Aussie Maths & Science Teachers: Save your time with SmarterEd
Each student in a class is given a packet of lollies. The teacher records the number of red lollies in each packet using a frequency table.
What is the relative frequency of a packet of lollies containing more than three red lollies?
What is `0.000\ 000\ 326` mm expressed in scientific notation?
The diagram below shows the location of three boats, `A`, `B` and `C`.
Boat `B` is on a bearing of 110° from boat `A`.
Boat `B` is also on a bearing of 035° from boat `C`.
Boat `A` is due north of boat `C`.
The angle `ABC` is
A. `35°`
B. `65°`
C. `70°`
D. `75°`
E. `110°`
| `/_CAB` | `= 180 – 110\ \ \ text{(straight angle at A)}` |
| `= 70°` |
| `:./_ABC` | `= 180 – (70 + 35)\ \ text{(Angle sum of}\ Delta ABC text{)}` |
| `= 75°` |
`=> D`
Consider the following sequence.
`2,\ 1,\ 0.5\ …`
Which of the following difference equations could generate this sequence?
| A. | `t_(n + 1) = t_n - 1` | `t_1 = 2` |
| B. | `t_(n + 1) = 3 - t_n` | `t_1 = 2` |
| C. | `t_(n + 1) = 2 × 0.5^(n – 1)` | `t_1 = 2` |
| D. | `t_(n + 1) = - 0.5t_n + 2` | `t_1 = 2` |
| E. | `t_(n + 1) = 0.5t_n` | `t_1 = 2` |
On day 1, Vikki spends 90 minutes on a training program.
On each following day, she spends 10 minutes less on the training program than she did the day before.
Let `t_n` be the number of minutes that Vikki spends on the training program on day `n`.
A difference equation that can be used to model this situation for `1 ≤ n ≤ 10` is
| A. `t_(n + 1) = 0.90t_n` | `t_1 = 90` |
| B. `t_(n + 1) = 1.10 t_n` | `t_1 = 90` |
| C. `t_(n + 1) = t_n - 0.10` | `t_1 = 90` |
| D. `t_(n + 1) = 1 - 10 t_n` | `t_1 = 90` |
| E. `t_(n + 1) = t_n - 10` | `t_1 = 90` |
A city has a population of 100 000 people in 2014.
Each year, the population of the city is expected to increase by 4%.
In 2018, the population is expected to be closest to
A. `108\ 000`
B. `112\ 000`
C. `115\ 000`
D. `117\ 000`
E. `122\ 000`
The seasonal indices for the first 11 months of the year, for sales in a sporting equipment store, are shown in the table below.
Part 1
The seasonal index for December is
A. `0.89`
B. `0.97`
C. `1.02`
D. `1.23`
E. `1.29`
Part 2
In May, the store sold $213 956 worth of sporting equipment.
The deseasonalised value of these sales was closest to
A. `$165\ 857`
B. `$190\ 420`
C. `$209\ 677`
D. `$218\ 322`
E. `$240\ 400`
The length of a type of ant is approximately normally distributed with a mean of 4.8 mm and a standard deviation of 1.2 mm.
Part 1
From this information it can be concluded that around 95% of the lengths of these ants should lie between
A. `text(2.4 mm and 6.0 mm)`
B. `text(2.4 mm and 7.2 mm)`
C. `text(3.6 mm and 6.0 mm)`
D. `text(3.6 mm and 7.2 mm)`
E. `text(4.8 mm and 7.2 mm)`
Part 2
A standardised ant length of `z = text(−0.5)` corresponds to an actual ant length of
A. `text(2.4 mm)`
B. `text(3.6 mm)`
C. `text(4.2 mm)`
D. `text(5.4 mm)`
E. `text(7.0 mm)`
The percentage histogram below shows the distribution of the fertility rates (in average births per woman) for 173 countries in 1975.
Part 1
In 1975, the percentage of these 173 countries with fertility rates of 4.5 or greater was closest to
A. `12text(%)`
B. `35text(%)`
C. `47text(%)`
D. `53text(%)`
E. `65text(%)`
Part 2
In 1975, for these 173 countries, fertility rates were most frequently
A. less than 2.5
B. between 1.5 and 2.5
C. between 2.5 and 4.5
D. between 6.5 and 7.5
E. greater than 7.5
Part 3
Which one of the boxplots below could best be used to represent the same fertility rate data as displayed in the percentage histogram?
The box plot below shows the distribution of the time, in seconds, that 79 customers spent moving along a particular aisle in a large supermarket.
Part 1
The longest time, in seconds, spent moving along this aisle is closest to
A. `40`
B. `60`
C. `190`
D. `450`
E. `500`
Part 2
The shape of the distribution is best described as
A. symmetric.
B. negatively skewed.
C. negatively skewed with outliers.
D. positively skewed.
E. positively skewed with outliers.
Part 3
The number of customers who spent more than 90 seconds moving along this aisle is closest to
A. `7`
B. `20`
C. `26`
D. `75`
E. `79`
Part 4
From the box plot, it can be concluded that the median time spent moving along the supermarket aisle is
A. less than the mean time.
B. equal to the mean time.
C. greater than the mean time
D. half of the interquartile range.
E. one quarter of the range.
`t_1=10`, `t_2=k` and `t_3=90` are the first three terms of a difference equation with the rule `t_n=t_(n - 1) + t_(n - 2)`.
The value of `k` is
A. `30`
B. `40`
C. `50`
D. `60`
E. `80`
A team of swimmers was training.
Claire was the first swimmer for the team and she swam 100 metres.
Every other swimmer in the team swam 50 metres further than the previous swimmer.
Jane was the last swimmer for the team and she swam 800 metres.
The total number of swimmers in this team was
A. `9`
B. `13`
C. `14`
D. `15`
E. `18`
Use the following information to answer Parts 1 and 2.
As part of a savings plan, Stacey saved $500 the first month and successively increased the amount that she saved each month by $50. That is, in the second month she saved $550, in the third month she saved $600, and so on.
Part 1
The amount Stacey will save in the 20th month is
A. `$1450`
B. `$1500`
C. `$1650`
D. `$1950`
E. `$3050`
Part 2
The total amount Stacey will save in four years is
A. `$13\ 400`
B. `$37\ 200`
C. `$58\ 800`
D. `$80\ 400`
E. `$81\ 600`
A poultry farmer aims to increase the weight of a turkey by 10% each month.
The turkey’s weight, `T_n`, in kilograms, after `n` months, would be modelled by the rule
A. `T_(n + 1) = T_n + 10`
B. `T_(n + 1) = 1.1T_n + 10`
C. `T_(n + 1) = 0.10T_n`
D. `T_(n + 1) = 10T_n`
E. `T_(n + 1) = 1.1T_n`
The prizes in a lottery form the terms of a geometric sequence with a common ratio of 0.95.
If the first prize is $20 000, the value of the eighth prize will be closest to
A. `$7000`
B. `$8000`
C. `$12\ 000`
D. `$13\ 000`
E. `$14\ 000`
The first three terms of a geometric sequence are
`0.125, 0.25, 0.5`
The fourth term in this sequence would be
A. `0.625`
B. `0.75`
C. `0.875`
D. `1`
E. `1.25`
The height (in cm) and foot length (in cm) for each of eight Year 12 students were recorded and displayed in the scatterplot below.
A least squares regression line has been fitted to the data as shown.
Part 1
By inspection, the value of the product-moment correlation coefficient `(r)` for this data is closest to
Part 2
The explanatory variable is foot length.
The equation of the least squares regression line is closest to
Part 3
The plot of the residuals against foot length is closest to
The histogram below displays the distribution of the percentage of Internet users in 160 countries in 2007.
Part 1
The shape of the histogram is best described as
A. approximately symmetric.
B. bell shaped.
C. positively skewed.
D. negatively skewed.
E. bi-modal.
Part 2
The number of countries in which less than 10% of people are Internet users is closest to
A. `10`
B. `16`
C. `22`
D. `32`
E. `54`
Part 3
From the histogram, the median percentage of Internet users is closest to
A. `10text(%)`
B. `15text(%)`
C. `20text(%)`
D. `30text(%)`
E. `40text(%)`
From a point `A` due south of a tower, the angle of elevation of the top of the tower `T`, is 23°. From another point `B`, on a bearing of 120° from the tower, the angle of elevation of `T` is 32°. The distance `AB` is 200 metres.
(i)
(ii) `text(Find)\ \ OT = h`
`text(Using the cosine rule in)\ Delta AOB :`
`200^2 = OA^2 + OB^2 – 2 * OA * OB * cos 60\ …\ text{(*)}`
`text(In)\ Delta OAT,\tan 23^@= h/(OA)`
`=> OA= h/(tan 23^@)\ …\ (1)`
`text(In)\ Delta OBT,\ tan 32^@= h/(OB)`
`=> OB= h/(tan 32^@)\ \ \ …\ (2)`
`text(Substitute)\ (1)\ text(and)\ (2)\ text(into)\ text{(*)}`
| `200^2` | `= (h^2)/(tan^2 23^@) + (h^2)/(tan^2 32^@) – 2 * h/(tan 23^@) * h/(tan 32^@) * 1/2` |
| `= h^2 (1/(tan^2 23^@) + 1/(tan^2 32^@) + 1/(tan23^@ * tan32^@) )` | |
| `= h^2 (4.340…)` | |
| `h^2` | `= (40\ 000)/(4.340…)` |
| `= 9214.55…` | |
| `:. h` | `= 95.99…` |
| `= 96\ text(m)\ \ \ text{(to nearest m)}` |
The parallel boxplots below summarise the distribution of population density, in people per square kilometre, for 27 inner suburbs and 23 outer suburbs of a large city.
Which one of the following statements is not true?
The following table shows the data collected from a sample of seven drivers who entered a supermarket car park. The variables in the table are:
distance – the distance that each driver travelled to the supermarket from their home
Part 1
The mean, `barx`, and the standard deviation, `s_x`, of the variable, distance, are closest to
A. `barx = 2.5\ \ \ \ \ \ \s_x = 3.3`
B. `barx = 2.8\ \ \ \ \ \ \s_x = 1.7`
C. `barx = 2.8\ \ \ \ \ \ \s_x = 1.8`
D. `barx = 2.9\ \ \ \ \ \ \s_x = 1.7`
E. `barx = 3.3\ \ \ \ \ \ \s_x = 2.5`
Part 2
The number of categorical variables in this data set is
A. `0`
B. `1`
C. `2`
D. `3`
E. `4`
Part 3
The number of female drivers with three children in the car is
A. `0`
B. `1`
C. `2`
D. `3`
E. `4`
The time spent by shoppers at a hardware store on a Saturday is approximately normally distributed with a mean of 31 minutes and a standard deviation of 6 minutes.
If 2850 shoppers are expected to visit the store on a Saturday, the number of shoppers who are expected to spend between 25 and 37 minutes in the store is closest to
A. 16
B. 68
C. 460
D. 1900
E. 2400
The table below shows the percentage of households with and without a computer at home for the years 2007, 2009 and 2011.
In the year 2009, a total of `5\ 170\ 000` households were surveyed.
The number of households without a computer at home in 2009 was closest to
| A. | `801\ 000` |
| B. | `1\ 153\ 000` |
| C. | `1\ 737\ 000` |
| D. | `3\ 433\ 000` |
| E. | `4\ 017\ 000` |
The lengths of the left feet of a large sample of Year 12 students were measured and recorded. These foot lengths are approximately normally distributed with a mean of 24.2 cm and a standard deviation of 4.2 cm.
Part 1
A Year 12 student has a foot length of 23 cm.
The student’s standardised foot length (standard `z` score) is closest to
A. –1.2
B. –0.9
C. –0.3
D. 0.3
E. 1.2
Part 2
The percentage of students with foot lengths between 20.0 and 24.2 cm is closest to
A. 16%
B. 32%
C. 34%
D. 52%
E. 68%
The passengers on a train were asked why they travelled by train. Each reason, along with the percentage of passengers who gave that reason, is displayed in the segmented bar chart below.
The percentage of passengers who gave the reason ‘no car’ is closest to
A. `text(14%)`
B. `text(18%)`
C. `text(26%)`
D. `text(74%)`
E. `text(88%)`
An animal study was conducted to investigate the relationship between exposure to danger during sleep (high, medium, low) and chance of attack (above average, average, below average). The results are summarised in the percentage segmented bar chart below.
The percentage of animals whose exposure to danger during sleep is high, and whose chance of attack is below average, is closest to
A. `4text(%)`
B. `12text(%)`
C. `28text(%)`
D. `72text(%)`
E. `86text(%)`
The back-to-back ordered stem plot below shows the female and male smoking rates, expressed as a percentage, in 18 countries.
Part 1
For these 18 countries, the lowest female smoking rate is
A. `5text(%)`
B. `7text(%)`
C. `9text(%)`
D. `15text(%)`
E. `19text(%)`
Part 2
For these 18 countries, the interquartile range (IQR) of the female smoking rates is
A. `4`
B. `6`
C. `19`
D. `22`
E. `23`
Part 3
For these 18 countries, the smoking rates for females are generally
A. lower and less variable than the smoking rates for males.
B. lower and more variable than the smoking rates for males.
C. higher and less variable than the smoking rates for males.
D. higher and more variable than the smoking rates for males.
E. about the same as the smoking rates for males.
The first time a student played an online game, he played for 18 minutes.
Each time he played the game after that, he played for 12 minutes longer than the previous time.
After completing his 15th game, the total time he had spent playing these 15 games was
A. `186` minutes
B. `691` minutes
C. `1206` minutes
D. `1395` minutes
E. `1530` minutes
The time series plot below displays the number of guests staying at a holiday resort during summer, autumn, winter and spring for the years 2007 to 2012 inclusive.
Part 1
Which one of the following best describes the pattern in the time series?
A. random variation only
B. decreasing trend with seasonality
C. seasonality only
D. increasing trend only
E. increasing trend with seasonality
Part 2
The table below shows the data from the times series plot for the years 2007 and 2008.
Using four-mean smoothing with centring, the smoothed number of guests for winter 2007 is closest to
A. `85`
B. `107`
C. `183`
D. `192`
E. `200`
The table below shows the hourly rate of pay earned by 10 employees in a company in 1990 and in 2010.
The value of the correlation coefficient, `r`, for this set of data is closest to
A. `0.74`
B. `0.86`
C. `0.92`
D. `0.93`
E. `0.96`
The time, in hours, that each student spent sleeping on a school night was recorded for 1550 secondary-school students. The distribution of these times was found to be approximately normal with a mean of 7.4 hours and a standard deviation of 0.7 hours.
Part 1
The time that 95% of these students spent sleeping on a school night could be
A. less than 6.0 hours.
B. between 6.0 and 8.8 hours.
C. between 6.7 and 8.8 hours.
D. less than 6.0 hours or greater than 8.8 hours.
E. less than 6.7 hours or greater than 9.5 hours.
Part 2
The number of these students who spent more than 8.1 hours sleeping on a school night was closest to
A. 16
B. 248
C. 1302
D. 1510
E. 1545
The heights of 82 mothers and their eldest daughters are classified as 'short', 'medium' or 'tall'. The results are displayed in the frequency table below.
Part 1
The number of mothers whose height is classified as 'medium' is
A. `7`
B. `10`
C. `14`
D. `31`
E. `33`
Part 2
Of the mothers whose height is classified as 'tall', the percentage who have eldest daughters whose height is classified as 'short' is closest to
A. `text(3%)`
B. `text(4%)`
C. `text(14%)`
D. `text(17%)`
E. `text(27%)`
The following ordered stem plot shows the percentage of homes connected to broadband internet for 24 countries in 2007.
Part 1
The number of these countries with more than 22% of homes connected to broadband internet in 2007 is
A. `4`
B. `5`
C. `19`
D. `20`
E. `22`
Part 2
Which one of the following statements relating to the data in the ordered stem plot is not true?
A. The minimum is 16%.
B. The median is 30%.
C. The first quartile is 23.5%
D. The third quartile is 32%.
E. The maximum is 38%.
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a.
b. `-2 <= x <= 4/3`
| a. | |
b. `text(Solving for)\ \ |\ 2x-1\ | <= |\ x-3\ |`
`text(Graph shows the statement is TRUE between the)`
`text(points of intersection.)`
`=>\ text(Intersection occurs when)`
| `(2x-1)` | `= (x-3)\ \ \ text(or)\ \ \ ` | `-(2x-1)` | `= x-3` |
| `x` | `= -2` | `-2x + 1` | `= x-3` |
| `-3x` | `= -4` | ||
| `x` | `= 4/3` |
`:.\ text(Solution is)\ \ {x: -2 <= x <= 4/3}`
The polynomial `p(x)` is given by `p(x) = ax^3 + 16x^2 + cx-120`, where `a` and `c` are constants.
The three zeros of `p(x)` are `-2, 3` and `beta`.
Find the value of `beta`. (3 marks)
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Evaluate `int_-1^1 1/sqrt(4 - x^2)\ dx`. (2 marks)
Differentiate `cos^(–1) (3x)` with respect to `x`. (2 marks)
The polynomial `x^3` is divided by `x + 3`. Calculate the remainder. (2 marks)
The graph shows the predicted population age distribution in Australia in 2008.
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Cecil invited 175 movie critics to preview his new movie. After seeing the movie, he conducted a survey. Cecil has almost completed the two-way table.
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What is the probability that the critic was less than 40 years old and did not like the movie? (2 marks)
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Will this movie be considered a box office success? Justify your answer. (1 mark)
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Bob is employed as a salesman. He is offered two methods of calculating his income.
\begin{array} {|l|}
\hline
\rule{0pt}{2.5ex}\text{Method 1: Commission only of 13% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\rule{0pt}{2.5ex}\text{Method 2: \$350 per week plus a commission of 4.5% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\end{array}
Bob’s research determines that the average sales total per employee per month is $15 670.
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You are organising an outside sporting event at Mathsville and have to decide which month has the best weather for your event. The average temperature must be between 20°C and 30°C, and average rainfall must be less than 80 mm.
The radar chart for Mathsville shows the average temperature for each month, and the table gives the average rainfall for each month.
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Danni is flying a kite that is attached to a string of length 80 metres. The string makes an angle of 55° with the horizontal.
How high, to the nearest metre, is the kite above Danni’s hand?
| `sin 55^@` | `= h/80` |
| `h` | `= 80 xx sin 55^@` |
| `= 65.532…\ text(m)` |
`=> B`
Luke’s normal rate of pay is $15 per hour. Last week he was paid for 12 hours, at time-and-a-half.
How many hours would Luke need to work this week, at double time, to earn the same amount?
Using the tax table, what is the tax payable on $43 561?
Which expression is equivalent to `12k^3 ÷ 4k`?
Consider the parabola `x^2 = 8(y\ – 3)`.
(i) `(0,3)`
(ii) `(0,5)`
| (iii) |  |
(iv) `10 2/3\ text(u²)`
| (i) | `text(Vertex)\ = (0,3)` |
| (ii) | `text(Using)\ \ \ x^2` | `= 4ay` |
| `4a` | `= 8` | |
| `a` | `= 2` |
`:.\ text(Focus) = (0,5)`
| (iii) | |
| (iv) | `text(Intersection when)\ y = 5` |
| `=> x^2` | `= 8 (5-3)` |
| `x^2` | `= 16` |
| `x` | `= +- 4` |
`text(Find shaded area)`
| `x^2` | `= 8 (y-3)` |
| `y – 3` | `= x^2/8` |
| `y` | `= x^2/8 +3` |
| `text(Area)` | `= int_-4^4 5\ dx – int_-4^4 x^2/8 + 3\ dx` |
| `= int_-4^4 5 – (x^2/8 + 3)\ dx` | |
| `= int_-4^4 2 – x^2/8\ dx` | |
| `= [2x – x^3/24]_-4^4` | |
| `= [(8 – 64/24) – (-8 + 64/24)]` | |
| `= 5 1/3 – (- 5 1/3)` | |
| `= 10 2/3\ text(u²)` |
Differentiate with respect to `x`:
`sinx/(x+4)`. (2 marks)
Differentiate with respect to `x`:
`x^2 log_e x` (2 marks)
Differentiate with respect to `x`:
`(x^2 + 3)^9` (2 marks)
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In the diagram, `ABCD` is a quadrilateral. The equation of the line `AD` is `2x- y- 1 = 0`.
Let `M` be the midpoint of `(-1, 4)` and `(5, 8)`.
Find the equation of the line through `M` with gradient `-1/2`. (2 marks)
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Find the sum of the first 21 terms of the arithmetic series 3 + 7 + 11 + ... (2 marks)
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Expand and simplify `(sqrt3-1)(2 sqrt3 + 5)`. (2 marks)
Solve `| 4x-3 | = 7`. (2 marks)
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Evaluate `2 cos (pi/5)` correct to three significant figures. (2 marks)
