If $\frac{1 - \cos x}{\cos x(1 + \cos x)} = \frac{\sin \alpha}{\cos x} - \frac{2}{1 + \cos x}$,then $\alpha = $

For $x > 1$,if $(2x)^{2y} = 4e^{2x - 2y}$,then $(1 + \log_e 2x)^2 \frac{dy}{dx}$ is equal to

Which of the following is not a Pteridophyte?

The number of integral values of $k$,for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution,is

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is

$\Delta \lambda$ is the difference between the wavelength of the $K_{\alpha}$ line and the minimum wavelength of the continuous $X$-ray spectrum when the $X$-ray tube is operated at a voltage $V$. If the operating voltage is changed to $V/3$,then the above difference is $\Delta \lambda^{\prime}$. Then: