Topic 1 All - Number & Algebra

[Maximum mark: 6]

After solving a problem, John has an exact answer of $z = 0.1475$.

1. Write down the exact value of $z$ in the form $a\times10^k$, where $1 \leq a < 10, k\in \mathbb{Z}$.[2]
   

2. State the value of $z$ given correct to $2$ significant figures. [1]
   

3. Calculate the percentage error if $z$ is given correct to $2$ significant figures. [3]
   

[Maximum mark: 6]

The Burns, Gordons and Longstaff families make meal plans for their households. The table below shows the amount of carbohydrate, fat and protein, all measured in grams, consumed by the family over a single day. The table also shows the daily calories, measured in kcal, this equates to.

Let $x$, $y$ and $z$ represent the amount of calories, in kcal, for $1$ g of carbohydrate, fat and protein respectively.

1. Write down a system of three linear equations in terms of $x$, $y$ and $z$ that represents the information in the table above. [2]
   

2. Find the values $x$, $y$ and $z$. [2]
   

The Howe family also plans meals. The table below shows the amount of carbohydrates, fat and protein consumed by the family over a single day.

3. Calculate the daily calories for the Howe family. [2]
   

[Maximum mark: 6]

Given that $z = \dfrac{10\sin \alpha}{3x+y}$, where $\alpha = \ang{30}$, $x = 6$ and $y = 46$.

1. Find the exact value of $z$. [2]
   

2. Write your answer to part (a)

   1. correct to $2$ decimal places;

   2. correct to $3$ significant figures;

   3. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$.[4]
      

[Maximum mark: 6]

Let $A = \sqrt{\dfrac{\sin \alpha - \sin \beta}{x^2 + 2y}}$, where $\alpha = \ang{54}$, $\beta = \ang{18}$, $x = 24$ and $y = 18.25$.

1. Find the value of $A$. Write down your full calculator display. [2]
   

2. Give your answer to part (a) correct to

   1. three significant figures;

   2. three decimal places. [2]
      

3. Give the answer to part (b) (i) in the form $a\times10^k$, where $1 \leq a < 10$, $k \in \mathbb{Z}$.[2]
   

[Maximum mark: 6]

Let $Q = \dfrac{(\sin 2x + b)(2\sin x - 1)}{a^2 - 4\tan x}$, where $x = \ang{45}$, $a = 18$ and $b = \sqrt{2}$.

1. Find the exact value of $Q$. [2]
   

2. Give your answer to part (a) correct to

   1. three decimal places;

   2. three significant figures. [2]
      

3. Calculate the percentage error if $Q$ is given to three decimal places. [2]
   

[Maximum mark: 6]

The volume of a hemisphere, $V$, is given by the formula

where $S$ is the total surface area.

The total surface area of a given hemisphere is $529$ cm$^2$.

1. Calculate the volume of this hemisphere in cm$^3$. Give your answer correct to one decimal place. [3]
   

2. Write down your answer to part (a) correct to the nearest integer. [1]
   

3. Write down your answer to part (b) in the form $a\times10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$.[2]
   

[Maximum mark: 6]

Four cement bags labelled, "5 kg", were delivered to a customer. The customer measured each bag to check their weights and recorded the following:

1. 1. Find the mean of the customer's measurements.

   2. Calculate the percentage error between the mean and the stated,
      approximate weight of $5$ kg. [3]
      

2. Calculate $\sqrt{2.15^8-5.12^{-0.8}}$, giving your answer

   1. correct to the nearest integer;

   2. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$. [3]
      

[Maximum mark: 6]

The distance between two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$ is equal to $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

1. Calculate the distance between points A$(40,-100)$ and B$(1,-2)$. Give your answer correct to three significant figures. [3]
   

2. Give your answer from part (a) correct to one decimal place. [1]
   

3. Write the answer to part (b) in the form $a\times10^k$, where $1 \leq a < 10$, $k \in \mathbb{Z}$. [2]
   

[Maximum mark: 6]

The following diagram shows a rectangle with sides of length $7.6\times10^2$ cm and $1.5\times10^3$ cm.

1. Write down the area of the rectangle in the form $a\times10^k$, where
   $1 \leq a < 10$ and $k \in \mathbb{Z}$. [3]
   

Natalie estimates the area of the rectangle to be $1\hspace{0.1em}200\hspace{0.15em}000$ cm$^2$.

