Find the common ratio of the sequence.

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.

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

Show that $p=\pm \frac{1}{\sqrt{3}}$.

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 $-3 \ln x$.

Find the value of $n$.

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

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

Find the common difference, $d$.

Find the common difference.

Find the tenth term.

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

Find the value of $a$.

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

Calculate the original price of the bicycle.

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

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 value of x.

Find the value of the common ratio for this sequence.

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.

Show that $a=8$.

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

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

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

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.

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

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

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

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

Show that ${\int }_0^{m}f\left(x\right)dx=\frac{32}{7}{\sin }^7 m$, where $m$ is a positive real constant.

It is given that ${\int }_m^{\frac{\mathrm{\pi }}{2}}f\left(x\right)dx=\frac{127}{28}$, where $0\le m\le \frac{\mathrm{\pi }}{2}$. Find the value of $m$.

Show that $b=21$.

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 $x$.

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

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

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

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

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

Calculate the volume of this hemisphere in cm³.

Give your answer correct to one decimal place.

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

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

Find the common difference.

Find the tenth term.

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

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

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

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 $3$.

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

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

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

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

Write your answer to part (a)

(i)     correct to two decimal places;

(ii)     correct to three significant figures.

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

Find the coordinates of $\text{B}$.

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

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

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

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

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

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.

Find an expression for r in terms of θ.

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

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

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

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

Write down the common ratio of the sequence.

Find the value of $u_5$.

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

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

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

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

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

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

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

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

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

Show that $u_1=4$.

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

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, $d$.

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

Write down your answer to part (a) correct to two decimal places;

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

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

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

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 $x$.

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

At point P, the graphs of $f$ and $g$ intersect for the 21st time. Find the coordinates of P.

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.

Find the common difference.

Find the first term.

Find the sum of the first $20$ terms.

Find u₈.

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

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 $b$.

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

Determine how many times the comet has been seen from Earth up to the year 2014.

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 $k$.

Find the common ratio.

Show that $\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)=\text{lo}{\text{g}}_3\sqrt{\text{cos}2x+2}$.

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 .

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 $k$.

Write down the radius of the planet.

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.

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

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

Write down the amount of money Jashanti saves per month.

Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.

Calculate how much extra money Jashanti needs.