Points `P` and `Q`, shown on the Cartesian plane diagram, are rotated 180° about the origin and become points `P^(′)` and `Q^(′)`.
Plot the points `P^(′)` and `Q^(′)` on the diagram. (3 marks)
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Points `P` and `Q`, shown on the Cartesian plane diagram, are rotated 180° about the origin and become points `P^(′)` and `Q^(′)`.
Plot the points `P^(′)` and `Q^(′)` on the diagram. (3 marks)
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`P^(′)(2,1)`
`Q^(′)(-4,2)`
Point `Q(3,1)` on the Cartesian plane is rotated 180° about the origin in a clockwise direction to become point `Q^(′)`.
What are the coordinates of `Q^(′)`. (2 marks)
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`Q^(′)(-3,-1)`
Gabby put 5 points on a grid and labelled them `A` to `E`, as shown on the diagram below.
Point `A` is 35 millimetres from point `D.`
Gabby adds a sixth point, `F` so that the arrangement of points has one line of symmetry.
How far is point `F` from point `B?` (3 marks)
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`42\ text{mm}`
The trapezium `ABCD` is moved to the new position shown by trapezium `SRQP.`
Which of these transformations resulted in the new position?
`C`
`text(Reflection in the)\ xtext(-axis:)`
`ABCD -> A^{prime}B^{prime}C^{prime}D^{prime}`
`text(Translate 8 units left:)`
`A^{prime}B^{prime}C^{prime}D^{prime} -> SRQP`
`=>C`
Rochelle drew a pattern which is pictured below.
Rochelle rotates the pattern.
How much does Rochelle to turn the pattern until it looks exactly the same?
`B`
`text(Outer pattern looks the same every)\ \ 1/8\ \ text(turn).`
`text(Inner cross pattern looks the same every)\ \ 1/4\ \ text(turn).`
`:.\ text(Whole pattern looks the same every)\ \ 1/4\ \ text(turn).`
`=>B`
This shape will be translated 4 units to the right and 2 units up.
Where will the image of the point `A` be located after the shape is translated? (2 marks)
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`(1,−3)`
`text(Current position)\ A(−3,−5)`
`text(Translate 4 units to right:)`
`A(−3 + 4,−5) \ -> \ (1,−5)`
`text(Translate 2 units up:)`
`(1,−5+2 ) \ -> \ (1,−3)`
`:. text(Image of)\ A\ text(is)\ (1,−3)`
The point `P(-1, -4)` lies on the Cartesian plane. It is reflected in the `x`-axis to form the point `P^(′)`.
Find the coordinates of `P^(′)`. (1 mark)
`(-1,4)`
`text(Reflections in the)\ xtext{-axis:}`
`ytext{-coordinate has opposite sign and}\ xtext{-coordinate is the same.}`
`P(-1,-4)\ ->\ P^(′)(-1,4)`
The point `P(-3, 7)` lies on the Cartesian plane. It is reflected in the `y`-axis to form the point `P^(′)`.
Find the coordinates of `P^(′)`. (1 mark)
`(7,5)`
`text(Reflections in the)\ ytext{-axis:}`
`xtext{-coordinate has opposite sign and}\ ytext{-coordinate is the same.}`
`P(-3,7)\ ->\ P^(′)(3,7)`
`P(2,3)` is translated 3 units up and 4 units left.
The new point is then reflected in `x`-axis to form point `P^(′)`.
Find the coordinates of `P^(′)`. (2 marks)
`(-2,-6)`
`text{1st transformation:}`
`(2,3)\ ->\ (2-4, 3+3)\ ->\ (-2,6)`
`text{2nd transformation (reflection):}`
`(-2,6)\ ->\ P^(′)(-2,-6)`
`P(-3,-5)` is reflected in the `x`-axis and then translated 3 units to the right to form point `P^(′)`.
