In a $\triangle ABC$,$a^2 \sin 2C + c^2 \sin 2A$ is equal to (in $Delta$)

Carbon,silicon,and germanium atoms have four valence electrons each. Their valence and conduction bands are separated by energy band gaps represented by $(Eg)_C$,$(Eg)_{Si}$,and $(Eg)_{Ge}$ respectively. Which one of the following relationships is true in their case?

$A$ motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver $n$-times the water through the same pipe in the same time,the power of the motor must be increased as follows:

The steps of a plant breeding program are given below. Arrange them in the correct sequence.
$(a)$ Cross-hybridization among the selected parents
$(b)$ Testing,release,and commercialization of new cultivars
$(c)$ Collection of variability
$(d)$ Selection and testing of superior recombinants
$(e)$ Evaluation and selection of parents

If $p$ and $q$ are distinct prime numbers and if the equation $x^2 - px + q = 0$ has positive integers as its roots,then the roots of the equation are

$A$ variable plane passes through a fixed point $(1, 2, 3)$. The foot of the perpendicular from the origin to the plane lies on: