Consider the following differential equation:

$$ t^2 \frac{d^2 y}{d t^2}+7 t \frac{d y}{d t}+8 t y=10 \sin (t) $$

Which one of the following options is correct?

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?