1)
Jan 21 2HR
15.5
The diagram shows a rectangle ABCD and a semicircle with diameter AB where
AB = 12 cm. The point E lies on DC and also on the semicircle.
Work out the area of the shaded region. Give your answer correct to 3 significant figures.
------------------------------------ cm (3)
2
2)
** Jan 16 4HR
66.8
The shaded shape is made by cutting a semicircle from a rectangular piece of
card, ABCF, as shown in the diagram.
FEDC is a straight line. The centre of the semicircle lies on ED.
AF = BC = 10 cm, AB = 20 cm, FE = DC = 4 cm.
Work out the perimeter of the shaded shape. Give your answer correct to 3 significant figures.
..................................................... cm (3)
1
3)
**Jan 19 1HR
10 𝝅
............................ (3)
4)
Jan 21 1H
54.3
The region, shown shaded in the diagram, is a path.
The boundary of the path is formed by two semicircles, with the same centre O, and two straight lines.
The inner semicircle has a radius of 7 metres. The path has a width of 2 metres.
Work out the perimeter of the path. Give your answer correct to one decimal place.
...................................................... m (3)
2
5)
Jan 14 / 4H
(b) 73.5
A rectangular lawn has a length of 3x metres and a width of 2x metres.
The lawn has a path of width 1 metre on three of its sides.
The total area of the lawn and the path is 100 m2
(a) Show that 6x 2 + 7x – 98 = 0
(2)
(b) Calculate the area of the lawn.
Show clear algebraic working.
............................ m2 (5)
3
6)
* Jan 13 / 3H
(b) x =20 ,
1200
The diagram shows a rectangular playground of width x metres and length 3x metres.
The playground is extended, by adding 10 metres to its width and 20 metres to its length, to form a
larger rectangular playground.
The area of the larger rectangular playground is double the area of the original playground.
(a) Show that 3x 2 − 50x − 200 = 0
(3)
(b) Calculate the area of the original playground.
.............................................................. m2 (5)
4
7)
Jan 19 2H
b)6.25
The diagram shows a trapezium.
All measurements shown on the diagram are in centimetres.
The area of the trapezium is 133 cm2
(a) Show that 8x2 – 6x – 275 = 0
(3)
(b) Find the value of x.
Show your working clearly.
............................ (3)
5
8)
Jan 20 2H
44.6
The diagram shows a shaded shape ABCD made from a semicircle ABC and a
right‑angled triangle ACD.
AC is the diameter of the semicircle ABC.
Work out the perimeter of the shaded shape.
Give your answer correct to 3 significant figures.
----------------------------------------- (5)
6
9)
* Jan 13 / 3H
6.01
A washing line is attached at points A and B on two vertical posts standing on horizontal ground.
Point A is 2.1 metres above the ground on one post.
Point B is 1.7 metres above the ground on the other post.
The horizontal distance between the two posts is 6 metres.
Calculate the distance AB.
Give your answer correct to 3 significant figures.
............................................... m (4)
10) *Jun 14 3H
44.4
The diagram shows a square ABCD drawn inside a circle, centre O.
A, B, C and D are points on the circle. The lengths of the sides of the square are 10 cm.
AC is a diameter of the circle. Calculate the circumference of the circle.
Give your answer correct to 3 significant figures.
...................................... cm (4)
7
11) Nov 20 2HR
$418
Markus makes a steel framework.
The framework is in the shape of the right‑angled triangle ABC shown in the diagram.
The steel that Markus uses costs $22 per metre.
The steel can only be bought in a length that is a whole number of metres.
Work out the total cost of the steel that Markus buys in order to make the framework.
$....................................................... (4)
12) Nov 06 a)
b)
The right-angled triangle in the diagram has sides of length 7x cm, 24x cm and 150 cm.
(a) Show that x2 = 36.
[2]
(b) Calculate the perimeter of the triangle.
Answer(b) ............................................cm [1]
8
13) *Nov 10 42 .
