DP Mathematics: Analysis and Approaches Questionbank

Find the term independent of $x$ in the expansion of $\frac{1}{x^3}{\left(\frac{1}{3x^2}-\frac{x}{2}\right)}^9$.

There are more males than females in the group.

Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.

Use de Moivre’s theorem and the result from part (a) to show that $\cot 4\theta =\frac{{\cot }^4 \theta -6 {\cot }^2 \theta +1}{4 {\cot }^3 \theta -4 \cot \theta }$.

Use the identity from part (b) to show that the quadratic equation $x^2-6x+1=0$ has roots ${\text{cot}}^2\frac{\mathrm{\pi }}{8}$ and ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

Consider the quartic equation $z^4+4z^3+8z^2+80z+400=0, z\in \mathrm{ℂ}$.

Two of the roots of this equation are $a+b\text{i}$ and $b+a\text{i}$, where $a, b\in \mathrm{\mathbb{Z}}$.

Find the possible values of $a$.

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.

Given that $1$ and $4+\text{i}$ are roots of the equation, write down the third root.

Verify that the mean of the two complex roots is $4$.

By varying the value of $q$ in the equation $x^3-7x^2+qx+1=0$, determine the smallest positive integer value of $q$ required to show that Noah is incorrect.

Using the result from part (c), show that when $q=17$, this equation has at least one complex root.

Explain why the equation will have at least one real root for all values of $q$.

By writing $x^4-9x^3+24x^2+22x-12$ as a product of one linear and one cubic factor, prove that the equation has at least one complex root.

Use your result from part (f)(ii) to show that the equation $x^4-2x^3+3x^2-4x+5=0$ has at least one complex root.

State what the result in part (f)(ii) tells us when considering this equation $x^4-9x^3+24x^2+22x-12=0$.

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 ${z_1}^2+{z_2}^2=z_1{z}_2$.

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.

Determine the roots of the equation $(z+2\text{i}{)}^3=216\text{i}$, $z\in \mathbb{C}$, giving the answers in the form $z=a\sqrt{3}+b\text{i}$ where $a,b\in \mathbb{Z}$.

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.

$r{\text{e}}^{\text{i}\theta }$ where $r$, $\theta \in \mathbb{R}$, $r>0$.

$a+\text{i}b$ where $a$, $b\in \mathbb{R}$.

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 $a$, $b\in \mathbb{Q}$.

Determine the value of

(i)     $1+\omega +{\omega }^2$;

(ii)     $1+\omega \text{*}+(\omega \text{*}{)}^2$.

Show that $(\omega -3{\omega }^2)({\omega }^2-3\omega )=13$.

Find the values of $x$ that satisfy the equation $\left|p\right|=\left|q\right|$.

Solve the inequality .

Solve the equation, giving the solutions in the form $a+\text{i}b$, where $a\text{, }b\in \mathbb{R}$.

Find, in terms of $a$, the probability that $X$ lies between 1 and 3.

Sketch the graph of $f$. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of $a$.

$a$.

$\text{E}\left(X\right)$.

the median of $X$.

Consider the expansion of ${\left(2+x\right)}^n$, where $n⩾3$ and $n\in \mathbb{Z}$.

The coefficient of $x^3$ is four times the coefficient of $x^2$. Find the value of $n$.

Let $P\left(z\right)=a{z}^3-37z^2+66z-10$, where $z\in \mathbb{C}$ and $a\in \mathbb{Z}$.

One of the roots of $P\left(z\right)=0$ is $3+\text{i}$. Find the value of $a$.

the girls do not sit together.

the girls do not sit on either end.

the girls do not sit on either end and do not sit together.

Sketch the graph of $y=x^4-6x^3-2x^2+58x-51$, stating clearly the coordinates of any maximum and minimum points and intersections with axes.

Consider the equation $z^4+a{z}^3+b{z}^2+cz+d=0$, where $a$, $b$, $c$, $d\in \mathbb{R}$ and $z\in \mathbb{C}$.

Two of the roots of the equation are log₂6 and $i\sqrt{3}$ and the sum of all the roots is 3 + log₂3.

Show that 6$a$ + $d$ + 12 = 0.

Show that $abc=8$.