Two springs have spring constants $K_A$ and $K_B$ such that $K_A > K_B$. If both springs are stretched by the same extension $x$,the work required will be:

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c}=\frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between the vectors $\overline{b}$ and $\overline{c}$,then the value of $\sin \theta$ is

Among the electrolytes $Na_2SO_4$,$CaCl_2$,$Al_2(SO_4)_3$ and $NH_4Cl$,the most effective coagulating agent for $Sb_2S_3$ sol is

$\sqrt{2+\sqrt{5}-\sqrt{6-3 \sqrt{5}+\sqrt{14-6 \sqrt{5}}}}$ is equal to

Two radioactive materials $X_1$ and $X_2$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei,then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $1 / e$ after a time:

In the diagram,$I_1$ and $I_2$ are the strengths of the currents in the loop and the infinitely long straight conductor,respectively. Given $OA = AB = R$. The net magnetic field at the centre $O$ is zero. Then the ratio of the currents in the loop and the straight conductor $(I_1/I_2)$ is: