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St Paul’s Catholic School Mathematics GCSE Revision
MAY HALF TERM
PACK 3 – GEOMETRY & MEASURES TOPICS TO GRADE 4/5
Name: ______________________________________ Maths Teacher:_______________________________
Page 2
Properties of Quadrilaterals and Triangles
Q1. Julie is drawing a quadrilateral with these properties. It has 4 equal sides.
Its diagonals intersect at 90°. She draws a square.
(a) Draw a different type of quadrilateral with these properties.
(1)
(b) What is the name of this quadrilateral?
Answer ................................................. (1)
(Total 2 marks)
Q2. Here is a list of quadrilaterals.
kite rectangle rhombus square trapezium
For each of the following descriptions, choose the correct name from the list.
(a) One pair of sides are parallel. The other two sides are not parallel.
Answer ................................................. (1)
(b) All the angles are the same size. Only opposite sides are equal.
Answer ................................................. (1)
(c) All the sides are the same length. The diagonals are not equal in length.
Answer ................................................. (1)
(Total 3 marks)
Q3. (a) Write down the name of this quadrilateral.
Answer ................................................. (1)
(b) Three of these statements are true for a kite. Draw arrows from the statements that are true to the picture of the kite. One of them has been done for you.
(2) (Total 3 marks)
Page 3
Q4. (a) The diagrams show the diagonals of two different quadrilaterals.
Write down the names of these quadrilaterals.
Quadrilateral A .......................... Quadrilateral B ............................ (2)
(b) (i) On the grid below draw a quadrilateral that has only one pair of parallel and exactly two right angles.
(1)
(ii) Write down the name of this quadrilateral.
Answer ................................................. (1)
(Total 4 marks)
Q5. ABCD is a rhombus and ABCE is a kite.
Work out the value of x.
..............................................................................................
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…...........................................................................................
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Answer ................................................. degrees (Total 4 marks)
Page 4
Q6. Frank draws two quadrilaterals on a seven-point triangular grid.
(a) (i) What special name is given to quadrilateral A?
Answer ................................................. (1)
(ii) What special name is given to quadrilateral B?
Answer ................................................. (1)
(b) By joining 4 dots on the seven-point grid below draw a rectangle.
(1)
(c) By joining 3 dots on the seven-point grid below draw an equilateral triangle.
(1)
(d) The perimeter of quadrilateral A can be found using the formula
Find P when a = 3 and b = 5.2
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Answer P = ................................................. (2)
(e) Frank now draws a quadrilateral and a triangle.
Explain why the areas of the two shapes are the same.
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........................................................................................................
........................................................................................................ (2)
(Total 8 marks)
Page 5
Q7. The diagram shows a triangle.
All the sides are equal in length.
(a) What is the name given to this special type of triangle?
Answer …………………................................................................... (1)
(b) The diagram shows a shape made up of two of these triangles.
(i) What is the mathematical name of this shape?
Answer …………........................................................................ (1)
(ii) Write down the order of rotational symmetry of this shape.
Answer ....................................................................... (1)
(iii) Draw the lines of symmetry on the shape. (2)
(Total 5 marks)
Q8. Joanne is making shapes using some of these rods.
Not drawn accurately
(a) She makes an isosceles triangle using three of the rods.
Draw a sketch to show how she could do this. Show the length on each side.
(1)
(b) She makes a quadrilateral using two 3 cm rods and two 5 cm rods.
Write down the names of 2 possible quadrilaterals that she could make.
Answer ………............... and ................................ (2)
(c) She tries to make a triangle using one rod of each length. Explain why she cannot do this.
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........................................................................................................ (1)
(Total 4 marks)
Page 6
Q9. (a) An isosceles triangle has one angle of 80°.
Write down the possible sizes of the other two angles.
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Answer ................ and ............... degrees
or ................ and ............... degrees (2)
(b) Triangle ABC is a right-angled triangle. BDC is an equilateral triangle.
Not drawn accurately
Show that triangle ABD is an isosceles triangle.
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..............................................................................................
.............................................................................................. (3)
(Total 5 marks)
Q10. ABC is a triangle. D is a point on AB such that BC = BD.
(a) Work out the value of x.
........................................................................................................
........................................................................................................
Answer .................................. degrees (2)
(b) Work out the value of y.
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Answer ................................ degrees (2)
(c) Does AD = DC? Give a reason for your answer.
.......................................................................................................
........................................................................................................ (1)
(Total 5 marks)
Page 7
Q11. ABC is an isosceles triangle. AB = BC
Not drawn accurately
Work out the values of x and y.
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........................................................................................................
Answer x = ...............degrees y = .................degrees (Total 3 marks)
Q12. (a) Triangle ABC is isosceles. AB = AC. Angle ACB = 80°.
Not drawn accurately
Work out angle BAC.
........................................................................................................
........................................................................................................
........................................................................................................
Answer .....................................degrees (2)
(b) In the diagram Angle QPS = 90° PQ = PR = PS Angle PSR = 70°
Not drawn accurately
Work out angle PRQ.
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.....................................................................................
.....................................................................................
Answer .....................................degrees (3)
(Total 5 marks)
Page 8
Q13. (a) Triangle PQR is isosceles. PQ = PR.
Work out the value of x.
……….............................................................................................
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Answer ................................................. degrees (3)
(b) Explain why the sum of the interior angles of any quadrilateral is 360°.
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........................................................................................................ (2)
(Total 5 marks)
Page 9
Interior and Exterior Angles
Q14. (a) Explain why the exterior angle of a regular pentagon, marked p on the diagram, is 72°.
........................................................................................................
........................................................................................................ (1)
(b) Two identical regular pentagons are joined as shown.
Not drawn accurately
Work out the size of angle x.
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Answer ...................................degrees (2)
(Total 3 marks)
Q15. ABCDE is a regular pentagon.
Not drawn accurately
Work out the value of x.
........................................................................................................
.......................................................................................................
……................................................................................................
Answer x = .................................................degrees (Total 4 marks)
Q16. The diagram shows part of a regular polygon. Each interior angle is 162°.
Calculate the number of sides of the polygon.
........................................................................................................
........................................................................................................
Answer ................................................. (Total 3 marks)
Page 10
Q17. The diagram shows a regular pentagon and a regular decagon joined at side XY.
Not drawn accurately
Show that the points A, B and C lie on a straight line.
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........................................................................................................ (Total 5 marks)
Q18. ABCDEF is a regular hexagon.
AFGH and AJKB are squares.
Not drawn accurately
Show that triangle AHJ is equilateral.
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………………………………………………...................................... (Total 4 marks)
Page 11
Q19. (a) Calculate the size of an interior angle of a regular octagon.
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Answer ................................................. degrees (3)
(b) Part of a tiled floor is shown.
The tiles labelled P, Q, R and S are regular octagons.
Explain why the tile labelled X is a square.
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........................................................................................................ (3)
(Total 6 marks)
Midpoint of a line
Q20. The diagram shows the points P (0, –4) and Q (5, 2).
Find the coordinates of the mid-point of the line segment PQ.
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Answer ( ........................... , ........................... ) (Total 2 marks)
Page 12
Parallel Lines
Q21. AC and DG are parallel lines. Angle ABE = 40° Angle BFG = 110°
Not drawn accurately
(a) Explain why angle BEF is 40°
........................................................................................................
........................................................................................................
........................................................................................................ (1)
(b) Show, giving reasons, that triangle BEF is isosceles.
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........................................................................................................ (3)
(Total 4 marks)
Q22. In the diagram AB is parallel to DC. Angle ABC = 70° Angle ACD = 55°
Not drawn accurately
Show that triangle ABC is isosceles. You must give reasons in your working.
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........................................................................................................ (Total 2 marks)
Page 13
Q23. In the diagram AB and CD are parallel.
Not drawn accurately
(a) Write down the value of x.
Answer ................................................... degrees (1)
(b) Work out the value of y.
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........................................................................................................
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Answer .................................................. degrees (2)
(Total 3 marks)
Area and Perimeter
Q24. (a) On each of these centimetre grids draw a different rectangle. Each rectangle must have an area of 12cm2.
Rectangle A
Rectangle B
(2)
(b) Work out the difference between the perimeter of rectangle A and rectangle B. You must show your working. .......................................................................................................
........................................................................................................
Answer ......................................... cm (2)
(Total 4 marks)
Page 14
Q25. (a) The diagram shows a rectangle drawn on a centimetre grid.
