Anda di halaman 1dari 79

EXTENDED MATHEMATICS

2002 2011

CLASSIFIEDS GEOMETRY
Compiled & Edited
By
Dr. Eltayeb Abdul Rhman

www.drtayeb.wordpress.com
First Edition
2011

4
7
NOT TO
SCALE
O

50
z
A

TA is a tangent at A to the circle, centre O.

Angle OAB = 50.
Find the value of
(a)

y,

[1]

[1]

[1]

(b) z,

(c) t.

Seismic shock waves travel at speed v through rock of density d.

v varies inversely as the square root of d.
v = 3 when d = 2.25.
Find v when d = 2.56.

UCLES 2011

0580/21/O/N/11

[3]

10
16

y
C (9,7)

NOT TO
SCALE

A (1,3)

B (3,0)

The co-ordinates of A, B and C are shown on the diagram, which is not to scale.
(a) Find the length of the line AB.

[3]

[3]

UCLES 2011

0580/21/O/N/11

11
22

4.40 cm

3.84 cm

NOT TO
SCALE

C
D

9.40 cm

A, B, C and D lie on a circle.

AC and BD intersect at X.
(a) Give a reason why angle BAX is equal to angle CDX.

[1]

(b) AB = 4.40 cm, CD = 9.40 cm and BX = 3.84 cm.

(i) Calculate the length of CX.

cm [2]

(ii) The area of triangle ABX is 5.41 cm2.

Calculate the area of triangle CDX.

UCLES 2011

0580/23/O/N/11

cm2 [2]

6
3

(a)
D

NOT TO
SCALE

40
A

30
B

AB is extended to E and angle EBC = 30.
AB = AD and BD = BC.
(i) Calculate angle BCD.

[3]

[1]

(b) A regular polygon has n sides.

5n
degrees.
Each exterior angle is
2
Find the value of n.

UCLES 2011

0580/43/O/N/11

[3]

7
(c)

NOT TO
SCALE
O

25

The diagram shows a circle centre O.

A, B and C are points on the circumference.
OC is parallel to AB.
Angle OCA = 25.
Calculate angle OBC.

UCLES 2011

0580/43/O/N/11

[3]

2
1

Javed says that his eyes will blink 415 000 000 times in 79 years.
(a) Write 415 000 000 in standard form.

[1]

(b) One year is approximately 526 000 minutes.

Calculate, correct to the nearest whole number, the average number of times his eyes will blink
per minute.

[1]

Luis and Hans both have their birthdays on January 1st.

In 2002 Luis is 13 and Hans is 17 years old.
(a) Which is the next year after 2002 when both their ages will be prime numbers?

[1]

(b) In which year was Hans twice as old as Luis?

[1]

Diagram 2

Diagram 1
(a) In Diagram 1, shade the area which represents AB.

[1]

0580/2, 0581/2 Jun02

[1]

3
4

NOT TO
SCALE

x
B

54

20
C

ABCD is a parallelogram and BCE is a straight line. Angle DCE = 54 and angle DBC = 20 .
Find x and y.

Calculate the length of the straight line joining the points ( 1, 4) and (5, 4).

15

[2]

(a)

[2]

(i) Complete quadrilateral ABCD so that the dotted

line is the only line of symmetry.
[1]

ABCD.

[1]

(b)

(i) Complete quadrilateral EFGH so that the dotted

line is one of two lines of symmetry.
[1]
(ii) Write down the order of rotational symmetry for

H

[1]

8
17

NOT TO
SCALE
4r

Two circles have radii r cm and 4r cm.

Find, in terms of and r.
(a) the area of the circle with radius 4r cm,

[1]

[1]

(b) the area of the shaded ring,

(c) the total length of the inner and outer edges of the shaded ring.

[2]

4
4

NOT TO
SCALE

B
b
c C

42
G

A sphere, centre C, rests on horizontal ground at A and touches a vertical wall at D.

A straight plank of wood, GBW, touches the sphere at B, rests on the ground at G and against the wall at W.
The wall and the ground meet at X.
Angle WGX = 42.
(a) Find the values of a, b, c, d and e marked on the diagram.

[5]

(b) Write down one word which completes the following sentence.
Angle CGA is 21 because triangle GBC and triangle GAC are .

[1]

(i)

[3]

(ii)

[1]

(iii)

[3]

(iv)

[1]

7
8

NOT TO
SCALE

20
6 cm

Calculate
(i)

[2]

(ii)

[2]

(b)
NOT TO
SCALE

20
6 cm

5 cm

A whole cheese is a cylinder, radius 6 cm and height 5 cm.

The diagram shows a slice of this cheese with sector angle 20.
Calculate
(i)

[2]

(ii)

the total surface area of the slice of cheese.

[4]

(c) The radius, r, and height, h, of cylindrical cheeses vary but the volume remains constant.
(i)

Which one of the following statements A, B, C or D is true?

A:

h is proportional to r.

B:

h is proportional to r2.

C:

h is inversely proportional to r.

D:

h is inversely proportional to r2.

(ii)

What happens to the height h of the cylindrical cheese when the volume remains constant but the
[2]

[2]

0580/4, 0581/4 Jun02

4
y

4
3
B

2
A

1
1

1
2
(a) Find the gradient of the line AB.
(b) Calculate the angle that AB makes with the x-axis.

0580/2, 0581/2 Jun 2003

5
12

3p

2p
q

NOT TO
SCALE

5q
D
E
A, B, C, D and E lie on a circle, centre O.
Find the value of

AOC is a diameter.

(a) p,

(b) q.

8
9

(a)
4

1
3
1
1

Diagram 1

3
Diagram 2

Diagram 3

Diagram 4

Diagram 1 shows a triangle with its base divided in the ratio 1 : 3.

Diagram 2 shows a parallelogram with its base divided in the ratio 1 : 3.
Diagram 3 shows a kite with a diagonal divided in the ratio 1 : 3.
Diagram 4 shows two congruent triangles and a trapezium each of height 1 unit.
For each of the four diagrams, write down the percentage of the total area which is shaded. [7]
(b)
1
2

80

80

Diagram 5

Diagram 6

Diagram 7

Diagram 5 shows a semicircle, centre O.

Diagram 6 shows two circles with radii 1 unit and 5 units.
Diagram 7 shows two sectors, centre O, with radii 2 units and 3 units.
For each of diagrams 5, 6 and 7, write down the fraction of the total area which is shaded. [6]

6
15

The area of triangle APQ is 99 cm2 and the area of triangle

ABC is 11 cm2. BC is parallel to PQ and the length of PQ is
12 cm.

NOT TO
SCALE

Q

UCLES 2004

0580/2, 0581/2 Jun/04

cm [3]

C
NOT TO
SCALE

D
11.1 cm
37o
E
A
9.5 cm

70o
B

AB = 9.5 cm, BC = 11.1 cm, angle ABC = 70o and angle CAD = 37o.
(a) Calculate the length of AC.

[4]

[1]

(c) Calculate the length of AD.

[4]

(d) A point E lies on the circle such that triangle ACE is isosceles, with EA = EC.
(i) Write down the size of angle AEC.

[1]

(ii) Calculate the area of triangle ACE.

[3]
0580/4, 0581/4 Jun/04

7
15 The points A(6,2) and B(8,5) lie on a straight line.
(a) Work out the gradient of this line.

[1]

(b) Work out the equation of the line, giving your answer in the form y = mx + c.

0580/02, 0581/02 Jun 05

[2]

11
21

C
O

NOT TO
SCALE

40o
5 cm

A
A, B and C are points on a circle, centre O.
Angle AOB = 40o.
(a) (i) Write down the size of angle ACB.

[1]

[1]

(b) The radius of the circle is 5 cm.

(i) Calculate the length of the minor arc AB.

cm [2]

UCLES 2005

cm2 [2]

18

22o
wo

NOT TO
SCALE

xo
E

yo

60o

zo

AD is a diameter of the circle ABCDE.

Angle BAC = 22 and angle ADC = 60.
AB and ED are parallel lines.
Find the values of w, x, y and z.

x=
y=
z=

UCLES 2006

0580/02, 0581/02 Jun 06

[4]

10

22 (a)

p cm

4 cm
q cm
B

4 cm

E
NOT TO
SCALE

5 cm
8 cm
A
In the diagram triangles ABE and ACD are similar.
BE is parallel to CD.
AB = 5 cm, BC = 4 cm, BE = 4 cm, AE = 8 cm, CD = p cm and DE = q cm.
Work out the values of p and q.

q=

[4]

(b) A spherical balloon of radius 3 metres has a volume of 36 cubic metres.

It is further inflated until its radius is 12 m.

UCLES 2006

m3 [2]

6
14

NOT TO
SCALE

70o
B

The diagram shows three touching circles.

A is the centre of a circle of radius x centimetres.
B and C are the centres of circles of radius 3.8 centimetres. Angle ABC = 70.
Find the value of x.

UCLES 2007

0580/02/J/07

[3]

8
19

R
Q

NOT TO
SCALE

w
S

64
O x

y
P

P, Q, R and S lie on a circle, centre O.

TP and TQ are tangents to the circle.
PR is a diameter and angle PSQ = 64.
(a) Work out the values of w and x.

[1]

x=

[1]

[2]

(b) Showing all your working, find the value of y.

UCLES 2007

0580/02/J/07

12
23

NOT TO
SCALE
A

B
12 cm

The largest possible circle is drawn inside a semicircle, as shown in the diagram.
The distance AB is 12 centimetres.

cm2 [4]

cm [2]

(b) Find the perimeter of the shaded area.

UCLES 2007

0580/02/J/07

4
3

(a)

A
55
40.3 cm

26.8cm
cm
26.8
92
X

NOT TO
SCALE

20.1 cm
cm
C

D
A, B, C and D lie on a circle.
AC and BD intersect at X.
Angle ABX = 55 and angle AXB = 92.
BX = 26.8 cm, AX = 40.3 cm and XC = 20.1 cm.
(i) Calculate the area of triangle AXB.

[2]

(ii) Calculate the length of AB.

[3]

(iii) Write down the size of angle ACD. Give a reason for your answer.

[2]

(iv) Find the size of angle BDC.

[1]

(v) Write down the geometrical word which completes the statement
Triangle AXB is

to triangle DXC.

(vi) Calculate the length of XD.

UCLES 2007

[1]

[2]

0580/04/M/J/07

6
5
F

NOT TO
SCALE
G

12 cm

A circle, centre O, touches all the sides of the regular octagon ABCDEFGH shaded in the diagram.
The sides of the octagon are of length 12 cm.
BA and GH are extended to meet at P. HG and EF are extended to meet at Q.
(a) (i) Show that angle BAH is 135.

[2]

(ii) Show that angle APH is 90.

[1]

(b) Calculate
(i) the length of PH,

[2]

[2]

[2]

(iv) the area of the octagon.

[3]

(c) Calculate
(i) the radius of the circle,

[2]

(ii) the area of the circle as a percentage of the area of the octagon.

[3]

UCLES 2008

0580/04/M/J/08

8
7

(a)

C
NOT TO
SCALE

40
D
z

O
130

x
y
A

A, B, C and D lie on a circle, centre O.

SCT is the tangent at C and is parallel to OB.
Angle AOB = 130, and angle BCT = 40.
Angle OBC = x, angle OBA = y and angle ADC = z.
(i) Write down the geometrical word which completes the following statement.
ABCD is a

[1]

[3]

[1]

[1]

UCLES 2008

0580/04/M/J/08

9
(b)

7 cm
NOT TO
SCALE

P
X

10 cm

P, Q, R and S lie on a circle.

PQ = 7 cm and SR = 10 cm.
PR and QS intersect at X.
The area of triangle SRX = 20 cm2.
(i)

Write down the geometrical word which completes the following statement.
Triangle PQX is

to triangle SRX.

[1]

[2]

[2]

UCLES 2008

0580/04/M/J/08

17

A
NOT TO
SCALE

38

C
B

AB is the diameter of a circle, centre O. C, D and E lie on the circle.

EC is parallel to AB and perpendicular to OD. Angle DOC is 38.
Work out
(a) angle BOC ,

[1]

[1]

[2]

UCLES 2010

0580/21/M/J/10

15
y
NOT TO
SCALE

y=4

0
2x + y = 8

3x + y = 18

(a) The line y = 4 meets the line 2x + y = 8 at the point A.

Find the co-ordinates of A.

[1]

[1]

[1]

(b) The line 3x + y = 18 meets the x axis at the point B.

Find the co-ordinates of B.

(c) (i) Find the co-ordinates of the mid-point M of the line joining A to B.

(ii) Find the equation of the line through M parallel to 3x + y = 18.

