The line joining the points $6a - 4b + 4c$ and $-4c$ and the line joining the points $-a - 2b - 3c$ and $a + 2b - 5c$ intersect at:

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the coordinate axes are rotated through an angle $\alpha$ keeping the origin fixed,the same line $L$ has intercepts $p$ and $q$ on the new axes. Then

What happens when
$(i)$ sodium metal is dropped in water?
$(ii)$ sodium metal is heated in free supply of air?
$(iii)$ sodium peroxide dissolves in water?

Two metal plates each of area $A$ form a parallel plate capacitor with air in between the plates. The distance between the plates is $d$. $A$ metal plate of thickness $d/2$ and of same area $A$ is inserted between the plates to form two capacitors of capacitances $C_1$ and $C_2$ as shown in the figure. If the effective capacitance of the two capacitors is $C^{\prime}$ and the capacitance of the capacitor initially is $C$,then $C^{\prime}/C$ is

$A$ body is executing simple harmonic motion. At a displacement $x$ its potential energy is $E_1$ and at a displacement $y$ its potential energy is $E_2$. The potential energy $(E)$ at displacement $(x+y)$ is

In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?