Topic 1 All

[Maximum mark: 6]

Given that $\log_a 2 = 5$.

1. Find the exact value of $\log_a 32$. [2]
   

2. Find the exact value of $\log_{\sqrt{a}} 2$. [2]
   

3. Find the value of $a$, giving your answer correct to $3$ significant figures. [2]
   

[Maximum mark: 6]

On Gary's $50$th birthday, he invests $\$P$ in an account that pays a nominal annual interest rate of $5$ %, compounded monthly.

The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, $\alpha$.

1. Find the value of $\alpha$, giving your answer to four significant figures. [3]
   

Gary makes no further deposits or withdrawals from the account.

2. Find the age Gary will be when the amount of money in his account will be double the amount he invested. [3]
   

[Maximum mark: 6]

Using mathematical induction, prove that $1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \in \mathbb{Z}^+$.

[Maximum mark: 5]

Solve the equation $\log_3(x^2-4x+4) = 1 + \log_3(x-2)$.

[Maximum mark: 6]

Consider the complex number $z = \dfrac{w_1}{w_2}$ where $w_1 = \sqrt{2} + \sqrt{6}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w_2 = 3 + \sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Express $w_1$ and $w_2$ in modulus-argument form and write down

   1. the modulus of $z$;

   2. the argument of $z$. [4]
      

2. Find the smallest positive integer value of $n$ such that $z^n$ is a real number. [2]
   

[Maximum mark: 6]

Prove by contradiction that the equation $3x^3-7x^2+5=0$ has no integer roots.

[Maximum mark: 5]

The third term of an arithmetic sequence is equal to $7$ and the sum of the first $8$ terms is $20$.

Find the common difference and the first term.

[Maximum mark: 6]

Ten students are to be arranged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shown in the following diagram.

1. Find the number of ways the ten students may be arranged in the lab. [1]
   

Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.

2. Find the number of ways the students may be arranged if Hugo and Leo must sit so that one is directly behind the other. For example, Desk $1$ and Desk $6$. [2]
   

3. Find the number of ways the students may be arranged if Hugo and Leo must not sit next to each other in the same row. [3]
   

[Maximum mark: 7]

Consider the complex numbers $u = 1 + 2 {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $v = 2 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Given that $\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{6\sqrt{2}}{w}$, express $w$ in the form $a + b {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [4]
   

2. Find $w^\ast$ and express it in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$. [3]
   

[Maximum mark: 6]

The $1$st, $5$th and $13$th terms of an arithmetic sequence, with common difference $d$, $d \neq 0$, are the first three terms of a geometric sequence, with common ratio $r$, $r \neq 1$. Given that the $1$st term of both sequences is $12$, find the value of $d$ and the value of $r$.

[Maximum mark: 7]

Consider the expansion of $\bigg(2x^6+\dfrac{x^2}{q}\bigg)^{10}$,  $q \neq 0$. The coefficient of the term

in $x^{40}$ is twelve times the coefficient of the term in $x^{36}$. Find $q$.

[Maximum mark: 5]

Use the extension of the binomial theorem for $n \in \mathbb{Q}$ to show that $\dfrac{x}{(1+x)^2} \approx x - 2x^2 + 3x^3$, $|x| < 1$.

[Maximum mark: 9]

Consider the following system of equations:

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

1. Find conditions on $a$ and $b$ for which

   1. the system has no solutions;

   2. the system has only one solution;

   3. the system has an infinite number of solutions. [6]
      

2. In the case where the number of solutions is infinite, find the general
   solution of the system of equations in Cartesian form. [3]
   

[Maximum mark: 7]

Given that $(5+nx)^2\bigg(1+\dfrac{3}{5}x\bigg)^n\hspace{-0.25em}=\hspace{0.05em}25+100x+\cdots$, find the value of $n$.

[Maximum mark: 18]

1. Express $-4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $- \pi < \theta \leq \pi$. [5]
   

Let the roots of the equation $z^3 = -4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ be $z_1$, $z_2$ and $z_3$.

2. Find $z_1$, $z_2$ and $z_3$ expressing your answers in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. [5]
   

On an Argand diagram, $z_1$, $z_2$ and $z_3$ are represented by the points A, B and C, respectively.