UCLES 2010

0580/22/M/J/10

[2]

10

17
Q

P
W

NOT TO
SCALE

8m
O

S
T

The diagram shows the junction of four paths.

In the junction there is a circular area covered in grass.
This circle has centre O and radius 8 m.
(a) Calculate the area of grass.

m2

[2]

(b)
Q
12 m
P

NOT TO
SCALE

45
O

The arc PQ and the other three identical arcs, RS, TU and VW are each part of a circle, centre O,
The angle POQ is 45.
The arcs PQ, RS, TU, VW and the circumference of the circle in part(a) are painted white.
Calculate the total length painted white.

UCLES 2010

0580/22/M/J/10

[4]

16

(a)
9 cm

NOT TO
SCALE
X
3 cm

6 cm

The lines AB and CDE are parallel.

AD and CB intersect at X.
AB = 9 cm, CD = 6 cm and DX = 3 cm.
(i) Complete the following statement.
Triangle ABX is

to triangle DCX.

[1]

cm

[2]

cm2

[2]

(iii) The area of triangle DCX is 6 cm2.

Calculate the area of triangle ABX.

(iv) Angle BAX = x and angle ABX = y.
Find angle AXB and angle XDE in terms of x and/or y.

Angle XDE =

UCLES 2010

0580/42/M/J/10

[2]

17

(b)
35

NOT TO
SCALE
Q

42
S

P, Q, R and S lie on a circle, centre O.

Angle OPS = 42 and angle PRQ = 35.
Calculate
(i) angle POS,

[1]

[1]

[1]

[1]

(iv) angle PSQ.

(c) The interior angle of a regular polygon is 8 times as large as the exterior angle.
Calculate the number of sides of the polygon.

UCLES 2010

0580/42/M/J/10

[3]

2
C
B

8 cm

NOT TO
SCALE

5 cm
3 cm
D
A

11 cm

In the quadrilateral ABCD, AB = 3 cm, AD = 11 cm and DC = 8 cm.

The diagonal AC = 5 cm and angle BAC = 90.
Calculate
(a) the length of BC,

cm

[2]

(b) angle ACD,

[4]

(c) the area of the quadrilateral ABCD.

UCLES 2010

0580/43/M/J/10

cm2

[3]

(a)
A
3 cm
Q

NOT TO
SCALE

4 cm

P
B
3.6 cm

The diagram shows two triangles ACB and APQ.

Angle PAQ = angle BAC and angle AQP = angle ABC.
AB = 4 cm, BC = 3.6 cm and AQ = 3 cm.
(i) Complete the following statement.
to triangle APQ.

Triangle ACB is

[1]

cm

[2]

cm2

[2]

(iii) The area of triangle ACB is 5.6 cm2.

Calculate the area of triangle APQ.

UCLES 2010

0580/43/M/J/10

(b)

H
S
R
NOT TO
SCALE
O

U
61
M

R, H, S, T and U lie on a circle, centre O.

HT is a diameter and MN is a tangent to the circle at T.
Angle RTM = 61.
Find
(i) angle RTH,

[1]

[1]

[1]

[1]

(c) ABCDEF is a hexagon.

The interior angle B is 4 greater than interior angle A.
The interior angle C is 4 greater than interior angle B, and so on, with each of the next interior
angles 4 greater than the previous one.
(i) By how many degrees is interior angle F greater than interior angle A?

[1]

[3]

(ii) Calculate interior angle A.

UCLES 2010

0580/43/M/J/10

[Turn over

4
9
A

2x

C
5x

NOT TO
SCALE
D

AB is parallel to CD.
Calculate the value of x.

UCLES 2011

0580/11/M/J/11

[3]

9
18
C
NOT TO
SCALE
O
B
24
A

A, B and C are points on a circle, centre O.

TA is a tangent to the circle at A and OBT is a straight line.
AC is a diameter and angle OTA = 24.
Calculate
(a) angle AOT,

[2]

[1]

[1]

UCLES 2011

0580/11/M/J/11

2
x
NOT TO
SCALE

52

A straight line intersects two parallel lines as shown in the diagram.

Find the value of x.

[1]

9
A

2x

C
5x

B
NOT TO
SCALE
D

AB is parallel to CD.
Calculate the value of x.

