Fourier Series Representation Questions and Answers

Q: The Fourier series of a real periodic function has only :P : cosine terms if it is evenQ : sine terms if it is oddR : cosine terms if it is oddS : sine terms if it is evenWhich of the above statements are correct ?

The correct statements are P and Q.An even function satisfies \(f(-t) = f(t)\). The sine terms are odd, so their coefficients \(b_{n}\) all vanish and the series contains only cosine terms (and possibly a constant). An odd function satisfies \(f(-t) = -f(t)\); now the cosine coefficients \(a_{n}\) and the constant vanish, and the series contains only sine terms.Statements R and S reverse these rules: an odd function cannot be built from cosines, which are even, and an even function cannot be built from sines, which are odd. So any option including R or S is wrong.

Q: The trigonometric Fourier series of a periodic time function can have only :

The keyed answer is cosine and sine terms.The trigonometric Fourier series of a periodic function is\[ \begin{aligned} f(t) &= a_{0} + \sum_{n}a_{n}\cos n\omega_{0}t \\ &\quad + \sum_{n}b_{n}\sin n\omega_{0}t \end{aligned} \]so in general it is built from cosine and sine terms (with a constant, the DC term). Only cosine terms, or only sine terms, appear just in the special cases of even and odd functions, and 'DC and cosine' is again the even-function case. Of the options, only 'cosine and sine terms' describes the general series.

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Assertion-Reason type questions :

The following items consist of two statements, one labelled as ‘Assertion A’ and the other labelled the ‘Reason (R)’. You are to examine these two statements and decide if the Assertion (A) and the Reason (R) are individually true and if so, whether the Reason is a correct explanation of the Assertion. Select your answers to these items using the codes given below and mark your answer sheet accordingly.

Assertion (A) : For a function to be odd \(f(-x) = -f(x)\).
Reason (R) : If a function is odd, its Fourier series only contains cosine terms.
Codes :

A.Both (A) and (R) are true and (R) is the correct explanation of (A).
B.Both (A) and (R) are true, but (R) is not the correct explanation of (A).
C.(A) is true and (R) is false.
D.(A) is false and (R) is true.

Exam Year(s)

UGC NET - 2013,

Answer: C

Assertion (A) is true, since the definition of an odd function is f(-x) = -f(x). Reason (R) is false: the Fourier series of an odd function has only sine terms, while the cosine terms belong to even functions. So (A) is true and (R) is false.

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The Fourier series of a real periodic function has only :

P : cosine terms if it is even
Q : sine terms if it is odd
R : cosine terms if it is odd
S : sine terms if it is even

Which of the above statements are correct ?

A.P and S
B.P and Q
C.P and R
D.Q and R

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: B

The correct statements are P and Q.

An even function satisfies \(f(-t) = f(t)\). The sine terms are odd, so their coefficients \(b_{n}\) all vanish and the series contains only cosine terms (and possibly a constant). An odd function satisfies \(f(-t) = -f(t)\); now the cosine coefficients \(a_{n}\) and the constant vanish, and the series contains only sine terms.

Statements R and S reverse these rules: an odd function cannot be built from cosines, which are even, and an even function cannot be built from sines, which are odd. So any option including R or S is wrong.

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The trigonometric Fourier series of a periodic time function can have only :

A.cosine terms
B.sine terms
C.DC and cosine terms
D.cosine and sine terms

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: D

The keyed answer is cosine and sine terms.

The trigonometric Fourier series of a periodic function is

\[ \begin{aligned} f(t) &= a_{0} + \sum_{n}a_{n}\cos n\omega_{0}t \\ &\quad + \sum_{n}b_{n}\sin n\omega_{0}t \end{aligned} \]

so in general it is built from cosine and sine terms (with a constant, the DC term). Only cosine terms, or only sine terms, appear just in the special cases of even and odd functions, and 'DC and cosine' is again the even-function case. Of the options, only 'cosine and sine terms' describes the general series.

Discuss in Forum

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Fourier Series Representation Questions and Answers

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