Aussie Maths & Science Teachers: Save your time with SmarterEd
Heidi’s funds in a superannuation scheme have a future value of $740 000 in 20 years time. The interest rate is 4% per annum and earnings are calculated six-monthly.
What single amount could be invested now to produce the same result over the same period of time at the same interest rate? (3 marks)
Which of the following expresses S25°E as a true bearing?
`text(S25°E)= 180-25= 155°`
`=>D`
Ita publishes and sells calendars for $25 each. The cost of producing the calendars is $8 each plus a set up cost of $5950.
How many calendars does Ita need to sell to breakeven? (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Worker A picks a bucket of blueberries in `a` hours. Worker B picks a bucket of blueberries in `b` hours.
--- 3 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
Simplify `(p/q)^3 ÷ (pq^(-2))`. (2 marks)
A ship sails 6 km from `A` to `B` on a bearing of 121°. It then sails 9 km to `C`. The
size of angle `ABC` is 114°.
--- 2 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
| a. |  |
`text(Let point)\ D\ text(be due North of point)\ B`
`/_ABD=180-121\ text{(cointerior with}\ \ /_A text{)}\ =59^@`
`/_DBC=114-59=55^@`
`:. text(Bearing of)\ \ C\ \ text(from)\ \ B\ \ text(is)\ 055^@`
b. `text(Using cosine rule:)`
| `AC^2` | `=AB^2+BC^2-2xxABxxBCxxcos/_ABC` |
| `=6^2+9^2-2xx6xx9xxcos114^@` | |
| `=160.9275…` | |
| `:.AC` | `=12.685…\ \ \ text{(Noting}\ AC>0 text{)}` |
| `=13\ text(km)\ text{(nearest km)}` |
c. `text(Need to find)\ /_ACB\ \ \ text{(see diagram)}`
| `cos/_ACB` | `=(AC^2+BC^2-AB^2)/(2xxACxxBC)` |
| `=((12.685…)^2+9^2-6^2)/(2xx(12.685..)xx9)` | |
| `=0.9018…` | |
| `/_ACB` | `=cos^(-1)0.9018…` |
| `=25.6^@\ text{(to 1 d.p.)}` |
`text(From diagram,)`
`/_BCE=55^@\ text{(alternate angle,}\ DB\ text(||)\ CE text{)}`
`:.\ text(Bearing of)\ A\ text(from)\ C`
| `=180+55+25.6=260.6` | |
| `=261^@\ text{(nearest degree)}` |
A yacht race follows the triangular course shown in the diagram. The course from `P` to `Q` is 1.8 km on a true bearing of 058°.
At `Q` the course changes direction. The course from `Q` to `R` is 2.7 km and `/_PQR = 74^@`.
--- 3 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
What is the area of this ‘no-go’ zone? (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
| i. |  |
`/_ PQS = 58^@ \ \ \ (text(alternate to)\ /_TPQ)`
`text(Bearing of)\ R\ text(from)\ Q= 180^@ + 58^@ + 74^@312^@`
ii. `text(Using Cosine rule:)`
| `RP^2` | `=RQ^2+PQ^2- 2 xx RQ xx PQ xx cos/_RQP` |
| `= 2.7^2+1.8^2-2 xx 2.7 xx 1.8 xx cos74^@` | |
| `=7.851…` | |
| `RP` | `= sqrt(7.851…)=2.8019…~~ 2.8\ text(km) (text(1 d.p.))` |
iii. `text(Using)\ A = 1/2 ab sinC:`
| `A` | `= 1/2` `xx 2.7` `xx 1.8` `xx sin74^@` |
| `= 2.3358…` | |
| `= 2.3\ text(km)^2` |
`:.\ text(No-go zone is 2.3 km)^2 .`
A plane flies on a bearing of 150° from `A` to `B`.
What is the bearing of `A` from `B`?
`/_TBA=30^@\ \ \ text{(angle sum of triangle)}`
`:.\ text(Bearing of)\ A\ text{from}\ B`
`=360-30`
`=330^@`
`=> D`
The diagram shows towns `A`, `B` and `C`. Town `B` is 40 km due north of town `A`. The distance from `B` to `C` is 18 km and the bearing of `C` from `A` is 025°. It is known that `∠BCA` is obtuse.
What is the bearing of `C` from `B`?
The following information is given about the locations of three towns `X`, `Y` and `Z`:
• `X` is due east of `Z`
• `X` is on a bearing of `145^@` from `Y`
• `Y` is on a bearing of `060^@` from `Z`.