UCLES 2011

0580/12/M/J/11

[3]

8
17 (a)

C
A

NOT TO
SCALE

B
Points A, B and C lie on the circumference of the circle shown above.
When angle BAC is 90 write down a statement about the line BC.

[1]

(b)

NOT TO
SCALE
O
54
A
D
B
C

O is the centre of a circle and the line ABC is a tangent to the circle at B.
D is a point on the circumference and angle BOD = 54.
Calculate angle DBC.

UCLES 2011

0580/12/M/J/11

[3]

8
16

k cm

The diagram shows a square of side k cm.

The circle inside the square touches all four sides of the square.
(a) The shaded area is A cm2.
Show that

4A = 4k2 k2.

[2]
(b) Make k the subject of the formula 4A = 4k2 k2.

UCLES 2011

0580/21/M/J/11

[3]

9
17
C
NOT TO
SCALE
O
B
24
A

A, B and C are points on a circle, centre O.

TA is a tangent to the circle at A and OBT is a straight line.
AC is a diameter and angle OTA = 24.
Calculate
(a) angle AOT,

[2]

[1]

[2]

(c) angle ABT.

UCLES 2011

0580/21/M/J/11

7
14
y
13
NOT TO
SCALE

1
x

The diagram shows the straight line which passes through the points (0, 1) and (3, 13).
Find the equation of the straight line.

[3]

cm [3]

15 A cylinder has a height of 12 cm and a volume of 920 cm3.

Calculate the radius of the base of the cylinder.

UCLES 2011

0580/23/M/J/11

10
20
V

U
NOT TO
SCALE

70
W

O
h

e
f
A

The diagram shows a circle, centre O.

VT is a diameter and ATB is a tangent to the circle at T.
U, V, W and X lie on the circle and angle VOU = 70.
Calculate the value of
(a) e,

[1]

[1]

[1]

[1]

(b) f,

(c) g,

(d) h.

UCLES 2011

0580/23/M/J/11

8
5

y
8

7
6
5
4
3
2
1
3

1
2
A

3
4

[2]

(ii) Write down the equation of the line AB in the form y = mx + c.

UCLES 2011

0580/31/M/J/11

[2]

7
5

(a) The table below shows how many sides different polygons have.
Complete the table.

Name of polygon

Number of sides
3

4
5

Hexagon

Heptagon

7
8

Nonagon

9
[3]

(b) Two sides, AB and BC, of a regular nonagon are shown in the diagram below.
C
NOT TO
SCALE
x
A

(i) Work out the value of x, the exterior angle.

[2]

(ii) Find the value of angle ABC, the interior angle of a regular nonagon.

UCLES 2011

0580/32/M/J/11

[1]

[Turn over

10
6

(a)
C
NOT TO
SCALE

140

p
B

The diagram shows a triangle ABC with BA extended to D.

AB = AC and angle CAD = 140.
Find the value of p.

[2]

(b)

72
NOT TO
SCALE
q

[2]

(c)
108

104

94

NOT TO
SCALE

UCLES 2011

0580/33/M/J/11

[1]

11
(d)
C
NOT TO
SCALE
22
B

In triangle ABC, angle A = 90 and angle B = 22.

Calculate angle C.

[1]

(e)
X

10 cm
NOT TO
SCALE

8 cm
Y

10 cm

In triangle XYZ, P is a point on XY and Q is a point on XZ.

PQ is parallel to YZ.
(i) Complete the statement.
Triangle XPQ is

to triangle XYZ.

[1]

(ii) PQ = 8 cm, XQ = 10 cm and YZ = 10 cm.

Calculate the length of XZ.

UCLES 2011

0580/33/M/J/11

cm [2]

3
(ii)
North
B 110
NOT TO
SCALE

110
C

4 km

AB = BC = 6 km.
Junior students follow a similar path but they only walk 4 km North from A, then 4 km on a
bearing 110 before returning to A.
Senior students walk a total of 18.9 km.
Calculate the distance walked by junior students.

km [3]

(c) The total amount, \$1380, raised in 2010 was 8% less than the total amount raised in 2009.
Calculate the total amount raised in 2009.

UCLES 2011

0580/41/M/J/11

[3]

6
4

8 cm

NOT TO
SCALE

6 cm
O

9 cm

The circle, centre O, passes through the points A, B and C.

In the triangle ABC, AB = 8 cm, BC = 9 cm and CA = 6 cm.
(a) Calculate angle BAC and show that it rounds to 78.6, correct to 1 decimal place.

[4]
(b) M is the midpoint of BC.
(i) Find angle BOM.

UCLES 2011

0580/43/M/J/11

[1]

17
(b)
F

G
2x

(x + 15)

NOT TO
SCALE

75
E

EFG is a triangle.
HJ is parallel to FG.
Angle FEG = 75.
Angle EFG = 2x and angle FGE = (x + 15).
(i) Find the value of x.

[2]

[1]

(ii) Find angle HJG.

UCLES 2011

0580/43/M/J/11

5
12
B

NOT TO
SCALE

95

109
D

E
In the hexagon ABCDEF, BC is parallel to ED and DC is parallel to EF.
Angle DEF = 109 and angle EFA = 95.
Angle FAB is equal to angle ABC.
Find the size of
(a) angle EDC,

[1]

[2]

0580/2/O/N/02

7
14

NOT TO
SCALE

S
P

21
X
33
80

PQRS is a cyclic quadrilateral. The diagonals PR and QS intersect at X.

Angle SPR = 21, angle PRS = 80 and angle PXQ = 33.
Calculate
(a) angle PQS,

[1]

[1]

[1]

15

Solve the simultaneous equations

4x + 5y = 0,
8x 15y = 5.

y = ......................................................
0580/2/O/N/02

[4]

11
22
P

NOT TO
SCALE

8cm

O
5.5cm
3cm
Q
K

M
8cm

In the circle, centre O, the chords KL and PQ are each of length 8 cm.
M is the mid-point of KL and R is the mid-point of PQ. OM = 3 cm.
(a) Calculate the length of OK.

[2]

Answer (b) Angle ROM = ..............................

0580/2/O/N/02

[3]

B
NOT TO SCALE
cm

10

6cm

8cm

5cm

5cm

F
The diagram shows a sketch of the net of a solid tetrahedron (triangular prism).
The right-angled triangle ABC is its base.
AC = 8 cm, BC = 6 cm and AB = 10 cm. FC = CE = 5 cm.
Show that BE = 61 cm.

[1]

(ii)

[1]

(iii)

[2]

(a) (i)

[4]

[3]

[3]

(e) Calculate the volume of the solid.

[The volume of a tetrahedron is 1 (area of the base) perpendicular height.]

[3]

0580/4,0581/4/O/N02

7
17
C
20
D

NOT TO
SCALE

62

A
B

AD is parallel to BC. The diagonals DB and AC meet at X.
Angle ACB # 62 and angle ACD # 20.
Calculate
(a) angle DBA,
Answer (a)Angle DBA # .................. [1]
(b) angle DAB,
Answer (b)Angle DAB # .................. [1]
(c) angle DAC,
Answer (c)Angle DAC # .................. [1]
(d) angle AXB,
Answer (d)Angle AXB # .................. [1]
(e) angle CDB.
Answer (e)Angle CDB # .................. [1]

0580/02/0581/02/O/N/03

9
19

35 m

35 m

NOT TO SCALE

The diagram shows an athletics track with six lanes.

The distance around the inside of the inner lane is 400 metres.
The radius of each semicircular section of the inside of the inner lane is 35 metres.
(a) Calculate the total length of the two straight sections at the inside of the inner lane.

m [3]

(b) Each lane is one metre wide.

Calculate the difference in the distances around the outside of the outer lane and the inside of
the inner lane.

UCLES 2004

0580/02/O/N/04

m [2]

8
10 Quadrilaterals P and Q each have diagonals which

are unequal,
intersect at right angles.

P has two lines of symmetry. Q has one line of symmetry.

Write down its geometrical name.

[2]

Write down its geometrical name.

[2]

(b) In quadrilateral P, an angle between one diagonal and a side is x.

Write down, in terms of x, the four angles of quadrilateral P.

[2]

(c) The diagonals of quadrilateral Q have lengths 20 cm and 12 cm.

Calculate the area of quadrilateral Q.

[2]

The lengths of the diagonals and sides of quadrilateral P are all integer values.
Find the length of a side of quadrilateral P.

[3]

UCLES 2004

0580/04/O/N/04

6
15

48o

E
O
NOT TO
SCALE
126o
C

A, B, C and D lie on a circle centre O. AC is a diameter of the circle.

