Expand and simplify the expression
`3a(4a-5)-2(a-3)` (2 marks)
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Expand and simplify the expression
`3a(4a-5)-2(a-3)` (2 marks)
`12a^2-17a+6`
`3a(4a-5)-2(a-3)` | `=12a^2-15a-2a+6` | |
`=12a^2-17a+6` |
Expand the expression `-2(5x^2-3x-4)` (2 marks)
`-10x^2+6x+8`
`-2(5x^2-3x-4)=-10x^2+6x+8`
Expand the expression `3(2y^2-3y+1)` (2 marks)
`6y^2-9y+3`
`3(2y^2-3y+1)=6y^2-9y+3`
Expand and simplify the expression `4a(a-3)-5(6-a)` (2 marks)
`4a^2-7a-30`
`4a(a-3)-5(6-a)` | `=4a^2-12a-30+5a` | |
`=4a^2-7a-30` |
Expand and simplify the expression `3x(x-2)+4(x-5)` (2 marks)
`3x^2-2x-20`
`3x(x-2)+4(x-5)` | `=3x^2-6x+4x-20` | |
`=3x^2-2x-20` |
Which of the following is always equal to `a-b`?
`D`
`a-b = -b + a`
`=>D`
Which expression is equivalent to `4x^2-12x + x^3?`
`D`
`x (4x-12 + x^2)`
`= 4x^2-12x + x^3`
`=>D`
What expression is equivalent to `-(y-6)`?
`D`
`-(y-6) = -y-(-6)=-y + 6`
`=>D`
Which expression is equal to `4x-8 + 3x + 2`?
`B`
`4x-8 + 3x + 2=7x-6`
`=>B`
Which expression is equivalent to `12x + 24`?
`A`
`3 (4x + 8)` | `= 3 xx 4x + 3 xx 8` |
`= 12x + 24` |
`=>A`
Which expression is equivalent to `5-6t`?
`D`
`-6t+5`
`=>D`
Which one of the following expressions is equivalent to `4 (3m-1)`?
`C`
`4 (3m-1)`
`= (4 xx 3m)-(4 xx 1)`
`= 12m-4`
`=>C`
The expression `3x + 7 + 8x + 11` can also be written as
`B`
`3x + 7 + 8x + 11 = 11x + 18`
`=>B`
Simplify the expression `(9h)/2 -: (h)/3` (2 marks)
`(27)/2`
`(9h)/2 -: (h)/3` | `=(9h)/2 xx 3/(h)` | |
`=(9h xx 3)/(2 xx h)` | ||
`=(27)/2` |
Simplify the expression `(13x)/15 -: (2x)/5` (2 marks)
`(13)/6`
`(13x)/15 -: (2x)/5` | `=(13x)/15 xx 5/(2x)` | |
`=(13x xx 5)/(15 xx 2x)` | ||
`=(13)/6` |
Simplify the expression `(3a)/4 -: (7a)/2` (2 marks)
`(3)/14`
`(3a)/4 -: (7a)/2` | `=(3a)/4 xx 2/(7a)` | |
`=(3a xx 2)/(4 xx 7a)` | ||
`=(3)/14` |
Simplify the expression `(3p)/4 xx (8p)/9` (2 marks)
`(2p^2)/3`
`(3p)/4 xx (8p)/9` | `=(3p xx 8p)/(4 xx 9)` | |
`=(p xx 2p)/3` | ||
`=(2p^2)/3` |
Simplify the expression `(2a)/7 xx (14a)/3` (2 marks)
`(4a^2)/3`
`(2a)/7 xx (14a)/3` | `=(2a xx 14a)/(7 xx 3)` | |
`=(2a xx 2a)/3` | ||
`=(4a^2)/3` |
Simplify the expression `(5)/6 xx c/4` (2 marks)
`(5c)/24`
`(5)/6 xx c/4` | `=(5 xx c)/(6 xx 4)` | |
`=(5c)/24` |
Simplify the expression `(5x)/9-x/6` (2 marks)
`(7x)/18`
`(5x)/9-x/6` | `=(10x)/18-(3x)/18` | |
`=(7x)/18` |
Simplify the expression `b/2-b/3` (2 marks)
`b/6`
`b/2-b/3` | `=(3b)/6-(2b)/6` | |
`=b/6` |
Simplify the expression `(4x)/5-(x)/3` (2 marks)
`(7x)/15`
`(4x)/5-(x)/3` | `=(12x)/15-(5x)/15` | |
`=(7x)/15` |
Simplify the expression `(3t)/5-(t)/2` (2 marks)
`(t)/10`
`(3t)/5-(t)/2` | `=(6t)/10-(5t)/10` | |
`=(t)/10` |
Simplify the expression `(3a)/5+(a)/4` (2 marks)
`(17a)/20`
`(3a)/5+(a)/4` | `=(12a)/20+(5a)/20` | |
`=(17a)/20` |
Simplify the expression `x/4+(2x)/3` (2 marks)
`(11x)/12`
`x/4+(2x)/3` | `=(3x)/12+(8x)/12` | |
`=(11x)/12` |
Simplify the expression `p/2+p/3` (2 marks)
`(5p)/6`
`p/2+p/3` | `=(3p)/6+(2p)/6` | |
`=(5p)/6` |
Manou purchased an oven that depreciates in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $6069, using the declining balance method.
