1.
OQP is a sector of a circle of radius 15 cm.
The angle of the sector is 120°.
O
120°
15 cm
Not drawn accurately
Q
P
(a)
Show that the length of the arc PQ is 10 cm.
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(2)
The sector is folded to form a cone.
O
Not to scale
P,Q
(b)
Calculate the radius of the base of the cone.
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Answer ......................................................... cm
(2)
(Total 4 marks)
The Robert Smyth School
1
2.
(a)
The diagram shows a sector of a circle of radius 9 centimetres.
Not drawn accurately
9 cm
80°
Find the perimeter of the sector.
Give your answer in terms of .
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Answer .................................................................. cm
(3)
(b)
The cross-section of a prism is a sector of a circle, of radius 9 centimetres, as shown.
The height of the prism is 10 centimetres.
Not drawn accurately
10 cm
80°
9 cm
Calculate the volume of the prism.
Give your answer in terms of .
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Answer .........................................................................
(4)
(Total 7 marks)
The Robert Smyth School
2
3.
(a)
A circle has a radius of 6 cm.
A sector has an arc length of 8.4 cm.
The angle at the centre of the sector is θ.
8.4 cm
O
6 cm
Not drawn accurately
Calculate the value of θ.
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Answer ....................................................... degrees
(3)
(b)
A cone has base radius 6 cm and height h cm.
A smaller cone of base radius 2 cm and height 3 cm is cut from the top.
The remaining frustum has dimensions as shown.
3 cm
2 cm
h cm
6 cm
Not drawn accurately
Calculate the volume of the frustum.
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Answer ................................................................ cm3
(5)
(Total 8 marks)
The Robert Smyth School
3
4.
AB is an arc of a circle, centre O,with radius 9.7cm.
Angle AOB = 110.
O
110°
not drawn accurately
9.7 cm
B
A
Calculate the area of the sector OAB.
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Answer.............................................................................................................................cm2
(Total 3 marks)
5.
AOB is a sector of a circle of radius 12 cm.
The area of the minor sector AOB is 98 cm2.
Not drawn accurately
O
12 cm
A
B
Calculate the size of angle AOB.
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Answer ............................................................ degrees
(Total 3 marks)
The Robert Smyth School
4
6.
The first diagram shows a cylindrical block of wood of diameter 24 cm and height 10 cm.
It is cut into six equal prisms as shown.
One of the prisms is shown in the second diagram.
F
B
E
E
C
D
10 cm
A
10 cm
A
24 cm
C
F
B
D
Not to scale
(a)
Calculate the area of sector BEC, the cross-section of the prism.
Give your answer in terms of π.
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Answer ..................................................................cm2
(2)
(b)
Calculate the area of CDFE, the curved surface of the prism.
Give your answer in terms of π.
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Answer ..................................................................cm2
(3)
(c)
Calculate the volume of the prism.
Give your answer in terms of π.
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Answer ..................................................................
(3)
(Total 8 marks)
The Robert Smyth School
5
7.
AB is a chord of a circle, centre O, radius 6 cm.
AB = 7 cm
Not drawn accurately
O
6 cm
A
7 cm
B
Calculate the area of the shaded segment.
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Answer .......................................................... cm2
(Total 6 marks)
The Robert Smyth School
6
1.
120
 215
360
(a)
Or
M1
30
3
oe
(b)
Cancelling to 10
2r = 10
Or
A1
M1
15
3
(r = )5A1
[4]
2.
(a)
2 ×× 9
M1
oe
80
× their (2 ×× 9)
360
M1
4 + 18
A1
oe
(b)
× 92
M1
80
 × 92
360
or × 92 × 10
180
M1 dep
A1
80
oe Must see some correct simplification of
× × 92 × 10
360
cm3
B1
Units mark
[7]
3.
(a)

360
 
 2  6  8.4
M1
8.4  360
2  6
A1
80.2(1...)
A1
r= 12 giving 40.1 is Ml, Al, A0
r = 3 giving 160.4 is Ml, Al, A0
(b)
h 3

6 2
M1
h = 9 (cm)
A1
h = 12 gives Ml, A0
1
1
  6 2  their 9    2 2  3
3
3
Ml for difference of two cone volumes
Al if all correct
The Robert Smyth School
M1,A1
7
(V) = 327 or 326.7 ............................(cm3)
Accept 330 if working seen, ft their h if both
M's awarded.
Alft
[8]
(b)
Alt
linear scale factor 1:3
Must be used. Just writing it down does not
qualify as a method unless progress is made.
M1
Volume scale factor 1:27
A1
Volume small cone
M1
1
  2 2  3  12.566 ...
3
Volume large cone 27 ×
(their 12.566)
339.292...
(V) = 327 or 326.7 ............................(cm3)
DM1
A1
Accept 330 if working seen.
(Scs) 12.566 only
B1
339.29 only
M1,A1,
M1
1
1
  36  h    4  3
3
3
M1,A1
[8]
4.
110
×  9.72
360
M2
M1 for
110
seen
360
90.3(199...)
[3]
5.

360
×  × 122 = 98
M1
or  × 122 + 98 or inverse
98 × 360  ( × 122)
360 + their (4.6) or inverse
M1
77.9 to 78.15
A1
[3]
6.
(a)
1/6 × π × 12 × 12
M1
24π
A1
not π24 ... only penalise notation on
the first occasion
(b)
2 × π × 12 × 10
The Robert Smyth School
M1
8
1/6 × (their)cylinder area
SC 3 marks out of the first 7
M1
40π
A1
for using r = 24 consistently
(c)
(their)24π × 10
Answers: 96π, 80π and 960π
240π
M1
A1ft
cm3
B1
units mark
[8]
7.
Angle at centre = 2 × sin–1( 3.5 6 )
Half angle
sin–1( 3.5
M1
6
) gets M1
= 71.(...)
A1
Area sector = ‘their 71’ ÷ 360 ×  × 62
M for use of area sector formula not for r2 ÷ 4 for example.
M1
Area sector = 22.4(….)_
A1
Their sector – their triangle area
Must make a valid attempt at calculating the area of the
triangle. (17.06…) and at least one of the previous M marks
must be awarded.
Area segment = 5.3…..
DM1
A1
[6]
The Robert Smyth School
9