Find the coordinates of `P^(′)`. (2 marks)
`(0,5)`
`text{1st transformation (reflection):}`
`P(-3,-5)\ ->\ (-3,5)`
`text{2nd transformation:}`
`(-3,5)\ ->\ (0,5)`
The point `A(-2, 5)` lies on the Cartesian plane. It is translated five units left and then reflected in the `y`-axis.
Find the coordinates of the final image of `A`. (2 marks)
`(7,5)`
`text(1st transformation:)`
`A(-2, 5)\ ->\ (-7,5)`
`text{2nd transformation (reflection):}`
`(-7,5)\ ->\ (7,5)`
The point `P(4, -3)` lies on the Cartesian plane. It is translated four units vertically up and then reflected in the `y`-axis.
Find the coordinates of the final image of `P`. (2 marks)
`(-4,1)`
`text(1st transformation:)`
`P(4,-3)\ ->\ (4,1)`
`text{2nd transformation (reflection):}`
`(4,1)\ ->\ (-4,1)`
Fiona and John are planning to hold a fund-raising event for cancer research. They can hire a function room for $650 and a band for $850. Drinks will cost them $25 per person.
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i. | `text(Fixed C) text(osts)` | `= 650 + 850` |
`= $1500` |
`text(Variable C) text(osts) = $25x`
`:.\ $C = 1500 + 25x`
ii. | `text(From the graph)` |
`text(C) text(osts = Income when)\ x = 60` | |
`text{(i.e. where graphs intersect)}` |
iii. `text(When)\ \ x = 80:`
`text(Income)` | `= 80 xx 50` | |
`= $4000` |
`$C` | `= 1500 + 25 xx 80` |
`= $3500` |
`:.\ text(Profit)` | `= 4000-3500` |
`= $500` |
The average height, `C`, in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with equation
`C = 6A + 79`
where `A` is the age in years. For this line, the gradient is 6.
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i. `text(It indicates that 6-11 year old girls, on average, grow)`
`text(6 cm per year.)`
ii. `text(Girls eventually stop growing, and the equation doesn’t)`
`text(factor this in.)`
Renee went bike riding on a holiday.
The hiring charges are listed in the table below:
\begin{array} {|l|c|c|}
\hline \text{Hours hired} \ (h) & 1 & 2 & 3 & 4 & 5 \\
\hline \text{Cost} \ (C) & 18 & 24 & 30 & 36 & 42 \\
\hline \end{array}
Which linear equation shows the relationship between `C` and `h`?
`A`
`text(Consider Option 1:)`
`12 + (6 xx 1) = 12+6=18`
`12 + (6 xx 2) = 12+12=24`
`12 + (6 xx 3) = 12+18=30\ \ \ \ text(etc …)`
`:.\ text(The linear equation is:)\ \ C = 12 + 6h`
`=>A`
The prices at an ice cream shop can be seen below.
Each extra scoop of ice cream costs the same amount of money.
How much will an ice cream with 5 scoops cost? (2 marks)
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`$8.80`
`text{Cost of 1 extra scoop}`
`= 6.25-5.40`
`= 0.85`
`:.\ text{Cost of an icecream with 5 scoops}`
`= $5.40 + 4\ text(extra scoops)`
`= 5.40 + (4 xx 0.85)`
`= 5.40 + 3.40`
`= $8.80`
This chart shows the longest run, in kilometres, that Deek ran each week over 5 weeks.
\begin{array} {|l|c|c|c|c|c|}
\hline \textbf{Week} & 1 & 2 & 3 & 4 & 5\\
\hline \textbf{Longest run (km)} & 8 & 11 & 14 & 17 & 20\\
\hline \end{array}
If the pattern continues, in which week is Deek's longest run 29 km?
`B`
`text(Deek’s longest run increases by 3 km each week.)`
`text(In week 6: Longest run = 23 km)`
`text(In week 7: Longest run = 26 km)`
`text(In week 8: Longest run = 29 km)`
`=>B`
At an apple orchard, apples are picked and put in a basket.
