Determine whether $f_n$ is an odd or even function, justifying your answer.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

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.

Write down the equation of the vertical asymptote.

Write down the equation of the horizontal asymptote.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

Hence, solve the inequality $0<\frac{2x-1}{x+1}<2$.

Solve the inequality $0<\frac{2\left|x\right|-1}{\left|x\right|+1}<2$.

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 $z_2$.

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.

Express $z_1$ in terms of $z_2$.

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

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

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.

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

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

Find the area of triangle UVW.

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

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

State the equation of the vertical asymptote on the graph of $y=f\left(x\right)$.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

Use an algebraic method to determine whether $f$ is a self-inverse function.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

Write down an expression for $f\text{'}\left(x\right)$.

Hence, given that $f^{-1}$ does not exist, show that $b^2-3ac>0$.

Show that $g^{-1}$ exists.

$g\left(x\right)$ can be written in the form $p(x-2{)}^3+q$ , where $p, q \in \mathrm{ℝ}$.

Find the value of $p$ and the value of $q$.

Hence find $g^{-1}\left(x\right)$.

State each of the transformations in the order in which they are applied.

Sketch the graphs of $y=g\left(x\right)$ and $y=g^{-1}\left(x\right)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Hence or otherwise, show that the series is convergent.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Show that $p=\frac{2}{3}$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

The sum of the first $n$ terms of the series is $\ln \left(\frac{1}{x^3}\right)$.

Find the value of $n$.

The cubic equation $x^3-k{x}^2+3k=0$ where $k>0$ has roots $\alpha , \beta$ and $\alpha +\beta$.

Given that $\alpha \beta =-\frac{k^2}{4}$, find the value of $k$.

The graph of $y=f\left(x\right)$ has a local maximum at A. Find the coordinates of A.

Show that there is exactly one point of inflexion, B, on the graph of $y=f\left(x\right)$.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=f(x)$;

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\frac{1}{f(x)}$;

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\left|\frac{1}{f(x)}\right|$.

Find $\int f(x)\cos x\text{d}x$.

By finding $g^{\prime }(x)$ explain why $g$ is an increasing function.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

the vertical asymptote of the graph of $f$.

the horizontal asymptote of the graph of $f$.

the $x$-axis.

the $y$-axis.

Sketch the graph of $f$ on the axes below.

The function $g$ is defined by $g\left(x\right)=\frac{ax+4}{3-x}$, where $x\in \mathrm{ℝ}, x\ne 3$ and $a\in \mathrm{ℝ}$.

Given that $g\left(x\right)=g^{-1}\left(x\right)$, determine the value of $a$.

Find the co-ordinates of all stationary points.

Write down the equation of the vertical asymptote.

With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

State the range of $g$.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

Find the coordinates of P.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

Show that $A=\frac{\sqrt{2}\mathrm{\pi }}{2}$.

Find the value of $k$.

Show that $m=-\frac{6x}{{\left(x^2+2\right)}^2}$.

Show that the maximum value of $m$ is $\frac{27}{32}\sqrt{\frac{2}{3}}$.

Sketch the graphs of $y=\frac{x}{2}+1$ and $y=\left|x-2\right|$ on the following axes.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

Show that $g^{-1}\left(x\right)=1+\frac{\sqrt{4x^2+x}}{x}$.

State the domain of $g^{-1}$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Find the largest value of $a$ such that $f$ has an inverse function.

For this value of $a$, find an expression for $f^{-1}\left(x\right)$, stating its domain.

Let f(x) = x⁴ + px³ + qx + 5 where p, q are constants.

The remainder when f(x) is divided by (x + 1) is 7, and the remainder when f(x) is divided by (x − 2) is 1. Find the value of p and the value of q.

Given that $q(x)$ has a factor $(x-4)$, find the value of $k$.

Hence or otherwise, factorize $q(x)$ as a product of linear factors.

Show that $f$ is an odd function.

The range of $f$ is $a\le y\le b$, where $a, b\in \mathrm{ℝ}$.

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

Show that $f\left(x\right)$ has no vertical asymptotes.

