The formula `C = 5/9 (F-32)` is used to convert temperatures between degrees Fahrenheit `(F)` and degrees Celsius `(C)`.
Convert 3°C to the equivalent temperature in Fahrenheit. (2 marks)
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Aussie Maths & Science Teachers: Save your time with SmarterEd
The formula `C = 5/9 (F-32)` is used to convert temperatures between degrees Fahrenheit `(F)` and degrees Celsius `(C)`.
Convert 3°C to the equivalent temperature in Fahrenheit. (2 marks)
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`37.4\ text(degrees)\ F`
`C` | `= 5/9(F-32)` |
`F-32` | `= 9/5C` |
`F` | `= 9/5C + 32` |
`text(When)\ \ C = 3,`
`F` | `= (9/5 xx 3) + 32` |
`= 37.4\ text(degrees)\ F` |
Ariana’s parents have given her an interest‑free loan of $4800 to buy a car. She will pay them back by paying `$x` immediately and `$y` every month until she has repaid the loan in full.
After 18 months Ariana has paid back $1510, and after 36 months she has paid back $2770.
This information can be represented by the following equations.
`x + 18y = 1510`
`x + 36y = 2770`
i.
`:.\ text(Solution is)\ \ x = 250, \ y = 70`
ii. `text(Let)\ \ A = text(the amount paid back after)\ n\ text(months)`
`A = 250 + 70n`
`text(Find)\ n\ text(when)\ A = 4800`
`250 + 70n` | `= 4800` |
`70n` | `= 4550` |
`n` | `= 65` |
`:.\ text(It will take Ariana 65 months to repay)`
`text(the loan in full.)`
At a particular time during the day, a tower of height 19.2 metres casts a shadow. At the same time, a person who is 1.65 metres tall casts a shadow 5 metres long.
What is the length of the shadow cast by the tower at that time? (2 marks)
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`58\ text{m}`
`text(Both triangles have right angles and a common)`
`text(angle to the ground.)`
`:.\ text{Triangles are similar (equiangular)}`
`text(Let)\ x =\ text(length of tower shadow)`
`x/5` | `= 19.2/1.65\ \ text{(corresponding sides of similar triangles)}` |
|
`x` | `= (5 xx 19.2)/1.65` | |
`= 58.1818…` | ||
`= 58\ text{m (nearest m)}` |
A family currently pays $320 for some groceries.
Assuming a constant annual inflation rate of 2.9%, calculate how much would be paid for the same groceries in 5 years’ time. (2 marks)
`$369.17\ \ text{(nearest cent)}`
`FV` | `= PV(1 + r)^n` |
`= 320(1.029)^5` | |
`= $369.1703…` | |
`= $369.17\ \ text{(nearest cent)}` |
Consider the equation `(2x)/3-4 = (5x)/2 + 1`.
Which of the following would be a correct step in solving this equation?
`B`
`(2x)/3-4` | `= (5x)/2 + 1` |
`(2x)/3` | `= (5x)/2 + 5` |
`=>B`
The area of the triangle shown is 250 cm².
What is the value of `x`, correct to the nearest whole number?
`D`
`text(Using)\ \ \ A = 1/2ab\ sin\ C`
`250` | `= 1/2 xx 30x\ sin\ 44^@` |
`250` | `= 15x\ sin\ 44 ^@` |
`:.x` | `= 250/(15\ sin\ 44^@)` |
`= 23.99…\ text(m)` |
`=>D`
What amount must be invested now at 4% per annum, compounded quarterly, so that in five years it will have grown to $60 000?
`C`
`text(Using)\ \ FV = PV(1 + r)^n`
`r` | `= text(4%)/4` | `= text(1%) = 0.01\ text(per quarter)` |
`n` | `= 5 xx 4` | `= 20\ text(quarters)` |
`60\ 000` | `= PV(1 + 0.01)^(20)` |
`:.PV` | `= (60\ 000)/1.01^(20)` |
`= $49\ 172.66…` |
`⇒ C`
The probability of winning a game is `7/10`.
Which expression represents the probability of winning two consecutive games?
`D`
`text{Since the two events are independent:}`
`P text{(W)}` | `= 7/10` |
`P text{(WW)}` | `= 7/10 xx 7/10` |
`=>D`
The length of a fish was measured to be 49 cm, correct to the nearest cm.
What is the percentage error in this measurement, correct to one significant figure?
`C`
`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 1 = 0.5\ text{cm}`
`text{% error}` | `=\ frac{text{absolute error}}{text{measurement}} xx 100%` | |
`=0.5/49 xx 100%` | ||
`=1.020… %` | ||
`=1%\ \ text{(to 1 sig fig)}` |
`=>C`
Which of the following is `3x^0 + 5x` in its simplest form?
`D`
`3x^0 + 5x` | `= 3 xx 1 + 5x` |
`= 3 + 5x` |
`=> D`
From the top of a cliff 67 metres above sea level, the angle of depression of a buoy is 42°.
How far is the buoy from the base of the cliff, to the nearest metre?
`B`
The Louvre Pyramid in Paris has a square base with side length 35 m and a perpendicular height of 22 m.
What is the volume of this pyramid, to the nearest m³?
`C`
`V` | `= 1/3Ah` |
`A` | `= 35 xx 35` |
`= 1225\ text(m)^2` |
`:.V` | `= 1/3 xx 1225 xx 22` |
`= 8983.33…\ text(m)^3` |
`=>C`
The times, in minutes, that a large group of students spend on exercise per day are presented in the box‑and‑whisker plot.
What percentage of these students spend between 40 minutes and 60 minutes per day on exercise?
`C`
`text{Q}_1 = 40, \ text(Median) = 60`
`:.\ text(% Students between 40 and 60)`
`= 50text{%}-25text{%}`
`=25 text{%}`
`=>C`
Which of the following is `4x + 3y-x-5y` in its simplest form?
`A`
`4x + 3y-x-5y`
`= 3x-2y`
`⇒ A`
What is 1 560 200 km written in standard form correct to two significant figures?
`D`
COMMENT: Incredibly, the first MC question in 2015 had the lowest mean mark of all MC questions in the exam!
`1\ 560\ 200`
`= 1.5602 xx 10^6`
`= 1.6 xx 10^6\ text(km)\ \ \ text{(2 sig fig)}`
`=> D`
In the diagram, `ABC` is an isosceles triangle with `AB = AC` and `/_BAC = 38^@`. The line `BC` is produced to `D`.
Find the size of `/_ACD`. Give reasons for your answer. (2 marks)
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`109^@`
Solve `3-5x <= 2`. (2 marks)
`x >= 1/5`
`3-5x` | `<= 2` |
`-5x` | `<= -1` |
`x` | `>= 1/5` |
Factorise `2x^2 + 5x-3`. (2 marks)
`(2x-1) (x + 3)`
`2x^2 + 5x-3= (2x-1) (x + 3)`
A new test has been developed for determining whether or not people are carriers of the Gaussian virus.
Two hundred people are tested. A two-way table is being used to record the results.
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What is the probability that the test results would show this? (2 marks)
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i. `A` | `= 200-(74 + 12 + 16)` |
`= 98` |
ii. `P` | `= text(# Positive carriers)/text(Total carriers)` |
`= 74/86` | |
`= 37/43` |
iii. `text(# People with inaccurate results)`
`= 12 + 16`
`= 28`
The diagram shows a parallelogram `ABCD` with `∠DAB = 120^@`. The side `DC` is produced to `E` so that `AD = BE`.
Prove that `ΔBCE` is equilateral. (3 marks)
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`text(See Worked Solutions)`
`BC` | `= AD\ text{(opposite sides of parallelogram}\ ABCD)` |
`∠BCD` | `= 120^@\ text{(opposite angles of parallelogram}\ ABCD)` |
`∠BCE` | `= 60^@\ (∠DCE\ text{is a straight angle)}` |
`∠CEB` | `= 60^@\ text{(base angles of isosceles}\ \Delta BCE)` |
`∠CBE` | `= 60^@\ text{(angle sum of}\ ΔBCE)` |
`:.ΔBCE\ text(is equilateral)`
Use the change of base formula to evaluate `log_3 7`, correct to two decimal places. (1 mark)
`1.77\ \ text{(to 2 d.p.)}`
`log_3 7` | `= (log_10 7)/(log_10 3)` |
`= 1.771…` | |
`= 1.77\ \ text{(to 2 d.p.)}` |
Express `((2x-3))/2-((x-1))/5` as a single fraction in its simplest form. (2 marks)
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`(8x-13)/10`
`((2x-3))/2-((x-1))/5`
`= (5(2x-3)-2(x-1))/10`
`= (10x-15-2x + 2)/10`
`= (8x-13)/10`
A sports car worth $150 000 is bought in December 2005.
In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.
In which year will the depreciated value first fall below $120 000? (2 marks)
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`text(The value falls below $120 000 in the third year)`
`text{which will be during 2008.}`
`text(Using)\ \ S = V_0(1-r)^n`
`text(where)\ \ V_0 = 150\ 000, r = text(10%)`
`text(If)\ \ n = 2,`
`S` | `= 150\ 000(1-0.1)^2` |
`= 121\ 500` |
`text(If)\ \ n= 3,`
`S` | `= 150\ 000(1-0.1)^3` |
`= 109\ 350` |
`:.\ text(The value falls below $120 000 in the third year)`
`text{which will be during 2008.}`
A 130 cm long garden rake leans against a fence. The end of the rake is 44 cm from the base of the fence.
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Simplify `(ab^2)/w xx (4w)/(3b).` (2 marks)
`(4ab)/3`
`(ab^2)/w xx (4w)/(3b)` | `=(4ab^2w)/(3bw)` | |
`=(4ab)/3` |
Make `L` the subject of the equation `T = 2piL^2`. (2 marks)
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`± sqrt(T/(2pi))`
`T` | `= 2piL^2` |
`L^2` | `= T/(2pi)` |
`:.L` | `= ±sqrt(T/(2pi))` |
David is paid at these rates:
His time sheet for last week is:
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i. `text{Pay (Fri)}` | `= text(4 hours) xx 18.00` |
`= $72.00` | |
`text{Pay (Sat)}` | `= 6\ text(hours) xx 1.5 xx 18.00` |
`= $162.00` | |
`text{Pay (Sun)}` | `= 5\ text(hours) xx 2 xx 18.00` |
`= $180.00` |
`:.\ text(Gross pay)` | `= 72 + 162 + 180` |
`= $414.00` |
ii. `text(Pay on Sat) = $162.00`
`text(Weekly equivalent hours)`
`= 162/18`
`= 9\ text(hours)`
`:.\ text(He will have to work 9 extra hours on)`
`text(a weekday for the same gross pay)`
This income tax table is used to calculate Evelyn’s tax payable.
Evelyn’s taxable income increases from $50 000 to $80 000.
What percentage of her increase will she pay in additional tax?
`B`
`text(Tax on $50 000)` | `= 2500 + 0.35 xx (50\ 000-45\ 000)` |
`= 2500 + 1750` | |
`= $4250` |
`text(Tax on $80 000)` | `= 11\ 250 + 0.52 xx (80\ 000-70\ 000)` |
`= 11\ 250 + 5200` | |
`= $16\ 450` |
`:.\ text(Extra tax)` | `= 16\ 450-4250` |
`= $12\ 200` |
`:.\ text(% Increase paid in tax)`
`= (12\ 200) / (30\ 000) xx 100`
`=\ text(40.66… %)`
`=> B`
A clay brick is made in the shape of a rectangular prism with dimensions as shown.
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Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.
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i. | `V` | `= l × b × h` |
`= 21 × 8 × 9` | ||
`= 1512\ text(cm)^3` |
ii. `text(Volume of each hole)`
`= pir^2h`
`= pi × 1.4^2 × 8`
`= 49.260…\ text(cm)^3`
`:.\ text(Volume of clay still in brick)`
`= 1512 − (3 × 49.260…)`
`= 1364.219…`
`= 1364\ text{cm}^3\ text{(nearest whole)}`
iii. `text(Percentage of clay removed)`
`= ((3 × 49.260…))/1512 × 100`
`= 9.773…`
`= 9.8 text{% (1 d.p.)}`
Two groups of people were surveyed about their weekly wages. The results are shown in the box-and-whisker plots.
Which of the following statements is true for the people surveyed?
`B`
`text{Option A: 50% of Under 21 group earned over $325 and 75%}`
`text{of Over 21 group did. NOT TRUE.}`
`text{Option B: 75% of Under 21 group earned below $350 is TRUE.}`
`text{Options C and D: can both be proven to be untrue using their}`
`text{median and quartile values.}`
`=> B`
The total cost, `$C`, of a school excursion is given by `C = 2n + 5`, where `n` is the number of students.
If three extra students go on the excursion, by how much does the total cost increase?
`A`
`C = 2n + 5`
`text(If)\ n\ text(increases to)\ n + 3`
`C` | `= 2(n + 3) + 5` |
`= 2n + 6 + 5` | |
`= 2n + 11` |
`:.\ text(Total cost increases by $6)`
`=> A`
On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table.
\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \textbf{Male viewers} & \textbf{Female viewers} \\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 1}\rule[-1ex]{0pt}{0pt} & 1372 & 3915\\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 2}\rule[-1ex]{0pt}{0pt} & 2054 & 3269\\
\hline
\end{array}
One male viewer was selected at random from all of the male viewers.
What is the probability that he voted for Contestant 1?