2. Find the percentage error in Natalie's estimate. [3]
   

[Maximum mark: 8]

A cuboid has the following dimensions: $\text{length} = 9.6\hspace{0.25em}$cm, $\text{width} = 7.4\hspace{0.25em}$cm, and $\text{height} = 5.2\hspace{0.25em}$cm, measured correct to the nearest millimetre.

1. Using these measurements, calculate the volume of the cuboid, in cm$^3$. Give your answer to two decimal places. [2]
   

The lower and upper bounds for the length of the cuboid can be expressed as $9.55 \leq l < 9.65$.

2. Write similar expressions for

   1. the width;

   2. the height. [2]
      

3. Hence, calculate the minimum volume of the cuboid. Give your answer to three significant figures. [2]
   

4. Write your answer to part (c) in the form $a\times10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$. [2]
   

[Maximum mark: 6]

Let $F = \dfrac{(4\sin 2z-1)(2\tan 3z+1)}{x^2-y^2}$, where $x = 12$, $y = 8$ and $z = \ang{15}$.

1. Calculate the exact value of $F$. [2]
   

2. Give your answer to $F$ correct to

   1. two significant figures;

   2. two decimal places. [2]
      

Sasha estimates the value of $F$ to be $0.03$.

3. Calculate the percentage error in Sasha's estimate. [2]
   

[Maximum mark: 7]

Brendan is training for a long distance bike race.

In week $1$ of his training he cycled $22\,$km. In week $2$ he cycled $34\,$km. This pattern continues, with him cycling an extra $12\,$km per week.

The distances Brendan cycled in the first $5$ weeks of training is shown in the following table.

1. Calculate how far he cycles in the $17$th week of his training. [3]
   

2. In total how far has he cycled after $17$ weeks? [2]
   

3. Find the mean distance per week he cycled over 17 weeks. [2]
   

[Maximum mark: 6]

The $15$th term of an arithmetic sequence is $21$ and the common difference is $-4$.

1. Find the first term of the sequence. [2]
   

2. Find the $29$th term of the sequence. [2]
   

3. Find the sum of the first $40$ terms of the sequence. [2]
   

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

1. State which sequence is arithmetic and find the common difference of the sequence. [2]
   

2. State which sequence is geometric and find the common ratio of the sequence.[2]
   

3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]
   

[Maximum mark: 6]

Given $r = 2a - \dfrac{\sqrt{b}}{c}$, $a = 0.975$, $b = 4.41$ and $c = 35$,

1. calculate the value of $r$. [2]
   

Albert first writes $a$, $b$ and $c$ correct to one significant figure and then uses these values to estimate the value of $r$.

2. 1. Write down $a$, $b$ and $c$ each correct to one significant figure.

   2. Find Albert's estimate of the value of $r$. [2]
      

3. Calculate the percentage error in Albert's estimate of the value of $r$. [2]
   

[Maximum mark: 6]

Jeremy invests $\$8000$ into a savings account that pays an annual interest rate of $5.5$ %, compounded annually.

1. Write down a formula which calculates that total value of the investment after $n$ years. [2]
   

2. Calculate the amount of money in the savings account after:

   1. $1$ year;

   2. $3$ years. [2]
      

3. Jeremy wants to use the money to put down a $\$10\hspace{0.15em}000$ deposit on an apartment. Determine if Jeremy will be able to do this within a $5$-year timeframe.[2]
   

[Maximum mark: 6]

A toy rocket is fired, from a platform, vertically into the air, its height above the ground after $t$ seconds is given by $s(t) = at^2 + bt + c$, where $a,b,c \in \mathbb{R}$ and $s(t)$ is measured in metres.

After $2$ second, the rocket is $28.3$ m above the ground; after $4$ seconds, $25.6$ m; after $5$ seconds, $14.7$ m.

1. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Hence find the values of $a$, $b$ and $c$. [3]
      

2. Find the height, above the ground, of the platform. [1]
   

3. Find the time it takes for the rocket to hit the ground. [2]
   

[Maximum mark: 6]

An arithmetic sequence has $u_1 = 40$, $u_2 = 32$, $u_3 = 24$.

1. Find the common difference, $d$. [2]
   

2. Find $u_8$. [2]
   

3. Find $S_8$. [2]
   

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

1. State which sequence is arithmetic and find the common difference of the sequence. [2]
   

2. State which sequence is geometric and find the common ratio of the sequence.[2]
   

3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
   

[Maximum mark: 6]

An arithmetic sequence has $u_1 = 12$, $u_2 = 21$, $u_3 = 30$.

1. Find the common difference, $d$. [2]
   

2. Find $u_{10}$. [2]
   

3. Find $S_{10}$. [2]
   

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits $4000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $6$ %, compounded semi-annually.

