The measure of the acute angle between the pair of lines $2x^2 + xy - y^2 - x + 2y - 1 = 0$ is:

The joint equation of a pair of lines passing through point $(1,4)$, one of which is parallel to X-axis and the other makes an angle of $45^\circ$ with the positive direction of X-axis, is

The line $(2 + k)x + (1 + k)y = 5 + 7k$ passes through the fixed point for different values of k. If 'd' is the distance of a fixed point from the origin, then $d^2 = \ldots$

If $\theta$ is the acute angle between the lines $2x^2 + 7xy + 3y^2 = 0$, then the value of $\dfrac{2\cos\theta - 3\sin\theta}{4\sin\theta + 5\cos\theta} = \ldots$