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 Chapter 9 Fourier Series and Fourier Transforms Previous Year Questions with Solutions

Pravegaa PYQ Series · Volume 1: Mathematical Physics

MP09: Fourier Series and Fourier Transforms

25 previous-year questions with complete solutions, from IIT JAM, GATE, CSIR NET-JRF, JEST and TIFR, arranged from IIT JAM and GATE through to TIFR and from the most recent paper backwards. Click Show solution under any question.

  • IIT JAM: 6
  • GATE: 4
  • CSIR NET: 6
  • JEST: 4
  • TIFR GS: 5
MP09.1IIT JAM 2026, Q24MCQ | 2 marks

A piecewise regular function \(f(x)\) is shown in the figure. It is expanded in Fourier series given by \[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+\sum_{n=1}^{\infty}b_n\sin(nx), \] where \(a_0\), \(a_n\), \(b_n\)'s are the Fourier coefficients.FigureWhich of the following options is correct?

  1. \(a_0=3\pi\)

  2. All \(a_n\) are zero

  3. All \(b_n\) are zero

  4. \(a_0=\dfrac{3\pi}{2}\)

Show solution

From the figure, \(f(x)=\dfrac{\pi}{2}-x\) for \(0<x<\pi\), repeated with period \(\pi\). For \(-\pi<x<0\), \(f(x)=f(x+\pi)=\dfrac{\pi}{2}-(x+\pi)=-\dfrac{\pi}{2}-x\). Hence, for \(0<x<\pi\), \[ f(-x)=-\frac{\pi}{2}+x=-f(x), \] so \(f\) is odd. The integrand \(f(x)\cos nx\) is then odd, and \[ a_n=\frac1\pi\int_{-\pi}^{\pi}f(x)\cos(nx)\,dx=0\qquad (n=0,1,2,\dots). \] In particular \(a_0=0\) (the average value is zero), so (A) and (D) are false, while the \(b_n\) are non-zero (only the sine terms survive), so (C) is false. \[ \boxed{\text{All } a_n=0} \]

Answer: (B) All \(a_n\) are zero

MP09.2IIT JAM 2023, Q32MSQ | 2 marks

A periodic function \(f(x)=x^2\) for \(-\pi<x<\pi\) is expanded in a Fourier series. Which of the following statement(s) is/are correct?

  1. Coefficients of all the sine terms are zero

  2. The first term in the series is \(\dfrac{\pi^2}{3}\)

  3. The second term in the series is \(-4\cos x\)

  4. Coefficients of all the cosine terms are zero

Show solution

\(f(x)=x^2\) is even, so all sine coefficients vanish: (A) is true. Write \(f(x)=\dfrac{a_0}{2}+\sum_{n\ge1}a_n\cos nx\). \[ a_0=\frac1\pi\int_{-\pi}^{\pi}x^2\,dx=\frac{2}{\pi}\cdot\frac{\pi^3}{3}=\frac{2\pi^2}{3}\quad\Rightarrow\quad\frac{a_0}{2}=\frac{\pi^2}{3}, \] so (B) is true. \[ a_n=\frac1\pi\int_{-\pi}^{\pi}x^2\cos nx\,dx=\frac2\pi\left[\frac{x^2\sin nx}{n}+\frac{2x\cos nx}{n^2}-\frac{2\sin nx}{n^3}\right]_0^\pi=\frac{4(-1)^n}{n^2}. \] For \(n=1\), \(a_1=-4\), so the second term is \(-4\cos x\) and (C) is true. The cosine coefficients are non-zero, so (D) is false. Thus \[ x^2=\frac{\pi^2}{3}-4\cos x+\cos2x-\frac49\cos3x+\cdots \] \[\boxed{\text{(A), (B), (C)}}\]

Answer: (A), (B), (C)

MP09.3IIT JAM 2021, Q28MCQ | 2 marks

In the Fourier series expansion of two functions \(f_1(t)=4t^2+3\) and \(f_2(t)=6t^3+7t\) in the interval \(-\dfrac T2\) to \(+\dfrac T2\), the Fourier coefficient \(a_n\) and \(b_n\) (\(a_n\) and \(b_n\) are coefficients of \(\cos(n\omega t)\) and \(\sin(n\omega t)\), respectively) satisfy

  1. \(a_n=0\) and \(b_n\neq0\) for \(f_1(t)\); \(a_n\neq0\) and \(b_n=0\) for \(f_2(t)\)

  2. \(a_n\neq0\) and \(b_n=0\) for \(f_1(t)\); \(a_n=0\) and \(b_n\neq0\) for \(f_2(t)\)

  3. \(a_n\neq0\) and \(b_n\neq0\) for \(f_1(t)\); \(a_n=0\) and \(b_n\neq0\) for \(f_2(t)\)

  4. \(a_n=0\) and \(b_n\neq0\) for \(f_1(t)\); \(a_n\neq0\) and \(b_n\neq0\) for \(f_2(t)\)

Show solution

On the symmetric interval, \[ a_n=\frac2T\int_{-T/2}^{T/2}f(t)\cos(n\omega t)\,dt,\qquad b_n=\frac2T\int_{-T/2}^{T/2}f(t)\sin(n\omega t)\,dt . \] \(f_1=4t^2+3\) is even, so \(f_1\sin(n\omega t)\) is odd and \(b_n=0\), while \(a_n\neq0\). \(f_2=6t^3+7t\) is odd, so \(f_2\cos(n\omega t)\) is odd and \(a_n=0\), while \(b_n\neq0\). \[ \boxed{f_1:\ a_n\neq0,\ b_n=0;\qquad f_2:\ a_n=0,\ b_n\neq0} \]

Answer: (B)

MP09.4IIT JAM 2018, Q23MCQ | 2 marks

The function \(f(x)=\begin{cases}x, & -\pi<x<0\\ -x, & 0<x<\pi\end{cases}\) is expanded as a Fourier series of the form \(a_0+\sum_{n=1}^{\infty}a_n\cos(nx)+\sum_{n=1}^{\infty}b_n\sin(nx)\). Which of the following is true?

  1. \(a_0\neq0,\ b_n=0\)

  2. \(a_0\neq0,\ b_n\neq0\)

  3. \(a_0=0,\ b_n=0\)

  4. \(a_0=0,\ b_n\neq0\)

Show solution

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