1. Find the amount of interest that Mia will earn over the next $2.5$ years. [3]
   

Ella also deposits AUD into a bank account. Her bank pays a nominal annual $\text{interest}$ rate of $4$ %, compounded monthly. In $2.5$ years, the total amount in Ella's account will be $4000$ AUD.

2. Find the amount that Ella deposits in the bank account. [3]
   

[Maximum mark: 6]

A geometric sequence has $u_1 =5$, $u_2 = -1$ and $u_3 = \dfrac{1}{5}$.

1. Find the common ratio, $r$. [2]
   

2. Find the exact value of $u_{7}$. [2]
   

3. Find the exact value of $S_{7}$. [2]
   

[Maximum mark: 6]

Emily starts reading Leo Tolstoy's War and Peace on the $1$st of February. The number of pages she reads each day increases by the same number on each successive day.

1. Calculate the number of pages Emily reads on the $14$th of February. [3]
   

2. Find the exact total number of pages Emily reads in the $28$ days of February.[3]
   

[Maximum mark: 6]

A geometric sequence has $20$ terms, with the first four terms given below.

1. Find $r$, the common ratio of the sequence. Give your answer as a fraction. [1]
   

2. Find $u_5$, the fifth term of the sequence. Give your answer as a fraction. [1]
   

3. Find the smallest term in the sequence that is an integer. [2]
   

4. Find $S_{10}$, the sum of the first $10$ terms of the sequence. Give your answer correct to one decimal place. [2]
   

[Maximum mark: 6]

A tennis ball bounces on the ground $n$ times. The heights of the bounces, $h_1, h_2, h_3, \dots,h_n,$ form a geometric sequence. The height that the ball bounces the first time, $h_1$, is $80$ cm, and the second time, $h_2$, is $60$ cm.

1. Find the value of the common ratio for the sequence. [2]
   

2. Find the height that the ball bounces the tenth time, $h_{10}$. [2]
   

3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to $2$ decimal places. [2]
   

[Maximum mark: 6]

The table shows the first four terms of three sequences: $u_n$, $v_n$, and $w_n$.

1. State which sequence is

   1. arithmetic;

   2. geometric. [2]
      

2. Find the sum of the first $50$ terms of the arithmetic sequence. [2]
   

3. Find the exact value of the $13$th term of the geometric sequence. [2]
   

[Maximum mark: 6]

Hannah buys a car for $\$24\hspace{0.15em}900$. The value of the car depreciates by $16$ % each year.

1. Find the value of the car after $10$ years. [3]
   

Patrick buys a car for $\$12\hspace{0.15em}000$. The car depreciates by a fixed amount each year, and after $6$ years it is worth $\$6200$.

2. Find the annual rate of depreciation of the car. [3]
   

[Maximum mark: 6]

Edward wants to buy a new car, and he decides to take out a loan of $70\hspace{0.15em}000$ Australian dollars from a bank. The loan is for $6$ years, with a nominal annual interest rate of $7.2\%$, compounded monthly. Edward will pay the loan in fixed monthly instalments.

1. Determine the amount Edward should pay each month. Give your answer to the nearest dollar.[3]
   

2. Calculate the amount Edward will still owe the bank at the end of the third year. [3]
   

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Elena invests in a retirement plan in which equal payments of €$1500$ are made at the beginning of each year. Interest is earned on each payment at a rate of $2.49$ % per year, compounded annually.

1. Find the value of the investment after $25$ years. [3]
   

2. Find the amount of interest Elena will earn on the investment over $25$ years.[3]
   

[Maximum mark: 6]

Maria invests $\$25\hspace{0.15em}000$ into a savings account that pays a nominal annual interest rate of $4.25$ %, compounded monthly.

1. Calculate the amount of money in the savings account after $3$ years. [3]
   

2. Calculate the number of years it takes for the account to reach $\$40\hspace{0.15em}000$. [3]
   

[Maximum mark: 6]

The table below shows the first four terms of three sequences: $u_n$, $v_n$, and $w_n$.

1. State which sequence is

   1. arithmetic;

   2. geometric. [2]
      

2. Find the exact value of the sum of the first $35$ terms of the arithmetic
   sequence. [2]
   

3. Find the exact value of the $10$th term of the geometric sequence. [2]
   

[Maximum mark: 6]

An owl takes off from from a tree branch and flies higher into the sky. Its height above the ground after $t$ seconds, where $t\geq 0$, is given by $s(t) = at^3 + bt^2 + ct+d$, where $a,b,c,d \in \mathbb{R}$ and $s(t)$ is measured in metres.