What was the purchase price of the oven? (2 marks)
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`$8400`
`S = V_0 (1-r)^n`
`6069` | `= V_0 (1-0.15)^2` |
`6069` | `= V_0 (0.85)^2` |
`V_0` | `= 6069/0.85^2` |
`= 8400` |
`:.\ text(The purchase price) = $8400`
A company purchases a machine for $50 000. The two methods of depreciation being considered are the declining-balance method and the straight-line method.
For the declining-balance method, the salvage value of the machine after `n` years is given by the formula
`S=V_(0)xx(0.80)^(n),`
where `S` is the salvage value and `V_(0)` is the initial value of the asset.
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
a. `text{Depreciation rate}\ = 1-0.8=0.2=20text{%}`
b. `text{Find}\ \ S\ \ text{when}\ \ n=3:`
`S` | `=V_0 xx (0.80)^n` | |
`=50\ 000 xx (0.80)^3` | ||
`=$25\ 600` |
Alan bought a light aircraft for $76 500. It will depreciate at 14% per annum.
Using the declining balance method, what will be the salvage value of the light aircraft after 6 years, to the nearest dollar?
`$30\ 949`
`S` | `= V_0 (1-r)^n` |
`= 76\ 500 (1-14/100)^6` | |
`= 76\ 500 (0.86)^6` | |
`= $30\ 949.39` | |
`=$30\ 949\ \ text{(nearest dollar)}` |
Marnus bought a cricket bowling machine two years ago that cost $3400. Its value has depreciated by 10% each year, based on the declining-balance method.
What is the salvage value today, to the nearest dollar? (2 marks)
`$2754`
`S` | `= V_0 (1-r)^n` |
`= 3400 (1-0.10)^2` | |
`= 3400 (0.90)^2` | |
`= $2754` |
Albert invests $3000 and earns interest at 4% per annum, compounded quarterly.
What is the future value of Albert's investment after 2.5 years? (3 marks)
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`$3313.87`
`text{Annual interest rate = 4%}`
`text(Quarterly interest rate) \ = frac(4%)(4)=1text{%}`
`n = 2.5 xx 4 = 10`
`FV` | `= PV(1 + r)^n` |
`= 3000 (1 + 0.01)^10` | |
`=3000(1.01)^10` | |
`= $3313.87` |
What amount must be invested now at 3% per annum, compounded annually, so that in two years it will have grown to $20 000? (2 marks)
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`$18\ 851.92`
`text(Using)\ \ FV = PV(1 + r)^n`
`r = 3text{%}, \ n=2`
`20\ 000` | `= PV(1 + 0.03)^(2)` |
`:.PV` | `= (20\ 000)/1.03^(2)` |
`= $18\ 851.92` |
Hugo is a professional bike rider.
The value of his bike will be depreciated over time using the flat rate method of depreciation.
The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. `$8000`
ii. `text(Value after 1 year) = $6500\ \ \ text{(from graph)}`
`:.\ text(Annual depreciation)` | `= 8000-6500` |
`= $1500` |
iii. `text(After 5 years:)`
`S` | `=V_0-Dn` |
`=8000-5 xx 1500` | |
`=$500` |
Rae paid $40 000 for new office equipment at the start of the 2019 financial year.
At the start of each following financial year, she used straight-line (flat rate) depreciation to revalue her equipment.
At the start of the 2022 financial year she revalued her equipment at $22 000.
Calculate the annual straight-line rate of depreciation she used, as a percentage of the purchase price. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`15text(%)`
`text(Depreciation over 3 years)`
`=40\ 000-22\ 000`
`=$18\ 000`
`:.\ text(Annual depreciation) = (18\ 000)/3 = $6000`
`:.\ text(Depreciation rate) = 6000/(40\ 000) = 0.15 = 15text(%)`
The table shows the income tax rates for the 2022-23 financial year.