Which diagram best represents this information?
`text(S)text(ince)\ X\ text(is due east of)\ Z`
`=> text(Cannot be)\ B\ text(or)\ D`
`text(The diagram shows we can find)`
`/_ZYX = 60 + 35^@ = 95^@`
`text(Using alternate angles)\ (60^@)\ text(and)`
`text(the)\ 145^@\ text(bearing of)\ X\ text(from)\ Y`
`=> C`
The diagram shows the position of `Q`, `R` and `T` relative to `P`.
In the diagram,
`Q` is south-west of `P`
`R` is north-west of `P`
`/_QPT` is 165°
What is the bearing of `T` from `P`?
`/_QPS=45^@\ \ \ text{(} Q\ text(is south west of)\ Ptext{)}`
`/_TPS = 165 – 45 = 120^@`
`:.\ /_NPT = 60^@\ \ text{(} 180^@\ text(in straight line) text{)}`
`=> A`
Town `B` is 80 km due north of Town `A` and 59 km from Town `C`.
Town `A` is 31 km from Town `C`.
What is the bearing of Town `C` from Town `B`?
A ship travels from Port A on a bearing of 050° for 320 km to Port B. It then travels on a bearing of 120° for 190 km to Port C.
--- 3 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
| i. |  |
`text(Let)\ \ D\ \ text(be south of)\ \ B`
`/_ ABD = 50^@ qquad text{(alternate angles)}`
`/_DBC = 60^@ qquad text{(180° in straight line)}`
`:. /_ ABC= 50 + 60= 110^@`
ii. `text(Using the cosine rule:)`
| `AC^2` | `= AB^2 + BC^2 – 2 *AB*BC* cos /_ ABC` |
| `= 320^2 + 190^2 – 2 xx 320 xx 190 xx cos 110^@` | |
| `= 180\ 089.64…` | |
| `:. AC` | `= 424.36…= 420\ text{km (nearest 10 km)}` |
Solve the equation `(2x)/5 + 1 = (3x + 1)/2`, leaving your answer as a fraction. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Joanna sits a Physics test and a Biology test.
Calculate the `z`-score for Joanna’s mark in this test. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Joanna’s `z`-score is the same in both the Physics test and the Biology test.
What is her mark in the Biology test? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
A cumulative frequency table for a data set is shown.
\begin{array} {|c|c|}
\hline
\ \ \ \ \ \ \ \textit{Score}\ \ \ \ \ \ \ & \ \ \ \ \ \textit{Cumulative}\ \ \ \ \ \\ & \textit{frequency} \\
\hline
\rule{0pt}{2.5ex} \text{1} \rule[-1ex]{0pt}{0pt} & 5 \\
\hline
\rule{0pt}{2.5ex} \text{2} \rule[-1ex]{0pt}{0pt} & 9 \\
\hline
\rule{0pt}{2.5ex} \text{3} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} \text{4} \rule[-1ex]{0pt}{0pt} & 20 \\
\hline
\rule{0pt}{2.5ex} \text{5} \rule[-1ex]{0pt}{0pt} & 34 \\
\hline
\rule{0pt}{2.5ex} \text{6} \rule[-1ex]{0pt}{0pt} & 42 \\
\hline
\end{array}
What is the interquartile range of this data set? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
The graph displays the mean monthly rainfall in Sydney and Perth.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
Ali made monthly deposits of $100 into an annuity for 5 years.
Calculate the total amount Ali deposited into the annuity over this period. (1 mark)
Jeremy rolled a biased 6-sided die a number of times. He recorded the results in a table.
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Number} \rule[-1ex]{0pt}{0pt} & \ \ 1 \ \ & \ \ 2 \ \ & \ \ 3 \ \ & \ \ 4 \ \ & \ \ 5 \ \ & \ \ 6 \ \ \\
\hline
\rule{0pt}{2.5ex} \text{Frequency} \rule[-1ex]{0pt}{0pt} & \ \ 23 \ \ & \ \ 19 \ \ & \ \ 48 \ \ & \ \ 20 \ \ & \ \ 21 \ \ & \ \ 19 \ \ \\
\hline
\end{array}
What is the relative frequency of rolling a 3? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
What is the `x`-intercept of the line `x + 3y + 6 = 0`?
A set of data is normally distributed with a mean of 48 and a standard deviation of 3.
Approximately what percentage of the scores lies between 39 and 45?