Initially the owl is $12\,$ metres above the ground.

1. Write down the value of $d$. [1]
   

After $1$ second, the owl is $19.8$ m above the ground; after $2$ seconds, $34.5$ m; after $4$ seconds, $22.8$ m.

2. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Hence find the values of $a$, $b$ and $c$. [3]
      

After some time the owl reaches a maximum height. At this time it spots some prey at ground level and then immediately swoops down to catch it.

3. 1. Find the maximum height of the owl above the ground as it spots the prey.

   2. Find the time it catches the prey. [2]
      

[Maximum mark: 6]

The third term, $u_3$, of an arithmetic sequence is $7$. The common difference of
the sequence, $d$, is $3$.

1. Find $u_1$, the first term of the sequence. [2]
   

2. Find $u_{60}$, the $60$th term of sequence. [2]
   

The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.

3. Calculate the sixth term of the geometric sequence. [2]
   

[Maximum mark: 6]

Isabella and Charlotte both receives $80\hspace{0.15em}000$ Australian dollars (AUD) on their $18$th birthday to invest for later in their life.

Isabella deposits her $80\hspace{0.15em}000$ AUD in a bank account that pays a nominal annual interest rate of $x$ %, compounded monthly.

1. The amount in a bank account after $6$ years will be $100\hspace{0.15em}000$ AUD. Find the nominal annual interest rate. Give your answer correct to two decimal places.[3]
   

Charlotte uses her $80\hspace{0.15em}000$ AUD to buy a house that increases in value at a rate of $3$ % per year.

2. Find the house price after $6$ years. [3]
   

[Maximum mark: 6]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower Replicas in different sizes. The height of the largest tower in this set is $64$ cm. The heights of successive smaller towers are $95$ % of the preceding larger tower, as shown in the diagram below.

1. Find the height of the smallest tower in this set. [3]
   

2. Find the total height if all $49$ towers were placed one on top of another. [3]
   

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Charlie deposits $8000$ Canadian dollars (CAD) into a bank account. The bank pays a nominal annual interest rate of $5$ %, compounded semi-monthly.

1. Find the amount of interest that Charlie will earn over the next $2$ years. [3]
   

Oscar also deposits CAD into a bank account. His bank pays a nominal annual interest rate of $6$ %, compounded quarterly. In $2$ years, the total amount in Oscar's account will be $\$8000$ CAD.

2. Find the amount that Oscar deposits in the bank account. [3]
   

[Maximum mark: 6]

Michael buys a second hand Tesla car for $\$18\hspace{0.15em}000$. The value of the car depreciates by $10\%$ each year.

1. Find the total amount the car will depreciate after 5 years, giving your answer correct to the nearest dollar. [3]
   

The price of a different used car depreciates by $5\%$ each year.

2. Find the value reduction of this car after $4$ years as a percentage, when compared to the original purchase price. [3]
   

[Maximum mark: 6]

The following transition diagram reflects the proportions of customers that Qatar Airways loses to its competitor airlines each year, and vice versa.

1. Construct a transition matrix $\bm{T}$ with elements in decimal form. [2]
   

2. Interpret the meaning of the elements with values

   1. $0.15$

   2. $0.75$ [2]
      

Assume that the initial state of the market share is

.

3. Determine the market share of Qatar Airways after $5$ years. [2]
   

[Maximum mark: 5]

There are $225$ senior students studying Computer Science and $169$ senior students studying Mathematics at a university. According to an academic survey, $20$ % of these $\text{students}$ tell they will pursue a postgraduate degree, $8$ % will start a business, $52$ % will get employed, $15$ % will start freelancing and the remaining students will become $\text{research}$ assistants.

The column matrix

represents the number of senior students studying

Computer Science and Mathematics.

1. Write down a row matrix, $\bm{R}$, to represent the percentages, in decimal form, of senior students who will choose one of the five routes after graduation. [1]
   

2. Hence calculate the product $\bm{A} = \bm{SR}$. Give each element $a_{ij}$ of the matrix $\bm{A}$ correct to the nearest whole number. [1]
   

3. In the context of this problem, explain what the element $a_{15}$ means. [1]
   

The cost for textbooks per year for a computer science student is $\$1245$ and for a mathematics student is $\$889$.

4. Write down a matrix calculation that gives the total cost for textbooks paid by all the senior students studying Computer Science and Mathematics. [1]
   

5. Hence calculate the total cost for all the textbooks. Give your answer correct to the nearest dollar. [1]