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The image contains three mathematics questions related to Fourier series. Here is a transcription of the text:

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**Class Test Examination 01 [EEE]**

**Mathematics I (MATH 1303)**

**Marks: 20**

**Time: 30 minutes**

**[Answer any two out of the following three questions]**

1. **Find the Fourier series expansion of \( f(x) = x \) in the interval \((- \pi, \pi)\) by using "Half-range Fourier series".**  
   Also show that:  
   \[
   1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{\pi}{4}
   \]

2. **Derive the Fourier Cosine series of \( f(x) = x^2 \) in the interval \((- \pi, \pi)\) and hence find the sum of the series:**
   \[
   \frac{1}{12} + \frac{1}{22} + \dots
   \]

3. **Compute the Fourier coefficients and hence Fourier series of the following function.**

\[
f(x) =
\begin{cases} 
0, & -\pi < x < -\frac{\pi}{2} \\
1, & -\frac{\pi}{2} < x < \frac{\pi}{2} \\
0, & \frac{\pi}{2} < x < \pi 
\end{cases}
\]

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Find the Fourier series expansion of f(x) = x in the interval (-π, π) by using the 'Half-range Fourier series'. Also, show that 1 - 1/3 + 1/5 - 1/7 + ... = π/4. Derive the Fourier Cosine series of f(x) = x^2 in the interval (-π, π) and hence find the sum of the series 1/12 + 1/22 + ... Compute the Fourier coefficients and hence Fourier series of the given piecewise function.

This problem set covers Fourier series expansion using half-range Fourier series for f(x) = x and cosine Fourier series for f(x) = x^2. Learn to compute Fourier coefficients and solve series summations. Suitable for advanced university-level mathematics and engineering students.

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