A time-limited waveform $g(x)$ is specified as follows:

$$ g(x)=\left\{\begin{array}{cc} -k, & -\pi

A new waveform $f(x)$ is constructed from $g(x)$ as follows:

$$ f(x)=\sum_{m=-\infty}^{\infty} g(x+2 \pi n), \text { for all } x \in R $$

The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\frac{2}{3 \pi}$. What is the value of $k$ ?

'The shopkeeper sells lemons.'

In this sentence, the word 'lemons' is the

The figure below is supposed to show three non-overlapping shapes - one oval and two triangles. Which one of the following figures P, Q, R, or S fits the missing portion indicated by '?' and completes the oval and the two triangles?

At how many points will the curves $y=x^2$ and $y=-x^2-2 x-1$ intersect in the real $(x, . y)$ plane?