Question1
(a) A fruit drink is made using $1.5$ litres of apple juice and $450$ millilitres of mango juice
Write the ratio apple juice: mango juice in its simplest form.
(b) One litre of fruit drink is shared between three cups.
The amount in the cups is in the ratio $9:6:10$
Calculate the number of millilitres in each cup
(c) A shop buys bottles of the fruit drink for \(\$3.20$ each
It sells them at a profit of $15\%$.
Calculate the selling price of each bottle of fruit drink
The number of bottles of fruit drink sold has grown exponentially at a constant rate of $2.5\%$ per year.
$5$ years ago, the shop sold $16620$ bottles.
Calculate the number of bottles sold this year.
(e)
The bottles of juice are $18.5$cm tall, correct to the nearest millimetre.
They are stored on shelves.
The distance between the shelves is $23$cm, correct to the nearest centimetre.
Calculate the lower bound for the distance, $d$ cm, between the top of a bottle and the shelf above it.
▶️Answer/Explanation
(a): $10:3$
(b): $360$, $240$, $400$
(c): $3.68$
(d): $18804.0$
(e): $3.95$
Question2
The diagram shows a straight line intersecting two parallel lines.
Find the value of a and the value of b.
(b) Calculate the interior angle of a regular 12-sided polygon.
The diagram shows a circle, centre $O.$
The points $M,N$ and $P$ lie on the circumference of the circle.
$AMB$ is a tangent to the circle at $M$
Find the value of $f$ and the value of $g$.
The diagram shows a cyclic quadrilateral.
Find the value of $k.$
▶️Answer/Explanation
(a): $142$, $142$
(b): $150$
(c): $56$, $34$
(d): $51$
Question3
(a) The table shows the time that each of 40 students takes to travel to school.
(i) Calculate an estimate of the mean.
(ii) On the grid, draw a histogram to show the information in the table.
(iii) Two students are selected at random from the $40$ students.
Calculate the probability that one student takes more than $25$ minutes and the other student takes 10 minutes or less to travel to school.
(b) This is some information about the time that $200$ people took to fill in a questionnaire:
• The longest time taken was 30 minutes.
• The median time was 22 minutes.
• The lower quartile was 8 minutes.
• The interquartile range was 19 minutes.
• The range was 25 minutes.
(i) Write down the shortest time taken.
(ii) On the grid, draw a box-and-whisker plot to show this information.
(iii) George says that $101$ of the $200$ people took more than $22$ minutes to fill in the questionnaire.
Explain why he is wrong.
▶️Answer/Explanation
(a)(i): $25.4375$
(a)(ii): Correct histogram
(a)(iii): $\frac{19}{260}$
(b)(i): $5$
(b)(ii):
(b)(iii): Correct explanation which states the median is 22 and correct reference to 100 or 101. e.g.
- Median is $22$ which is $50\%$ of the people and $101$ is more than $50\%$.
- The median is $22$ which is the 100th number (accept $100.5$th number).
Question4
The diagram shows a tank in the shape of a half-cylinder of radius 12 cm and length 1 metre
The tank is fixed horizontally and is completely filled with water.
$\begin{array}{ll}{\mathbf{i})}&{\text{Calculate the volume of water in the tank.}}\\&{\text{Give your answer correct to the nearest 10 cm}^{3}.}\end{array}$
(ii)
Water is removed from the tank until the level of water is 6cm below the top of the tank
The diagram shows the cross-section of the tank
Calculate the volume of water that is now in the tank.
$\mathbf{(b)}$ A rectangular fish tank with length 42 cm and width $35$cm is full of water.
A stone lies at the bottom of the tank.
When the stone is removed from the tank, the depth of the water decreases by $0.2$ cm
The density of the stone is $2.2$ g/cm3 .
Calculate the mass of the stone in grams.
[Density=mass$\div$volume]
The diagram shows a cuboid, $ABCDEFGH.$
Calculate the angle that AG makes with the base of the cuboid.
▶️Answer/Explanation
(a)(i): $22620$
(a)(ii):
$8840$ or $8850$ or $8836$ to $8850$
(b): $647$ or $646.8$
(c): $46.1$ or $46.12$ to $46.14$
Question5
$(\mathbf{a})$ Simplify $(25x^6)^{\frac32}.$
$( \mathbf{b} )$ These are the first five terms of a sequence.
$\frac16\quad1\quad6\quad36\quad216$
Find the $n$th term of the sequence.