Work out the perimeter of the rectangle.
........................................................................................................
Answer ................................................................. cm (1)
(b) The perimeter of a square is 12 cm. Draw the square on the grid below.
(2) (Total 3 marks)
Q26. (a) Plot four points A, B, C and D on the grid to make a rectangle ABCD of length 6 cm and width 4 cm. (2)
(b) Tick whether each statement is always true, sometimes true or never true.
(i) Rectangles with an area of 24 cm2 have a length of 6 cm.
Always true Sometimes true Never true
(ii) Rectangles with a perimeter of 20 cm have a length of 12 cm.
Always true Sometimes true Never true
(iii) Rectangles with length 6 cm and width 4 cm have area 24cm2 and perimeter 20 cm.
Always true Sometimes true Never true (Total 5 marks)
Page 15
Q27. A rectangle has an area of 40 cm2 and a perimeter of 26cm. Find the length and width of the rectangle. You may use the grid to help you.
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Answer Length ........................................... cm
Width ............................................. cm (Total 2 marks)
Q28. Large areas can be measured in hectares. 1 hectare is 10 000 m2.
(a) Explain why the diagram represents 1 hectare.
.............................................................................................................
............................................................................................................. (1)
(b) This L–shape has an area of one hectare.
All lengths are a whole number of metres.
Not drawn accurately
Work out the value of a.
Give your answer in metres.
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Answer ........................................... m (3) (Total 4 marks)
Page 16
Q29. The diagram shows five shapes, A, B, C, D and E drawn on a grid.
Put the shapes in order of area, starting with the smallest. The smallest and largest are done for you.
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Answer D ................ , ................ , ................, A (Total 2 marks)
Q30. A circle of diameter 60 cm is cut out of a square of side 80 cm.
Not drawn accurately
Calculate the shaded area.
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Answer ............................................cm2
(Total 3 marks)
Q31. A and B are two interlocking shapes as shown.
Complete the following using greater than or less than or equal to
(a) The perimeter of A is ............................................. the perimeter of B.
(b) The area of A is ...................................................... the area of B. (Total 2 marks)
Page 17
Q32. ABCD is a square. PQRS is a square with vertices on the sides of ABCD. AS = DR = CQ = BP = 9 cm PA = SD = RC = QB = 1 cm
What is the area of the shaded square PQRS?
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Answer ................................................................ cm2
(Total 3 marks)
Q33. A shop sells square carpet tiles in two different sizes.
(a) What is the area of a small carpet tile?
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Answer ................................................. cm2
(2)
(b) What is the length of a side of a large carpet tile?
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Answer ................................................. cm (1)
(c) The floor of a rectangular room is 300 cm long and 180 cm wide.
How many small tiles are needed to carpet the floor?
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Answer ................................................. (3)
(Total 6 marks)
Page 18
Q34. Some shapes are drawn on a 1 centimetre triangular grid.
(a) Find the perimeter of shape D.
........................................................................................................
Answer ................................................. cm (1)
(b) Which two shapes have the same perimeter?
........................................................................................................
Answer ................................................. (1)
(c) Which two shapes have the same area?
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Answer ................................................. (2) (Total 4 marks)
Q35.(a) Two squares of side 4cm are removed from a square of side 12cm.
Work out the shaded area.
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Answer ................................................. (3)
(b) Two squares of side x cm are removed from a square of side 3x cm.
Work out the fraction of the large square which remains. Give your answer in its simplest form. You must show your working.
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Answer ................................................. (3) (Total 6 marks)
Page 19
3D Solids
Q36. The diagram shows a cube of side 2 cm.
(a) How many faces does a cube have?
Answer ................................................. (1)
(b) Draw an accurate net of this cube on the grid below.
(3)
(Total 4 marks)
Q37. Which three of the following are nets of a cube?
Answer ................................................. (Total 2 marks)
Q38. Three faces of this cube have shaded triangles on them. The other three faces are blank.
Draw the shaded triangles on the net above. (Total 3 marks)
Page 20
Q39. This solid is made from five cubes. The plan view shows the number of cubes in each stack.
(a) These solids are also made from five cubes.
Complete the numbers in each stack for each solid.
(2)
(b) Draw the solid for this plan view.
