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Shape and Space Higher GCSE
43 minutes
46 marks
Page 1 of 23
Q1. ABCD is a square.
Not drawn accurately
Work out the size of angle x. You must show your working, which may be on the diagram.
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Answer ........................................................ degrees (Total 4 marks)
Page 2 of 23
Q2. AB is parallel to CD. LMN and PMQ are straight lines. MQ = MN
Not drawn accurately
Work out the value of x.
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Answer ........................................................ degrees (Total 3 marks)
Page 3 of 23
Q3. AB is parallel to CD. LMN and PMQ are straight lines. MQ = MN
Not drawn accurately
Work out the value of x.
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Answer ........................................................ degrees (Total 3 marks)
Page 4 of 23
Q4. The dotted line is one side of an enlargement of the hexagon.
(a) Complete the enlarged hexagon. (2)
(b) What is the scale factor of the enlargement?
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Answer ...................................................................... (1)
(Total 3 marks)
Page 5 of 23
Q5. PQR is an enlargement of ABC.
Not drawn accurately
(a) Work out the scale factor of the enlargement.
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Answer ...................................................................... (1)
(b) Write down the size of angle P.
Answer ........................................................ degrees (1)
(c) Work out the length AB.
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Answer ................................................................ cm (2)
(Total 4 marks)
Page 6 of 23
Q6. Work out the area of a semi-circle of radius 6 cm. Give your answer in terms of π.
Not drawn accurately
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Answer ...................................................................... cm2
(Total 2 marks)
Q7. (a) A, B and C are points on the circumference of a circle with centre O.
Not drawn accurately
Work out the size of angle x.
Answer ........................................................ degrees (1)
Page 7 of 23
(b) P, Q and R are points on the circumference of a circle with centre O.
Not drawn accurately
Work out the size of angle y. Give a reason for your answer.
Answer ........................................................ degrees
Reason ...........................................................................................................
......................................................................................................................... (2)
(Total 3 marks)
Q8. A triangle, square and pentagon have a total area of 48 cm2.
The areas of the shapes are in the ratio of their number of sides.
Work out the area of the pentagon.
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Answer ............................................................... cm2
(Total 3 marks)
Page 8 of 23
Q9. Reflect the triangle in the line y = 2
(Total 2 marks)
Q10. The dotted line is one side of an enlargement of the hexagon.
Page 9 of 23
(a) Complete the enlarged hexagon. (2)
(b) What is the scale factor of the enlargement?
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Answer ...................................................................... (1)
(Total 3 marks)
Q11. A scooter is travelling at a constant speed of 75 kilometres per hour.
(a) The scooter travels at this speed for 20 minutes.
How many kilometres has the scooter travelled in this time?
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Answer ...................................................................... km (2)
(b) The speed limit is 50 miles per hour.
Is the scooter travelling faster or slower than the speed limit?
You must show your working.
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......................................................................................................................... (2)
(Total 4 marks)
Faster
Slower
Q12. A scooter is travelling at a constant speed of 75 kilometres per hour.
(a) The scooter travels at this speed for 20 minutes.
How many kilometres has the scooter travelled in this time?
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Answer ...................................................................... km (2)
Page 10 of 23
(b) The speed limit is 50 miles per hour.
Is the scooter travelling faster or slower than the speed limit?
You must show your working.
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......................................................................................................................... (2)
(Total 4 marks)
Faster
Slower
Q13. Work out the area of a semi-circle of radius 6 cm. Give your answer in terms of π.
Not drawn accurately
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Answer ...................................................................... cm2
(Total 2 marks)
Page 11 of 23
Q14. The cone below has radius 3 cm and slant height l cm.
The total surface area, including the base, is 24π cm2.
Work out the length l.
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Answer ...................................................................... cm (Total 3 marks)
Q15. A circle is drawn inside a square as shown.
Page 12 of 23
Show that the area of the circle is more than 75% of the area of the square.
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Page 13 of 23
Q16. ABCD is a triangular based pyramid. The base BCD is a right-angled triangle.
A is directly above B. BC = BD AB = 2 × BC
The volume of the pyramid is 72 cm3.
The formula for the volume of a pyramid is
Calculate the length of BC, labelled x in the diagram.
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Answer ................................................................ cm (Total 3 marks)
Page 14 of 23
M1.
M1 for showing, or stating (right angles may be implied by subtraction) enough angles to solve the problem ie. An obtuse angle written at A or two obtuse angles written at B and C in same quadrant A1 for A = 120° or A1 for B = 100 and C = 110 in same quadrant
M1,A1
180 − their A
or 360 − (90 + their B + their C) M1dep
60° 60 no working SC3
A1 [4]
M2. Enough angles (at least 2) marked or stated to complete the proof with no incorrect angles marked or stated
180 − (62 + 62) M1
56 A1
Complete method, showing 2 angles of 62 and subtraction from 180 Strand (ii)
Q1
Alternative method
AMQ and AML and PMB and NMB marked (stated) as 62
(360 − 4 × 62) ÷ 2 M1
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56 A1
Complete method, showing 4 angles of 62 and subtraction from 360 and division by 2 Strand (ii)
Q1 [3]
M3. Enough angles (at least 2) marked or stated to complete the proof with no incorrect angles marked or stated
180 − (62 + 62) M1
56 A1
Complete method, showing 2 angles of 62 and subtraction from 180 Strand (ii)
Q1
Alternative method
AMQ and AML and PMB and NMB marked (stated) as 62
(360 − 4 × 62) ÷ 2 M1
56 A1
Complete method, showing 4 angles of 62 and subtraction from 360 and division by 2 Strand (ii)
Q1 [3]
M4. (a) Fully correct diagram with vertices within 1mm
B1 for 2 or 3 sides correct from a full hexagon. B1 for symmetrical diagram (about vertical line) with bottom vertex correct. Ignore any internal lines.
