If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

$A$ bomb moving with velocity $(40 \hat{i} + 50 \hat{j} - 25 \hat{k}) \text{ m/s}$ explodes into two pieces of mass ratio $1:4$. After the explosion,the smaller piece moves away with velocity $(200 \hat{i} + 70 \hat{j} + 15 \hat{k}) \text{ m/s}$. The velocity of the larger piece after the explosion is:

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

Read the following five statements $(1-5)$ carefully and answer the question that follows.
$(1)$ In $Equisetum$,the female gametophyte is retained on the parent sporophyte.
$(2)$ In $Ginkgo$,the male gametophyte is not independent.
$(3)$ The sporophyte of $Polytrichum$ is more developed than that of $Riccia$.
$(4)$ Sexual reproduction in $Volvox$ is isogamous.
$(5)$ Spores of slime molds lack a cell wall.
How many of the above statements are correct?

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi \left( \frac{y}{x} \right)}{\phi' \left( \frac{y}{x} \right)}$ is

In which of the following reactions is $H_2O_2$ acting as a reducing agent?