(2)
(Total 4 marks)
Page 21
Q40. The diagram represents a solid made from 9 small cubes.
The view of the solid from direction A is shown below.
On the grid below, draw the view of the solid from direction B.
(Total 2 marks)
Page 22
Volume & Surface Area Q41. Centimetre cubes are fitted together to make a solid as
shown on the left.
The solid is packed into a box as shown on the right.
The box is a cuboid.
Work out the volume of the box.
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Answer ........................................... cm3 (Total 3 marks)
Q42. The diagram shows a can of Baked Beans.
Not drawn accurately
The cans are delivered to shops in cardboard boxes. Each box contains 48 cans.
Not drawn accurately
Work out suitable dimensions for one of these cardboard boxes.
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Answer ........... mm by ........... mm by .......... mm (Total 3 marks)
Page 23
Q43. This cuboid is made from centimetre cubes.
(a) Explain why the total number of cubes needed to make this cuboid is 60.
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........................................................................................................ (1)
(b) The 60 centimetre cubes are separated. They are then used to make cubes, 2 cm by 2 cm by 2 cm.
How many of these complete cubes can be made?
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Answer ................................................ (2) (Total 3 marks)
Q44. A school hall is in the shape of a cuboid.
(a) The school hall is 30 m long, 12 m wide and 4 m high. Calculate the volume of the hall.
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Answer ................................................. m3
(2)
(b) The school buys ten 5 litre tins of paint to paint the hall. The area to be painted is 279 m2. Each tin covers 30 m2. Calculate the percentage of paint used.
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Answer .................................................. % (3)
(Total 5 marks)
Page 24
Q45. A large carton contains 4 litres of orange juice. Cylindrical glasses of height 10 cm and radius 3 cm are to be filled from the carton.
How many glasses can be filled? You must show all your working.
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Answer ...................................... glasses (Total 5 marks)
Q46. A water container is in the shape of a cuboid.
Its base is 20 cm by 20 cm and the depth of the water in the container is 15 cm.
Tony adds 1000 cm3 of water to the container.
Not drawn accurately
Calculate the new depth, d, of the water, in centimetres.
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Answer ........................................... cm (Total 4 marks)
Page 25
Q47. (a) The diagram shows a circle of radius three metres.
Not drawn accurately
Work out the area of the circle. Give your answer in terms of π.
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Answer ............................................ m2 (2)
(b) The diagram shows a cylindrical water tank. The cross-section of the tank is a circle of radius three metres.
The depth of water in the tank is 0.5 metres.
Not drawn accurately
Calculate the volume of water in the tank. Give your answer in terms of π.
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Answer .............................................. m3 (2) (Total 4 marks)
Page 26
Transformations
Q48. Triangle T is drawn on this grid. Draw the image of triangle T after a 90° anticlockwise rotation about O.
(3)
(b) The triangle T is reflected to form a new triangle S. The coordinates of S are (–4, 4), (–3, 3), and (–4, 1). Work out the equation of the mirror line.
.......................................................................................................
Answer .............................................. (2) (Total 5 marks)
Q49. Square A can be transformed onto Square B by
(a) a translation or (b) a reflection or (c) a rotation.
Describe each of these transformations fully.
(a) Square B is a translation of Square A by................................. (1)
(b) Square B is a reflection of Square A in .................................... (1)
(c) Square B is a rotation of Square A of...........................................
………………...................................................................................... (2)
(Total 4 marks)
Page 27
Q50. (a)
(i) Describe fully the single transformation that takes the shaded triangle to triangle A.
........................................................................................................
........................................................................................................ (2)
(ii) On the grid above translate the shaded triangle by 2 squares to the right and 4 squares down.
(1)
(b) Triangle P is an enlargement of the shaded triangle.
(i) What is the scale factor of the enlargement?
Answer ................................................. (1)
(ii) What is the centre of enlargement?
Answer (......................., ......................) (1)
(Total 5 marks)
Page 28
Q51. The diagram shows two identical shapes, A and B.
Describe fully the single transformation which takes shape A to shape B.
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........................................................................................................ (Total 3 marks)
Q52. The diagram shows a shaded flag.
