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

The area of a triangle formed by a line with the coordinate axes is $49$ sq. units. If the perpendicular drawn from the origin to this line makes an angle of $45^\circ$ with the positive X-axis, then the equation of line is....

If the equation $16x^2 - 24xy + 9y^2 - 8x + 6y - 35 = 0$ represents a pair of straight lines, then the equation of the locus of points equidistant from these two lines is..........