In the right-angled triangle ABC, AB = x cm, BC = (x + 7) cm and AC = 17 cm.
(i) Show that x2 + 7x – 120 = 0.
Answer(a)(i)
[3]
(ii) Factorise x2 + 7x – 120.
Answer(a)(ii)------------------------- [2]
(iii) Write down the solutions of
x2 + 7x – 120 = 0.
Answer(a)(iii) x = ------------------------ or x = ------------------------ [1]
(iv) Write down the length of BC.
Answer(a)(iv) BC = -------------------------cm [1]
9
(b)
The rectangle and the square shown in the diagram above have the same area.
(i) Show that 2x2 – 15x – 9 = 0.
Answer(b)(i)
[3]
(ii) Solve the equation 2x2 – 15x – 9 = 0.
Show all your working and give your answers correct to 2 decimal places.
Answer(b)(ii) x =------------------------ or x =------------------------ [4]
(iii) Calculate the perimeter of the square.
Answer(b)(iii) ------------------------cm [1]
10
14) *June 06.
(a)(i) x2 + 4x – 96 = 0 (ii) 8 and –12 (iii) 8 (iv) 176
PYTH
The diagram shows a trapezium.
Two of its angles are 90o.
The lengths of the sides are given in terms of x.
The perimeter is 62 units.
(i) Write down a quadratic equation in x to show this information. Simplify your equation.
-------------------------------------------- [2]
(ii) Solve your quadratic equation.
-----------------------, --------------------[2]
(iii) Write down the only possible value of x.
-----------------------------------------[1]
(iv) Calculate the area of the trapezium.
-------------------------------------- [2]
11
(b)(ii) 4.35 and –0.35 (iii) 13.8
(b) The diagram shows a right-angled triangle.
The lengths of the sides are given in terms of y.
(i) Show that 2y2 – 8y – 3 = 0.
[3]
(ii) Solve the equation 2y – 8y – 3 = 0, giving your answers to 2 decimal places.
2
-------------------------------------------[4]
(iii) Calculate the area of the triangle.
------------------------------------------- [2]
12
15) ** Jan 12 3H
b)30
The diagram shows a trapezium ABCD with AD parallel to BC.
AB = x cm, BC = (x + 5) cm and AD = (x + 8) cm.
The area of the trapezium is 42 𝑐𝑚2
(a) Show that
2x 2 + 13x – 84 = 0
(2)
(b) Calculate the perimeter of the trapezium.
.............................................................. cm (5)
13
16) *** Jan 19 3HR
x = 5.5, A = 67.5
............................ (6)
14
17) * Jun 11/3H
20.2
Triangle ABC is right-angled at B.
Angle BAC = 32°. AC = 47 m. D is the point on AB such that angle BDC = 51°
Calculate the length of BD. Give your answer correct to 3 significant figures.
................................ m (5)
18) Jan 18 / 3HR
13.2
Here is a right-angled triangle ABC. Angle ABC = 90°. AC = 20 cm. AB = 10 cm
D is the midpoint of BC. Work out the length of AD. Give your answer correct to 1 decimal place.
..............................cm (4)
15
19) Jun 17 4HR
24
The diagram shows triangle ABC
AB = 9 cm BC = 15 cm D is the point on AC such that AD = 5 cm. Angle BAC = 90°
Calculate the size of angle x. Give your answer to the nearest degree
…………………………(4)
20) * Jan 21 2HR
195.4
The diagram shows two hot air balloons.
A is a point on the base of one of the balloons and B is a point on the base of the other balloon.
The distance between A and B is 500 metres. The angle of depression of B from A is 23°
Calculate the vertical height of A above B. Give your answer correct to one decimal place.
------------------------------------ metres (3)
16
21) Jun 18 4H
49.1
............................ (5)
17
22) Nov 20 1H
108.4
The diagram shows trapezium ABCD in which BC and AD are parallel.