(a) Rotate the shaded flag 90° anticlockwise about the origin. Label this new flag with the letter A. (3)
(b) Translate the original shaded flag 2 units to the right and 3 units down. Label this new flag with the letter B. (1)
(c) Reflect the original shaded flag in the line y = 1. Label this new flag with the letter C.
(2) (Total 6 marks)
Page 29
Q53. Here are seven shapes made from small squares.
(a) Which two shapes are congruent?
Answer .............................. and .............................. (1)
(b) (i) Which shape is an enlargement of shape C?
Answer ....................................................................... (1)
(ii) What is the scale factor of the enlargement?
Answer ....................................................................... (1)
(c) On the grid, draw an enlargement of shape E by scale factor 3.
(3) (Total 6 marks)
Q54. Here are two rectangles, A and B.
Not drawn accurately
Is rectangle B an enlargement of rectangle A?
Tick the correct box.
Yes No
Explain your answer.
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........................................................................................................ (Total 3 marks)
Page 30
Q55. The vertices of triangle T are (1, 1), (1, 2) and (4, 1).
Enlarge triangle T by scale factor 2, with (0, 0) as the centre of enlargement. (Total 3 marks)
Page 31
Pythagoras
Q56. The diagram shows a right-angled triangle.
Not drawn accurately
Calculate the length x.
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Answer ......................................... cm (Total 3 marks)
Q57. The diagram shows two triangles, P and Q.
Not drawn accurately
In which of these triangles does a2 + b2 = c2? Explain your answer.
Triangle .................................. Explanation..............................
........................................................................................................
.......................................................................... (Total 2 marks)
Q58. A support for a flagpole is attached at a height of 3 m and is fixed to the ground at a distance of 1.2 m from the base.
Calculate the length of the support (marked x on the diagram).
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Answer ........................................... m (Total 3 marks)
Page 32
Q59. (a) Ali uses this method to estimate the height of a flag pole.
▪ He stands, as shown, so that his angle of sight is 45° when he
looks up to the top of the flag pole.
▪ He then measures his distance from the flagpole.
▪ Finally he measures the distance that his eyes are above the ground.
This sketch shows Ali’s measurements.
Not drawn accurately
Use Ali’s measurements to calculate the height of the flag pole, explaining why he uses an angle of 45°.
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Answer ........................................... m (2)
(b) Ben uses this method to estimate the height of a building.
▪ He tapes a 1 metre ruler to the building.
▪ He takes a photograph of the building and the metre ruler.
On the graph he measures the height of the building and the length of the ruler. The measurements are shown on this sketch.
Use this information to estimate the height_ of the building. Give your answer to the nearest metre.
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Answer ..................................... metres (3) (Total 5 marks)
Page 33
Q60. (a) The right-angled triangle has sides shown.
Not drawn accurately
Show that x = 9 cm
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........................................................................................................ (2)
(b) This right-angled triangle has sides n, m and n + 1. m and n are integers.
Prove that m must be an odd number
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...................................................................................................... (5) (Total 7 marks)
Q61. Triangle ABC has a right angle at B.
Angle BAC = 38° AB = 7.21 cm
Not drawn accurately
Calculate the length of BC.
Give your answer to an appropriate degree of accuracy.
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Answer ............................................ cm (Total 4 marks)
Page 34
Measures & Scales
Q67. This scale shows pints and litres.
(a) Draw an arrow on the scale to show 2.5 pints. (1)
(b) Use the scale to estimate how many pints are in 1 litre
Answer ................................................. pints (1)
(c) Estimate the number of litres in 8 pints.
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Answer ................................................. litres (2) (Total 4 marks)
Q68. The weight of a 2p coin is 7g. Find the weight of £10 worth of 2p coins. Give your answer in kilograms.
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Answer ................................................. kilograms (Total 4 marks)
Q69. Give the values shown by the arrows on these scales.
(a)
Answer ................................................. cm (1)
(b)
Answer ................................................. kg (1)
(c)
Answer ................................................. mph (1) (Total 3 marks)
Page 35
Q70. Which metric unit would you use to measure the following?
(a) The length of a pencil
Answer ................................................. (1)
(b) The amount of petrol in a car’s tank
Answer ................................................. (1)
(c) The area of a football pitch
Answer ..................................................... (1)
(d) The weight of a bus
Answer ................................................. (1) (Total 4 marks)
Page 36
Speed
Q71. Susan completes a journey in two stages. In stage 1 of her journey, she drives at an average speed of 80 km/h and takes 1 hour 45 minutes.
