Question1
(a) ( i) Alain and Beatrice share $\$750$ in the ratio Alain: Beatrice$= 8: 7$.
Show that Alain receives $\$400$.
(ii) (a) Alain spends $\$150.$
Write $150 as a percentage of $\$400$..
$(\mathbf{b})$ He invests the remaining $\$250$. at a rate of $2\%$ per year simple interest
Calculate the amount Alain has at the end of $5$ years
(iii) Beatrice invests her $\$350$. at a rate of $0.25\%$ per month compound interest
Calculate the amount Beatrice has at the end of $5$ years.
Give your answer correct to the nearest dollar.
(b) Carl, Dina and Eva share $100$ oranges.
$\begin{array}{ll}\text{The’ratio}&\text{Carl’s oranges:Dina’s oranges}=3:5.\\\text{The ratio}&\text{Carl’s oranges:Eva’s oranges}=2:3.\end{array}$
Find the number of oranges Carl receives.
(c) Fred buys a house.
At the end of the first year, the value of the house increases by $5\%$
At the end of the second year, the value of the house increases by $3\%$ of its value at the end of the
first year.
The value of Fred’s house at the end of the second year is $\$60564$
Calculate how much Fred paid for the house.
$(\mathbf{d})$ Gabrielle invests $\$500$ at a rate of $r\%$ per year compound interest
At the end of $8$ years the value of Gabrielle’s investment is $\$609.20$ .
Find the value of $r.$
▶️Answer/Explanation
60
Question2
(a) 100 students take part in a reaction test.
The table shows the results.
(i) Write down the mode.
(ii) Find the median.
(iii) Calculate the mean.
(iv) Two students are chosen at random.
Find the probability that both their reaction times are greater than or equal to $9$ seconds.
(b) The box-and-whisker plot shows the heights, h cm, of some students.
(i) Find the range.
(ii) Find the interquartile range.
(c) The mass of each of 200 potatoes is measured.
The table shows the results.
(i) Calculate an estimate of the mean.
(ii) Complete the histogram to show the information in the table.
▶️Answer/Explanation
(a) 3750 cao
(b) 3800 cao
Question3
\(\begin{aligned}&\text{The diagam shows a cylinder contanting water.}\\&\text{There is a solid metal sphere touching the base of the cylinder.}\\&\text{Halif of the sphere is in the water.}\\&\text{The radus of the cylinder is 12 cm and the radius of the sphere is 3 cm.}\\&\mathrm{(a)}\\&\text{Show that }h=0.125\\&[\text{The volume, }V,\text{ of a sphere with radius }r\mathrm{~is~}V=\frac{4}{3}\pi r^3.]\end{aligned}\)
$\mathbf{(b)}$ The water in the cylinder is poured into another cylinder of radius $R$ cm
The depth of the water in this cylinder is 18cm.
Calculate the value of $R.$
(c) The sphere is melted down and some of the metal is used to make $30$ cubes with edge length $1.5$cm.
Calculate the percentage of metal not used
[The volume, $V$, of a sphere with radius $r$ is $V=\frac43\pi r^{3}.]$
▶️Answer/Explanation
4 1.4[0]
Question4
(a) 
(i) Enlarge triangle $T$ by scale factor $3,$ centre $(0,2)$.
$( \mathbf{ii} )$ $( \mathbf{a} )$ Rotate triangle $T$ about (4,2) by $90^{\circ}$ clockwise.
Label the image $P.$
(b) Reflect triangle $T$ in the line $x+y=6.$
Label the image $Q$
(c) Describe fully the single transformation that maps triangle $P$ onto triangle $Q$
(b)
The diagram shows triangle $OHK$, where $O$ is the origin
The position vector of $H$ is a and the position vector of $K$ is b.
$Z$ is the point on $HK$ such that $HZ:ZK=2:5.$
Find the position vector of $Z$, in terms of a and b
Give your answer in its simplest form.
▶️Answer/Explanation
Parallelogram
Question5
$( \mathbf{a} )$ Expand and simplify
$(2p^{2}-3)(3p^{2}-2)$
$( \mathbf{b} )$ $s= \frac 12( u+ \nu ) t$
$( \mathbf{i} )$ Find the value of s when $u=20,\nu=30$ and $t=7$
$( \textbf{ii})$ Rearrange the formula to write $\nu$ in terms of $s,u$ and $t$
(c) Factorise completely.
