Question1
(a) The table shows the areas, in km2, of the four largest rainforests in the world.
(i) Find the area of the Valdivian rainforest as a percentage of the area of the Amazon rainforest.
(ii) Write, in its simplest form, the ratio of the areas of the rainforests $\text{Valdivian : Atlantic : Congo.}$
$( \textbf{iii})$ The Amazon rainforest has $60\%$ of its area in Brazil and $10\%$ of its area in Colombia
$43\frac{1}{3}\%$ of the remaining area of the rainforest is in Peru.
Find the percentage of the Amazon rainforest that is in Brazil, Colombia and Peru.
Calculate the total area of rainforest in the world
Give your answer correct to the nearest $100000km^2$.
$( \mathbf{v} )$ In the world, $60.7$ hectares of rainforest are lost every minute
Calculate the total area, in hectares, of rainforest that is lost in $365$ days
Give your answer in standard form.
$\mathbf{b} )$ The Amazon river has a length of 6440km, correct to the nearest 10km
The Congo river has a length of 4400km, correct to the nearest 100 km
Calculate the upper bound of the difference between the lengths of the Amazon river and the
Congo river.
▶️Answer/Explanation
(a)(i): $4.55$ or $4.545…$
(a)(ii): $50:263:400$
(a)(iii): $83$
(a)(iv): $10\,200\,000$
(a)(v): $3.19 \times 10^7$ or $3.190… \times 10^7$
(b): $2095$
Question2
On the grid, draw the image of
(i) triangle $T$ after a reflection in the x-axis
(ii) triangle $T$ after a translation by the vector $\binom{-5}{-2}$
$( \textbf{iii})$ triangle $T$ after an enlargement by scale factor $-\frac12$ with centre (-1,1).
(b) A shape $P$ is enlarged by scale factor 3 to give shape $Q$
Shape $Q$ is then enlarged by scale factor $\frac{2}{5}$ to give shape $R.$
The area of shape $P$ is 10 cm$^{2}.$
Calculate the area of shape $R.$
▶️Answer/Explanation
(a)(i): Triangle at $(2, 1)$, $(1, 3)$, $(5, 3)$
(a)(ii): Triangle at $(-4, -5)$, $(-3, -3)$, $(0, -5)$
(a)(iii): Triangle at $(-2.5, 2)$, $(-4, 3)$, $(-2, 3)$
(b): $14.4$
Question3
$(\mathbf{a})$
$C=\frac{1}{4}xy^2$
$( \mathbf{i} )$ Find $C$ when $x=5$ and $y=8.$
$( \textbf{ii) }$ Find the positive value of $y$ when $C=15$ and $x=2.4.$
$( \mathbf{b} )$ Write as a single fraction in its simplest form.
$\frac4{x-1}-\frac3{2x+5}$
$( \mathfrak{c} )$ Expand and simplify
$\begin{pmatrix}2x+3\end{pmatrix}\begin{pmatrix}4-x\end{pmatrix}^2$
\((\mathbf{d})\quad\mathrm{Simplify.}\left(\frac{y^8}{16x^{16}}\right)^{-\frac{3}{4}}\)
▶️Answer/Explanation
(a)(i): $80$
(a)(ii): $5$
(b): $\frac{5x+23}{(x-1)(2x+5)}$ or $\frac{5x+23}{2x^2+3x-5}$
(c): $2x^3 – 13x^2 + 8x + 48$
(d): $\frac{8x^{12}}{y^6}$ or $8x^{12}y^{-6}$
Question4
(a) Jianyu records the time, in seconds, that some cars take to travel $195$m.
The box and whisker plot shows this information.
(i) Find the median time.
(ii) Find the interquartile range.
(iii) Find the difference between the average speed of the fastest car and the average speed of the slowest car.
Give your answer in kilometres per hour.
(b) Matilda records the distances that $80$ different cars can travel with a full tank of fuel.The table shows this information.
(i) Write down the class interval that contains the median.
(ii) Calculate an estimate of the mean.
(iii) A histogram is drawn to show the information in the table. The height of the bar for the interval $250 \leqslant d \leqslant 300$ is $2.8$ cm.
Calculate the height of the bar for each of the following intervals.
(iv) Two of the $80$ cars are chosen at random. Find the probability that, with a full tank of fuel, one of the cars can travel more than $450$km and the other car can travel not more than $300$ km.
▶️Answer/Explanation
(a)(i): $9.3$
(a)(ii): $3.4$
(a)(iii): $63$
(b)(i): $420 < d < 450$
(b)(ii): $411.25$
(b)(iii): $2.6$, $19$, $14$
(b)(iv): $\frac{7}{158}$
Question5
$\begin{array}{ll}{(\mathbf{a})}&{P\text{ is the point }(1,7).}\\&{Q\text{ is the point }(5,-5).}\end{array}$
$( \mathbf{i} )$ Find$\overrightarrow{P Q}$ .
$( \mathbf{ii})$ Show that$\left|\overrightarrow {OP}\right|=\left|\overrightarrow{OQ}\right|.$
(iii) $PQ$ is a chord of a circle with centre $O.$
Calculate the circumference of this circle
(iv) $PQ$ is the diameter of a different circle with centre $R.$
Find the coordinates of $R$.
(v) Find the equation of the perpendicular bisector of $PQ$.
Give your answer in the form $y = mx+c.$
$\mathbf{( b) }$ The position vector of$A$ is a.
The position vector of $B$ is b.
$M$ is a point on $AB$ such that $AM:MB=2:3.$
Find, in terms of a and b, the position vector of $M.$
Give your answer in its simplest form.
