The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$,where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis,are

$A$ metallic wire of length $L$ is fixed between two rigid supports. If the wire is cooled through a temperature difference $\Delta T$ ($Y$ = Young's modulus,$\rho$ = density,$\alpha$ = coefficient of linear expansion),then the frequency of transverse vibration is proportional to:

In the circuit shown in the figure, what is the value of $I_1$ just after pressing the key $K$?

Which of the following is a heterosporous pteridophyte?

Calculate the heat of combustion (in $kJ$) of methane from the following data:
$(I)$ $C_{(graphite)} + 2H_{2(g)} \rightarrow CH_{4(g)} \quad \Delta H = -74.8 \ kJ$
$(II)$ $C_{(graphite)} + O_{2(g)} \rightarrow CO_{2(g)} \quad \Delta H = -393.5 \ kJ$
$(III)$ $H_{2(g)} + 1/2 O_{2(g)} \rightarrow H_2O_{(l)} \quad \Delta H = -286.2 \ kJ$

$\int \frac{dx}{\sqrt{x}(x+9)}$ is equal to