$( \mathbf{i} )$ $2qt- 3t- 6+ 4q$
(ii) $x^3- 25x$
▶️Answer/Explanation
20
Question6
$A$ is the point (0,4) and $B$ is the point (8,0).
The line $L_1$ is parallel to the x-axis.
The line $L_2^1$ passes through $A$ and $B.$
(a) Write down the equation of $L_1.$
$\mathbf{( b) }$ Find the equation of$L_2.$
Give your answer in the form $y=mx+c.$
$( \mathbf{c} )$ $C$ is the point (2,3).The line $L_{3}$ passes through $C$ and is perpendicular to $L$,
$\mathbf{( i) }$ Show that the equation of $L_3$ is $y=2x-1.$
$( \mathbf{ii} )$ $L_{3}$crosses the x-axis at $D$
Find the length of $CD$
▶️Answer/Explanation
06 15 or 6:15am
Question7
$\mathscr{E}=\{$students in a class$\}$ $P= \{$students who study Physics$\}$ $C= \{$students who study Chemistry)
n$( \mathscr{E} )= 24$ n$( P) = 17$ n$( C) = 14$ n$(P\cap C)=9$
$( \mathbf{a} )$ Complete the Venn diagram.
$( \mathbf{b} )$ $( \mathbf{i} )$ Find n$(P\cap C^{\prime}).$
$( \mathbf{ii})$ Find n$(P\cup C^{\prime}).$
$(\mathbf{c})$ Two students are picked from the class at random
Find the probability that one student studies both subjects and one student studies Chemistry but not Physics.
$(\mathbf{d})$ Two of the students who study Physics are picked at random
Find the probability that they both study Chemistry
▶️Answer/Explanation
$7 \% \quad \frac{15}{213} \quad 0.071 \quad 0.7$
Question8
(a)
Calculate the area of the triangle
$(b)$
$AB=(2x+3)$cm and $h=(x+5)$cm The area of triangle $ABC=50\mathrm{cm}^{2}.$
Find the value of $x$, giving your answer correct to 2 decimal places.
You must show all your working.
▶️Answer/Explanation
$\frac{2}{7}$
Question9
$$\mathrm{f}(x)=x^3-3x^2-4$$
$(\mathbf{a})$ Find the gradient of the graph of $y=$f( x) where x= 1.
$\mathbf{(b)}$ Find the coordinates of the turning points of the graph of $y=$f( x) .
$\mathbf{(c)}$ Sketch the graph of $y = f( x)$.
▶️Answer/Explanation
$2a – 11b$
Question10
The diagram shows a quadrilateral $ABCD.$
$AC=12.3$cm and $AD=16.5$ cm
Angle $BAC=31°$, angle $ABC=90°$ and angle $ACD=90^{\circ}.$
$(\mathbf{a})$ Show that $AB=10.54$ cm, correct to $2$ decimal places
$(\mathbf{b})$ Show that angle $DAC=41.80^{\circ}$ correct to $2$ decimal places
$(\mathbf{c})$ Calculate $BD.$
$(\mathbf{d})$ Calculate angle $CBD.$
$(\mathbf{e})$ Calculate the shortest distance from $C$ to $BD$
▶️Answer/Explanation
$2a – 11b$
Question11
$$\mathrm{f}(x)=2x-1\quad\mathrm{g}(x)=3x+2\quad\mathrm{h}(x)=\frac{1}{x},x\neq0\quad\mathrm{j}(x)=x^2$$
$(\mathbf{a})$ Find $j(-1)$.
$\mathbf{(b)}$ Find $x$ when $f(x)+g(x)=0$
$( \mathbf{c} )$ Find $gg(x)$, giving your answer in its simplest form.
$( \mathbf{c} )$ Find $gg(x)$, giving your answer in its simplest form
$( \mathbf{d} )$ Find $hf(x)+gh(x)$,giving your answer as a single fraction in its simplest form
$(\mathbf{e})$ When $pp(x)=x$, $p(x)$ is a function such that $\mathfrak{p}^-1(x)=\mathfrak{p}(x).$
Draw a ring around the function that has this property
$$\mathrm{f}(x)=2x-1\quad\mathrm{g}(x)=3x+2\quad\mathrm{h}(x)=\frac{1}{x},x\neq0\quad\mathrm{j}(x)=x^2$$
▶️Answer/Explanation
$2a – 11b$
Question12
▶️Answer/Explanation
$2a – 11b$