The value of $\cos 15^{\circ} \cos 7.5^{\circ} \sin 7.5^{\circ}$ is

Let $f: R \to R$,$f(x) = \max \{ |\tan^{-1} x|, \cot^{-1} x \}$. Consider the following statements:
$I$. The function is continuous and derivable $\forall x \in R$.
$II$. The range of the function is $[\frac{\pi}{4}, \pi]$.
$III$. $f(x)$ is many-one into.
Identify the correct option.

What is the chemical name of Vitamin $B_1$?

Two liquids $A$ and $B$ are at $32\,^{\circ}C$ and $24\,^{\circ}C$. When mixed in equal masses,the temperature of the mixture is found to be $28\,^{\circ}C$. Their specific heats are in the ratio of:

$A$ transverse wave is described by the equation $y = y_0 \sin 2\pi \left( ft - \frac{x}{\lambda} \right)$. The maximum particle velocity is equal to four times the wave velocity if:

The period of $\left(\tan \theta - \frac{1}{3} \tan^3 \theta\right) \left(\frac{1}{3} - \tan^2 \theta\right)^{-1}$,where $\tan^2 \theta \neq \frac{1}{3}$,is