DP Mathematics: Analysis and Approaches Questionbank

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 $p\in \mathrm{\mathbb{Q}}$.

Use mathematical induction to prove that $\frac{{\text{d}}^n}{\text{d}{x}^n}\left(x{\text{e}}^{px}\right)=p^{n-1}\left(px+n\right){\text{e}}^{px}$ for $n \in {\mathrm{\mathbb{Z}}}^{+}, p \in \mathrm{\mathbb{Q}}$.

There are more males than females in the group.

Find the volume of the smallest slice of pie.

The apple pie has a volume of $61 425 {\text{cm}}^3$.

Find the total number of slices Mia can cut from this pie.

Find the amount of fuel pumped into the tank in the $\text{13th}$ hour.

Solve the equation $2 \ln x= \ln 9+4$. Give your answer in the form $x=p{\mathrm{e}}^q$ where $p, q\in {\mathrm{\mathbb{Z}}}^{+}$.

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

Give a reason why the series expansion found in part (b) is not valid for $x=\frac{3}{4}$.

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

Find the value of Amelia’s investment after $5$ years to the nearest hundred dollars.

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

Find the modulus of $zw$.

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_n$.

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

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

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 $x^4$.

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

Find $S_{\infty }$.

The equation $2x^2-5x+1=0$ has two real roots, $\alpha$ and $\beta$.

Consider the equation $x^2+mx+n=0$, where $m, n\in \mathrm{\mathbb{Z}}$ and which has roots $\frac{1}{{\alpha }^3}$ and $\frac{1}{{\beta }^3}$.
Without solving $2x^2-5x+1=0$, determine the values of $m$ and $n$.

$y$-axis.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

Hence state a condition in terms of $p$ and $q$ that would imply $x^4+p{x}^3+q{x}^2+rx+s=0$ has at least one complex root.

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.

The expression $\frac{3\sqrt{x}-5}{\sqrt{x}}$ can be written as $3-5x^p$. Write down the value of $p$.

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

State the value of the first term, $u_1$.

Given that the $n$^th term of this sequence is $-33$, find the value of $n$.

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.

Find the value of $r$ required so that the amount in David’s account after $10$ years will be equal to the amount in Sam’s account.

Find the interest David will earn over the $10$ years.

the digits are distinct.

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

By solving the differential equation, show that $y=\frac{8x+x^4}{4-x^3}$.

Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.

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

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

Write down the power series for $\frac{1}{1+x^2}$.

By differentiating both sides of the expression and then substituting $x=0$, find the value of $a_1$.

For that day find how much weight was added after each lift.

Calculate the café’s total profit for the first 12 weeks.

Find the distance from the base of this ladder to the top rung.

Find the value of the common ratio for this sequence.

Find the sum of the first 30 terms of the sequence.

Find the tenth term.

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.

Express the binomial coefficient $\left(\begin{array}{c}3n+1\\ 3n-2\end{array}\right)$ as a polynomial in $n$.

Use mathematical induction to prove that $2^{n+1}>n^2$ for $n\in \mathbb{Z}$, $n⩾3$.

Find the common ratio of this sequence.

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

Write down your answer to part (a) correct to the nearest integer.

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 $n\in {\mathbb{Z}}^{+}$.

Find the common difference.

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 $n\in \mathbb{Z},n⩾3$.

Calculate the percentage of the population of Ottawa that speak English but not French.

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 $v$.

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.

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

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

Hence or otherwise find the coefficient of the term in $x^9$ in the expansion of ${\left(x+3\right)}^{11}$.

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 $P_{n-1}$.

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

Find an expression for $s$ in terms of $t$.

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

Hence find the area of the shaded region.

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

Find the values of $a$ for which the sum to infinity of the series exists.

Find the value of $a$ when $S_{\infty }=76$.

Write down the common difference, $d$.

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 $\cos \theta$.

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

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

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.

Find the value of $A$ and the value of $B$.

Consider the expansion of $(3+x^2{)}^{n+1}$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that the coefficient of $x^4$ is $20 412$, find the value of $n$.

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

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

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

The oblique asymptote of the graph of $f$ can be written as $y=ax+b$ where $a, b\in \mathrm{\mathbb{Q}}$.

