The power dissipated in the circuit shown in the figure is $30 \, W$. The value of $R$ is ............. $\Omega$.

The position of a projectile launched from the origin at $t=0$ is given by $\vec{r}=(40 \hat{i}+50 \hat{j}) \text{ m}$ at $t=2 \text{ s}$. If the projectile was launched at an angle $\theta$ from the horizontal,then $\theta$ is (take $g = 10 \text{ m/s}^2$)

If $[x]$ denotes the greatest integer not exceeding $x$ and if the function $f$ defined by $f(x)= \begin{cases} \frac{a+2 \cos x}{x^2} & , x < 0 \\ b \tan \frac{\pi}{[x+4]} & , x \geq 0 \end{cases}$ is continuous at $x=0$,then the ordered pair $(a, b)$ is equal to

The products formed in the reaction of phenol with $Br_2$ dissolved in $CS_2$ at $0^{\circ} C$ are

Two conducting circular loops of radii $R_1$ and $R_2$ are placed in the same plane with their centres coinciding. If $R_1 > R_2$,the mutual inductance $M$ between them will be directly proportional to

If $\alpha, \beta, \gamma$ are the roots of $2x^3 - 2x - 1 = 0$,then $(\Sigma \alpha \beta)^2$ is equal to