`text(Given)\ \ mu = 48, \ sigma = 3`♦ Mean mark 47%.
| `ztext{-score (39)}` | `= (x-mu)/sigma` |
| `= (39-48)/3` | |
| `=-3` |
| `ztext{-score (45)}` | `= (45-48)/3` |
| `=-1` |
`:.\ text(Scores between 39 and 45)`
`~~ 16text(%)`
`=>A`
`text(Note that % of scores below 39 is very small.)`
During a year, the maximum temperature each day was recorded. The results are shown in the table.
From the days with a maximum temperature less than 25°C, one day is selected at random.
What is the probability, to the nearest percentage, that the selected day occurred during winter?
The table shows the compounded values of $1 at different interest rates over different periods.
Amy hopes to have $21 000 in 2 years to buy a car. She opens an account today which pays interest of 4% pa, compounded quarterly.
Using the table, which expression calculates the minimum single sum that Amy needs to invest today to ensure she reaches her savings goal?
A set of data is summarised in this frequency distribution table.
Which of the following is true about the data?
The diagram shows the positions of towns `A`, `B` and `C`.
Town `A` is due north of town `B` and `angleCAB = 34°`
What is the bearing of town `C` from town `A`?
`text(Bearing of Town)\ C\ text(from Town)\ A:`
`text(Bearing)= 180 + 34= 214^@`
`=>C`
A set of data is displayed in this dot plot.
Which of the following best describes this set of data?
A survey asked the following question.
'How many brothers do you have?'
How would the responses be classified?
A set of scores has the following five-number summary.
lower extreme = 2
lower quartile = 5
median = 6
upper quartile = 8
upper extreme = 9
What is the range?
What is the gradient of the line `2x + 3y + 4 = 0`?
The diagram shows the location of three schools. School `A` is 5 km due north of school `B`, school `C` is 13 km from school `B` and `angleABC` is 135°.
--- 4 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
i. `text(Using cosine rule:)`
| `AC^2` | `= AB^2 + BC^2-2 xx AB xx BC xx cos135^@` |
| `= 5^2 + 13^2-2 xx 5 xx 13 xx cos135^@` | |
| `= 285.923…` | |
| `:. AC` | `= 16.909…= 17\ text{km (nearest km)}` |
| ii. |  |
`text(Using sine rule, find)\ angleBAC:`
| `(sin angleBAC)/13` | `= (sin 135^@)/17` |
| `sin angleBAC` | `= (13 xx sin 135^@)/17` |
| `= 0.5407…` | |
| `angleBAC` | `= 32.7^@` |
`:. text(Bearing of)\ C\ text(from)\ A`
`= 180 + 32.7= 212.7^@= 213^@`
A set of data has a lower quartile (`Q_L`) of 10 and an upper quartile (`Q_U`) of 16.
What is the maximum possible range for this set of data if there are no outliers? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
All the students in a class of 30 did a test.
The marks, out of 10, are shown in the dot plot.
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
A group of Year 12 students was surveyed. The students were asked whether they live in the city or the country. They were also asked if they have ever waterskied.
The results are recorded in the table.
--- 4 WORK AREA LINES (style=lined) ---
Is this true, based on the survey results? Justify your answer with relevant calculations. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
A table of future value interest factors for an annuity of $1 is shown.
An annuity involves contributions of $12 000 per annum for 5 years. The interest rate is 4% per annum, compounded annually.
--- 1 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.
`220, 105, 101, 450, 37, 338, 151, 205`
--- 1 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
The heights of Year 12 girls are normally distributed with a mean of 165 cm and a standard deviation of 5.5 cm.
What is the `z`-score for a height of 154 cm?
Which of the data sets graphed below has the largest positive correlation coefficient value?
| A. |  |
B. |  |
| C. |  |
D. |  |
A single amount of $10 000 is invested for 4 years, earning interest at the rate of 3% per annum, compounded monthly.
Which expression will give the future value of the investment?
What is the value of `x` in the equation `(5-x)/3 = 6`?
A factory’s quality control department has tested every 50th item produced for possible defects.
What type of sampling has been used?
In a survey of 200 randomly selected Year 12 students it was found that 180 use social media.
Based on this survey, approximately how many of 75 000 Year 12 students would be expected to use social media?
The graph shows the relationship between infant mortality rate (deaths per 1000 live births) and life expectancy at birth (in years) for different countries.
What is the life expectancy at birth in a country which has an infant mortality rate of 60?