The table below shows the total number of apples in the basket after each minute.
\begin{array} {|c|c|c|}
\hline \textbf{Minutes} & \textbf{Total number of apples} \\
\hline 1 & 4 \\
\hline 2 & 8 \\
\hline 3 & 12 \\
\hline 4 & 16 \\
\hline \end{array}
How many apples are in the basket after 10 minutes?
`D`
`text(4 apples are put into the basket each minute.)`
`:.\ text(Apples in basket after 10 minutes)`
`=4 xx 10`
`= 40\ text(apples)`
`=>D`
Jeremy sold ice creams out of his ice cream truck.
He drew the graph below to show how the number of ice creams he sells in a week is related to their price.
Which statement best describes the graph?
`A`
`text(As the ice cream price goes up, the number)`
`text(sold goes down.)`
`=>A`
An equilateral triangle has vertices `O(0,0)` and `A(8,0)` as shown in the diagram below.
Find `k` if the coordinates of the third vertex are `B(4,k)`. (4 marks)
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`text{Proof (See worked solutions)}`
`ΔOAB\ text{is equilateral}\ \ =>\ \ OA=AB=OB=8`
`text{Let}\ C=(0,4)`
`text{Consider}\ ΔOCB:`
`OB^2` | `=OC^2+CB^2` | |
`64` | `=16+CB^2` | |
`CB^2` | `=48` | |
`CB` | `=sqrt(48)` | |
`=4sqrt(2)` |
`B=(4,4sqrt(2))`
`:.k=4sqrt(2)`
Prove the points `(1,-1), (-1,1)` and `(-sqrt3,-sqrt3)` are the vertices of a equilateral triangle. (4 marks)
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`text{Proof (See worked solutions)}`
`text{Let points be:}\ A(1,-1), B(-1,1) and C(-sqrt3,-sqrt3)`
`text(Using the distance formula):`
`d_(AB)` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`=sqrt{(1-(-1))^2+(-1-1)^2}` | ||
`=sqrt8` |
`d_(BC)` | `=sqrt{(-1-(-sqrt3))^2+(1-(-sqrt3))^2}` | |
`=sqrt{(-1+sqrt3)^2+(1+sqrt3)^2}` | ||
`=sqrt(1-2sqrt3+3 +1+2sqrt3+3)` | ||
`=sqrt8` |
`d_(AC)` | `=sqrt{(1-(-sqrt3))^2+(-1-(-sqrt3))^2}` | |
`=sqrt{(1+sqrt3)^2+(-1+sqrt3)^2}` | ||
`=sqrt(1+2sqrt3+3 +1-2sqrt3+3)` | ||
`=sqrt8` |
`text{Since}\ AB=BC=AC`
`:. ΔABC\ text{is equilateral.}`
A straight line passes through points `Q(3,-2)` and `R(-1,4)` .
Find the equation of `QR` and express in general form. (3 marks)
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`2y+3x-5=0`
`text{Line goes through}\ (3,-2) and (-1,4).`
`text(Using the gradient formula):`
`m` | `=(y_2-y_1)/(x_2-x_1)` | |
`=(-2-4)/(3-(-1))` | ||
`=-3/2` |
`text{Find equation through}\ (3,-2), m=-3/2:`
`y-y_1` | `=m(x-x_1)` | |
`y-(-2)` | `=-3/2(x-3)` | |
`2(y+2)` | `=-3(x-3)` | |
`2y+4` | `=-3x+9` | |
`2y+3x-5` | `=0` |
A straight line passes through points `A(-2,-2)` and `B(1,5)` .
Find the equation of `AB` and express in form `y=mx+b`. (3 marks)
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`y=7/3x+8/3`
`text{Line goes through}\ (-2,-2) and (1,5).`
`text(Using the gradient formula):`
`m` | `=(y_2-y_1)/(x_2-x_1)` | |
`=(5-(-2))/(1-(-2))` | ||
`=7/3` |
`text{Find equation through}\ (1,5), m=7/3:`
`y-y_1` | `=m(x-x_1)` | |
`y-5` | `=7/3(x-1)` | |
`y-5` | `=7/3x-7/3` | |
`y` | `=7/3x+8/3` |
Albert drew a straight line through points `P` and `Q` as shown on the graph below.