AD, BE and CF are parallel lines. Angle ABE = 48 and angle ACF = 126.
Find
(a) angle DAE,

[1]

[1]

[1]

UCLES 2005

7
17
B

NOT TO
SCALE

ABCDE is a regular pentagon.

DEF is a straight line.
Calculate
(a) angle AEF,

[2]

[1]

[2]

[2]

18 Simplify

x 27
(a)
27

x 2
(b)
4

3
,

2
.

UCLES 2005

8
19
y
NOT TO
SCALE

5
l

10

[2]

[2]

UCLES 2005

10
21

B
NOT TO
SCALE

O
D

A,B,C and D lie on a circle, centre O, radius 8 cm.

AB and CD are tangents to a circle, centre O, radius 4 cm.
ABCD is a rectangle.
(a) Calculate the distance AE.

cm [2]

UCLES 2005

0580/02, 0581/02 Nov 2005

cm2 [3]

6
12
P

T
38

38

S
NOT TO
SCALE

In the diagram PT and QR are parallel. TP and TR are tangents to the circle PQRS.
Angle PTR = angle RPQ = 38.
(a) What is the special name of triangle TPR. Give a reason for your answer.
reason

[1]

(b) Calculate
(i) angle PQR,

[1]

[1]

13 A statue two metres high has a volume of five cubic metres.

A similar model of the statue has a height of four centimetres.
(a) Calculate the volume of the model statue in cubic centimetres.

cm3 [2]

m3 [1]

UCLES 2006

0580/02/N/06

8
16
A

13x cm

NOT TO
SCALE

5x cm
y

17x cm

12x cm

ABCD is a trapezium.
(a) Find the area of the trapezium in terms of x and simplify your answer.

cm2 [2]

(b) Angle BCD = y. Calculate the value of y.

UCLES 2006

0580/02/N/06

[2]

11
21

y
2x + 3y = 17

4x y = 6
NOT TO
SCALE
A

C (4,3)
x
B (1, 2)

In the diagram, the line AC has equation 2x + 3y = 17 and the line AB has equation 4x y = 6.
The lines BC and AB intersect at B(1, 2).
The lines AC and BC intersect at C(4, 3).
(a) Use algebra to find the coordinates of the point A.

[3]

[3]

(b) Find the equation of the line BC.

UCLES 2006

0580/02/N/06

8
18

y
NOT TO
SCALE

10
l

The line l passes through the points (10, 0) and (0, 8) as shown in the diagram.
(a) Find the gradient of the line as a fraction in its simplest form.

[1]

(b) Write down the equation of the line parallel to l which passes through the origin.

[1]

(c) Find the equation of the line parallel to l which passes through the point (3, 1).

UCLES 2007

0580/02/O/N/07

[2]

10
20
A

NOT TO
SCALE
B
90
O
50
123

The points A, B, C and D lie on a circle centre O.

Angle AOB = 90, angle COD = 50 and angle BCD = 123.
The line DT is a tangent to the circle at D.
Find
(a) angle OCD,

[1]

[1]

[1]

[1]

(d) reflex angle AOC.

UCLES 2007

0580/02/O/N/07

9
8

D
y
E

C
x

NOT TO
SCALE
O
36
78

ABCDE is a pentagon.
A circle, centre O, passes through the points A, C, D and E.
Angle EAC = 36, angle CAB = 78 and AB is parallel to DC.
(a) Find the values of x, y and z, giving a reason for each.

[6]

[1]

(c) Find the value of angle EOC.

[1]

(d) AB = AC.
Find the value of angle ABC.

[1]

UCLES 2008

0580/04/O/N/08

8
18

NOT TO
SCALE

y = 2x + 4

y = mx + c
A

6 units

The line y = mx + c is parallel to the line y = 2x + 4.

The distance AB is 6 units.
Find the value of m and the value of c.

UCLES 2009

0580/21/O/N/09

and c =

[4]

9
19

NOT TO
SCALE

68

Points A, B and C lie on a circle, centre O, with diameter AB.

BD, OCE and AF are parallel lines.
Angle CBD = 68.
Calculate
(a) angle BOC,

[2]

[2]

UCLES 2009

0580/21/O/N/09

3
D

NOT TO
SCALE

50

O is the centre of the circle.