The trapezium has exactly one line of symmetry.
BC = 8.4 cm
AD = 17.6 cm
The trapezium has area 179.4 cm2
Work out the size of angle ABC.
Give your answer correct to 1 decimal place.
....................................................... ° (6)
18
23) * Nov 20 2HR
6.75
The diagram shows four congruent right‑angled triangles ABJ, BCI, CDH and DEG.
The diagram also shows the straight line ABCDEF.
Angle BAJ = 35°
AF = 80 cm
Work out the length of EF.
Give your answer correct to 3 significant figures.
....................................................... cm (5)
19
24) Jun 15 3H
122
ABCD is a trapezium. AB = 25 cm. BC = 24 cm. CD = 10 cm.
Angle ABC = angle BCD = 90° . Calculate the size of angle CDA.
Give your answer correct to 3 significant figures.
....................................................... °(4)
25) * Jan 14 / 3H
57.3
ABCD is a parallelogram. AB = 8.9 cm. AD = 6.7 cm. Angle BAD = 74°.
Calculate the area of parallelogram ABCD.
Give your answer correct to 3 significant figures.
............................ c𝑚2 (3)
20
26) Nov 14 /22
13.5 sine rule
Angle BAC = 110° and the area of the triangle is 85 cm2 .
Calculate AC.
Answer AC = .......................................... cm [3]
21
27) Jan 15 3HR
56.3
Here is triangle LMN, where angle LMN is an obtuse angle.
ML = 13.8 cm
MN = 8.5 cm
Angle MNL = 47°
Work out the area of triangle LMN.
Give your answer correct to 3 significant figures.
.........................................c𝑚2 (6)
22
28) Jan 16 4HR 𝟓𝟗°
𝟏𝟐𝟏°
**ABC is a triangle.
AB = 12 cm. AC = 14 cm
The area of triangle ABC is 72 c𝑚2
Find, in degrees, the two possible sizes of angle BAC.
Give your answers correct to the nearest degree.
.....................................................(4)
29) * Jun 18 1HR
91.4
............................ (6)
23
30) Jun 18 2H
101.5
A triangle has sides of length 8 cm, 10 cm and 14 cm.
Work out the size of the largest angle of the triangle.
Give your answer correct to 1 decimal place.
............................ (3)
31) **Jun 17 4H
LMNP is a quadrilateral
67.5
Work out the size of angle MLP.
Give your answer to correct 3 significant figures
……………………….(6)
24
32) Jan 19 2HR
65.9 to 66.1
............................ (5)
25
33) **** Jun 09 3H
10
The diagram shows the length, in centimeters, of each side of triangle ABC.
Angle BAC = 60°.
Find the value of x.
x = ..................... (5)
26
34) * Jun 14 4HR
068
A, B and C are 3 villages.
B is 6.4 km due east of A.
C is 3.8 km from A on a bearing of 210°
Calculate the bearing of B from C.
Give your answer correct to the nearest degree.
Show your working clearly.
............................ ° (6)
27
35) Jan 21 1H
140
The diagram shows the positions of three ships, A, B and C.
Ship B is due north of ship A. The bearing of ship C from ship A is 120°
Calculate the bearing of ship C from ship B. Give your answer correct to the nearest degree.
...................................................... (5)
28
36) Jun 13 3HR
35.3
The diagram shows a cube ABCDEFGH.
The sides of the cube are of length 5 cm.
Calculate the size of the angle between the diagonal AH and the base EFGH.
Give your answer correct to 1 decimal place.
°
…………………………………..(4)
29
37) Nov 12/23
(a) 12.7 (b) 28.2
The diagram shows a triangular prism.
ABCD is a horizontal rectangle with DA = 10 cm and AB = 5 cm.
BCQP is a vertical rectangle and BP = 6 cm.
Calculate
(a) the length of DP,
Answer(a) DP =-------------------------- cm [3]
(b) the angle between DP and the horizontal rectangle ABCD.