\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{ Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{ Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$45 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$45 001 – \$120 000}\rule[-1ex]{0pt}{0pt} & \text{\$5092 plus 32.5 cents for each \$1 over \$45 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$120 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$29 467 plus 37 cents for each \$1 over \$120 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$51 667 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}
Boonie is a professional cricketer and has a gross income of $145 000. During the financial year, he has allowable tax deductions of $1300 for cricket bats and pads.
What is Boonie's total amount of tax payable for the financial year? (3 marks)
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`$38\ 236`
`text(Taxable income)\ =145\ 000-1300=$143\ 700`
`text{Tax payable}`
`= 29\ 467 + 0.37 (143\ 700-120\ 000)`
`= 29\ 467 + 8769`
`= $38\ 236`
\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{ Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{ Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$45 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$45 001 – \$120 000}\rule[-1ex]{0pt}{0pt} & \text{\$5092 plus 32.5 cents for each \$1 over \$45 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$120 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$29 467 plus 37 cents for each \$1 over \$120 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$51 667 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}
Using the tax table, what is the tax payable on $47 580?
`B`
`text(Tax Payable)`
`= 5092 + 0.325 (47\ 580-45\ 000)`
`= 5092 + 838.50`
`= 5930.50`
`=> B`
George makes a single deposit of $9000 into an account that pays simple interest.
After 4 years, George's account has a balance of $10 350.
What simple interest rate did George receive on his investment? (2 marks)
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`3.75text(%)`
`text(Interest earned)` | `= 10\ 350-9000` |
`= $1350` |
`text(Using)\ \ I = Prn,`
`1350` | `= 9000 xx r xx 4` |
`:. r` | `= 1350/(4 xx 9000)` |
`= 0.0375` | |
`= 3.75text(%)` |
$6000 is invested in an account that earns simple interest at the rate of 3.5% per annum.
The total interest earned in the first four years is
`D`
`P = 6000,\ \ r = 3.5text(%),\ \ n = 4`
`I` | `= Prn` |
`= 6000 xx 3.5/100 xx 4` | |
`= 840` |
`=> D`
Pamela deposits $2000 into a savings account which earns simple interest at the rate of 2.5% per annum.
No deposits or withdrawals are made from this account.
After 2 years, how much is in Pamela's savings account? (2 marks)
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`$2100`
`text(Interest earned)` | `= Prn` |
`= 2000 xx 2.5/100 xx 2` | |
`= $100` |
`:.\ text{Account balance}` | `= 2000 + 100` |
`= $2100` |
Dante deposits $5000 into a savings account which earns simple interest.
No deposits or withdrawals are made from this account.
After 4 years, Dante notices there is $5600 in the account.
What is the annual rate of interest for the account? (2 marks)
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`3text(%)`
`text(Interest per year)` | `= (5600-5000)÷4` |
`= 600 ÷ 4` | |
`= $150` |
`:.\ text{Interest rate (p.a.)}` | `= 150/5000 xx 100` |
`= 3 text(%)` |
Mr. Soros put $500 into a simple interest account for a year.
He did not take any money out or add any money to the account.
At the end of the year he had $530 in the account.
What was the annual percentage interest rate? (2 marks)
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`text(6%)`
`text(Interest earned)` | `= 530-500` |
`= $30` |
`:.\ text{Interest rate (annual)}`
`= 30/500`
`= 0.06`
`= 6text(%)`
Cassie opens a savings account and deposits $900 into it.
She makes no more deposits and earns simple interest on her original deposit at 3.5% each year.
How much interest will Cassie earn after 4 years? (2 marks)
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`$126`
`text(Interest earned)` | `=Prn` |
`= 900 xx 3.5/100 xx 4` | |
`= $126` |
On a weekend, Abbey works 8 hours at her normal pay rate and 12 hours at time and a half of her normal pay.
Abbey was paid $707.20 in total for this work.
What is her normal pay rate per hour? (2 marks)
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`$27.20\ text(per hour)`
`text(Total normal hours)`
`= 8 + 1.5 xx 12`
`= 26`
`:.\ text(Abbey’s normal pay rate)`
`= 707.20/26`
`= $ 27.20\ text(per hour.)`
John, Olivia and Louis are picking grapes to earn money.
Their pay is based on the number of tonnes of grapes they pick.
Tonnes picked | |
John | 4 |
Olivia | 3 |
Louis | 1 |
Their total pay is $640.
How much does Olivia earn? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`$240`
`text(Payment per tonne)`
`= 640/8`
`= $80`
`:.\ text(Olivia earns)` | `= 3 xx 80` |
`= $240` |
David has a job selling mobile phone plans. His weekly salary, `W` dollars, is calculated using the rule below:
`W = 300 + 0.05 P`
where `P` is the total value in dollars of the mobile phone plans he sells that week.
David sold $32 000 worth of mobile phone plans in a given week.
What was David's salary in the week?
`C`
`W` | `= 300 + 0.05 xx 32\ 000` |
`= 300 + 1600` | |
`= 1900` |
`=>C`
Bella is an electrician.
She charges $150 to attend a job and then a fixed price for each minute she spends on the job.
The graph below shows Bella's charge based on the number of minutes she spends at the job.
How much will Bella charge for a job that takes 60 minutes? (2 marks)
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`$330`
`text(Charge to attend = $150)`
`text(From the graph,)`
`=>\ text(50 minute job costs $300)`
`:.\ text(C)text(ost per minute)` | `= (300-150)/50 = $3` |
`:.\ text(C)text(ost of 60 minute job)`
`= 150 + 60 xx 3`
`= $330`
Olivia earned $17.24 per hour working at a pizza store.
This week she worked for `8 1/4` hours.
She used the money she earned this week to buy concert tickets for herself and her friends.
Each concert ticket cost $14.15.
Calculate the maximum number of concert tickets Olivia could have bought? (2 marks)
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`10`
`text(Money earned)` | `= 8 1/4 xx 17.24` |
`= $142.23` |
`:.\ text(Maximum tickets)` | `= 142.23/14.15` |
`= 10.05…` | |
`= 10` |
The container shown is initially full of water.
Water leaks out of the bottom of the container at a constant rate.
Which graph best shows the depth of water in the container as time varies?
A. | B. | ||
C. | D. |
`D`
`text(Depth will decrease slowly at first and accelerate.)`
`=> D`
Given the formula `C = (A(y + 1))/24`, calculate the value of `y` when `C = 120` and `A = 500`. (3 marks)
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`4.76`
`text(Make)\ \ y\ \ text(the subject:)`
`C` | `= (A(y + 1))/24` |
`24C` | `= A(y + 1)` |
`y + 1` | `= (24C)/A` |
`y` | `= (24C)/A-1` |
`= (24 xx 120)/500-1` | |
`= 4.76` |
A teacher surveyed the students in her Year 8 class to investigate the relationship between the average number of hours of phone use per day and the average number of hours of sleep per day.
The results are shown on the scatterplot below.
\begin{array} {|l|c|c|}
\hline
& \textit{Average hours of} & \textit{Average hours of} \\ & \textit{phone use per day} & \textit{sleep per day} \\
\hline
\rule{0pt}{2.5ex} \text{Alinta} \rule[-1ex]{0pt}{0pt} & 4 & 8 \\
\hline
\rule{0pt}{2.5ex} \text{Birrani} \rule[-1ex]{0pt}{0pt} & 0 & 10.5 \\
\hline
\end{array}
--- 2 WORK AREA LINES (style=lined) ---
a. \(\text{New data points are marks with a × on the diagram below.}\)
b. \(\text{9 hours (see LOBF in diagram above)}\)
`C`
`text{Scale factor}\ =3/2 =1.5`
`:.\ x = 1.5 xx 12 = 18`
`text{Alternate solution}`
`text{Using sides of similar figures in the same ratio:}`
`x/12` | `=3/2` | |
`x` | `=12 xx (3/2)` | |
`x` | `=18` |
`=> C`
Let `P(x)` be a polynomial of degree 5. When `P(x)` is divided by the polynomial `Q(x)`, the remainder is `2x+5`.
Which of the following is true about the degree of `Q`?
`D`
`text{Given}\ \ P(x)\ \ text{has degree 5}`
`P(x) -: Q(x)\ \ text{has remainder}\ \ 2x+5`
`text{Consider examples to resolve possibilities:}`
`text{eg.}\ \ x^5+2x+5 -: x^3 = x^2+\ text{remainder}\ 2x+5`
`:.\ text{Degree must be 2 is incorrect}`
`Q(x)\ \ text{can have a degree of 2, 3 or 4}`
`=>D`
A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
--- 2 WORK AREA LINES (style=lined) ---
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & & & \\
\hline \end{array}
--- 0 WORK AREA LINES (style=lined) ---
a. `M=180/T`
b.
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}
a. | `M` | `prop 1/T` |
`M` | `=k/T` | |
`12` | `=k/15` | |
`k` | `=15 xx 12` | |
`=180` |
`:.M=180/T`
b.