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

Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

By solving the differential equation $\frac{dy}{dt}=y$, show that $y=A{\text{e}}^t$ where $A$ is a constant.

Hence show that $x=-\frac{B}{2}{\text{e}}^{2t}+C$, where $C$ is a constant.

Find the two values for $\lambda$ that satisfy $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$.

Show that $p^2-2q={\alpha }^2+{\beta }^2+{\gamma }^2$.

Given that $p^2<3q$, deduce that $\alpha , \beta$ and $\gamma$ cannot all be real.

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.

Expand and simplify ${(1-a)}^3$ in ascending powers of $a$.

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

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

Prove that the sum of these three integers is always divisible by $3$.

Prove that the sum of the squares of these three integers is never divisible by $3$.

Find the number of ways these five people can be seated in this row.

the digits are distinct and are in increasing order.

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

Find the value of r, giving your answer to four significant figures.

Show that ${\left(2n-1\right)}^2+{\left(2n+1\right)}^2=8n^2+2$, where $n\in \mathbb{Z}$.

By substituting $x=\frac{1}{9}$ find a rational approximation to $\sqrt[3]{9}$.

the number of years it will take for the population to triple.

Calculate the original price of the bicycle.

Find the value of $k$.

Find the distance that the post is driven into the ground by the eighth strike of the hammer.

Find the total depth that the post has been driven into the ground after 10 strikes of the hammer.

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

Write down the common ratio of the sequence.

Find the year in which the comet was seen from Earth for the fifth time.

Determine the mean of X.

Hence find the least value of $n$ for which $\left(\begin{array}{c}3n+1\\ 3n-2\end{array}\right)>{10}^6$.

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

John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.

Calculate the tea-shop’s total profit for the first 12 weeks.

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

Use the regression equation to estimate the value of y when x = 3.57.

Determine the value of

(i)     $1+\omega +{\omega }^2$;

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

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

Boxes of mixed fruit are on sale at a local supermarket.

Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.

Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.

Box C contains 5 bananas and 4 kiwifruit and costs $3.00.

Find the cost of each type of fruit.

Write your answer to part (b) in the form $a\times {10}^k$, where 1 ≤ $a$ < 10 , $k\in \mathbb{Z}$.

Write down your answer to part (b) in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$.

Given that ${\log }_{10}\left(\frac{1}{2\sqrt{2}}\left(p+2q\right)\right)=\frac{1}{2}\left({\log }_{10}p+{\log }_{10}q\right),p>0,q>0$, find $p$ in terms of $q$.

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

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

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

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 $\text{CDBA}$.

Calculate the time, in minutes, it takes for light from the Sun to reach the Earth.

Calculate the number of people in Ottawa that speak both English and French.

Find the remainder when ${15}^{1207}$ is divided by $13$.

Convert ${\left(7A2\right)}_{16}$ to base $5$, where ${\left(A\right)}_{16}={\left(10\right)}_{10}$.

Consider the equation ${\left(1251\right)}_n+{\left(30\right)}_n={\left(504\right)}_n+{\left(504\right)}_n$.

Find the value of $n$.

Write down the coordinates of $\text{B}$.

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

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

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

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

Find an expression for r in terms of θ.

Find the common ratio in terms of $a$.

Find the total amount of fuel pumped into the tank in the first $8$ hours.

Prove by contradiction that ${\log }_2 5$ is an irrational number.

Show that $H_n=2400n+67 600$.

Hence, expand $\frac{2+7x}{\left(1+2x\right)\left(1-x\right)}$ in ascending powers of $x$, up to and including the term in $x^2$.

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 minimum value of $k$.

in the position immediately after Andrea.

in any position after Andrea.

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

Show that $a=8$.

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 $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Find the first term of the sequence, $u_1$.

Hence find $y$ as a function of $t$.

A polygonal number, $P_r\left(n\right)$, can be represented by the series

$\underset{m=1}{\overset{n}{\text{Σ}}}\left(1+\left(m-1\right)\left(r-2\right)\right)$ where $r\in {\mathrm{\mathbb{Z}}}^{+}, r\ge 3$.

Use mathematical induction to prove that $P_r\left(n\right)=\frac{\left(r-2\right)n^2-\left(r-4\right)n}{2}$ where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Deduce from part (d)(i) that the complex roots of the equation $\left(z-r\right)\left(z^2-2az+a^2+b^2\right)=0$ can be expressed as $a\pm \text{i}\sqrt{g\text{'}\left(a\right)}$.