Find the equation of Albert's line and express in general form. (3 marks)
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`3y-5x+2=0`
`text{Line goes through}\ (-2,-4) and (1,1).`
`text(Using the gradient formula):`
`m` | `=(y_2-y_1)/(x_2-x_1)` | |
`=(1-(-4))/(1-(-2))` | ||
`=5/3` |
`text{Find equation through}\ (1,1), m=5/3:`
`y-y_1` | `=m(x-x_1)` | |
`y-1` | `=5/3(x-1)` | |
`3y-3` | `=5x-5` | |
`3y-5x+2` | `=0` |
Calculate the value(s) of `p` given that the points `(p,3)` and `(1,p)` are exactly 10 units apart. (3 marks)
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`p=9\ text{or}\ -5`
`(p,3),\ \ (1,p)`
`text{Using the distance formula:}`
`d` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`10` | `=sqrt{(p-1)^2+(3-p)^2}` | |
`10` | `=sqrt{p^2-2p+1+9-6p+p^2}` | |
`10` | `=sqrt{2p^2-8p+10}` | |
`100` | `=2p^2-8p+10` | |
`0` | `=2p^2-8p-90` | |
`0` | `=p^2-4p-45` | |
`0` | `=(p-9)(p+5)` |
`:.p=9\ text{or}\ -5`
Calculate the distance between the points `(2,-3)` and `(-5,4)`.
Round your answer to the nearest tenth. (2 marks)
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`9.9\ text{units}`
`(2,-3),\ \ (-5,4)`
`text{Using the distance formula:}`
`d` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`=sqrt{(2-(-5))^2+(-3-4)^2}` | ||
`=sqrt{49+49}` | ||
`=sqrt{98}` | ||
`=9.899…` | ||
`=9.9\ text{units (nearest tenth)}` |
Calculate the distance between the points `(6,-5)` and `(0,3)`. (2 marks)
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`10\ text{units}`
`(6,-5),\ \ (0,3)`
`text{Using the distance formula:}`
`d` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`=sqrt{(6-0)^2+(-5-3)^2}` | ||
`=sqrt{36+64}` | ||
`=sqrt{100}` | ||
`=10\ text{units}` |
Calculate the distance between the point `(-6,2)` and the origin.
Give your answer in exact form. (2 marks)
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`2sqrt{10}\ \ text{units}`
`(-6,2),\ \ (0,0)`
`text{Using the distance formula:}`
`d` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`=sqrt{(-6-0)^2+(2-0)^2}` | ||
`=sqrt{36+4}` | ||
`=sqrt{40}` | ||
`=2sqrt{10}\ \ text{units}` |
The point `C(-2,3)` is the midpoint of the interval `AB`, where `B` has coordinates `(-1,0).`
What are the coordinates of `A`? (3 marks)
`(-3,6)`
`text(Using the midpoint formula):`
`(x_A + x_B)/2` | `= x_C` | `(y_A + y_B)/2` | `= y_C` |
`(x_A-1)/2` | `= -2` | `(y_A + 0)/2` | `= 3` |
`x_A` | `= -3` | `y_A` | `= 6` |
`:. A\ text(has coordinates)\ (-3,6).`
Given `C(-3,-5)` and `D(-5,1)`, find the midpoint of `CD`. (2 marks)
`(-4, -2)`
`C(-3,-5),\ \ \ D(-5,1)`
`M` | `= ( (x_1 + x_2)/2, (y_1 + y_2)/2)` |
`= ( (-3-5)/2, (-5+1)/2)` | |
`= (-4, -2)` |
Find `M`, the midpoint of `PQ`, given `P(2, -1)` and `Q(5, 7)`. (2 marks)
`M(7/2, 3)`
`P(2,-1)\ \ \ Q(5,7)`
`M` | `= ( (x_1 + x_2)/2, (y_1 + y_2)/2)` |
`= ( (2+5)/2, (-1+7)/2)` | |
`= (7/2, 3)` |
On the Cartesian plane below, graph the equation `y-1=-1/2x`.
