Question 1:
1 A grocer sells potatoes, mushrooms and carrots.
(a) A customer buys 3 kg of mushrooms at $1.04$ per kg and 4kg of carrots at $1.28 per kg$.
Calculate the total cost.
$ …………………………………………
(b) In one week, the ratio of the masses of vegetables sold by the grocer is
potatoes : mushrooms : carrots = 11 : 8 : 6.
(i) Work out the mass of mushrooms sold as a percentage of the total mass.
……………………………………… %
(ii) The total mass of potatoes, mushrooms and carrots sold is 1500kg.
Find the mass of carrots the grocer sells this week.
…………………………………….. kg
(iii) The profit the grocer makes selling 1kg of carrots is $0.75 .
Find the total profit the grocer makes selling carrots this week.
$ ………………………………………… $
(iv) On the last day of the week, the grocer reduces the price of 1kg of potatoes by 8% to $1.15$ .
Calculate the original price of 1kg of potatoes.
$ ………………………………………… $
(c) The grocer buys 620kg of onions, correct to the nearest 20kg.
He packs them into bags each containing 5kg of onions, correct to the nearest 1kg.
Calculate the upper bound for the number of bags of onions that he packs.
▶️Answer/Explanation
Ans :
1(a) 8.24 cao
1(b)(i) 32
1(b)(ii) 360
1(b)(iii) 270
1(b)(iv) 1.25 cao
1(c) 140 nfww
Question 2:
A, B, C and D are points on a circle.
ADX and BCX are straight lines.
Angle BAD = x° and angle DCX = y°.
(a) Explain why x = y.
Give a geometrical reason for each statement you make.
(b) Show that triangle ABX is similar to triangle CDX.
(c) AD = 15cm, DX = 9 cm and CX = 12cm.
(i) Find BC.
BC = ……………………………………. cm
(ii) Complete the statement.
The ratio area of triangle ABX : area of triangle CDX = ………….. : 1.
▶️Answer/Explanation
Ans :
2(a) y+BCD = angle 180 oe
AND
angles on a straight line
AND
x+ angle BCD = 180 oe
AND
opposite angles of a cyclic quadrilateral
are supplementary
OR
angles in opposite segments are supplementary
leading to x = y with no errors
2(b) Allow any two statements from:
CXD is common angle
or angle AXB = angle CXD
x = y or angle BAX = angle DCX
angle ABX= angle CDX
States all three equal pairs of angles
OR
2/all angles equal so triangles similar
2(c)(i) 6 nfww
2(c)(ii) 4
Question 3:
3 (a) The table shows information about the marks gained by each of 10 students in a test.
(i) Calculate the range.
(ii) Calculate the mean.
(iii) Find the median.
(iv) Write down the mode.
(b) Paulo’s mean mark for 7 homework tasks is 17.
After completing the 8th task, his mean mark is 17.5 .
Calculate Paulo’s mark for the 8th task.
(c) The table shows the percentage scored by each of 100 students in their final exam.
On the grid, draw a histogram to show this information.
▶️Answer/Explanation
Ans :
3(a)(i) 5
3(a)(ii) 16.8
3(a)(iii) 16.5
3(a)(iv) 15
3(b) 21
3(c) 5 correct blocks, with correct widths,
heights 0.8cm, 1.8cm 7cm, 4cm, 1cm
Question 4:
4 (a)
The diagram shows a pyramid with a square base BCDE.
The diagonals CE and BD intersect at M, and the vertex F is directly above M.
BE = 12 cm and FM = 9 cm.
(i) Calculate the volume of the pyramid.
[The volume, V, of a pyramid with base area A and height h is V=$\frac{1}{3}Ah$.]
………………………………….. cm$^{3}$
(ii) Calculate the total surface area of the pyramid.
………………………………….. cm$^{2}$
(b)
The diagram shows a toy made from a cone and a hemisphere.
The base radius of the cone and the radius of the hemisphere are both r cm.
The slant height of the cone is 3r cm.
The total surface area of the toy is 304 cm$^{2}$
Calculate the value of r.
[The curved surface area, A, of a cone with radius r and slant height l is A r = r l.]
[The curved surface area, A, of a sphere with radius r is A 4πr$^{2}$ .]
r = …………………………………………
▶️Answer/Explanation
Ans :
4(a)(i) 432
4(a)(ii) 404 or 403.5 to 403.7
4(b) 4.4[0] or 4.398 to 4.399… nfww
Question 5:
5 (a) (i) Factorise.
$x^{2}-x-12$
(ii) Simplify.
$\frac{x^{2}-16}{x^{2}-x-12}$
(b) Simplify.
(2x-3) $^{2}$-(x+1) $^{2}$
(c) Write as a single fraction in its simplest form.
$\frac{2x+4}{x-3}-\frac{x}{x-3}$
(d) Expand and simplify.
(x-3) (x -5) ( )(2x+1)
(e) Solve the simultaneous equations.
You must show all your working.