▶️Answer/Explanation
(a)(i): $\begin{pmatrix} 4 \\ -12 \end{pmatrix}$
(a)(ii): $1^2 + 7^2$, $5^2 + (-15)^2$. Both $\sqrt{50}$
(a)(iii): $44.4$ or $44.42[8…]$ to $44.43$
(a)(iv): $(3, 1)$
(a)(v): $y = \frac{1}{3}x$
(b): $\frac{3}{5}\mathbf{a} + \frac{2}{5}\mathbf{b}$
Question6
\(\begin{aligned}&\text{The diagram shows the positions of two lightouses }A\mathrm{~and~}B,\text{a boat }C\text{ and a harbour }H.\\&C\text{ is due east of }B.\\&(\mathbf{a})\quad\text{Find the bearing of the harbour from boat }C.\\&(\mathbf{b})\quad(\mathbf{i})\quad\text{Show that angle }CBH=100°.\\&(\mathbf{i}\mathbf{i})\quad\text{Show that }BH=13.7\text{ km, correct to l decimal place.}\end{aligned}\)
(c) Calculate the bearing of $A$ from $B.$
$(\mathbf{d})$ At 1 pm boat $C$ sails 32 km directly to the harbour at a speed of 10 knots
$\begin{array}{ll}{(\mathbf{i})}&{\text{Calculate the time when boat }C\text{ arrives at the harbour.}}\\&{\text{Give this time correct to the nearest minute.}}\end{array}$
$1~ knot= 1.852 ~km/ h$
$\begin{array}{ll}(\mathbf{i})&\text{Calculate the distance of boat }C\text{ to the harbour when boat }C\text{ is at the shortest distance from}\\&\text{lighthouse }B.\end{array}$
▶️Answer/Explanation
(a): $245$
(b)(i): $180 – (55 + 25) = 100$
(b)(ii): $\frac{32 \times \sin 25}{\sin 100} = 13.73…$
(c): $258$ or $257.9$ to $258.0…$
(d)(i): $2:44$ pm or $14:44$
(d)(ii): $7.857$ to $7.88$
Question7
(a)
The diagram shows a box in the shape of a cuboid.
The box is open at the top.
(i) Work out the surface area of the inside of the open box.
(ii) Cylinders with height 20cm and diameter $15$cm are placed in the box.
Work out the maximum number of these cylinders that can completely fit inside the box.
$(\mathbf{b})$ A solid bronze cone has a mass 750g
The density of the bronze is 8.9g/cm$^3.$
The ratio $\textbf{radius of cone: height of cone=1:3}$.
$( \mathbf{i} )$ Show that the radius of the cone is $2.99$cm, correct to $3$ significant figures.
[Density=mass$\div$volume]
$[$The volume, $V$,of a cone with radius $r$ and height $h$ is $V=\frac13\pi r^2h.]$
(ii) Calculate the total surface area of the cone.
[The curved surface area, $A$, of a cone with radius $r$ and slant height $l$ is $A=\pi rl.]$
▶️Answer/Explanation
(a)(i): $10\,100$
(a)(ii): $16$
(b)(i): $\frac{1}{3}\pi r^2 \times 3r = \text{their}(750 \div 8.9)$, $r^3 = \frac{\text{their}(750 \div 8.9)}{\pi}$, $r = 2.993…$
(b)(ii): $117$ or $116.9$ to $117.2$
Question8
(a) On the axes, sketch the graph of $y=x^2+7x-18$
On your sketch, write the values where the graph meets the x-axis and the y-axis.
$( \mathbf{b} )$ $( \mathbf{i} )$ Find the derivative of $y=x^2-3x-28.$
$( \textbf{ii})$ Find the coordinates of the turning point of $y= x^2- 3x- 28.$
$(\mathbf{c})$ The line $y=5-2x$ intersects the graph of $y=x^2-3x-28$ at point $P$ and point $Q$
Find the coordinates of $P$ and $Q.$
You must show all your working and give your answers correct to 2 decimal places.
▶️Answer/Explanation
(a): Correct sketch with roots indicated at $x = -9$ and $x = 2$ and y intercept = $-18$. Minimum should be in 3rd quadrant.
(b)(i): $2x – 3$
(b)(ii): $(1.5, -30.25)$
(c): $x^2 – x – 33 = 0$ seen. $\frac{[-]1 \pm \sqrt{([-]1)^2 – 4(1)(-33)}}{2 \times 1}$. $-5.27$ or $-5.267$ to $-5.266$ and $6.27$ or $6.266$ to $6.267$. $(-5.27, 15.53 \text{ or } 15.54)$ and $(6.27, -7.53 \text{ or } -7.54)$
Question9
$$\mathrm{f}(x)=4x+1\quad\mathrm{g}(x)=6-2x\quad\mathrm{h}(x)=3^{x-2}$$
$( \mathbf{a} )$ Find
$( \mathbf{i} )$ f(3)
$( \mathbf{ii})$ gf(3).
$( \mathbf{b} )$ Find g$^-1(x).$
$( \mathbf{c} )$ Find $x$ when f$(x)=$g$(2x-7).$
$( \mathbf{d} )$ Find the value of hh(2)
$( \mathbf{e} )$ Find $x\text{when h}^-1(x)=10.$
▶️Answer/Explanation
(a)(i): $13$
(a)(ii): $-20$
(b): $\frac{6-x}{2}$
(c): $2.375$
(d): $\frac{1}{3}$ or $0.333…$
(e): $6561$