The line among the following which touches the parabola $y^2=4ax$ is

If $f(x) = \begin{cases} \frac{1-\sqrt{2} \sin x}{\pi-4x} & \text{if } x \neq \frac{\pi}{4} \\ a & \text{if } x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then $a$ is equal to

Two simple harmonic motions are represented by $y_{1} = 5[\sin 2 \pi t + \sqrt{3} \cos 2 \pi t]$ and $y_{2} = 5 \sin (2 \pi t + \frac{\pi}{4})$. The ratio of their amplitudes is

The normal freezing point $(f.p.)$ of nitrobenzene is $278.82\, K$. If a non-volatile solute is dissolved in it and a solution of $0.25\, m$ (molality) is prepared,the observed $f.p.$ of the solution is $276.82\, K$. Evaluate $K_f$ of nitrobenzene in $K\, kg\, mol^{-1}$.

Let $f:[-1, 3] \to R$ be defined as $f(x) = \begin{cases} |x| + [x], & -1 \le x < 1 \\ x + |x|, & 1 \le x < 2 \\ x + [x], & 2 \le x \le 3 \end{cases}$ where $[t]$ denotes the greatest integer function. Then $f$ is discontinuous at:

Which of the following has the lowest thermal stability?