DP Mathematics: Analysis and Approaches Questionbank

Use the binomial theorem to expand ${\left(\cos \theta +\mathrm{i} \sin \theta \right)}^4$. Give your answer in the form $a+b\mathrm{i}$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

Find $\gamma$.

Hence show that $1+4\text{i}$ is a Gaussian prime.

Determine whether $\frac{\gamma }{\alpha }$ is a Gaussian integer.

On an Argand diagram, plot and label all Gaussian integers that have a norm less than $3$.

Show that $N\left(\alpha \beta \right)=N\left(\alpha \right)N\left(\beta \right)$.

Given that $\alpha =a+b\text{i}$ where $a, b\in \mathrm{\mathbb{Z}}$, show that $N\left(\alpha \right)=a^2+b^2$.

Verify that $2$ is not a Gaussian prime.

By expressing the positive integer $n=c^2+d^2$ as a product of two Gaussian integers each of norm $c^2+d^2$, show that $n$ is not a Gaussian prime.

Write down another prime number of the form $c^2+d^2$ that is not a Gaussian prime and express it as a product of two Gaussian integers.

Use proof by contradiction to prove that a prime number, $p$, that is not of the form $a^2+b^2$ is a Gaussian prime.

Find the argument of $zw$ in terms of $k$.

Find the modulus of $zw$.

Find the minimum value of $k$.

For the value of $k$ found in part (i), find the value of $zw$.

Hence, deduce that ${\omega }^2+\omega +1=0$.

Hence, write down an expression for ${\text{P}}_0{\text{P}}_{n-1}$ in terms of $n$ and $\omega$.

Show that $(\omega -1)({\omega }^2+\omega +1)={\omega }^3-1$.

Show that ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2\times {\text{P}}_0{\text{P}}_3=4$.

Suggest a value for ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2\times \dots \times {\text{P}}_0{\text{P}}_{n-1}$.

Express $z^{n-1}+ z^{n-2}+ \dots +z+1$ as a product of linear factors over the set $\mathrm{ℂ}$.

By factorizing $z^4-1$, or otherwise, deduce that ${\omega }^3+{\omega }^2+\omega +1=0$.

Write down expressions for ${\text{P}}_0{\text{P}}_2$ and ${\text{P}}_0{\text{P}}_3$ in terms of $\omega$.

Show that ${\text{P}}_0{\text{P}}_1\times {\text{P}}_0{\text{P}}_2=3$.

Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).

Find $\mathrm{AC}$.

Plot the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ on an Argand diagram.

Find ${\omega }_2$ and ${\omega }_3$, expressing these in the form $a+{\mathrm{e}}^{\mathrm{i}\theta }$, where $a\in \mathrm{ℝ}$ and $\theta >0$.

By using de Moivre’s theorem, show that $\alpha =\frac{1}{1-{\mathrm{e}}^{\mathrm{i}{\displaystyle \frac{\mathrm{\pi }}{6}}}}$ is a root of this equation.

Determine the value of $\text{Re}\left(\alpha \right)$.

Verify that ${\omega }_1=1+{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{6}}$ is a root of this equation.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

$f\left(\text{i}u\right)$.

$g\left(\text{i}u\right)$.

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

Consider the equation $z^2+az+12=0$, where $z\in \mathrm{ℂ}$ and $a\in \mathrm{ℝ}$.

Given that $0<\alpha -\theta <\pi$, deduce that only one equilateral triangle ${\text{Z}}_1{\text{OZ}}_2$ can be formed from the point $\text{O}$ and the roots of this equation.

Given that $\text{Re}\left(z_1{z_2}^{\ast }\right)=0$, show that ${\text{Z}}_1{\text{OZ}}_2$ is a right-angled triangle.

Use the result from part (c)(ii) to show that $a^2-3b=0$.

Show that $z_1{z_2}^{\ast }=r_1{r}_2{\text{e}}^{\text{i}\left(\alpha -\theta \right)}$ where ${z_2}^{\ast }$ is the complex conjugate of $z_2$.

Express $z_1$ in terms of $z_2$.

Hence show that ${z_1}^2+{z_2}^2=z_1{z}_2$.

Find the area of triangle UVW.

Find $u$, $v$ and $w$ expressing your answers in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

By considering the sum of the roots $u$, $v$ and $w$, show that

$\text{cos}\frac{5\pi }{18}+\text{cos}\frac{7\pi }{18}+\text{cos}\frac{17\pi }{18}=0$.

Express $-3+\sqrt{3}\text{i}$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

Hence show that $\underset{\alpha \to 0}{\text{lim}}\frac{\alpha }{\text{sin}\alpha }=1$.

By considering the area of two triangles and the area of the sector show that .

Show that the length of the line segment $P_0{P}_1$ is $2\text{sin}\frac{\pi }{n}$.

Find the total area of this $n$ sided polygon.

Let $z^n=1\text{,}z\in \mathbb{C}\text{,}n\in \mathbb{N}\text{,}n⩾5$. Working in modulus/argument form find the $n$ solutions to this equation.

Represent these $n$ solutions on an Argand diagram. Let their positions be denoted by $P_0\text{,}{P}_1\text{,}{P}_2\text{,}\dots P_{n-1}$ placed in order in an anticlockwise direction round the circle, starting on the positive $x$-axis. Show the positions of $P_0\text{,}{P}_1\text{,}{P}_2$ and $P_{n-1}$.

Using part (b) find the limit of this area as $n\to \mathrm{\infty }$.

Using part (b) find the limit of this perimeter as $n\to \mathrm{\infty }$.

Hence, write down the total length of the perimeter of the regular $n$ sided polygon $P_0{P}_1{P}_2\dots P_{n-1}{P}_0$.

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.

Express $z$ in the form $a+\text{i}b$, where $a,b\in \mathbb{Q}$.

Find the exact value of the modulus of $z$.

Find the argument of $z$, giving your answer to 4 decimal places.

Find the roots of the equation $w^3=8\text{i}$, $w\in \mathbb{C}$. Give your answers in Cartesian form.

One of the roots $w_1$ satisfies the condition $\text{Re}\left(w_1\right)=0$.

Given that $w_1=\frac{z}{z-\text{i}}$, express $z$ in the form $a+b\text{i}$, where $a$, $b\in \mathbb{Q}$.

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

Find an expression in terms of θ for $\text{arg}\left(z-2a\right)$.

Find an expression in terms of θ for $\text{arg}\left(\frac{z}{z-2a}\right)$.

Hence or otherwise find the value of θ for which $\text{Re}\left(\frac{z}{z-2a}\right)=0$.

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.

Find the roots of $z^{24}=1$ which satisfy the condition , expressing your answers in the form $r{e}^{\text{i}\theta }$, where $r$, $\theta \in {\mathbb{R}}^{+}$.

Hence, or otherwise, show that S = $\frac{1}{2}\left(1+\sqrt{2}\right)\left(1+\sqrt{3}\right)\left(1+\text{i}\right)$.

Show that Re S = Im S.

By writing $\frac{\pi }{12}$ as $\left(\frac{\pi }{4}-\frac{\pi }{6}\right)$, find the value of cos $\frac{\pi }{12}$ in the form $\frac{\sqrt{a}+\sqrt{b}}{c}$, where $a$, $b$ and $c$ are integers to be determined.