`C`
`text(Total male viewers)\ = 1372 + 2054= 3426`
`P\ text{(Male viewer chosen voted for C1)}`
`= text(Males who voted for C1)/text(Total male viewers)`
`= 1372/3426`
`=> C`
Lie detector tests are not always accurate. A lie detector test was administered to 200 people.
The results were:
• 50 people lied. Of these, the test indicated that 40 had lied;
• 150 people did NOT lie. Of these, the test indicated that 20 had lied.
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Find integers `a` and `b` by showing working to expand and simplify
`(3-sqrt2)^2 = a-b sqrt2`. (2 marks)
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`a = 11,\ b = 6`
`(3-sqrt2)^2` | `= 9-6 sqrt2 + (sqrt2)^2` |
`= 9-6 sqrt2 + 2` | |
`= 11-6 sqrt2` | |
`:.\ a = 11, \ \ b = 6` |
Solve `(x-5)/3-(x+1)/4 = 5`. (2 marks)
`83`
`(x-5)/3-(x+1)/4` | `= 5` |
`12((x-5)/3)-12((x+1)/4)` | `= 12 xx 5` |
`4x-20-3x-3` | `= 60` |
`x-23` | `= 60` |
`:. x` | `= 83` |
A salesman earns $200 per week plus $40 commission for each item he sells.
How many items does he need to sell to earn a total of $2640 in two weeks?
`B`
`text(Let items sold) = n`
`text{Wages over 2 weeks}\ (w)`
`= (2 xx 200) + 40n`
`= 400 + 40n`
`text(Find)\ n\ text(when)\ w = 2640:`
`2640` | `= 400 + 40n` |
`40n` | `= 2240` |
`n` | `= 56` |
`=> B`
The angle of depression of the base of the tree from the top of the building is 65°. The height of the building is 30 m.
How far away is the base of the tree from the building, correct to one decimal place?
`B`
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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i. `text(The claim is incorrect because each of her friends bought)`
`text(a different number of tickets and therefore their chances of)`
`text(winning are different.)`
ii. `text(Number of tickets not sold to K or H)`
`= 45 + 10 + 14`
`= 69`
`:.\ text(Probability 1st prize NOT won by K or H)`
`= 69/100`
Simplify `2m^2 × 3m p^2`
`D`
`2m^2 × 3m p^2` | `= 6m^((2+1))p^2` | |
`=6m^3p^2` |
`=> D`
Which formula should be used to calculate the distance between Toby and Frankie?
`A`
`text(The triangle is not a right-angled triangle,)`
`:.\ text(Not)\ B`
`text(Given the information on the diagram provides)`
`text(2 angles and 1 side, the sine rule will work best.)`
`a/sinA = b/sinB`
`=> A`
What is the value of `(a-b)/4`, if `a = 240` and `b = 56`?
`B`
`(a-b)/4` | `= (240-56)/4` |
`= 46` |
`=> B`
If `d = 6t^2`, what is a possible value of `t` when `d = 2400`?
`B`
`d` | `= 6t^2` |
`t^2` | `= d/6` |
`t` | `= +- sqrt(d/6)` |
`text(When)\ \ d = 2400:`
`t` | `= +- sqrt(2400/6)` |
`= +- 20` |
`=> B`
In a stack of 10 DVDs, there are 5 rated PG, 3 rated G and 2 rated M.
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Grant chooses two DVDs at random from the stack. Copy or trace the tree diagram into your writing booklet.
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Sandy travels to Europe via the USA. She uses this graph to calculate her currency conversions.
She converts all of her money to euros.How many euros does she have to spend in Europe? (3 marks)
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`text(A$, then 1 A$ will buy more euros than)`
`text(before and the gradient used to convert)`
`text{the currencies will steepen (increase).}`
i. `text(From graph:)`
`75\ text(US$)` | `=\ text(100 A$)` |
`=> 150\ text(US$)` | `=\ text(200 A$)` |
`:.\ text(Sandy has a total of 800 A$)`
`text(Converting A$ to €:)`
`text(100 A$)` | `= 60\ €` |
`:.\ text(800 A$)` | `= 8 xx 60` |
`= 480\ €` |
ii. `text(If the value of the euro falls against the)`
`text(A$, then 1 A$ will buy more euros than)`
`text(before and the gradient used to convert)`
`text{the currencies will steepen (increase).}`
The distance in kilometres (`D`) of an observer from the centre of a thunderstorm can be estimated by counting the number of seconds (`t`) between seeing the lightning and first hearing the thunder.
Use the formula `D = t/3` to estimate the number of seconds between seeing the lightning and hearing the thunder if the storm is 1.2 km away. (1 mark)
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`3.6\ text(seconds)`
`D = t/3`
`text(When)\ \ D = 1.2,`
`t/3` | `= 1.2` |
`t` | `= 3.6\ text(seconds)` |
Which of the following correctly expresses `T` as the subject of `B = 2pi (R + T/2)`?
`A`
`B` | `= 2pi (R + T/2)` |
`B/(2pi)` | `= R + T/2` |
`T/2` | `= B/(2pi)-R` |
`T` | `= B/pi-2R` |
`=> A`
Leanne copied a two-way table into her book.
Leanne made an error in copying one of the values in the shaded section of the table.
Which value has been incorrectly copied?
`D`
`text(By checking row and column total, the number)`
`text(of females part-time work is incorrect)`
`=> D`
If pressure (`p`) varies inversely with volume (`V`), which formula correctly expresses `p` in terms of `V` and `k`, where `k` is a constant?
`A`
`p prop 1/V`
`p = k/V`
`=> A`
Which of the following would be most likely to have a positive correlation?
\(A\)
\(\text{Positive correlation means that as one variable increases,}\)
\(\text{the other tends to increase also.}\)
\(\Rightarrow A\)
`A`
`text(Take two corresponding sides)`
`text(In)\ Delta A:\ 3\ text(cm)`
`text(In)\ Delta B:\ 1 \frac{1}{2}\ text(cm)`
`:.\ text(Scale factor converting)\ Delta A\ text(to)\ Delta B = frac{1}{2}`
`=> A`
Joe is about to go on holidays for four weeks. His weekly salary is $280 and his holiday loading is 17.5% of four weeks pay.
What is Joe’s total pay for the four weeks holiday?
`D`
`text(Salary)\ text{(4 weeks)}` | `= 4 xx 280` |
`= $1120` |
`text(Holiday loading)` | `= 1120 xx 17.5%` |
`= $196` |
`:.\ text(Total pay)` | `= 1120 + 196` |
`= $1316` |
`=> D`
The polynomial `x^3` is divided by `x + 3`. Calculate the remainder. (2 marks)
`-27`
`P(-3)` | `= (-3)^3` |
`= -27` |
`:.\ text(Remainder when)\ x^3 -: (x + 3) = -27`
A plasma TV depreciated in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $2023, using the declining balance method.
What was the purchase price of the plasma TV? (2 marks)
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`$2800`
`S = V_0 (1-r)^n`
`2023` | `= V_0 (1-0.15)^2` |
`2023` | `= V_0 (0.85)^2` |
`V_0` | `= 2023/0.85^2` |
`= 2800` |
`:.\ text(The purchase price) = $2800`
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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i. `text{Critics liked and}\ >= 40`
`= 102-65`
`= 37`
`:. A = 37+31=68`
ii. `text{Critics did not like and < 40}`
`= 175-65-37-31`
`= 42`
`:.\ P text{(not like and < 40)}`
`= 42/175`
`= 6/25`
iii. `text(Critics liked) = 102`
`text(% Critics liked)` | `= 102/175 xx 100` |
`= 58.28…%` |
`:.\ text{Movie NOT a box office success (< 65% critics liked)}`
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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i. `text(Method 1)`
`text(Yearly sales)` | `= 12 xx 15\ 670` |
`= 188\ 040` |
`:.\ text(Earnings)` | `= text(13%) xx 188\ 040` |
`= $24\ 445.20` |
ii. `text(Method 2)`
`text(In 1 Year, Weekly Wage)` | `= 350 xx 52` |
`= 18\ 200` |
`text(Commission)` | `= text(4.5%) xx 188\ 040` |
`= 8461.80` |
`text(Total earnings)` | `= 18\ 200 + 8461.80` |
`= $26\ 661.80` |
`:.\ text(Bob should choose Method 2.)`
A point `P` lies between a tree, 2 metres high, and a tower, 8 metres high. `P` is 3 metres away from the base of the tree.
From `P`, the angles of elevation to the top of the tree and to the top of the tower are equal.
What is the distance, `x`, from `P` to the top of the tower?
`D`
`text(Triangles are similar)\ \ text{(equiangular)}`
`text(In smaller triangle:)`
`h^2` | `= 2^2 + 3^2` |
`= 13` | |
`h` | `= sqrt 13` |
`x/sqrt13` | `= 8/2\ \ \ text{(sides of similar Δs in same ratio)}` |
`x` | `= (8 sqrt 13)/2` |
`= 14.422…` |
`=> D`
A scatterplot is shown.
Which of the following best describes the correlation between \(R\) and \(T\)?
\(A\)
\(\text{Correlation is positive.}\)
\(\text{NB. The skew does not directly relate to correlation.}\)
\(\Rightarrow A\)