DP Mathematics: Applications and Interpretation Questionbank

Write down $z_1$ and $z_2$ in exponential form, with a constant modulus.

Find, in hours, the shortest time from sunrise to sunset at point $\text{A}$ that is predicted by this model.

Hence or otherwise find an equation for $L$ in the form $L\left(t\right)=p \sin (qt+r)+d$, where $p, q, r, d\in \mathrm{ℝ}$.

Write down $2+5\text{i}$ in exponential form.

An equilateral triangle is to be drawn on the Argand plane with one of the vertices at the point corresponding to $2+5\text{i}$ and all the vertices equidistant from $0$.

Find the points that correspond to the other two vertices. Give your answers in Cartesian form.

$z=2\text{i}$.

$z=1+\text{i}$.

Describe these two transformations and give the order in which they are applied.

Hence, or otherwise, find the value of $z$ when $w=2-\text{i}$.

Find the value of ${\left(\frac{z_1}{z_2}\right)}^4$ for $n=2$.

Find the value of ${z_1}^3$.

Find the least value of $n$ such that $z_1{z}_2\in {\mathrm{ℝ}}^{+}$.

Hence find $\text{Re}\left(w_1+w_2\right)$ in the form $A \cos \left(x-a\right)$, where $A>0$ and $0<a\le \frac{\mathrm{\pi }}{2}$.

Express $z$ in the form $z=a{\text{e}}^{\text{i}b}$, where $a, b\in \mathrm{ℝ}$, giving the exact value of $a$ and the exact value of $b$.

Find the phase shift of $I_{\text{T}}$.

Find $w_1+w_2$ in the form ${\text{e}}^{\text{i}x}\left(c+\text{i}d\right)$.

Find the maximum value of $I_{\text{T}}$.

Plot the position of $z$ on an Argand Diagram.

$\theta =\mathrm{\pi }$.

$\theta =\frac{\mathrm{\pi }}{2}$.

$\theta =\frac{3\mathrm{\pi }}{2}$.

Find this value of $\theta$.

For this value of $\theta$, plot the approximate position of $z_{\theta }$ on the Argand diagram.

Find an expression for $V_T$ in the form $A \cos (Bt+C)$, where $A, B$ and $C$ are real constants.

Hence write down the maximum voltage in the circuit.

find the values of $w^2$, $w^3$, and $w^4$.

draw $w$, $w^2$, $w^3$, and $w^4$ on the following Argand diagram.

Let $z=\frac{w}{2-\text{i}}$.

Find the value of $a$ for which successive powers of $z$ lie on a circle.

Express w² and w³ in modulus-argument form.

Sketch on an Argand diagram the points represented by w⁰ , w¹ , w² and w³.

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$.

Let $z=2\left(\text{cos}\frac{\pi }{n}+\text{i}\text{sin}\frac{\pi }{n}\right),n\in {\mathbb{Z}}^{+}$. The points represented on an Argand diagram by $z^0,z^1,z^2,\dots ,z^n$ form the vertices of a polygon $P_n$.

Show that the area of the polygon $P_n$ can be expressed in the form $a\left(b^n-1\right)\text{sin}\frac{\pi }{n}$, where $a,b\in \mathbb{R}$.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the modulus of $w$;

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$.

Find the smallest positive integer value of $n$, such that $w^n$ is a real number.

Show the points represented by $z$ and $z-2a$ on the following Argand diagram.

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

Hence find the cube roots of $z$ in modulus-argument form.

In the following Argand diagram the point A represents the complex number $-1+4\text{i}$ and the point B represents the complex number $-3+0\text{i}$. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.