Clearly label the coordinates of the intercepts with both the `x` and `y`-axes. (2 marks)
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On the Cartesian plane below, graph the equation `y=3x+2`.
Clearly label the coordinates of the intercepts with both the `x` and `y`-axes. (2 marks)
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The graph of `y = 2x-3` will be drawn on this grid.
Which two points will the straight line pass through?
`D`
`text(Solution 1)`
`y = 2x-3\ text(passes through)\ (0, -3)`
`text(with a gradient of 2.)`
`:. A and C`
`text(Solution 2)`
`text{Substitute the coordinates of each point into the equation:}`
`A(-1, -5), \ B(1, -5), \ C(3, 3), \ D(-2, 1)`
`text(Only)\ A and C\ text(satisfy the equation) \ \ y = 2x-3.`
`=>D`
A Cartesian plane is shown.
Select the correct statement below.
`D`
`R\ text(occurs when)`
`x < 0 and y < 0`
`=>D`
Leo drew a straight line through the points (0, 5) and (3, -2) as shown in the diagram below.
What is the gradient of the line that Leo drew?
`-7/3`
`text{Line passes through (0, 5) and (3, – 2)}`
`text(Gradient)` | `= (y_2-y_1)/(x_2-x_1)` |
`= (5-(-2))/(0-3)` | |
`= -7/3` |
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`
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}` |
A plumber charges a call-out fee of $90 as well as $2 per minute while working.
Suppose the plumber works for `t` hours.
Which equation expresses the amount the plumber charges ($`C`) as a function of time (`t` hours)?
`D`
`text(Hourly rate)\ = 60 xx 2=$120`
`:. C = 90 + 120t`
`=>D`
Last Saturday, Luke had 165 followers on social media. Rhys had 537 followers. On average, Luke gains another 3 followers per day and Rhys loses 2 followers per day.
If `x` represents the number of days since last Saturday and `y` represents the number of followers, which pair of equations model this situation?
A. | `text(Luke:)\ \ y = 165x + 3`
`text(Rhys:)\ \ y = 537x - 2` |
B. | `text(Luke:)\ \ y = 165 + 3x`
`text(Rhys:)\ \ y = 537 - 2x` |
C. | `text(Luke:)\ \ y = 3x + 165`
`text(Rhys:)\ \ y = 2x - 537` |
D. | `text(Luke:)\ \ y = 3 + 165x`
`text(Rhys:)\ \ y = 2 - 537x` |
`B`
`text(Luke starts with 165 and adds 3 per day:)`
`y = 165 + 3x`
`text(Rhys starts with 537 and loses 2 per day:)`
`y = 537 – 2x`
`=> B`
What is the `x`-intercept of the line `x + 3y + 6 = 0`?
`A`
`x text(-intercept occurs when)\ y = 0:`
`x + 0 + 6` | `= 0` |
`x` | `= -6` |
`:. x text{-intercept is}\ (-6, 0)`
`=> A`
The point `R(9, 5)` is the midpoint of the interval `PQ`, where `P` has coordinates `(5, 3).`
What are the coordinates of `Q`?
`C`
`text(Using the midpoint formula):`
`(x_Q + x_P)/2` | `= x_R` | `(y_Q + y_P)/2` | `= y_R` |
`(x_Q + 5)/2` | `= 9` | `(y_Q + 3)/2` | `= 5` |
`x_Q` | `= 13` | `y_Q` | `= 7` |
`:. Q\ text(has coordinates)\ (13, 7).`
`=> C`
What is the gradient of the line `2x + 3y + 4 = 0`?
`A`
`2x + 3y + 4` | `= 0` |
`3y` | `= -2x-4` |
`y` | `= -2/3 x-4/3` |
`:.\ text(Gradient)` | `= -2/3` |
`=> A`
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?
\(A\)