(a) How far does Susan travel in stage 1 of her journey?
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Answer ................................................. km (2)
(b) Altogether, Susan drives 190 km and takes a total time of 2 hours 15 minutes. What is her average speed, in km/h, in stage 2 of her journey?
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Answer ................................................. km/h (2) (Total 4 marks)
Q72. (a) Change a speed of 72 kilometres per hour into miles per hour.
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Answer .............................................. miles per hour (2)
(b) A car travels 200 kilometres in 3 hours 30 minutes. Calculate its average speed in kilometres per hour. Give your answer to an appropriate degree of accuracy.
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Answer ..................................... kilometres per hour (4) (Total 6 marks)
Q73. Harry drives 182 miles. His average speed is 35 miles per hour.
How long does the journey take? Give your answer in hours and minutes.
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Answer ...................... hours ..................... minutes (Total 4 marks)
Page 37
Q74. The diagram shows a map of three paths AB, AC and AD through a wood.
(a) A rambler wants to walk towards her house from point A. Her house is to the north-west of the wood. Which path should she take?
Answer .................................................. (1)
(b) A warden wants to know the length of the path from D to A. He walks along the path. It takes him 40 minutes. He knows that he walks at 3 miles an hour.
How long is the path?
........................................................................................................
Answer ........................................ miles (2)
(c) Measure the three-figure bearing of D from A.
Answer ............................................... ° (1) (Total 4 marks)
Bearings, Loci and Construction
Q75. There are two TV transmitters on an island. The transmitter at A has a range of 40 km. The transmitter at B has a range of 60 km.
Show clearly the area in which the signal from both transmitters can be received.
(Total 3 marks)
Page 38
Q76. The diagram shows an island with North lines drawn at points A and B.
(a) Treasure is buried on a bearing of 037° from A and 290° from B. Mark, with a ×, the position of the treasure.
(3)
(b) Find the real distance between the points A and B.
.................................................................................................................
Answer ................................................. km (3)
(Total 6 marks)
Q77. A is due North of B. The bearing of C from A is 115°. The bearing of C from B is 075°.
Mark the position of C on the diagram. (Total 3 marks)
Page 39
Q78. The diagram shows a port P and two lighthouses A and B on the coast.
(a) (i) A fishing boat sails to A from P. What is the three-figure bearing of A from P?
Answer .......................................................... degrees (1)
(ii) A yacht sails to B from P.
What is the three-figure bearing of B from P?
Answer .......................................................... degrees (1)
(b) A ship leaves the port, P, on a course that is an equal distance from PA and PB. Using ruler and compasses only, construct the course on the diagram. You must show your construction arcs. (2)
(Total 4 marks)
Q79. (a) The line LM is drawn below.
Use ruler and compasses to construct the perpendicular bisector of LM. You must show clearly all your construction arcs.
(2)
(b) Complete the sentence.
The perpendicular bisector of LM is the locus of points which are…
................................................................................................................. (1)
(Total 3 marks)
Q80. Using ruler and compasses only, construct the bisector of angle PQR.
(Total 2 marks)
Page 40
Q81. In the space below, make an accurate drawing of this triangle. The base line has been drawn for you.
(Total 2 marks)
Q82. (a) Using a ruler and compasses only, construct an angle of 60°. Show all your construction lines and arcs.
(2)
(b) Two lifeboat stations A and B receive a distress call from a boat. The boat is within 6 kilometres of station A. The boat is within 8 kilometres of station B. Shade the possible area in which the boat could be.
(2)
(Total 4 marks)
Page 41
Page 42
M1. (a) Draws any rhombus
Accuracy of 3 mm. Angle between sides must not be 90
B1
(b) Rhombus
Not square, diamond, oblong B1 ft
[2]
M2. (a) Trapezium B1
(b) Rectangle B1
(c) Rhombus B1
[3]
M3. (a) Rhombus B1
(b) Diagonals cross at right angles;
One pair of opposite angles equal.