Answer(b) -------------------------- [3]
30
38) * Jan 13 / 3H
57.8
The diagram shows a pyramid with a horizontal rectangular base PQRS. PQ = 16 cm. QR = 10 cm.
M is the midpoint of the line PR. The vertex, T, is vertically above M. MT = 15cm
Calculate the size of the angle between TP and the base PQRS.
Give your answer correct to 1 decimal place.
..............................................................° (4)
31
39) Jun 13 4H
29.5
The diagram shows a triangular prism with a horizontal rectangular base ABCD.
AB = 10 cm.
BC = 7 cm.
M is the midpoint of AD.
The vertex T is vertically above M.
MT = 6 cm.
Calculate the size of the angle between TB and the base ABCD.
Give your answer correct to 1 decimal place.
………………………° (4)
32
40) *Jun 14 4H
29.1
ABCDEF is a triangular prism.
AB = 9 cm, BC = 15 cm and AE = 12 cm.
Angle ABC = 90°
M is the midpoint of CD.
Calculate the size of the angle between AM and the plane BCDF.
Give your answer correct to 1 decimal place.
............................ ° (5)
33
41) *Jan 17 4H
a) 46.8 b) 33.1
The diagram shows a cuboid ABCDEFGH.
AB = 21 cm and CH= 9 cm
K is the point on EH such that angle AKB = 68° and BK = 16.5 cm.
(a) Calculate the size of angle BAK.
Give your answer correct to 1 decimal place.
....................................................... °(3)
(b) *** Calculate the size of the angle between the line BK and the plane ABCD.
Give your answer correct to 1 decimal place.
....................................................... °(2)
34
42) Jun 18 2H
21.6
The diagram shows a triangular prism.
AF = 10 cm, AB = 24 cm and BC = 8 cm.
Angle FAB = angle ADC = angle BCD = 90o
Work out the size of the angle between the line BE and the plane ABCD.
Give your answer correct to 1 decimal place.
............................ (3)
35
43) Jun 17 / 21
165
The diagram shows a regular octagon joined to an equilateral triangle.
Work out the value of x.
x = ....................................... [3]
44) * Nov 10 4H
24
The size of each interior angle of a regular polygon is 11 times the size of each exterior angle.
Work out the number of sides the polygon has.
--------------------------- [4]
45) Jan 14 4H
16
The diagram shows part of a regular polygon.
The interior angle and the exterior angle at a vertex are marked.
The size of the interior angle is 7 times the size of the exterior angle.
Work out the number of sides of the polygon.
......................................... (3)
36
46) Jun 18 2H
110
The diagram shows a hexagon ABCDEF. BC is parallel to ED.
Work out the size of the obtuse angle DEF.
............................ (5)
47) Jun 18 3H
126
ABCDE is a regular pentagon. AQE and ALB are straight lines.
ALMNPQ is a hexagon with two angles of size y° and four angles of size x°
Work out the value of x.
............................ (4)
37
48) Jun 16 3H
The diagram shows a pentagon.
188
Work out the area of the pentagon.
Give your answer correct to 3 significant figures.
.......................................................(6)
38
49) Jan 17 3HR
76.9
ABCDE is a regular pentagon with sides of length 10cm
Calculate the area of triangle ACD.
Give your answer correct to 3 significant figures
…………………………….. 𝑐𝑚2 (6)
39
50) *Jan 14 3HR
52.1
The diagram shows a regular pentagon inside a circle, centre O.
The points A and B lie on the circle such that AB is a side of the pentagon.
OA = 7 cm. TA is a tangent to the circle and OBT is a straight line.
Calculate the area of triangle ABT. Give your answer correct to 3 significant figures.
........................... c𝑚2 (5)
40
51) Jun 08 3H
12
The diagram shows part of a tiling pattern.
The tiling pattern is made from three shapes.
Two of the shapes are squares and regular hexagons.