Use this diagram to determine the roots of the corresponding equation of the form $\left(z-r\right)\left(z^2-2az+a^2+16\right)=0$ for $z\in \mathrm{ℂ}$.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

The third term in the expansion is the mean of the second term and the fourth term in the expansion.

Find the possible values of $x$.

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.

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

Express $z_1$ in terms of $z_2$.

Find the amount that Sam will have in his account after $10$ years.

The complex numbers $w$ and $z$ satisfy the equations

$\frac{w}{z}=2\text{i}$

${\text{z}}^{\ast }-3w=5+5\text{i}$.

Find $w$ and $z$ in the form $a+b\text{i}$ where $a$, $\text{b}\in \mathbb{Z}$.

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

Write down and simplify the first three terms, in ascending powers of $x$, in the Extended Binomial expansion of ${\left(1-x\right)}^{\frac{1}{3}}$.

the number of years it will take for the population to triple.

Show that the general solution of this differential equation is $P=A{\text{e}}^{kt}$, where $A\in \mathbb{R}$.

$\underset{t\to \mathrm{\infty }}{\text{lim}}P$

Write down the first three non-zero terms of $w_n$.

Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.

Calculate the difference between the original price of the bicycle and the total amount Juan will pay.

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.

Let $f(x)=(x^2+3{)}^7$. Find the term in $x^5$ in the expansion of the derivative, $f^{\prime }(x)$.

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

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

Show that there will be approximately 2645 fish in the lake at the start of 2020.

$a+\text{i}b$ 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 value of $a$ for which the system of equations does not have a unique solution.

Find the tenth term.

Find the value of x.

Write your answer to part (a)

(i)     correct to two decimal places;

(ii)     correct to three significant figures.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

Given that $q=8d^2$, find the other two real roots.

Find the sum of the first 8 terms.

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

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, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.

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

Karl decides to donate this interest to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = t EUR.

Calculate the value of t.

Calculate the amount of money he has in the account after 5 years.

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.

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

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

Find the maximum displacement of $P$, in metres, from its initial position.

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

At the end of the $5$ years, Imon withdrew $x \text{SGD}$ from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of $5.7\%$, compounded half-yearly.

The value of the super-savings account increased to $20 000 \text{SGD}$ after $18$ months.

Find the value of $x$.

Show that the volume of the tank is $624 000 {\text{m}}^3$, correct to three significant figures.

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

A biased coin is weighted such that the probability, $p$, of obtaining a tail is $0.6$. The coin is tossed repeatedly and independently until a tail is obtained.

Let $E$ be the event “obtaining the first tail on an even numbered toss”.

Find $\text{P}\left(E\right)$.

Find $\gamma$.

Write down the radius of the planet.

Consider an arithmetic sequence where $u_8=S_8=8$. Find the value of the first term, $u_1$, and the value of the common difference, $d$.

Determine the number of years required for Amelia’s investment to reach the target.

Bill invests his $\$9000$ in an account that offers an interest rate of $r\%$ per annum compounded monthly, where $r$ is set to two decimal places.

Find the minimum value of $r$ needed for Bill to reach the target after $10$ years.

Hence, find an expression for $f\left(x\right)$.

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

Show that $2x-3-\frac{6}{x-1}=\frac{2x^2-5x-3}{x-1}, x\in \mathrm{ℝ}, x\ne 1$.

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

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

Write down the equation of the vertical asymptote of the graph of $f$.

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

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

Show that $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

For $n=4$, sketch a diagram clearly showing your answer to part (b)(ii).

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 expanding $\left(x-\alpha \right)\left(x-\beta \right)\left(x-\gamma \right)$ show that:

$p=-\left(\alpha +\beta +\gamma \right)$

$q=\alpha \beta +\beta \gamma +\gamma \alpha$

$r=-\alpha \beta \gamma$.

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

Show that $b=21$.

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

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

By using a suitable substitution for $a$, show that $1-3 \cos 2x+3 {\cos }^2 2x- {\cos }^3 2x=8 {\sin }^6 x$.

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

Find the actual value of $y$ when $x=1.5$.

