The first term in an arithmetic sequence is $4$ and the fifth term is ${\log }_2 625$.

Find the common difference of the sequence, expressing your answer in the form ${\log }_2 p$, where $p\in \mathrm{\mathbb{Q}}$.

Very briefly, explain why the value of this integral must be negative.

Express the function $f\left(x\right)=\frac{-1}{x+x^2}$ in partial fractions.

Use parts (a) and (b) to show that .

Three planes have equations:

$2x-y+z=5$

$x+3y-z=4$     , where $a\text{, }b\in \mathbb{R}$.

$3x-5y+az=b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

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

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

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

Find $\mathrm{AC}$.

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 $\text{Re}\left(\alpha \right)$.

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 $a,b\in \mathbb{R}$.

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 $n$.

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

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 $v$.

Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the same pen.

Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which share a boundary.

Find how many sets of four points can be selected which can form the vertices of a quadrilateral.

Find how many sets of three points can be selected which can form the vertices of a triangle.

Verify that $\text{P}$ is the point of intersection of the two lines.

Write down the value of $\lambda$ corresponding to the point $\text{A}$.

Write down $\overrightarrow{\text{PA}}$ and $\overrightarrow{\text{PB}}$.

Let $\text{C}$ be the point on $l_1$ with coordinates (1, 0, 1) and $\text{D}$ be the point on $l_2$ with parameter $\mu =-2$.

Find the area of the quadrilateral $\text{CDBA}$.

Use the method of mathematical induction to prove that $4^n+15n-1$ is divisible by 9 for $n\in {\mathbb{Z}}^{+}$.

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.

Find the value of $a$ for which the system of equations does not have a unique solution.

Find the solution of the system of equations when $a=2$.

Find the number of different possible teams that could be chosen.

Find the number of different possible teams that could be chosen, given that the team must include at least one girl and at least one boy.

Find the solution of ${\log }_{2}x-{\log }_{2}5=2+{\log }_{2}3$.

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

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.

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

Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.

Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

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

Show that $\frac{1-\cos 2x}{2\sin x}\equiv \sin x,x\ne k\pi$ where $k\in \mathbb{Z}$.

Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.

Prove by mathematical induction that $\left(\begin{array}{c}2\\ 2\end{array}\right)+\left(\begin{array}{c}3\\ 2\end{array}\right)+\left(\begin{array}{c}4\\ 2\end{array}\right)+\dots +\left(\begin{array}{c}n-1\\ 2\end{array}\right)=\left(\begin{array}{c}n\\ 3\end{array}\right)$, where $n\in \mathbb{Z},n⩾3$.

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

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

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 integers $a$ and $b$ such that $a^2+b^2$ is exactly divisible by $4$. Prove by contradiction that $a$ and $b$ cannot both be odd.

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

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

Find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

Show that the three planes do not intersect.

Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

Find the distance between $L$ and $\prod _3$.

Prove by contradiction that the equation $2x^3+6x+1=0$ has no integer roots.

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

State the restriction which must be placed on $x$ for this expansion to be valid.

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.

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

Use the principle of mathematical induction to prove that

$1+2\left(\frac{1}{2}\right)+3{\left(\frac{1}{2}\right)}^2+4{\left(\frac{1}{2}\right)}^3+\dots +n{\left(\frac{1}{2}\right)}^{n-1}=4-\frac{n+2}{2^{n-1}}$, where $n\in {\mathbb{Z}}^{+}$.

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.

Consider the function $f\left(x\right)=x{\text{e}}^{2x}$, where $x\in \mathbb{R}$. The $n^{\text{th}}$ derivative of $f\left(x\right)$ is denoted by $f^{\left(n\right)}\left(x\right)$.

Prove, by mathematical induction, that $f^{\left(n\right)}\left(x\right)=\left(2^{n}x+n{2}^{n-1}\right){\text{e}}^{2x}$, $n\in {\mathbb{Z}}^{+}$.

Express $f(x)$ in partial fractions.

Use part (a) to show that $f(x)$ is always decreasing.

Use part (a) to find the exact value of $\underset{-1}{\overset{0}{\int }}f(x)dx$, giving the answer in the form $\text{ln}q$,   $q\in \mathbb{Q}$.

Find an expression for $z_1{z}_2$ in terms of $b$.

Hence, given that $\text{arg}\left(z_1{z}_2\right)=\frac{\mathrm{\pi }}{4}$, find the value of $b$.

Show that the probability that Chloe wins the game is $\frac{3}{8}$.

Determine the mean of X.

Determine the variance of X.

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 $a, b \in \mathrm{\mathbb{Z}}$.

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 .

find the value of $d$.

determine the value of $\sum _{r=1}^N{u}_r$.

Solve the equation $4^x+2^{x+2}=3$.

Solve the simultaneous equations

$\text{lo}{\text{g}}_{2}6x=1+2\text{lo}{\text{g}}_{2}y$

$1+\text{lo}{\text{g}}_{6}x=\text{lo}{\text{g}}_6\left(15y-25\right)$.

Solve the equation ${\log }_2(x+3)+{\log }_2(x-3)=4$.

the value of $d$;

the value of $r$;

Solve .

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.

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

Let $f(x)=\frac{1}{1-x^2}$ for . Use partial fractions to find $\int f\left(x\right)dx$.

It is given that $2 \cos A \sin B\equiv \sin \left(A+B\right)-\sin \left(A-B\right)$. (Do not prove this identity.)

Using mathematical induction and the above identity, prove that $\underset{r=1}{\overset{n}{\mathrm{\Sigma }}}\cos \left(2r-1\right)\theta =\frac{\sin {\displaystyle }{\displaystyle 2}{\displaystyle n}{\displaystyle \theta }}{{\displaystyle 2}{\displaystyle }\sin \theta }$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Consider the equation $\frac{2z}{3-z*}=\text{i}$, where $z=x+\text{i}y$ and $x, y\in \mathrm{ℝ}$.

Find the value of $x$ and the value of $y$.

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 mathematical induction to prove that $\sum _{r=1}^{n}r\left(r\text{!}\right)=\left(n+1\right)\text{!}-1$, for $n\in {\mathbb{Z}}^{+}$.

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 $\text{P}(A\cup B)=\text{P}(A)+\text{P}(A^{\prime }\cap B)$.

(i)     show that $\text{P}(A)=\frac{1}{3}$;

(ii)     hence find $\text{P}(B)$.

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

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

Consider the expansion of ${\left(8x^3-\frac{1}{2x}\right)}^n$ where $n\in {\mathrm{\mathbb{Z}}}^{+}$. Determine all possible values of $n$ for which the expansion has a non-zero constant term.

Show that $f\text{'}\text{'}\left(x\right)=-\frac{1}{4\sqrt{{\left(1+x\right)}^3}}$.

Use mathematical induction to prove that $f^{\left(n\right)}\left(x\right)={\left(-\frac{1}{4}\right)}^{n-1}\frac{\left(2n-3\right)!}{\left(n-2\right)!}{\left(1+x\right)}^{\frac{1}{2}-n}$ for $n\in \mathrm{\mathbb{Z}}, n\ge 2$.

Let $g\left(x\right)={\text{e}}^{mx}, m\in \mathrm{\mathbb{Q}}$.

Consider the function $h$ defined by $h\left(x\right)=f\left(x\right)\times g\left(x\right)$ for $x>-1$.

It is given that the $x^2$ term in the Maclaurin series for $h\left(x\right)$ has a coefficient of $\frac{7}{4}$.

Find the possible values of $m$.