–1 eeoo
SCI if only two more lines are drawn and one is correct
B2
[3]
M4. (a) A Parallelogram B1
B Rhombus
Allow parallelogram for B if not given for A
B1
(b)
Page 43
oe B1
Trapezium
No ft for square or rectangle No ft for parallelogram or rhombus if given in (a)
B1 ft
[4]
M5. Note Mark the method that gives the best score Do not award M1 if either B1 clearly comes from incorrect assumptions
(e.g. BAE = 90) SC3 complete method with 1 arithmetic error
( BAD) → 70 B1
( BAE) → 100 B1
(their 100) – (their 70) M1
30
A1
( ADC) → 250 B1
360 – 250 – 50 or 60 B1
(their 60) ÷ 2 M1
30
A1
( ADB) → 55 or ( ADE) → 125 B1
( AED) → 25 B1
(their 55) – (their 25) or 180 – (their 125) – (their 25)
Page 44
M1
30
A1
( CAD) → 35 B1
( CAE) → 65 B1
(their 65) – (their 35) M1
30
A1
[4]
M6. (a) (i) Kite B1
(ii) Trapezium B1
(b) Rectangle drawn B1
(c) Equilateral triangle drawn
2 possible sizes B1
(d) P = 2 × 3 + 2 × 5.2
6 + 10.4, 2 × 8.2 M1
16.4 A1
(e) Method 1 Attempt to compare using equilateral triangles/rhombi
Method 2 Using formulae
Method 1 eg, 2 bottom halves equal and lines drawn Method 2 eg, b × h for
rhombus or for triangle
B1
Complete argument Method 1 Show that both top
Page 45
halves are of a rhombus or are the same
Method 2 Using both formulae and triangle has double the base (or height_) oe
B2 Complete hexagon on diagram and show each is 1/3 of hexagon
B1
[8]
M7. (a) Equilateral (triangle) B1
(b) (i) Rhombus B1
(ii) 2
Accept in words B1
(iii) 2 diagonals drawn
–1 eeoo B2
[5]
M8. (a) Correct sketch with sides marked
Do not accept equilateral triangles
B1
(b) Any 2 of rectangle, parallelogram, arrowhead or kite
B1 for 1 correct B2
(c) The 3 cm rod and the 5 cm rod would not meet
oe eg, 3 + 5 < 9 B1
[4]
M9. (a) 80° and 20° B1
50° and 50° B1
(b) ∠BAD = 30° or
Page 46
any angle in Δ BCD = 60° B1
∠ABD = 30° B1
Isosceles because ∠BAD = ∠ABD
oe B1
[5]
M10. (a) (180 – 34) ÷ 2 M1
73 A1
(b) 180 – (38 + 34 their x) M1
35
Their 73 – 38 A1
(c) No, because 38 ≠ 35
oe angles are not the same ft their answer to y but not 38
B1
[5]
M11. M1
x = 27
A1
153
126 + their x, 180 – their x B1 ft
[3]
M12. (a) CBA = 80° B1
BAC = 20° B1
(b) RPS = 40°
Page 47
B1
QPR = 50°
90 – Their RPS B1ft
65°
(180 – Their 50) ÷ 2 B1ft
[5]
M13. (a) 180 – 48 ( = 132)
Provided that the candidate has not used R = 48°
M1
Their 132 ÷ 2 (= 66) DM1
180 – 66 = 114
66 + 48 = 114 scores A1 A1 cao
(b) Angle sum of triangle = 180° B1
Quadrilateral = 2 triangles
Quadrilateral = 4 triangles –
360° or 4 × st. lines – sum of ext angles i.e. 4 ×180 – 360
B1
[5]
M14 (a) 5 (equal) exterior angles must total 360° and 360 ÷ 5 = 72 or 5 × 72 = 360
360 ÷ 5 = 72 is not enough … there must be some reference to exterior angles
B1
(b) 2 × 72 or 360 – (2 × 108)
oe M1
(x =) 144 A1
[3]
Page 48
M15. 360/5
540/5 M1
72 or 108 seen A1
(180 – their 108)/2
108 – 72 or 180 – 72 – 72 M1
36 A1
[4]
M16. 180 – 162 or 18
(n – 2) × 180 = 162n M1
360 ÷ their 18 M1 dep
x = 20 A1
[3]
M17. 360 ÷ 10 or 360 ÷ 5
or 36 or 72 or 144 or 108
NB Angles may be marked on diagram