3. Find the area of the triangle ABC. [4]
   

4. By considering the sum of the roots $z_1$, $z_2$ and $z_3$, show that

   $$\cos\Big(\dfrac{2\pi}{9}\Big) + \cos\Big(\dfrac{4\pi}{9}\Big) + \cos\Big(\dfrac{8\pi}{9}\Big) = 0$$

   [4]
   

[Maximum mark: 7]

Given that $(1 + x)^3(1 + px)^4 = 1 + qx + 93x^2 + \dots + p^4x^7$, find the possible values of $p$ and $q$.

[Maximum mark: 6]

The barcode strings of a new product are created from four letters A, B, C, D and ten digits $0,1,2,\dots,9$. No three of the letters may be written consecutively in a barcode string. There is no restriction on the order in which the numbers can be written.

Find the number of different barcode strings that can be created.

[Maximum mark: 19]

1. 1. Expand $(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)^4$ by using the binomial theorem.

   2. Hence use de Moivre's theorem to prove that

      $$\begin{aligned} \cos 4\theta = \cos^4 \theta - 6\cos^2 \theta\sin^2 \theta + \sin^4 \theta. \\ \end{aligned}$$

   3. State a similar expression for $\sin 4 \theta$ in terms of $\cos \theta$ and $\sin \theta$. [6]
      

Let $z = r(\cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)$, where $\alpha$ is measured in degrees, be the solution
of $z^4 - {\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$ which has the smallest positive argument.

2. Find the modulus and argument of $z$. [4]
   

3. Use (a) (ii) and your answer from (b) to show that $8\cos^4\alpha - 8\cos^2 \alpha + 1 = 0$. [4]
   

4. Hence express $\cos \ang{22.5}$ in the form $\dfrac{\sqrt{a + b\sqrt{c}}}{d}$ where $a,b,c,d \in \mathbb{Z}$. [5]
   

[Maximum mark: 15]

Bill takes out a bank loan of $\$100\hspace{0.15em}000$ to buy a premium electric car, at an annual interest rate of $5.49$%. The interest is calculated at the end of each year and added to the amount outstanding.

1. Find the amount of money Bill would owe the bank after $10$ years. Give your answer to the nearest dollar. [3]
   

To pay off the loan, Bill makes quarterly deposits of $\$P$ at the end of every quarter in a savings account, paying a nominal annual interest rate of $3.2$%. He makes his first deposit at the end of the first quarter after taking out the loan.

2. Show that the total value of Bill's savings after $10$ years is $P\bigg[\dfrac{1.008^{40}-1}{1.008-1}\bigg]$. [3]
   

3. Given that Bill's aim is to own the electric car after $10$ years, find the value for $P$ to the nearest dollar. [3]
   

Melinda visits a different bank and makes a single deposit of $\$\hspace{0.05em}Q$, the annual $\text{interest}$ rate being $3.5$%.

4. 1. Melinda wishes to withdraw $\$8000$ at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

      $$\dfrac{8000}{1.035} + \dfrac{8000}{1.035^2} + \dfrac{8000}{1.035^3} + \cdots + \dfrac{8000}{1.035^n}.$$

   2. Hence, or otherwise, find the minimum value of $Q$ that would permit Melinda to withdraw annual amounts of $\$8000$ indefinitely. Give your answer to the nearest dollar. [6]
      

[Maximum mark: 17]

1. Solve the equation $z^3 = 27$, $z \in \mathbb{C}$, giving your answer in the form
   $z = r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$ and in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [6]
   

2. Consider the complex numbers $z_1 = -1 + {\mathrm{\hspace{0.05em}i}\mkern 1mu} \text{ and } z_2 =\dfrac{1}{\sqrt{2}}\bigg[\mathrm{cos}\bigg(\dfrac{\pi}{3}\bigg)+\mathrm{i}\,\mathrm{sin}\bigg(\dfrac{\pi}{3}\bigg)\bigg]$ .

   1. Write $z_1$ in the form $r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$.

   2. Calculate $z_1z_2$ and write in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$.

   3. Hence find the value of $\tan\left(\dfrac{\pi}{12}\right)$ in the form $c + d\sqrt{3}$ where $c,d \in \mathbb{Z}$.

   4. Find the smallest $p \in \mathbb{Q}^+$ such that $(z_1z_2)^p$ is a positive real number. [11]