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

By substituting $x=0$, find the value of $a_0$.

Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left(1+x\right)}^q\text{,}q\in \mathbb{Q}$.

the population after 10 years

Find the value of $r$, the common ratio of the sequence.

Find the common difference.

Find the value of $n$ for which $u_n=2$.

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 $k\in \mathbb{Z}$.

Solve the simultaneous equations

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

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

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

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

Solve the inequality $x^2>2x+1$.

Find u₈.

Calculate, in years, when the bicycle value will be less than 50 USD.

Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.

Hence find the value of n such that $\sum _{k=1}^n{x}_k=861$.

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

Give your answer to part (a) correct to three significant figures.

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

Write your answer to part (b)(ii) in the form $a\times {10}^k$, where .

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

Find the area of $R$.

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

Find the common difference.

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

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

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

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.

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

The following diagram shows [CD], with length $b\text{ cm}$, where $b>1$. Squares with side lengths $k\text{ cm},k^2\text{ cm},k^3\text{ cm},\dots$, where , are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.

The total sum of the areas of all the squares is $\frac{9}{16}$. Find the value of $b$.

Show that $abc=8$.

Consider the expansion of ${\left(3x^2-\frac{k}{x}\right)}^9$, where $k>0$.

The coefficient of the term in $x^6$ is $6048$. Find the value of $k$.

Find $h$, the height of the tank.

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

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

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

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

Consider $z=\cos \theta +\text{i} \sin \theta$ where $z\in \mathrm{ℂ}, z\ne 1$.

Show that $\text{Re}\left(\frac{1+z}{1-z}\right)=0$.

Write down the first three terms of the binomial expansion of $(1+t{)}^{-1}$ in ascending powers of $t$.

Hence or otherwise, solve the equation  $2 \sin 2\theta -3-\frac{6}{\sin 2\theta -1}=0$  for  $0\le \theta \le \mathrm{\pi }, \theta \ne \frac{\mathrm{\pi }}{4}$.

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

Find the least value of $n$ such that $S_{\infty }-S_n<0.001$.

$x$-axis.

Show that $\frac{dx}{dt}-x=-A{\text{e}}^t$.

For triangular numbers, verify that $P_3\left(n\right)=\frac{n\left(n+1\right)}{2}$.

State the coordinates of ${\text{C}}_2$.

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.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Hence or otherwise solve $\text{lo}{\text{g}}_3\left(\text{2}\text{sin}x\right)=\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)$ for .

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

Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for ${\left(1+x\right)}^{-2}$.

Calculate, in CAD, the total amount John pays for the bicycle.

the value of $d$;

Find the total number of sticks used by Tomás for all 24 diagrams.

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.

Write your answer to part (b)(ii) in the form $a\times {10}^k$, where .

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

On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.

Calculate the number of positive terms in the sequence.

Find the value of $m$.

Find the value of a and of b.

Use this model to calculate the circumference of the Moon in kilometres. Give your full calculator display.

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

Use mathematical induction to prove that ${\left(1-a\right)}^n>1-na$ for $\left\{n\text{:}n\in {\mathbb{Z}}^{+},n⩾2\right\}$ where .

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

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 $a$.

Find the value of $a$.

Find the least value of n for which S_n > 163.

Find the values of θ which give the greatest value of the sum.

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

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

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.

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

Show that the total value of Phil’s savings after 20 years is $\frac{({1.02}^{20}-1)P}{(1.02-1)}$.

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

Show that Re S = Im S.

Find the common ratio.

Find the common ratio.

Show that one of the real roots is equal to 1.

David wishes to withdraw $5000 at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

$\frac{5000}{1.028}+\frac{5000}{{1.028}^2}+\dots +\frac{5000}{{1.028}^n}$.

Consider a geometric sequence where the first term is 768 and the second term is 576.

Find the least value of $n$ such that the $n$th term of the sequence is less than 7.

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 $y$.

Show that, at these times, $\tan 6t=2$.

Hence show that $\frac{v_2}{v_1}=\frac{v_3}{v_2}=-{\text{e}}^{-\frac{\mathrm{\pi }}{2}}$.

Determine the values of $p$, $q$ and $r$.

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

Write down the remainder when ${14}^{2022}$ is divided by $7$.

Use Fermat’s little theorem to find the remainder when ${14}^{2022}$ is divided by $17$.