$( \mathbf{c} )$ Expand and simplify
$(x+4)(x-3)(3x-1)$
$( \mathbf{d} )$ $( \mathbf{i} )$ Show that $( 3x+ 5) + \frac 7{x- 2}= x$ simplifies to $2x^{2}+ x- 3= 0$
(ii) Solve by factorisation $2x^2+x-3=0.$
$(\mathbf{e})$ A solid cylinder has base radius $x$ and height 3$x.$
The total surface area of the cylinder is the same as the total surface area of a solid hemisphere of radius $5y.$
Show that $x^2=\frac{75y^2}8.$
$[$The surface area, $A$, of a sphere with radius $r$ is $A=4\pi r^2.]$
▶️Answer/Explanation
(a): $125x^9$
(b): $6^{n-2}$
(c): $3x^3 + 2x^2 – 37x + 12$
(d)(i): Eliminates the fraction correctly e.g. $(3x+5)(x-2) + 7 = x(x-2)$, leading to $3x^2 + 5x – 6x – 10 + 7 = x^2 – 2x$, leading to $2x^2 + x – 3 = 0$
(d)(ii): $(2x+3)(x-1)$, $-1.5$ and $+1$
(e): [TSA cylinder] $= 2\pi x^2 + 2\pi x \times 3x$, [TSA hemisphere] $= \pi (5y)^2 + \frac{4\pi (5y)^2}{2}$. Leading to $2\pi x^2 + 6\pi x^2 = \pi (5y)^2 + \frac{4\pi (5y)^2}{2}$, $x^2 = \frac{75y^2}{8}$
Question6
$ABCD$ is a quadrilateral with $AB=10.4$ cm and $AD=6.5$ cm Angle $DAB=64^{\circ}$,angle $BDC=26^{\circ}$ and angle $DBC=42^{\circ}.$ $(\mathbf{a})$ Show that $BD=9.55$ cm, correct to 2 decimal places
$( \mathbf{b} )$ $( \mathbf{i} )$ Show that angle $BCD=112^{\circ}.$
(ii) Calculate $CD.$
(c) Find the shortest distance from $D$ to $AB.$
▶️Answer/Explanation
(a): $\sqrt{10.4^2 + 6.5^2 – 2 \times 10.4 \times 6.5 \times \cos 64} = 9.546 \text{ to } 9.547$
(b)(i): $180 – (26 + 42)$
(b)(ii): $6.89$ or $6.888$ to $6.892…$
(c): $5.84[2…]$
Question7
$(\mathbf{a})$ Solve $3x-8=6-4x.$
(b) Factorise fully $10a^2+5a.$
$( \mathbf{c} )$ Factorise fully $( 2x- 3) ^2- 9.$
$(\mathbf{d})$
$\mathrm f(x)=\dfrac{1}{4x-1},\:x\neq\dfrac{1}{4}\quad\mathrm g(x)=3^x$
$( \mathbf{i} )$ Find $f(4)$.
$( \mathbf{ii})$ Find $gg(2)$.
(iii) Find $k$ when g$(k)=$f(7).
▶️Answer/Explanation
(a): $2$
(b): $5a(2a+1)$
(c): $4x(x-3)$
(d)(i): $\frac{1}{15}$
(d)(ii): $19683$
(d)(iii): $-3$
Question8
A baker decorates $x$ small cakes and $y$ large cakes.
In one day, he decorates:
$\begin{array}{ll}\bullet&\text{not more than 16 small cakes}\\\bullet&\text{less than 10 large cakes}\\\bullet&\text{more small cakes than large cakes}\\\bullet&\text{a total of not more than 24 cakes}\end{array}$
One of the inequalities that shows this information is $x\leqslant16.$
(a) Write down the other three inequalities in $x$ and/or y.
$\mathbf{(b)}$ On the grid, draw four straight lines and shade the unwanted regions to show these inequalities
Label the region, R, which satisfies the four inequalities.
(c) The baker earns $8 for decorating a small cake and $\$12$ for decorating a large cake.
Use your diagram to find the largest amount the baker can earn in one day by decorating cakes.
▶️Answer/Explanation
(a): $y < 10$, $y < x$, $x + y \le 24$
(b): Correct lines and region indicated (region R)
(c): $228$
Question9
$OAB$ is a sector of a circle, centre $O$, radius 17cm. $OCD$ is a sector of a circle, centre $O$, radius 10cm. $OCA$ and $ODB$ are straight lines and angle $AOB=60^{\circ}.$
The perimeter of the shaded shape $ABDC$ can be written in the form $(a\pi+b)$ cm.
Find the value of $a$ and the value of $b.$
The diagram shows a regular hexagon. The area of the hexagon is $127.3 cm^2$.
(i) Show that the length of one side of the hexagon is $7.0$ cm, correct to 1 decimal place
(ii) The hexagon is the cross-section of a prism of length $10$cm
$( \mathbf{a} )$ Find the volume of the prism.
$(\mathbf{b})$ Calculate the surface area of the prism.
▶️Answer/Explanation
(a): $a = 9$, $b = 14$
(b)(i): $60^\circ$ at centre or interior angle $= 120^\circ$. $[6 \times] \frac{1}{2} \times d^2 \times \sin 60$. $[d^2 =] \frac{127.3}{6 \times \frac{1}{2} \times \sin 60}$. $6.99[9…] \text{ to } 7.00[…] $
(b)(ii)(a): $1273$
(b)(ii)(b): $675$ or $674.5$ to $674.6$
Question10
$( \mathbf{a} )$ $A$ is the point $(6,2)$ and $B$ is the point $(3,-4)$
$( \mathbf{i} )$ Find the coordinates of the midpoint of $AB$
(ii) Calculate the length $AB.$
$(\mathbf{b})$ The equation of line $l$ is $4x+3y-12=0.$
$( \mathbf{i} )$ Find the gradient of $l$
$( \mathbf{ii} )$ Find the coordinates of the point where $l$ crosses the y-axis
(iii) Line $p$ is perpendicular to $l$ and passes through (6,5).
Find the equation of $p$ in the form $y=mx+c.$
▶️Answer/Explanation
(a)(i): $(4.5, -1)$
(a)(ii): $6.71$ or $6.708…$
(b)(i): $-\frac{4}{3}$
(b)(ii): $(0, 4)$
(b)(iii): $y = \frac{3}{4}x + \frac{1}{2}$
Question11
$( \mathbf{a} )$ The point $(-1,6)$ lies on a curve.
This curve has the derived function $\frac{\mathrm{d}y}{\mathrm{d}x}=-4x^{3}-9x^{2}+5.$
Show that (-1,6) is a stationary point of the curve
$( \mathbf{b} )$ A different curve has equation $y=2x^3-6x+8.$
$( \mathbf{i} )$ Calculate the gradient of the tangent to this curve at the point $(-2,2)$.
(ii) Find the x-coordinates of the stationary points of this curve.
▶️Answer/Explanation
(a): $-4(-1)^3 – 9(-1)^2 + 5 = 0$ [so stationary point]
(b)(i): $18$
(b)(ii): $1$ and $-1$