Find the value of $b$ and of $c$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

The area of the shaded region is 12 cm². Find the value of θ.

Find the range of $g$.

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

Determine an expression for $f^{\prime }(x)$ in terms of $x$.

Find the value of $\text{tan}\theta$.

Solve the equation $(f\circ g)\left(x\right)=2 \cos 2x$ where $0\le x\le \pi$.

Given that the area of the logo is $13.4 {\text{cm}}^2$, find the value of $\theta$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

Hence show that $f(x)=0$ can be expressed as $u^3-7u^2+15u-9=0$.

Show that $\sin 2x+\cos 2x-1=2 \sin x\left(\cos x-\sin x\right)$.

Find the period of $g$.

The equation $g\left(x\right)=12$ has two solutions where $\pi$ ≤ $x$ ≤ $\frac{4\pi }{3}$. Find both solutions.

Find the area of the shaded region, in terms of θ.

Hence, solve the equation $2 {\cos }^2 x+5 \sin x=4, 0\le x\le 2\mathrm{\pi }$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

The range of $f$ is $k$ ≤ $f\left(x\right)$ ≤ $m$. Find $k$ and $m$.

Find the distance that the rocket travels during the first stage.

Let .

Find, in terms of b, the solutions of $\text{sin}2x=-a,0⩽x⩽\pi$.

Find the least positive value of $x$ for which $\cos \left(\frac{x}{2}+\frac{\mathrm{\pi }}{3}\right)=\frac{1}{\sqrt{2}}$.

Show that $2x-3-\frac{6}{x-1}=\frac{2x^2-5x-3}{x-1}, x\in \mathrm{ℝ}, x\ne 1$.

Hence or otherwise, solve the equation  $2 \sin 2\theta -3-\frac{6}{\sin 2\theta -1}=0$  for  $0\le \theta \le \mathrm{\pi }, \theta \ne \frac{\mathrm{\pi }}{4}$.

Let $f\left(x\right)=4\text{cos}\left(\frac{x}{2}\right)+1$, for $0⩽x⩽6\pi$. Find the values of $x$ for which $f\left(x\right)>2\sqrt{2}+1$.

Sketch a graph of $y=f^{\prime }(x)$ for .

Find an expression for the velocity, $v$ m s⁻¹, of the rocket during the first stage.

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

Hence or otherwise, solve $\sin 2x+\cos 2x-1+\cos x-\sin x=0$ for $0<x<2\mathrm{\pi }$.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

Express $\sin x$ in terms of $u$.

Express $\sin 2x$ in terms of $u$.

Show that the equation $2 {\cos }^2 x+5 \sin x=4$ may be written in the form $2 {\sin }^2 x-5 \sin x+2=0$.