Two loops $P$ and $Q$ are made from a uniform wire. The radii of $P$ and $Q$ are $r_1$ and $r_2$ respectively. Their moments of inertia are $I_1$ and $I_2$ respectively. If $I_2/I_1 = 4$,then $r_2/r_1 = ?$

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$,$\overrightarrow{b}=\hat{i}+\hat{j}$,$\overrightarrow{c}=\hat{i}$ and $(\overrightarrow{a} \times \overrightarrow{b}) \times \overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$,then $\lambda+\mu$ is equal to:

If $A(2,-1)$ and $B(6,5)$ are two points,the ratio in which the foot of the perpendicular from $P(4,1)$ to $AB$ divides $AB$ is:

$A$ sample of $0.1 \text{ g}$ of water at $100^{\circ} \text{C}$ and normal pressure $(1.013 \times 10^5 \text{ Nm}^{-2})$ requires $54 \text{ cal}$ of heat energy to convert to steam at $100^{\circ} \text{C}$. If the volume of the steam produced is $167.1 \text{ cc}$,the change in internal energy of the sample is ....... $\text{J}$.

If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}+\vec{b}|=\sqrt{29}$ and $\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}$,then a possible value of $(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is

If the temperature of the sun were to increase from $T$ to $2T$ and its radius from $R$ to $2R$,then the ratio of the radiant energy received on Earth to what it was previously will be