In the nuclear reaction,${ }_{90}^{234} Th \longrightarrow{ }_{91}^{234} Pa + X$,$X$ is:

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the number of ways to select at least one book is $63$,then the value of $n$ is:

When a $p-n$ diode is reverse biased,then

$(b \times c) \times (c \times a) = $

In a quadrilateral $ABCD$,the point $P$ divides $DC$ in the ratio $1:2$ and $Q$ is the midpoint of $AC$. If $\overrightarrow{AB}+2\overrightarrow{AD}+\overrightarrow{BC}-2\overrightarrow{DC}=k\overrightarrow{PQ}$,then $k$ is equal to

The coefficient of $x^{24}$ in the expansion of $(1+x^2)^{12}(1+x^{12})(1+x^{24})$ is