If the function $f$ defined by $f(x) = K(x - x^2)$ for $0 < x < 1$ and $f(x) = 0$ otherwise,is the probability density function (p.d.f.) of a random variable $X$,then the value of $P(X < \frac{1}{2})$ is

$A$ closed pipe is suddenly opened and changed to an open pipe of the same length. The fundamental frequency of the resulting open pipe is less than that of the $3^{rd}$ harmonic of the earlier closed pipe by $55 ~Hz$. Then,the value of the fundamental frequency of the closed pipe is: (in $~Hz$)

Which of the following halides does not undergo hydrolysis?

Assertion $(A)$: Propagation of light through an optical fibre is due to total internal reflection taking place at the core-clad interface.
Reason $(R)$: Refractive index of the material of the core of the optical fibre is greater than that of air.

If $f(x) = \begin{cases} \frac{x-1}{2x^2-7x+5}, & \text{for } x \neq 1 \\ -\frac{1}{3}, & \text{for } x=1 \end{cases}$, then $f'(1)$ is equal to:

Fungal herbicides are produced by which of the following?