$x-3y=13$
$2x^{2}-9y=116$
▶️Answer/Explanation
Ans :
5(a)(i) ( x-4)(x+3 ) final answer
5(a)(ii) $\frac{x+4}{x+3}$ final answer
5(b) 3x $^{2}$-14x+8 or (x-4) (3x-2) final answer
5(c) $\frac{x^{2}-3x-12}{(x+1) (x-3)}$ or $\frac{x^{2}-3x-12}{(x^{2}-2x-3)}$ final answer
5(d) $2x^{3}-15x^{2}+22x+15$
5(e) $2x^{2}-3x-77[=0] \mathbf{oe}$
$(6x^{2}9x-231[=0])$
$\mathbf{or}$
$18y^{2}+147y+222[=0] \mathbf{oe}$
$(6y^{2}+49y+74[=0])$
$(2x+11)(x-7)[=0]$
$\mathbf{oe}$
$\mathbf{or}\frac{[–]3+\sqrt{([-]3)^{2}-4\times 2\times -77}}{2\times 2}\mathbf{oe}$
$\mathbf{or}(6y+37)(3y+6)[=0]$
$\mathbf{or} \frac{-147\sqrt{147^{2}-4\times 18\times 222}}{2\times 18}\mathbf{oe}$
$x=7\mathbf{ and } y=-2$
$x=-5\frac{1}{2} \mathbf{oe}\; \mathbf{and} y=-6\frac{1}{6}\mathbf{oe}$
Question 6:
The diagram shows triangle ABC with AB = 17.2cm.
Angle ABC = 54° and angle ACB = 68°.
(a) Calculate AC.
AC = ……………………………………. cm
(b) M lies on BC and MC = 12.8 cm.
Calculate AM.
AM = ……………………………………. cm
(c) Calculate the shortest distance from A to BC.
………………………………………cm
▶️Answer/Explanation
Ans:
6(a) 15[.0] or 15.00 to 15.01
6(b) 15.7 or 15.65 to 15.66
6(c) 13.9 or 13.90 to 13.92
Question 7:
7 (a) $p=\binom{8}{-5}q=\binom{-4}{5}$
(i) Find 3q.
(ii) (a) Find p q – .
(b) Find p q – .
(b)
In triangle OMN, O is the origin, $\overrightarrow{OM}=a \overrightarrow{ON}$ .
S is a point on MN such that MS : : SN = 5 3 .
Find, in terms of a and/or b, the position vector of S.
Give your answer in its simplest form.
▶️Answer/Explanation
Ans :
7(a)(i) $\binom{-12}{15}$
7(a)(ii)(a) $\binom{12}{-10}$
7(a)(ii)(b) 15.6 or 15.62…
7(b)$ \frac{3}{8}a+\frac{5}{8}b$ final answer
Question 8:
8 (a) On the axes, sketch the graph of y x = -4 3 .
(b) On the axes, sketch the graph of y=-x $^{2}$
(c) (i) Find the coordinates of the turning points of the graph of y=10+9x $^{2}$-2x $^{3}$
You must show all your working.
( ………….. , ………….. ) and ( ………….. , ………….. )
(ii) Determine whether each turning point is a maximum or a minimum.
Show how you decide.
▶️Answer/Explanation
Ans :
8(a) Ruled line with negative gradient and positive y-intercept
8(b) Negative quadratic, with vertex at origin
8(c)(i) 18x – 6x$^{2}$ isw
setting their derivative = 0 or $\frac{dy}{dx}=0$
(0, 10) and (3.37)
8(c)(ii) (0, 10) minimum with correct reason
AND
(3, 37) maximum with correct reason
Question 9:
9 (a) Janna and Kamal each invest $8000$.
At the end of 12 years, they each have $12800$.
(i) Janna invests in an account that pays simple interest at a rate of r% per year.
Calculate the value of r.
r = …………………………………………
(ii) Kamal invests in an account that pays compound interest at a rate of R% per year.
Calculate the value of R.
R = …………………………………………
(b) The population of a city is growing exponentially at a rate of 1.8% per year.
The population now is 260000.
Find the number of complete years from now when the population will first be more than 300000.
▶️Answer/Explanation
Ans:
9(a)(i) 5
9(a)(ii) 4[.0] or 3.99…
9(b) 9 nfww
Question 10:
The table shows some values for y =2x$^{3}$+ 6x$^{2}-2.5$
(a) Complete the table.
(b) On the grid, draw the graph of y =2x$^{3}$+ 6x$^{2}-2.5$= for -3\leqslant x \leqslant 1
(c) By drawing a suitable line on the graph, solve the equation 2x$^{3}$+6x$^{2}$=4.5
x = ……………….. or x = ……………….. or x = ………………..
(d) The equation 2x$^{3}$+6x$^{2}$-2.5 =k has exactly two solutions.
Write down the two possible values of k.
k = ………………………… or k = …………………………
▶️Answer/Explanation
Ans :
10(a) −2.5 −1.25 5.5
10(b) Correct graph
10(c) y = 2 drawn
−2.75 to –2.65
–1.1 to −1.05
0.75 to 0.85
10(d) –2.5 5.5
Question 11:
$f(x)=\frac{1}{x^{,}}x\neq0$
g(x)=3x-5
h(x)=2$^{x}$
(a) Find.
(i) gf(2)
(ii) g$^{-1}$(x)
g$^{-1}$(x) = – …………………………..
(b) Find in its simplest form g(x-2)
(c) Find the value of x when
(i) fg(x) = 0.1
(ii) h(x) – g(7)= 0
▶️Answer/Explanation
Ans :
11(a)(i) −3.5 oe
11(a)(ii) $\frac{x+5}{3}$ oe final answer
11(b) 3 x-11 final answer
11(c)(i) 5
11(c)(ii) 4 nfww
Question 12:
12 (a)
The diagram shows a circle of radius 12cm, with a sector removed.
Calculate the perimeter of the remaining shaded shape.
…………………………………….. cm
(b) The diagram in part(a) shows the top of a cylindrical cake with a slice removed.
The volume of cake that remains is 3510 cm $^{3}$
Calculate the height of the cake.
…………………………………….. cm
▶️Answer/Explanation
Ans :
12(a) 88.9 or 88.92 to 88.93…
12(b) 9.01 or 9.009 to 9.010…