1. No category

MSc-Mathematics
Sample Question Paper
Instructions:
Total Marks: 100
Pass Marks: 40
Time: 3 Hrs
Section A:
1. Contains total 30 Multiple Choice Questions (MCQ)
2. All Questions are mandatory
3. Each question carries 2 marks
4. No Negative Marks
Section B:
1. Contains total 02 Essay type Questions
2. All Questions are mandatory
3. Each question carries 20 marks
Note: 1. Only One Question will be displayed on screen at a time
2. Student will move to the next Question only after attempting the current
3. Student can’t go back and change the answer attempted previously.
Section A
Attempt all Questions
Total Questions= 30
Total Marks: 30 X 2 = 60 Marks
Complex Analysis
1. Find the Fourier series of the function f (x) = x, − π ≤ x ≤ π .
(
(a) f ( x ) : 2 sin ( x )
⎛
)
(c ) f ( x ) 2 ⎜ sin ( x ) −
⎝
(
(b) f ( x ) : sin ( x )
)
sin ( 2 x ) sin ( 3x ) ⎞
sin ( 2 x )
+
... ⎟ (d) f ( x ) :
2
2
3
⎠
Correct Answer (c)
2. Consider the figure. Identify which is incorrect.
(a) The function sin x has periods 2π , 4π , 6π ,...
(b) all are equal
(c) sin ( x + 2π ) ,sin ( x + 4π ) ,sin ( x + 6π ) ,...
(d) all sin x.
Correct Answer (a)
3. Find the amplitude spectrum and the phase spectrum of the 2π -periodic function f (x)
defined over interval as [ −π ; π ] as f (x) = ex.
(a) ϕ k = arg ck = arctan k
(c) ϕ k = arctan k
(b) ϕ k = arg ck
(d) ϕ k =
arg ck
arctan k
Correct Answer (a)
4. Let f (x) be defined in the interval [0, T] and determined outside of this interval by its periodic
extension, i.e. assume that f (x) has the period T.
(a) bn =
2 T
f ( x ) sin
T ∫0
(c) bn = 2
∫ f ( x ) sin nwx dx
T
0
2 T
f ( x ) sin nwx dx
T ∫0
2 n
(d) bn = ∫ f ( x ) sin nwx dx
T 0
(b) bn =
Correct Answer (b)
5. Let an and bn be the Fourier coefficients of f. What will be the Phase angle?
⎛ bn ⎞
⎟
⎝ an ⎠
⎛ bn ⎞
⎟:
⎝ an ⎠
−1
(a) δ n = tan ⎜ −
−1
(b) δ n = tan ⎜ −
−1
(c) δ n = tan ( −bn ) :
−1
(d) δ n = tan ( −an )
Section B
Attempt all Questions
Total Questions= 02
Total Marks: 2 X 20 = 40 Marks