The third shape is a regular n-sided polygon A.
Work out the value of n.
n = ...........................(5)
52) * Nov 06
15
The sides of an equilater triangle ABC and two regular polygons meet at the point A.
AB and AD are adjacent sides of a regular 10-sided polygon.
AC and AD are adjacent sides of a regular n-sided polygon.
Work out the value of n.
n = ...........................(5)
41
53) Jan 21 1HR
96
AEJ and HIJ are straight lines.
Work out the size of the angle marked x.
Show your working clearly.
------------------------------- (5)
42
54) * Jun 12 3H
4.46 or 4.45
The diagram shows a sector OAPB of a circle, centre O.
AB is a chord of the circle.
OA = OB = 5.4 cm. Angle AOB = 72°.
Calculate the area of the shaded segment APB.
Give your answer correct to 3 significant figures.
............................................................... cm2 (5)
43
55) * Jun 09 3H
15.7
The diagram shows a sector OAPB of a circle, centre O.
AB is a chord of the circle. The radius of the circle is 6 cm. Angle AOB = 78°.
Calculate the perimeter of the shaded segment APB.
Give your answer correct to 3 significant figures.
..................... cm (6)
44
56) **Jun 17 4HR
23.3
The diagram shows triangle KLM.
KLP is a sector of a circle with centre L and radius 10.4 cm.
The region of the triangle outside the sector is shown shaded in the diagram.
Calculate the area of the shaded region.
Give your answer correct to 3 significant figures
……………………….𝑐𝑚2 (5)
45
57) Jan 19 3HR
4.6
............................ (5)
46
58) Jun 19 2HR
20.1
---------------------------------(6)
47
59) Jan 15 4HR
a) 42.5 b) 15
c) 189
In the diagram ABC and ADE are straight lines.
BD is parallel to CE.
AB = 9 cm, BC = 13.5 cm, AD = 10 cm, BD = 17 cm
(a) Calculate the length of CE.
............................... cm (2)
(b) Calculate the length of DE.
............................... cm (2)
The area of triangle ABD is 36 c𝑚2
(c) *Calculate the area of quadrilateral BDEC.
............................... c𝑚2 (3)
48
60) *Jun 14 3H
32
The diagram shows two regular hexagons, OABCDE and OFGHIJ.
OAF and OEJ are straight lines.
OF = 3 OA.
The area of OABCDE is 4 c𝑚2 .
Calculate the area of the shaded region.
...................................... c𝑚2 .(3)
49
61) *Nov 08 3H
a) 3.4 b)9/16 oe
ABC and AED are two straight lines.
BE is parallel to CD.
AE = 5.1 cm, BE = 6 cm, CD = 10 cm.
(a) Calculate the length of DE
.............. cm(3)
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑛𝑔𝑙𝑒 𝐴𝐵𝐸
(b) Calculate the value of𝐴𝑅𝐸𝐴 𝑂𝐹 𝑇𝑅𝐴𝑃𝐸𝑍𝐼𝐴𝑀 𝐵𝐶𝐷𝐸
3)
..............(3)
50
62) ** Jun 19 1H a) 2160
b) 8V/ 27
51
63) Nov 20 1H
625
R and S are two similar solid shapes.
Shape R has surface area 108 cm2 and volume 135 cm3
Shape S has surface area 300 cm2
Work out the volume of shape S.
....................................................... cm 3 (3)
64) Jan 21 1H
630
A and B are two similar solids.
A has a volume of 1836 cm3.
B has a volume of 4352 cm3
B has a total surface area of 1120 cm2.
Work out the total surface area of A.
...................................................... cm 2 (3)
52
65) *** Jan 20 2HR
583.2
The diagram shows two similar vases, A and B.
The height of vase A is 9 cm and the height of vase B is 13 cm.
Given that
surface area of vase A + surface area of vase B = 1800 cm2
calculate the surface area of vase A.
....................................................... cm 2 (4)
53