DA is the tangent to the circle at A and DB is the tangent to the circle at C.
AOB is a straight line. Angle COB = 50.
Calculate
(a) angle CBO,

[1]

[1]

J
NOT TO
SCALE

50 m

20 m

JGR is a right-angled triangle. JR = 50m and JG = 20m.

Calculate angle JRG.

[2]

[1]

[1]

Write 0.00658
(a) in standard form,

UCLES 2010

0580/22/O/N/10

5
10
A

140
E

140

140
C

NOT TO
SCALE

The pentagon has three angles which are each 140.

The other two interior angles are equal.
Calculate the size of one of these angles.

UCLES 2010

0580/22/O/N/10

[3]

6
13
NOT TO
SCALE

The diagram shows a circle of radius 5cm in a square of side 18cm.

cm2

[3]

14

Draw, accurately, the locus of all the points outside the triangle which are 3 centimetres away from
the triangle.
[3]

UCLES 2010

0580/22/O/N/10

4
9
A
NOT TO
SCALE

8 cm
Q
P
B

10 cm

12 cm

APB and AQC are straight lines. PQ is parallel to BC.

AP = 8 cm, PQ = 10 cm and BC = 12 cm.
Calculate the length of AB.

UCLES 2010

0580/23/O/N/10

cm

[2]

11
23

D
50
NOT TO
SCALE

O
86

30
C

The points A, B, C and D lie on the circumference of the circle, centre O.

Angle ABD = 30, angle CAD = 50 and angle BOC = 86.
(a) Give the reason why angle DBC = 50.

[1]

(b) Find

[1]

[1]

[2]

UCLES 2010

0580/23/O/N/10

10
6

(a)
A
P
19.5 cm

16.5 cm

11 cm

NOT TO
SCALE

The diagram shows a toy boat.

AC = 16.5 cm, AB = 19.5 cm and PR = 11 cm.
Triangles ABC and PQR are similar.
(i) Calculate PQ.

cm [2]

cm [3]

(iii) Calculate angle ABC.

UCLES 2010

0580/41/O/N/10

[2]

11
(iv) The toy boat is mathematically similar to a real boat.
The length of the real boat is 32 times the length of the toy boat.
The fuel tank in the toy boat holds 0.02 litres of diesel.
Calculate how many litres of diesel the fuel tank of the real boat holds.

litres

[2]

(b)
E
F
32
143
NOT TO
SCALE

67 m

105 m
70
D

The diagram shows a field DEFG, in the shape of a quadrilateral, with a footpath along the
diagonal DF.
DF = 105 m and FG = 67 m.
Angle EDF = 70U, angle EFD = 32U and angle DFG = 143U.
(i) Calculate DG.

[4]

[4]

UCLES 2010

0580/41/O/N/10

12
7

(a)
A
y

NOT TO
SCALE

62

w
O

z
C

A, B, C and D are points on the circumference of a circle centre O.

AC is a diameter.
BD = BC and angle DBC = 62U.
Work out the values of w, x, y and z.
w=

because

[2]

x=

because

[2]

y=

because

[2]

z=

because

[2]

UCLES 2010

0580/41/O/N/10

3
2
R

4 km

Q
NOT TO
SCALE

7 km
4.5 km
S

85
40
P

The diagram shows five straight roads.

PQ = 4.5 km, QR = 4 km and PR = 7 km.
Angle RPS = 40 and angle PSR = 85.
(a) Calculate angle PQR and show that it rounds to 110.7.

[4]
(b) Calculate the length of the road RS and show that it rounds to 4.52 km.

[3]
(c) Calculate the area of the quadrilateral PQRS.
[Use the value of 110.7 for angle PQR and the value of 4.52 km for RS.]

UCLES 2010

0580/43/O/N/10

km2

[5]

4
9
topicAngleproperties

NOT TO
SCALE

t
104
117

91

In the pentagon the two angles labelled t are equal.

Calculate the value of t.

topicEquationsandInequalities

topicNumbersetnotationandlanguage

0580/2 W00

[3]

EXTENDED MATHEMATICS
2002 2011
CLASSIFIEDS GEOMETRY
Compiled & Edited
By
Dr. Eltayeb Abdul Rhman

www.drtayeb.wordpress.com
First Edition
2011
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}