1.
A, B and C are points on the circumference of a circle with centre O.
BD and CD are tangents.
Angle BDC = 40°
Not drawn accurately
B
A
q
O p
40°
D
C
Not drawn accurately
(a)
Work out the value of p.
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Answer .......................................................... degrees
(2)
(b)
Hence write down the value of q.
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Answer .......................................................... degrees
(1)
(Total 3 marks)
2.
(a)
O is the centre of the circle.
52°
O
x
Not drawn
accurately
y
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1
(i)
Find the value of x.
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Answer x = .................................................... degrees
(1)
(ii)
Find the value of y.
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Answer y = .................................................... degrees
(1)
(b)
PQ and PR are tangents to the circle centre O.
 QPR is 65.
Q
Not drawn
O z
65°
P
accurately
R
Calculate the size of angle QOR (marked z on the diagram).
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Answer .................................................... degrees
(2)
(Total 4 marks)
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3.
A, B, C and D are four points on the circumference of a circle.
The lines AB and DC are produced to meet at E.
Angle CBE = 67° and angle BEC = 35°
A
Not drawn accurately
x
B
67°
35°
D
(a)
C
E
What is the special name for the quadrilateral ABCD?
Answer ………………………………….........
(1)
(b)
Work out the value of x.
You must show your working.
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Answer ………………………………… degrees
(3)
(Total 4 marks)
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4.
In the diagram below points Q and S lie on a circle centre O.
SR is a tangent to the circle at S.
Angle QRS = 40° and angle SOQ = 80°
O
80°
Q
40°
S
R
Not drawn accurately
Prove that triangle QSR is isosceles.
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(Total 3 marks)
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4
5.
(a)
(i)
The diagram shows a circle with centre O.
50°
Not drawn accurately
O
x
Work out the size of the angle marked x.
.....……………………………………………………………………………..
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Answer ............................. degrees
(1)
(ii)
The diagram shows a different circle with centre O.
50°
O
Not drawn accurately
y
Work out the size of the angle marked y.
.....……………………………………………………………………………..
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Answer ............................. degrees
(1)
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(b)
The diagram shows a cyclic quadrilateral ABCD.
The straight lines BA and CD are extended and meet at E.
EA = AC
Angle ABC = 3x°
Angle ADC = 9x°
Angle DAC = 2x°
Not drawn accurately
B
3x°
C
2x°
A
9x°
D
E
(i)
Show that x = 15
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(2)
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(ii)
Calculate the size of angle EAD.
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Answer ............................... degrees
(4)
(Total 8 marks)
6.
Not drawn accurately
40º
xº
yº
(i)
Write down the value of x.
Answer ............................................. degrees
(1)
(ii)
Calculate the value of y.
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Answer ............................................. degrees
(1)
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(b)
A and C are points on the circumference of a circle centre B.
AD and CD are tangents.
Angle ADB = 40°.
A
Not drawn accurately
40º
B
D
C
Explain why angle ABC is 100°.
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(2)
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(c)
P is a point on the circumference of a circle with centre O.
PQ is a tangent of length 8 cm.
The area of triangle OPQ is 24 cm2.
Not drawn accurately
O
P
8 cm
Q
Calculate the area of the circle.
Give your answer in terms of .
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Answer .................................................... cm2
(3)
(Total 7 marks)
7.
(a)
(b)
360 – (90 + 90 + 40)
oe
M1
140
A1
70
B1ft
[3]
8.
(a)
(b)
(i)
x = 104°
B1
(ii)
y = 128°
B1
115°
M1,A1
M1 for sight of R or Q = 90
A1 for 360 – 90 – 90 – 65
[4]
9.
(a)
Cyclic (quadrilateral)
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B1
9
(b)
 BCE = 78° or  C = 78°
May be seen on diagram
B1
 BCD= 102°
May be seen on diagram
B1
(x =) 78
B1
Alternatives to be marked as follows
Use of cyclic quad properties  finding a correct angle
eg  D = 67
B1
Finding third angle in a triangle
eg  ADE
B1
Correct answer
B1
[4]
10.
angle OSQ = angle OQS = 50°
Isosceles triangle OQS
B1
Penalise ‘no reason’
angle OSR = 90°  angle QSR = 40°
Tangent-radius property first time only
B1
angle QSR = angle QRS (Isosceles)
B1
[3]
11.
(a)
(i)
100
(ii)
130
B1
B1
Allow angels marked clearly on the diagram
(b)
(i)
9x + 3x = 180
x = 15
(ii)
angle DAC(2x) = 30°
angle DCA = 15°
angle DEA = 15°
angle EAD = 120°
allow subst. of x = 15 as long as opp.
angles of cyc. (answer given)
quad. clearly used
Allow angles marked clearly on the diagram
M1
A1
B1
B1 f.t.
B1 f.t.
B1
[8]
12.
(a)
(b)
(i)
40
(ii)
140 or 180 – (their x)
Do not ft if answer = 140 in (a)(i)
Logical and precise explanation
(either written or as calculation)
B1 for 1 angle labelled or stated correctly, no reason
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B1
B1ft
B2
10
(c)
24 ÷ 8 × 2
M1
or OP = 6
(or 3.14) × (their 6)2
M1
36 or 36 ×  or  × 36
allow 36
SC2 108 to 114 or  × 9 oe
SC1 27 to 28.5
A1
[7]
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