B2
(b) (×) 3 (×) or 1 : 3 Accept − 3 or both
B1 [3]
M5. (a) 2.5 or
oe accept 1 : 2.5 or 2 : 5 Incorrect cancelling of 15 / 6 is B0.
B1
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(b) 60 B1
(c) 20 ÷ their 2.5, oe eg AB × their 2.5 = 20 ft from their (a)
M1
8 Accuracy to 1 dp or better
A1ft [4]
M6. π × 6 × 6 ÷ 2
oe accept a numerical value for π M1
18 π or a numerical value
[55.8, 56.57]
Accept π × 18 or π18 A1
[2]
M7. (a) 35 B1
(b) 100 B1
Angle at centre twice angle on circumference Must use words ‘centre’ and ‘circumference’ (or ‘perimeter’) Allow poor spelling even though both words given oe (strand) (i)
Q1 [3]
M8. 3 + 4 + 5 (= 12)
3 and 4 must be used B1
48 ÷ their 12 (= 4) ‘Their 12’ means their addition of 3 + 4 + 5 or their total if they think that a pentagon does not have 5 sides
M1
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20 ft on B0 Accept 12 : 16 : 20
A1ft [3]
M9.
B2 correct B1 reflection in x = 2 or y = 2 drawn with no other lines drawn
B2 [2]
M10. (a) Fully correct diagram with vertices within 1mm
B1 for 2 or 3 sides correct from a full hexagon. B1 for symmetrical diagram (about vertical line) with bottom vertex correct. Ignore any internal lines.
B2
(b) (×) 3 (×) or 1 : 3 Accept − 3 or both
B1 [3]
M11. (a) 75 ÷ 3
75 ÷ 60 × 20 or 1.25 km per minute M1
25 A1
(b) Any correct conversion between miles and km seen, eg 5 miles = 8 km or 1 mile = 1.6 km or 1km = 5 / 8 mile
75 × M1
Page 18 of 23
Slower as limit is 80 km Slower as 46.875 < 50
A1 [4]
M12. (a) 75 ÷ 3
75 ÷ 60 × 20 or 1.25 km per minute M1
25 A1
(b) Any correct conversion between miles and km seen, eg 5 miles = 8 km or 1 mile = 1.6 km or 1km = 5 / 8 mile
75 × M1
Slower as limit is 80 km Slower as 46.875 < 50
A1 [4]
M13. π × 6 × 6 ÷ 2
oe accept a numerical value for π M1
18 π or a numerical value
[55.8, 56.57]
Accept π × 18 or π18 A1
[2]
M14. πrl + πr2 = 24π
15π M1
3l + 9 = 24
oe e.g. 3πl = 15π M1
5
SC1 for 8 from πrl = 24π Must see working
SC1 for 6 from πrl + 2πr = 24π Must see working
NB if height calculated after 5 seen ignore A1
[3]
Page 19 of 23
M15. Any side chosen for square and squared, eg 102 = 100
M2 is for both square and circle areas attempted with correct numerical values (eg if 10 chosen for side of square, then 5 must be used as radius of circle, or if 4 chosen as radius then 8 used as side of square) M1 if both square and circle area attempted with one incorrect numerical value (eg if 10 chosen for side of square, then 10 used as radius of circle, or if 4 chosen as radius then 4 used as side of square)
Half the side squared and multiplied by π, eg π × 25
π must be [3.1, 3.142] or
M2 is for both square and circle areas attempted with correct numerical values (eg if 10 chosen for side of square, then 5 must be used as radius of circle, or if 4 chosen as radius then 8 used as side of square) M1 if both square and circle area attempted with one incorrect numerical value (eg if 10 chosen for side of square, then 10 used as radius of circle, or if 4 chosen as radius then 4 used as side of square)
M2
Work out 75% of their square and a correct calculation of the circle area, or works out what percentage the circle area is of the square area
This can be awarded even if only M1 awarded.
Allow π used if a clear comparison, eg π × 25 > 3 × 25 A1
A method mark gained and correct conclusion based on 75% of their square with their circle
Strand (ii). Do not award if their circle area > square area, eg 78.5 > 25
Q1
Page 20 of 23
Alternative Method
2r length of side of square giving 4r2 as area
M2 is for both square and circle area attempted with correct
numerical values (eg if r chosen for side of square, then must
be used as radius of circle, or if r chosen as radius then 2r used as side of square) M1 if both square and circle area attempted with one incorrect numerical value (eg if x chosen for side of square, then x used as radius of circle, or if x chosen as radius then x used as side of square)
r as radius of circle giving πr2 as area of circle
M2 is for both square and circle area attempted with correct
numerical values (eg if r chosen for side of square, then must
be used as radius of circle, or if r chosen as radius then 2r used as side of square) M1 if both square and circle area attempted with one incorrect numerical value (eg if x chosen for side of square, then x used as radius of circle, or if x chosen as radius then x used as side of square)
M2
75% of their square (= 3r2) and correct expression for area of circle with their chosen radius
A1
A method mark gained and correct conclusion based on 75% of their square with their circle. eg π > 3
Strand (ii). Do not award if their circle area > square area, eg πr2 >
r2 Q1
[4]
M16. × × x × x × 2x
or × × CB × DB × AB (2 BC)
× x × x × 2x = 24 is M1 by implication. M1
x3 = 216
M1
6 6 from T&I is 3 marks 6 without verification or working is 1 mark.
A1 [3]
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