value of fourier coefficient  is
A)

B)

C)

D)

Correct Answer - B
Question No: 79
If f  and f is continuous at  then f(x) can be expanded as a fourier series
A)

B)

C)

D)

Correct Answer - B
Question No: 80
If f & g are in  having same fourier coefficient then
A)
g = 0
B)
f = 0
C)
f ≠ g
D)
f = g almost everywhere
Correct Answer - D
Question No: 81
The fourier transform which is completely symmetrical was first given by
A)
Euler
B)
Cauchy
C)
Plancherel
D)
Fisher
Correct Answer - C
Question No: 82
The value of fourier coefficient  to the function f(x) = x in the interval 
is
A)

B)

C)
0
D)

Correct Answer - B
Question No: 83
If f(x) is an even function in  the value of the fourier coefficient  is
A)
1
B)
0
C)

D)

Correct Answer - B
Question No: 84
Fourier sine transform  is
A)
 
B)

C)
 
D)

Correct Answer - D
Question No: 85
The fourier series for f(x) = x in (- ) is
A)
2{ . . .}
B)
2{ . . .}
C)
{ . . .}
D)
{  . . .}
Correct Answer - A
Question No: 86
A function f(x) is said to be an odd function if
A)

B)

C)

D)

Correct Answer - A
Question No: 87
Let f(x) = 1 if  and if the fourier cosine transform for f(x) is

A)

B)

C)

D)

Correct Answer - C
Question No: 88
If F(s) is the fourier transform of f(x) then -∞∞f(x)2 dx = 0
A)

B)

C)

D)

Correct Answer - C
Question No: 89
If the series f(x) =  is defined in the interval (0,  ) then the value of 
is

A)


B)

C)

D)

Correct Answer - A
Question No: 90
find
A)

B)

C)

D)

Correct Answer - D
Question No: 91
If cosx is a periodic function then the period
A)

B)
2
C)
3
D)
4
Correct Answer - B
Question No: 92
If F(s) is the fourier transform of f(x) then  is equal to
A)
 [f(s+a) + f(s-a)]
B)
 [f(s+a) - f(s-a)]
C)
 [F(s+a) + F(s-a)]
D)
 [F(s+a) - F(s-a)]
Correct Answer - A
Question No: 93
If a function f(x) is even function then the fourier coefficient  in the
interval (- ) is
A)
1
B)
2
C)
3
D)
0
Correct Answer - D
Question No: 94
The inverse fourier cosine transform of  is
A)

B)

C)

D)


Correct Answer - C
Question No: 95
If f(x) =  in  then the fourier coefficient  is
A)
0
B)

C)
2
D)

Correct Answer - B
Question No: 96
If f(x) =  in  then the fourier coefficient  is
A)
0
B)

C)
2
D)

Correct Answer - B
Question No: 97
In the fourier transformation  =

A)
 
B)
 
C)

D)

Correct Answer - A
Question No: 98
In the fourier series if f(x) is continuous at x=a then
A)

B)

C)

D)

Correct Answer - A
Question No: 99
If f(x) = is defined in the interva then the fourier coefficient an is
A)

B)

C)

D)

Correct Answer - D
Question No: 100
The inverse fourier transform of  is
A)

B)

C)

D)

Correct Answer - D
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