Find the equation of the horizontal asymptote. 

Find the exact value of $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$, giving the answer in the form $\text{ln}q\text{,}q\in \mathbb{Q}$.

Find the possible values for $p$.

Consider the case when $p=4$. The roots of the equation can be expressed in the form $x=\frac{a\pm \sqrt{13}}{6}$, where $a\in \mathrm{\mathbb{Z}}$. Find the value of $a$.

Find $f^{\prime }\left(x\right)$.

Show that, if $f^{\prime }\left(x\right)=0$, then $x=2-\sqrt{3}$.

find the coordinates of the $y$-intercept.

show that there are no $x$-intercepts.

sketch the graph, showing clearly any asymptotic behaviour.

Show that $\frac{3}{x+1}-\frac{1}{x-1}=\frac{2x-4}{x^2-1}$.

The area enclosed by the graph of $y=f\left(x\right)$ and the line $y=4$ can be expressed as $\text{ln}v$. Find the value of $v$.

The following diagram shows the graph of $y=f\left(x\right)$. The graph has a horizontal asymptote at $y=-1$. The graph crosses the $x$-axis at $x=-1$ and $x=1$, and the $y$-axis at $y=2$.

On the following set of axes, sketch the graph of $y={\left[f\left(x\right)\right]}^2+1$, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

Write down the least value of $a$ such that $g$ has an inverse.

For the value of $a$ found in part (b), write down the domain of $g^{-1}$.

For the value of $a$ found in part (b), find an expression for $g^{-1}\left(x\right)$.

Consider the function $f\left(x\right)=x^4-6x^2-2x+4$, $x\in \mathbb{R}$.

The graph of $f$ is translated two units to the left to form the function $g\left(x\right)$.

Express $g\left(x\right)$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ where $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

Express $g\left(x\right)$ in the form $A+\frac{B}{x-2}$ where A, B are constants.

Sketch the graph of $y=\frac{x-4}{2x-5}$, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Factorize $x^2+3x+2$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

Find all the intercepts of the graph of $f\left(x\right)$ with both the $x$ and $y$ axes.

Write down the equation of the vertical asymptote.

As $x\to \pm \mathrm{\infty }$ the graph of $f\left(x\right)$ approaches an oblique straight line asymptote.

Divide $2x^2-5x-12$ by $x+2$ to find the equation of this asymptote.

Write down the range of $f$.

Find an expression for $f^{-1}(x)$.

Write down the domain and range of $f^{-1}$.

Solve the equation ${\log }_3 \sqrt{x}=\frac{1}{2 {\log }_2 3}+{\log }_3\left(4x^3\right)$, where $x>0$.

The quadratic equation $x^2-2kx+(k-1)=0$ has roots $\alpha$ and $\beta$ such that ${\alpha }^2+{\beta }^2=4$. Without solving the equation, find the possible values of the real number $k$.

Solve .

Describe a sequence of transformations that transforms the graph of $y=\text{arctan\hspace{0.05em}}x$ to the graph of $y=\text{arctan}\left(2x+1\right)+\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}$.

Show that $\text{arctan} p+\text{arctan} q\equiv \text{arctan}\left(\frac{p+q}{1-pq}\right)$ where $p, q>0$ and $pq<1$.

Verify that $\text{arctan\hspace{0.05em}}\left(2x+1\right)=\text{arctan\hspace{0.05em}}\left(\frac{x}{x+1}\right)\mathrm{+}\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}, x>0$.

Using mathematical induction and the result from part (b), prove that $\underset{r=1}{\overset{n}{\text{Σ}}}\text{arctan}\left(\frac{1}{2r^2}\right)=\text{arctan}\left(\frac{n}{n+1}\right)$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

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 $\cos \theta$.

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 ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

Hence find the exact value of ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

Deduce a quadratic equation with integer coefficients, having roots ${\mathrm{cosec}}^2 \frac{\mathrm{\pi }}{8}$ and ${\mathrm{cosec}}^2 \frac{3\mathrm{\pi }}{8}$.