Quadratics

A curve has equation $y = 5 + 3x - 2x^2$ and a straight line has equation $y = kx + 13$, where $k$ is a constant.

Find the set of values of $k$ for which the curve and the line do not meet.

Solve the equation $6\sin\theta = 1 + \frac{2}{\sin\theta}$ for $-180^\circ < \theta < 180^\circ$.

Find the coordinates of the points of intersection of the curve and the line with equations

The first two terms of a geometric progression are

where $\theta$ is an angle such that $0 < \theta < \frac{1}{6}\pi$.

Given that the sum to infinity of the progression is $\frac{1}{2}$, find the value of $\theta$. Give your answer in the form $\sin^{-1} k$, where $k$ is a rational number.

Solve the equation

for $-180^\circ < \theta < 180^\circ$.

In the expansion of $(3 + ax)^5 + (6 - x)^4$, the coefficient of $x^2$ is six times the coefficient of $x$.

Find the possible values of the constant $a$.

Find the set of values of the constant $k$ for which the quadratic equation

has two distinct real roots.

In the expansion of

the coefficient of $x^4$ is 216.

Find the value of the positive constant $p$.