Prove that a number in base $13$ is divisible by $6$ if, and only if, the sum of its digits is divisible by $6$.

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

Write down the number of hours that the pump was pumping fuel into the tank.

Show that the tank will never be completely filled using this pump.

Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number.

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 $n\in {\mathrm{\mathbb{Z}}}^{+}$.

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

For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.

Find Jane’s total earnings at the start of her $10$th year of employment. Give your answer correct to the nearest dollar.

Verify that $2$ is not a Gaussian prime.

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

Find the amount Chris needs to deposit initially in order to reach the target after $5$ years. Give your answer to the nearest dollar.

In the expansion of $(x+k{)}^7$, where $k\in \mathrm{ℝ}$, the coefficient of the term in $x^5$ is $63$.

Find the possible values of $k$.

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

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

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

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.

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

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

Let the two values found in part (c)(ii) be ${\lambda }_1$ and ${\lambda }_2$.

Verify that $y=F{\text{e}}^{{\lambda }_{1}t}+G{\text{e}}^{{\lambda }_{2}t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

By differentiating $\frac{dy}{dt}=-x+y$ with respect to $t$, show that $\frac{d^{2}y}{d{t}^2}=2\frac{{\displaystyle dy}}{{\displaystyle dt}}$.

State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.

Hence show that $P_5\left(n\right)=\frac{n\left(3n-1\right)}{2}$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$, that is both a triangular number and a pentagonal number.

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

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

Find the value of $n$.

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

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

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

Laurie makes no further deposits to or withdrawals from the account.

Find the year in which the amount of money in Laurie’s account will become double the amount she invested.

Hence prove that the square of any integer can be written in the form $4t$ or $4t+1$, where $t\in {\mathbb{Z}}^{+}$.

Hence, using integration, find the power series for $\text{arctan}x$, giving the first four non-zero terms.

Consider the power series $1-x+x^2-x^3+x^4-...$

By considering the ratio of consecutive terms, explain why this series is equal to ${\left(1+x\right)}^{-1}$ and state the values of $x$ for which this equality is true.

Solve the differential equation $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, giving your answer in the form $P=g\left(t\right)$.

Find the exact value of $S_k$.

Calculate the total distance Carlos runs in the first year.

Find the sum of the first ten terms of the sequence.

the value of $r$;

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

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

Find the two smallest non-zero values of $x$ for which $f(x)=g(x)$.

Calculate $\sqrt[3]{\frac{p}{q}}$. Give your full calculator display.

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

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

For that day find the weight of Sergei’s first lift.

Find the total amount John has paid to insure his bicycle for the first 5 years.

Given that x_{k + 1} = x_k + a, find a.

Hence find the cube roots of $z$ in modulus-argument 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 $a$, $b\in \mathbb{Q}$.

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

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 $\text{Re}\left(\frac{z}{z-2a}\right)=0$.

Find the length, in cm, of this line. Give your answer in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$.

Write down the length of the Amazon River in cm.

Find the distance from the Earth to Polaris in millions of km. Give your answer in the form $a\times {10}^k$ with and $k\in \mathbb{Z}$.

In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.

Find the number of different orders in which these 18 examination papers may be taken.

Calculate the value of $p$. Write down your full calculator display.

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

The region $R$ is now rotated about the $y$-axis, through $2\pi$ radians, to form a solid.

By writing $\text{si}{\text{n}}^{3}y$ as $\left(1-\text{co}{\text{s}}^{2}y\right)\text{sin}y$, show that the volume of the solid formed is $\frac{2\pi }{3}$.

State Fermat’s little theorem.

A geometric sequence has $u_4=-70$ and $u_7=8.75$. Find the second term of the sequence.

the girls do not sit together.

Find the first term.

Find the sum of the first $20$ terms.

Find the value of $r$.

Find the area of triangle UVW.

Express w² and w³ in modulus-argument form.

Write down the amount of interest he earned after 5 years.

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

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

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

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 base $13$ number $1y93y25$ is divisible by $6$. Find the possible values of the digit $y$.

Show that $u_1=4$.

Given that $J_n$ follows a geometric sequence, state the value of the common ratio, $r$.

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.