\(\text{When infant mortality rate is 60, life expectancy}\)
\(\text{at birth is 68 years (see below).}\)
\(\Rightarrow A\)
The box-and-whisker plot for a set of data is shown.
What is the median of this set of data?
The graph shows the life expectancy of people born between 1900 and 2000.
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
The ages of members of a dance class are shown in the back-to-back stem-and-leaf plot.
Pat claims that the women who attend the dance class are generally older than the men.
Is Pat correct? Justify your answer by referring to the median and skewness of the two sets of data. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
The mass `M` kg of a baby pig at age `x` days is given by `M = A(1.1)^x` where `A` is a constant. The graph of this equation is shown.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
The table gives the contribution per period for an annuity with a future value of $1 at different interest rates and different periods of time.
Margaret needs to save $75 000 over 6 years for a deposit on a new apartment. She makes regular quarterly contributions into an investment account which pays interest at 3% pa.
How much will Margaret need to contribute each quarter to reach her savings goal? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
The heights of 400 students were measured. The results are displayed in this cumulative frequency polygon.
Use the polygon to estimate the interquartile range. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`text(See graph for values:)`
`IQR= Q_3-Q_1= 172-136=36\ text(cm)`
A small population consists of three students of heights 153 cm, 168 cm and 174 cm. Samples of varying sizes can be taken from this population.
What is the mean of the mean heights of all the possible samples? Justify your answer. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
The diagram shows towns `A`, `B` and `C`. Town `B` is 40 km due north of town `A`. The distance from `B` to `C` is 18 km and the bearing of `C` from `A` is 025°. It is known that `∠BCA` is obtuse.
What is the bearing of `C` from `B`?
A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.
A person is selected at random from the group.
What is the approximate probability that the person selected is a smoker OR is male?
The box-and-whisker plots show the results of a History test and a Geography test.
In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.
How many students completed the Geography test?
A grouped data frequency table is shown.
\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Class Interval} \rule[-1ex]{0pt}{0pt} & \ \ \ \ \ \textit{Frequency}\ \ \ \ \ \\
\hline
\rule{0pt}{2.5ex} \text{1 – 5} \rule[-1ex]{0pt}{0pt} & 3 \\
\hline
\rule{0pt}{2.5ex} \text{6 – 10} \rule[-1ex]{0pt}{0pt} & 6 \\
\hline
\rule{0pt}{2.5ex} \text{11 – 15} \rule[-1ex]{0pt}{0pt} & 8 \\
\hline
\rule{0pt}{2.5ex} \text{16 – 20} \rule[-1ex]{0pt}{0pt} & 9 \\
\hline
\end{array}
What is the mean for this set of data?
A soccer referee wrote down the number of goals scored in 9 different games during the season.
`2, \ 3, \ 3, \ 3, \ 5, \ 5, \ 8, \ 9, \ ...`
The last number has been omitted. The range of the data is 10.
What is the five-number summary for this data set?
The speed limit outside a school is 40 km/h. Year 11 students measured the speed of passing vehicles over a period of time. They found the set of data to be normally distributed with a mean speed of 36 km/h and a standard deviation of 2 km/h.
What percentage of the vehicles passed the school at a speed greater than 40 km/h?
| `z` | `= (x-mu)/sigma` |
| `= (40-36)/2` | |
| `= 2` |
`=> A`
The table shows the future value of an investment of $1000, compounding yearly, at varying interest rates for different periods of time.
Based on the information provided, what is the future value of an investment of $2500 over 3 years at 4% pa?
Which set of data is classified as categorical and nominal?
The graph shows a scatterplot for a set of data.
Which of the following is the best approximation for the correlation coefficient of this set of data?
The pulse rates of a large group of 18-year-old students are approximately normally distributed with a mean of 75 beats/minute and a standard deviation of 11 beats/minute.
The percentage of 18-year-old students with pulse rates less than 53 beats/minute or greater than 86 beats/minute is closest to
`mu=75,\ \ \ sigma=11`
`z text{-score (53)}=(x-mu) /sigma=(53-75)/11= -2`
`z text{-score (86)}= (86-75)/11=1`
`text(From the diagram, the % of students that have a)`
`z text(-score below –2 or above 1)`
`=2.5+16=18.5 text(%)`
`=>D`
The head circumference (in cm) of a population of infant boys is normally distributed with a mean of 49.5 cm and a standard deviation of 1.5 cm.
Four hundred of these boys are selected at random and each boy’s head circumference is measured.
The number of these boys with a head circumference of less than 48.0 cm is closest to