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}
A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
--- 4 WORK AREA LINES (style=lined) ---
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}
--- 0 WORK AREA LINES (style=lined) ---
a. | `M` | `prop 1/T` |
`M` | `=k/T` | |
`12` | `=k/15` | |
`k` | `=15 xx 12` | |
`=180` |
`:.M=180/T`
b.
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ 30\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}
a. | `M` | `prop 1/T` |
`M` | `=k/T` | |
`12` | `=k/15` | |
`k` | `=15 xx 12` | |
`=180` |
`:.M=180/T`
b.
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ 30\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}
Which of the following could be the graph of `y= -2 x+2`?
`A`
`text{By elimination:}`
`y text{-intercept = 2 → Eliminate}\ B and C`
`text{Gradient is negative → Eliminate}\ D`
`=>A`
A composite solid is shown. The top section is a cylinder with a height of 3 cm and a diameter of 4 cm. The bottom section is a hemisphere with a diameter of 6 cm. The cylinder is centred on the flat surface of the hemisphere.
Find the total surface area of the composite solid in cm², correct to 1 decimal place. (4 marks)
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`122.5\ text{cm}^2`
`text{S.A. of Cylinder}` | `=pir^2+2pirh` | |
`=pi(2^2)+2pi(2)(3)` | ||
`=16pi\ text{cm}^2` |
`text{S.A. of Hemisphere}` | `=1/2 xx 4pir^2` | |
`=2pi(3^2)` | ||
`=18pi\ text{cm}^2` |
`text{Area of Annulus}` | `=piR^2-pir^2` | |
`=pi(3^2)-pi(2^2)` | ||
`=5pi\ text{cm}^2` |
`text{Total S.A.}` | `=16pi+18pi+5pi` | |
`=39pi` | ||
`=122.522…` | ||
`=122.5\ text{cm}^2\ \ text{(to 1 d.p.)}` |
A dam is in the shape of a triangular prism which is 50 m long, as shown.
Both ends of the dam, `A B C` and `D E F`, are isosceles triangles with equal sides of length 25 metres. The included angles `B A C` and `E D F` are each `150^@`.
Calculate the number of litres of water the dam will hold when full. (4 marks)
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`7\ 812\ 500\ text{L}`
`V=Ah`
`text{Use sine rule to find}\ A:`
`A` | `=1/2 ab\ sinC` | |
`=1/2 xx 25 xx 25 xx sin150^@` | ||
`=156.25\ text{m}^2` |
`:.V` | `=156.25 xx 50` | |
`=7812.5\ text{m}^3` |
`text{S}text{ince 1 m³ = 1000 litres:}`
`text{Dam capacity}` | `=7812.5 xx 1000` | |
`=7\ 812\ 500\ text{L}` |
The diagram shows two right-angled triangles, `ABC` and `ABD`,
where `AC=35 \ text{cm},BD=93 \ text{cm}, /_ACB=41^(@)` and `/_ADB=theta`.
Calculate the size of angle `theta`, to the nearest minute. (4 marks)
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`19^@6^{′}`
`text{In}\ Delta ABC:`
`cos 41^@` | `=35/(BC)` | |
`BC` | `=35/(cos 41^@)` | |
`=46.375…` |
`angle BCD = 180-41=139^@`
`text{Using sine rule in}\ Delta BCD:`
`sin theta/(46.375)` | `=sin139^@/93` | |
`sin theta` | `=(sin 139^@ xx 46.375)/93` | |
`:.theta` | `=sin^(-1)((sin 139^@ xx 46.375)/93)` | |
`=19.09…` | ||
`=19^@6^{′}\ \ text{(nearest minute)}` |
The formula `C=100 n+b` is used to calculate the cost of producing laptops, where `C` is the cost in dollars, `n` is the number of laptops produced and `b` is the fixed cost in dollars.
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a. `text{Find}\ \ C\ \ text{given}\ \ n=1943 and b=20\ 180`
`C` | `=100 xx 1943 + 20\ 180` | |
`=$214\ 480` |
b. `text{Find}\ \ n\ \ text{given}\ \ C=97\ 040 and a=26`
`C` | `=100 n+a n+20\ 180` | |
`97\ 040` | `=100n + 26n +20\ 180` | |
`126n` | `=76\ 860` | |
`n` | `=(76\ 860)/126` | |
`=610 \ text{laptops}` |
Which of the following correctly expresses `x` as the subject of `y=(ax-b)/(2)` ?
`A`
`y` | `=(ax-b)/(2)` | |
`2y` | `=ax-b` | |
`ax` | `=2y+b` | |
`:.x` | `=(2y+b)/a` |
`=>A`