The volume of the planet can be expressed in the form $\mathrm{\pi }\left(a\times {10}^k\right) {\text{km}}^3$ where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

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

Consider two consecutive positive integers, $n$ and $n+1$.

Show that the difference of their squares is equal to the sum of the two integers.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of $n$ for the value of the grand prize in the $n\text{th}$ week of the lottery.

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 $n\in {\mathrm{\mathbb{Z}}}^{+}$.

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

Write down an expression for $f^{-1}\left(x\right)$.

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

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

Prove the identity ${\left(p+q\right)}^3-3pq\left(p+q\right)\equiv p^3+q^3$.

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 ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.

Show that $8P_3\left(n\right)+1$ is the square of an odd number for all $n\in {\mathrm{\mathbb{Z}}}^{+}$.

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

Consider the curve with equation $y=(2x-1){\text{e}}^{kx}$, where $x\in \mathrm{ℝ}$ and $k\in \mathrm{\mathbb{Q}}$.

The tangent to the curve at the point where $x=1$ is parallel to the line $y=5{\text{e}}^{k}x$.

Find the value of $k$.

Find the common difference, $d$.

Show that the three planes do not intersect.

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

Explain why any integer can be written in the form $4k$ or $4k+1$ or $4k+2$ or $4k+3$, where $k\in \mathbb{Z}$.

Hence, by recognising the pattern, deduce the first four terms in a power series for ${\left(1+x\right)}^{-n}$, $n\in {\mathbb{Z}}^{+}$.

Repeat this process to find the first four terms in a power series for ${\left(1+x\right)}^{-3}$.

Given that the initial population is 1000, $L=10000$  and $m=0.003$, find the number of years it will take for the population to triple.

Find the value of $r$.

Find the approximate number of fish in the lake at the start of 2042.

Write down the distance Rosa runs in the third training session;

Write down the distance Rosa runs in the $n$th training session.

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

An infinite geometric series is given as $S_{\mathrm{\infty }}=a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\dots$, $a\in {\mathbb{Z}}^{+}$.

Find the largest value of $a$ such that .

Diagram $n$ is formed with 52 sticks. Find the value of $n$.

Find the value of $u_5$.

Find the smallest value of $n$ for which $u_n$ is less than ${10}^{-3}$.

find the value of $d$.

The following diagram shows part of the graph of $g$ reflected in the $x$-axis. It also shows part of the graph of $f$ and the point P.

Find an expression for the area of the shaded region. Do not calculate the value of the expression.

Determine the variance of X.

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

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

The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.

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

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

Solve the inequality .

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

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

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

Write down your answer to part (b) in the form $a\times {10}^k$ where and k $\in \mathbb{Z}$.

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

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

the median of $X$.

Find the least value of $n$ such that $S_n>5543$.

The coefficient of $x^2$ in the expansion of ${\left(\frac{1}{x}+5x\right)}^8$ is equal to the coefficient of $x^4$ in the expansion of ${\left(a+5x\right)}^7,a\in \mathbb{R}$. Find the value of $a$.

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 $a,b\in \mathbb{R}$.

Given that Phil’s aim is to own the house after 20 years, find the value for $P$ to the nearest dollar.

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

Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.

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

Find the times when $P$ comes to instantaneous rest.

Find the common ratio of the sequence.

Calculate the value of Imon’s investment after $5$ years.

Find the value of $n$ such that $u_n=0$.

Find the value of $N$.

Show that Chris will never reach the target if his initial deposit is $\$9000$.

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 $n\in \mathrm{\mathbb{Z}}, n\ge 2$.

The expression for $f\prime \left(x\right)$ can be written in the form $\frac{a}{x}+\frac{b}{k-x}$, where $a, b\in \mathrm{ℝ}$. Find $a$ and $b$ in terms of $k$.

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

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

Given that the $k$th term of the sequence is zero, find the value of $k$.

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

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 $k$.

By substituting $Y=\frac{dy}{dt}$, show that $Y=B{\text{e}}^{2t}$ where $B$ is a constant.

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

Show that $P_3\left(n\right)+P_3\left(n+1\right)\equiv {\left(n+1\right)}^2$.

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.

Hence show that ${\left(\alpha -\beta \right)}^2+{\left(\beta -\gamma \right)}^2+{\left(\gamma -\alpha \right)}^2=2p^2-6q$.