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Unit 6 - Practice Quiz

MTH165 50 Questions
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1 Which of the following represents the trigonometric Fourier series of a periodic function defined in the interval ?

A.
B.
C.
D.

2 For a function defined in , Euler's formula for the coefficient () is given by:

A.
B.
C.
D.

3 For a function defined in , Euler's formula for the coefficient is given by:

A.
B.
C.
D.

4 If the Fourier series is expressed as , what is the formula for in the interval ?

A.
B.
C.
D.

5 Which of the following is NOT one of Dirichlet's conditions for the existence of a Fourier series for ?

A. must be differentiable everywhere in the interval.
B. must have a finite number of maxima and minima in any one period.
C. must have a finite number of discontinuities in any one period.
D. must be periodic, single-valued, and finite.

6 If a function satisfies Dirichlet's conditions, at a point of finite discontinuity , the Fourier series converges to:

A.
B.
C.
D.

7 The value of the integral (where are integers) is:

A. if
B. $1$
C. if
D. $0$ for all integer

8 For an even function defined in , which Fourier coefficients are always zero?

A.
B. None of the coefficients
C.
D.

9 If is an odd function in , then is given by:

A.
B.
C.
D.

10 Which of the following functions is an Even function?

A.
B.
C.
D.

11 In the Fourier expansion of in , the value of is:

A.
B.
C. $0$
D.

12 For a function defined in the interval , the coefficient is calculated as:

A.
B.
C.
D.

13 What is the Fourier Series expansion of in the Half Range Cosine Series for ?

A.
B.
C.
D.

14 To expand as a Half Range Sine Series in , we extend the function to the interval as:

A. A periodic function
B. An odd function
C. An even function
D. A constant function

15 The coefficient in a Half Range Sine Series for in is:

A.
B.
C.
D.

16 Parseval's formula for a function with period in states that:

A.
B.
C.
D.

17 In the Complex form of Fourier Series defined in , the coefficient is given by:

A.
B.
C.
D.

18 The relationship between the complex coefficient and the real coefficients (for ) is:

A.
B.
C.
D.

19 For a real-valued function , the complex coefficients satisfy the property:

A.
B. (Complex Conjugate)
C.
D.

20 The value of where is an integer is:

A.
B.
C.
D.

21 The value of where is an integer is:

A.
B.
C.
D.

22 If in , which coefficients are zero?

A. No coefficients are zero
B. only
C. only
D. and

23 What is the period of the function ?

A.
B.
C.
D.

24 In the Fourier series of the square wave defined by for and for , the value of is:

A.
B.
C.
D.

25 Which term represents the 'fundamental harmonic' in a Fourier series?

A. The term
B. The term with
C. The term with
D. The term with

26 Parseval's identity is useful for:

A. Calculating the derivative of .
B. Determining if a function is even or odd.
C. Finding the phase angle of the harmonics.
D. Finding the sum of infinite series involving squares of integers.

27 If in , the value of is:

A.
B.
C.
D.

28 The complex coefficient corresponds to:

A. Zero always.
B. The value of the function at .
C. The average value of the function over the period.
D. The amplitude of the fundamental frequency.

29 If the interval of a function is changed from to , the basis functions change from to:

A.
B.
C.
D.

30 Which of the following conditions ensures that the Dirichlet's integral converges?

A. The function is strictly increasing.
B. The function is differentiable everywhere.
C. The function is continuous everywhere.
D. The integral of over the period is finite (Absolute Integrability).

31 For a function expanded in a Fourier series, the term represents:

A. The amplitude of the -th harmonic.
B. The power of the -th harmonic.
C. The phase angle of the -th harmonic.
D. The frequency of the -th harmonic.

32 In the Fourier series of in , the coefficients are:

A.
B.
C.
D.

33 What happens to the Fourier series of a discontinuous function near the discontinuity?

A. It becomes infinite.
B. It becomes zero.
C. It becomes a straight line.
D. Gibbs Phenomenon (overshoot)

34 For the Half Range Sine Series of in , the value of is:

A.
B.
C.
D.

35 The constant term in the Fourier Series of in is:

A.
B.
C.
D.

36 If is periodic with period , then equals:

A.
B.
C.
D.

37 The complex form of the Fourier series is often preferred in signal processing because:

A. It simplifies algebraic manipulations involving derivatives and integrals.
B. It uses only real numbers.
C. It eliminates the DC component.
D. It only works for even functions.

38 When expanding as a half-range cosine series in , the period of the extended function is:

A.
B.
C.
D.

39 The value of for is:

A.
B.
C.
D.

40 For a function defined in , the coefficient is:

A.
B.
C.
D.

41 If is even, then is:

A. Odd
B. Constant
C. Periodic but neither even nor odd
D. Even

42 Identify the relation used to derive the Fourier coefficients:

A. Orthogonality of trigonometric functions.
B. Pythagorean theorem.
C. Laplace transform.
D. Taylor series expansion.

43 Which series contains only terms with odd harmonics ()?

A. Odd functions.
B. Functions with Half-wave symmetry: .
C. Even functions.
D. Functions with period .

44 The RMS (Root Mean Square) value of a function over is related to Parseval's theorem by:

A.
B.
C.
D.

45 In the Fourier series of in , the function is:

A. Odd
B. Zero
C. Even
D. Neither even nor odd

46 If is the complex Fourier coefficient, then represents:

A. The derivative of the signal.
B. The time domain signal.
C. The power spectrum at frequency .
D. The phase spectrum.

47 For a half-range cosine series of (constant) in :

A. for all .
B. , all other .
C. for all .
D. All coefficients are non-zero.

48 The sum of the series derived from the Fourier series of a square wave is:

A.
B.
C.
D.

49 If is continuous and piecewise smooth, the Fourier series converges:

A. Only at .
B. Pointwise but not uniformly.
C. To infinity.
D. Uniformly to .

50 In the expansion of in , the value of is:

A.
B.
C.
D.
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