Limestone is present in the blast furnace production of iron in order to:
$(I)$ provide a source of $CaO$
$(II)$ remove some impurities
$(III)$ supply $CO_2$

$A$ random variable $X$ has the probability distribution given below. Its variance is

$X$	$1$	$2$	$3$	$4$	$5$
$P(X=x)$	$k$	$2k$	$3k$	$2k$	$k$

Two trains travelling on the same track are approaching each other with equal speeds of $40\, m/s$. The drivers of the trains begin to decelerate simultaneously when they are just $2.0\, km$ apart. Assuming the decelerations to be uniform and equal,the value of the deceleration to barely avoid collision should be.........$m/s^2$.

Consider a gas phase reaction which occurs in a closed vessel: $2A \rightarrow 4B + C$. The concentration of $B$ is found to be increased by $5 \times 10^{-3} \ mol \ L^{-1}$ in $10 \ s$. The rate of disappearance of $A$ (in $mol \ L^{-1} \ s^{-1}$) is:

If $X$ is a Poisson variate such that $P(X=1)=P(X=2)$,then $P(X=4)$ is equal to

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ $(a \neq 1, b \neq 1, c \neq 1, a \neq b \neq c)$ are coplanar,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is: