Aussie Maths & Science Teachers: Save your time with SmarterEd
Kirbee is shopping for computer software. Novirus costs `$115` more than
Funmaths. Let `x` dollars be the cost of Funmaths.
In 2004 there were 13.5 million registered motor vehicles in Australia. The number of registered motor vehicles is increasing at a rate of 2.3% per year.
Which expression represents the number (in millions) of registered motor vehicles, if `y` represents the number of years after 2004?
What is the bearing of `A` from `B`?
`text(Bearing of)\ A\ text(from)\ B`
`= 180 +120`
`= 300^@`
`=> D`
The diagram shows the points `A(text(−1) , 3)` and `B(2, 0)`.
The line `l` is drawn perpendicular to the `x`-axis through the point `B`.
| (i) | `A(-1,3)\ \ \ \ \ B(2,0)` |
| `AB` | `= sqrt( (x_2 – x_1)^2 + (y_2 – y_1)^2 )` |
| `= sqrt( (2+1)^2 + (0-3)^2 )` | |
| `= sqrt(9+9)` | |
| `= sqrt 18` | |
| `= 3 sqrt 2\ text(units)` |
| (ii) | `text(Gradient of)\ AB` | `= (y_2 – y_1)/(x_2 – x_1)` |
| `= (0 – 3)/(2 + 1)` | ||
| `= – 1` |
| (iii) | `text(S) text(ince Gradient)\ AB = – 1` |
`/_ABO = 45^@`
`:.\ text(Angle between)\ AB\ text(and)\ l`
`= 90 – 45`
`= 45^@`
| (iv) | `text(Equation of)\ AB\ text(has)\ m = -1,\ text(through)\ \ (2,0)` |
| `y – y_1` | `= m(x – x_1)` |
| `y – 0` | `= -1 (x – 2)` |
| `y` | `= -x + 2` |
`:.\ x + y – 2 = 0\ \ \ …\ text(as required)`
(v)
`text{Origin (0,0) satisfies the inequality}`
`:.\ text(Shaded area is below)\ x + y – 2 = 0`
| (vi) | `text(Equation of)\ l` |
| `x = 2` |
| (vii) | `m_(AC) xx m_(AB)` | `= -1\ \ \ (AC⊥AB)` |
| `m_(AC) xx -1` | `= -1` | |
| `m_(AC)` | `= + 1` |
`text(Equation of)\ AC\ text(has)\ \ m = 1,\ text(through)\ (-1, 3),`
| `y – y_1` | `= m(x – x_1)` |
| `y – 3` | `= 1 (x + 1)` |
| `y` | `= x + 4` |
`=>C\ text(lies on)\ y = x + 4`
`text(When)\ \ x = 2,\ \ y = 6`
`:. C\ (2, 6)`
The mean of a set of 5 scores is 62.
What is the new mean of the set of scores after a score of 14 is added?
Peter rides his bike at a speed of 27 km/h.
What is this speed in m/s?
Kay randomly selected a marble from a bag of marbles, recorded its colour and returned it to the bag. She repeated this process a number of times.
Based on these results, what is the best estimate of the probability that Kay will choose a green marble on her next selection?
Which equation represents the relationship between `x` and `y` in this table?
Find integers `a` and `b` by showing working to expand and simplify
`(3-sqrt2)^2 = a-b sqrt2`. (2 marks)
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The radius of Mars is approximately `3\ 397\ 000` metres. Write this number in scientific notation, correct to two significant figures. (2 marks)
Marcella is planning to roll a standard six-sided die 60 times.
How many times would she expect to roll the number 4?
A set of scores is displayed in a stem-and-leaf plot.
What is the median of this set of scores?
Stan worked for 24 hours as shown on his pay slip.
What was his hourly rate of pay?
There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
Give a reason why Justine’s statement is NOT correct. (1 mark)
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Rita purchased a camera for $880 while on holidays in Australia. This price included 10% GST. When she left Australia she received a refund of the GST.
What was Rita’s refund?
The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.
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Calculate the volume of straw required. Give your answer in cubic metres. (2 marks)
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How many bags does she need to buy? (2 marks)
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Find the length of this fence to the nearest metre. (2 marks)
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The table is used to calculate monthly loan repayments.
Samantha has borrowed $70 000 at 8% per annum for 15 years.
What is her monthly loan repayment?
During a ten-minute period, Kath is exercising and Jim is resting.
How much more air would Kath breathe than Jim during this time?
Simplify `2m^2 × 3m p^2`
Janet’s gross income last year was $60 000. She had allowable tax deductions of $5000. Janet paid 1.5% of her taxable income for the Medicare levy.
How much was Janet’s Medicare levy?
Which formula should be used to calculate the distance between Toby and Frankie?
Four radio stations reported the probability of rain as shown in the table.
Which radio station reported the highest probability of rain?
If `d = 6t^2`, what is a possible value of `t` when `d = 2400`?
Using the tax table, determine the tax payable on a taxable income of $47 000.
Use the set of scores 1, 3, 3, 3, 4, 5, 7, 7, 12 to answer Questions 6 and 7.
Question 6
What is the range of the set of scores?
Question 7
What are the median and the mode of the set of scores?
In 2005, Peter’s annual salary was $35 000.
At the start of each subsequent year, his annual salary increases by 4.75%.
In 2010 his salary will be closest to
A. $36 663
B. $42 140
C. $43 310
D. $44 140
E. $55 670
A new air-conditioning unit was purchased for $5000 on 1 January 2009.
On 1 January of each year after 2009 its value is depreciated by 20% using the reducing balance method.
The value of the air conditioner will be below $1500 for the first time on 1 January
A. 2012
B. 2013
C. 2014
D. 2015
E. 2016
Sam and Charlie each invest $5000 for three years.
Sam’s investment earns simple interest at the rate of 7.5% per annum.
Charlie’s investment earns interest at the rate of 7.5% per annum compounding annually.
At the conclusion of three years, correct to the nearest cent, Sam will have
A. $86.48 less than Charlie.
B. $86.48 more than Charlie.
C. $132.23 less than Charlie.
D. $132.23 more than Charlie.
E. the same as Charlie.
Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments over one year.
A flat interest rate of 6% per annum is charged.
Part 1
The amount of each monthly instalment is
Part 2
The annual effective interest rate that Sandra pays under this agreement is closest to
Pia invests $800 000 in an ordinary perpetuity to provide an ongoing fortnightly pension for her retirement.
The interest rate for this investment is 5.8% per annum.
Assuming there are 26 fortnights per year, the amount she will receive at the end of each fortnight is closest to
A. $464
B. $892
C. $1422
D. $1785
E. $3867
The price of a property purchased in 2006 was $200 000.
Stamp duty was paid on this purchase according to the schedule below.
The amount of stamp duty paid was
A. $2560
B. $2800
C. $5100
D. $7660
E. $9460
A sum of money is invested in an account paying simple interest at a rate of 8% per annum.
The total interest earned on this investment over 6 years is $27 000.
The sum of money invested is
A car is valued at $30 000 when new.
Its value is depreciated by 25 cents for each kilometre it travels.
The number of kilometres the car travels before its value depreciates to $8000 is
A. 32 000
B. 55 000
C. 88 000
D. 120 000
E. 550 000
A photocopier is depreciated by $0.04 for each copy it makes.
Three years ago the photocopier was purchased for $48 000.
Its depreciated value now is $21 000.
The total number of copies made by the photocopier in the three years is
A. `108\ 000`
B. `192\ 000`
C. `276\ 000`
D. `525\ 000`
E. `675\ 000`
A bank statement for the month of October is shown below.
Interest on this account is calculated at a rate of 0.15% per month on the minimum monthly balance.
The interest payment for the month of October will be
A. $0.19
B. $0.57
C. $0.71
D. $0.92
E. $1.28
Sally planned to repay a loan fully with six equal monthly repayments of `$800`.
Interest was calculated monthly on the reducing balance.
Sally missed the third payment, but made a double payment of `$1600` in the fourth month.
Which of the following statements is true?
A. The same amount of interest is paid each month.
B. The amount owing after three months is the same as the amount owed after two months.
C. The amount owing after three months is less than the amount owed after two months.
D. To fully repay the loan, Sally will pay less than $4800.
E. To fully repay the loan, Sally will pay more than $4800.
Nathan bought a $2500 bedroom suite on a contract that involves no deposit and an interest-free loan for a period of 48 months.
He has to pay an initial set-up fee of $25.
In addition, he pays an administration fee of $3.95 per month.
The total amount that Nathan will have to pay in fees for the entire 48 months, as a percentage of the original price of $2500, is closest to
A. 1.6%
B. 4.0%
C. 7.6%
D. 8.5%
E. 8.6%
The following graph shows the decreasing value of an asset over eight years.
Let `P` be the value of the asset, in dollars, after `n` years.
A rule for evaluating `P` could be
A. `P = 250\ 000 xx (1 + 0.14)^n`
B. `P = 250\ 000 xx 1.14 xx n`
C. `P = 250\ 000 xx (0.14)^n`
D. `P = 250\ 000 xx (1 - 0.14)^n`
E. `P = 250\ 000 xx (1 - 0.14) xx n`
The graph below shows the growth in value of a $1000 investment over a period of four years.
A different amount of money is invested under the same investment conditions for eight years.
In total, the amount of interest earned on this investment is $600.
The amount of money invested is
A. `$500`
B. `$600`
C. `$1500`
D. `$2000`
E. `$2400`
The purchase price of a car is $15 000.
A deposit of $3000 is paid and the balance will be repaid with 36 monthly payments of $400.
The annual flat rate of interest charged is closest to
A. 1.3%
B. 4.0%
C. 5.3%
D. 6.7%
E. 20.0%
The closing price of a share on Monday was $30.
The closing price of the share on Tuesday was 5% more than its closing price on Monday.
The closing price of the share on Wednesday was 5% less than its closing price on Tuesday.
Which one of the following calculations will give the closing price of the share, in dollars, on Wednesday?
A. `30 xx 1.05 xx 0.95`
B. `30 xx 1.05 xx –0.05`
C. `30 + 1.05 xx 0.95`
D. `30 + 0.05 xx 30 - 0.05 xx 30`
E. `30 + 1.05 - 0.95`
The constraints of a linear programming problem are given by the following set of inequalities.
| `x + y` | `<= 8` |
| `3x + 5y` | `<= 30` |
| `x` | `>= 0` |
| `y` | `>= 0` |
The coordinates of the points that define the boundaries of the feasible region for this linear programming problem are
A. `(0, 0), (0, 6), (3, 5), (8, 0)`
B. `(0, 0), (0, 6), (5, 3), (8, 0)`
C. `(0, 0), (0, 6), (5, 3), (10, 0)`
D. `(0, 0), (0, 8), (5, 3), (8, 0)`
E. `(0, 0), (0, 8), (5, 3), (10, 0)`
`x + y = 8\ \ text{cut the axes at (0, 8), (8, 0)}`
`3x + 5y = 30\ \ text{cut the axes at (0, 6), (10, 0)}`
`text(From the diagram, 3 of the boundary points clearly)`
`text{lie at (0, 6), (0, 0), and (8, 0).}`
`:. text(Eliminate)\ C,\ D,\ text(and)\ E`
`text(Choosing between)\ A\ text(and)\ B,\ text{only (5, 3) lies}`
`text(on both lines and is therefore the intersection.)`
`:. text(Eliminate)\ A`
`=> B`
Consider the following statements that relate to the solution of linear programming problems.
Which one of the following statements is true?
A. Only one point can be a solution.
B. No point outside the feasible region can be a solution.
C. To have a solution, the feasible region must be bounded.
D. Only the corner points of a feasible region can be a solution.
E. Only the corner points with integer coordinates can be a solution.
The distance–time graph below shows a train’s journey between two towns.
During the journey, the train stopped for 30 minutes.
The average speed of the train, in kilometres per hour, for the journey is closest to
A. `45`
B. `50`
C. `60`
D. `65`
E. `80`
A line passes through the points `(–1, 1)` and `(3, 5)`.
Another point that lies on this line is
A. `(0, 1)`
B. `(1, 3)`
C. `(2, 6)`
D. `(3, 4)`
E. `(4, 7)`
The point `(2, 20)` lies on the graph of `y = k/x`, as shown below.
The value of `k` is
A. `5`
B. `10`
C. `20`
D. `40`
E. `80`
Robert invested $6000 at 4.25% per annum with interest compounding quarterly.
Immediately after interest is paid at the end of each quarter, he adds $500 to his investment.
The value of Robert’s investment at the end of the third quarter, after his $500 has been added, is closest to
A. $6193
B. $7569
C. $7574
D. $7709
E. $8096
New furniture was purchased for an office at a cost of $18 000.
Using flat rate depreciation, the furniture will be valued at $5000 after four years.
The expression that can be used to determine the value of the furniture, in dollars, after one year is
A. `18\ 000 - (4 xx 5000)`
B. `18\ 000 - ({18\ 000 - 5000}/4)`
C. `18\ 000 - 5000/4`
D. `(18\ 000)/4 - 5000`
E. `18\ 000 xx 0.726`
Amy invests $15 000 for 150 days.
Interest is calculated at the rate of 4.60% per annum, compounding daily.
Assuming that there are 365 days in a year, the value of her investment after 150 days is closest to
A. `$15\ 279`
B. `$15\ 284`
C. `$15\ 286`
D. `$15\ 690`
E. `$16\ 776`
An internet car market site charges $120 to advertise a car for sale.
The car is sold for $15 000.
The $120 charge as a percentage of the selling price of the car is
A. 0.008%
B. 0.08%
C. 0.80%
D. 1.20%
E. 1.25%
The cost of manufacturing a number of frying pans consists of a fixed cost of $400 plus a cost of $50 per frying pan.
The manufacturer could break even by selling
A. 10 frying pans at $90 each.
B. 10 frying pans at $45 each.
C. 15 frying pans at $60 each.
D. 15 frying pans at $30 each.
E. 20 frying pans at $50 each.
In a linear programming problem involving animal management on a farm
• `x` represents the number of cows on the farm
• `y` represents the number of sheep on the farm.
The feasible region (with boundaries included) for the problem is indicated by the shaded region on the diagram below.
One of the constraints defining the feasible region indicates that
A. there must be 20 cows and 60 sheep.
B. there must be 40 cows and 40 sheep.
C. the number of sheep cannot exceed 40.
D. the number of cows must be at least 60.
E. the total number of cows and sheep cannot exceed 80.
The graph above represents the depth of water in a channel (in metres) as it changes over time (in hours).
During the time interval shown, the number of times the depth of the water in the channel is 10 metres is
A. `0`
B. `1`
C. `2`
D. `3`
E. `4`
`text(From the graph above, the depth of water in the)`
`text(channel is 10m on 4 occasions.)`
`=> E`
On the graph above the equation of the line passing through the point `(1, 2)` is
A. `x = 1`
B. `y = 1`
C. `x = 2`
D. `y = 2`
E. `y = x + 1`
Mapupu and Minoha are two towns on the equator.
The longitude of Mapupu is `text(16°E)` and the longitude of Minoha is `text(52°W)`.
How far apart are these two towns if the radius of Earth is approximately `6400\ text(km)`?
(A) `4000\ text(km)`
(B) `7600\ text(km)`
(C) `1\ 447\ 600\ text(km)`
(D) `2\ 734\ 400\ text(km)`
The following five constraints apply to a linear programming problem.
`x>= 0,\ \ y>= 0,\ \ x + y>=50,\ \ x + y<=100,\ \ y<=x`
In the diagram below, the shaded region (with boundaries included) represents the feasible region for this linear programming problem.
The aim is to maximise the objective function `Z = 2x + ky`.
If the maximum value of `Z` occurs only at the point `(100, 0)`, then a possible value for `k` is
A. `1`
B. `2`
C. `3`
D. `4`
E. `5`
Which one of the following pairs of simultaneous linear equations has no solution?
| A. | `3x - y = 5` |
| `4x + y = 9` |
| B. | `2x−y = 1` |
| `4x−2y = 3` |
| C. | `x + 3y = 0` |
| `2x - y = 7` |
| D. | `x - 3y = 10` |
| `3x + 2y = 8` |
| E. | `4x + y = - 6` |
| `2x - y = 0` |
The cost of hiring one motorbike for up to 4 hours is shown in the graph above.
Two motorbikes were hired.
The total charge for hiring the two motorbikes was $45.
The time for which each motorbike was hired could have been
A. 1 hour and 2 hours.
B. 1 hour and 3 hours.
C. 1.5 hours and 2 hours.
D. 1.5 hours and 3 hours.
E. 2 hours and 3.5 hours.
The line above passes through the origin and the point `(2, 1)`.
The slope of this line is
A. `− 2`
B. `− 1`
C. `− 1/2`
D. `1/2`
E. `2`
