The oxoanion which contains all equivalent $M-O$ bonds is:
$(I)$ $CrO_4^{2-}$
$(II)$ $MnO_4^{-}$
$(III)$ $Cr_2O_7^{2-}$

The equation of a stationary wave is $y = 0.8 \cos \left( \frac{\pi x}{20} \right) \sin (200 \pi t)$,where $x$ is in $cm$ and $t$ is in $sec$. The separation between consecutive nodes will be..... $cm$.

Identify the blank spaces $a, b, c$ and $d$ given in the following table and select the correct answer.

Type of microbes	Scientific name	Commercial product
Bacterium	$a$	Clot buster
$b$	Aspergillus niger	Citric acid
Fungus	Trichoderma polysporum	$c$
Bacterium	$d$	Butyric acid

If the pair of straight lines given by $A x^2+2 H x y+B y^2=0$ $(H^2>A B)$ forms an equilateral triangle with the line $a x+b y+c=0$,then $(A+3 B)(3 A+B)$ is equal to :

When $25 \ g$ of a non-volatile solute is dissolved in $100 \ g$ of water,the vapour pressure is lowered by $0.225 \ mm$. If the vapour pressure of water at $20^{\circ}C$ is $17.5 \ mm$,what is the molecular weight of the solute?

$\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x$ is equal to