DP Mathematics: Analysis and Approaches Questionbank

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $u$.

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

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

Let .

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

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

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 range of $g$.

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

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.

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

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

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.

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 $f\left(x\right)>2\sqrt{2}+1$.

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

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

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.