The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct?

The correct answer is: A. P and S

A real-valued periodic function $f(x)$ can be represented as a Fourier series, which is a sum of sine and cosine functions with frequencies that are integer multiples of the fundamental frequency. The Fourier series of a real-valued even function $f(x)$ has only cosine terms, while the Fourier series of a real-valued odd function $f(x)$ has only sine terms.

Here is a more detailed explanation of each option:

• P. Cosine terms if it is even

A real-valued even function $f(x)$ is a function that satisfies $f(-x) = f(x)$. For example, the function $f(x) = x^2$ is an even function. The Fourier series of an even function has only cosine terms. This is because the cosine function is an even function, and the Fourier series is a sum of sine and cosine functions.

• Q. Sine terms if it is even

A real-valued odd function $f(x)$ is a function that satisfies $f(-x) = -f(x)$. For example, the function $f(x) = x$ is an odd function. The Fourier series of an odd function has only sine terms. This is because the sine function is an odd function, and the Fourier series is a sum of sine and cosine functions.

• R. Cosine terms if it is odd

This statement is incorrect. A real-valued odd function $f(x)$ has only sine terms, not cosine terms.

• S. Sine terms if it is odd

This statement is correct. A real-valued odd function $f(x)$ has only sine terms.