If $\sec^{-1}\left(\dfrac{x^2 + y^2}{x^2 - y^2}\right) = 2a$ such that $y\dfrac{dy}{dx} = x \cdot f(a)$ then the value of $f\left(\dfrac{2\pi}{3}\right)$ is

The value of $\cos 15^\circ - \sin 15^\circ$ is ......

The general solution of $\cos x + \sin x = \cos 2x + \sin 2x$ is $x = np\pi$ or $x = \dfrac{nq\pi}{3} + \dfrac{\pi}{6}$ for $n \in Z$ then $p : q = $

The number of common solutions of the pair of equations $2\sin^2\theta - 2\cos^2\theta + 1 = 0$ and $2\sin^2\theta + 3\sin\theta - 2 = 0$ in the interval $[0, 2\pi]$, is