M1
144 and 108
or 36 and 72 A1
∠BXC = 360 – (144 + 108)
or ∠BXD = 36 + 72 or 108
(X is point where decagon and pentagon meet between B and C)
M1
∠XBC = ∠XCB = (180 – 108) ÷ 2
or ∠XBC = 36 M1
∠ABX + ∠XBC = 144 + 36 ( = 180)
oe
eg, ∠CBX calculated from
ΔBXC equals exterior angle of decagon
A1
[5]
Page 49
M18. BAF = 120°
This can just be stated
or exterior angle of hexagon = 60°
or reflex FAB = 240° B1
360 – (120 + 90 + 90) = 60°
oe
HAJ must be shown to be 60° by calculation
B1
AH = AJ
This can just be stated or shown on diagram
B1
AJH = AHJ = (180 – 60) ÷ 2
Dep on first B2 B1dep
[4]
M19. (a) 360 ÷ 8
or 45 seen or 6 × 180 or 1080 or (2 × 8 – 4) right angles
M1
180 – (their 45)
(their 1080) ÷ 8 M1 dep
135
135 A1
(b) 360 – (their 135 + 135) or 2 × 45 M1
90° in X A1
Sides of X are equal
or (regular) octagons so sides are equal
4 lines of symmetry or rotational symmetry of order 4 scores 3 marks Other symmetry scores B1
B1
[6]
Page 50
M20.
or evidence of good use of grid
M1
2.5, –1
Take one or other value correct as evidence for the M1 SC1 for (–1, 2.5)
A1
[2]
M21. (a) Alternate
Do not accept ‘Z angle’ B1
(b) Full explanation
Angle BFE = 70 (straight line)
Angle EBF = 70 angles in a triangle
Hence isosceles as angles same
E3
Partial explanation missing one salient point
Angle BFE = 70 (straight line)
Angle EBF = 70
Hence isosceles as angles same
E2
Partial explanation missing two salient points
Angle BFE = 70 (straight line)
Angle EBF = 70
Hence isosceles E1
[4]
M22. ∠BAC = 55° and resaon
∠BCA = 55° and reason B2
∠BCA = 55°
∠BAC = 55° B1
B1 For each angle and B1 for complete reason why one of them is 55°.
For example:
Page 51
• ∠BCA and ∠ACB are alternate (not Z
angles)
• ∠ABC and ∠BC? are alternate and stating
that sum of ∠s on straight
line = 180 to find ∠BCA from 180 – 70 – 55
• stating that sum of ∠s in Δ = 180 to find either
angle from 180 – 70 – 55
Assuming that Δ ABC is isosceles scores zero [2]
M23. (a) 41 B1
(b) 180 – 67
71 + their 41 oe 360 – 41 – 67 – 139
M1
113 A1
[3]
M24. (a) Two different rectangles drawn with area 12 cm2
eg, 1 by 12 or 2 by 6 or 3 by 4 Allow eg, 8 by 1.5
B1 Any one rectangle drawn with an area of 12 cm2
or
Two different rectangles drawn with the same area
B2
(b) Both (their) perimeters correct
eg, 26, 16 or 14 B1 ft
Correct subtraction of (their) perimeters
eg, 10, 12 or 2
Award this mark only if one or both of (their) perimeters are correct.
B1 ft
[4]
M25. (a) 20 B1
(b) 3 by 3 square drawn
B1 12 ÷ 4 (= 3)
or 5 by 1 or 4 by 2 rectangle drawn
B2
[3]
Page 52
M26. (a) Fully correct rectangle
B1 for one correct side B2
(b) (i) Sometimes true B1
(ii) Never true B1
(iii) Always true B1
[5]
M27. Length 8 and width_ 5
allow8 by 5 rectangle drawn or B1 rectangle with area 40 or B1 rectangle with perimeter 26 cm
B2
[2]
M28. (a) 1 km = 1000 m or area = 1000 × 10 = 10 000 m2
B1
(b) 200 or 7000 seen B1
7000 ÷ 200 M1
35 A1
[4]
M29. C, B, E
Any two in order B1
ie, BEC, ECB, CEB, BCE B2
[2]
M30. π × 302 (2827) M1
80 × 80 – ‘Their 900π’ M1dep
3570 to 3574 A1
[3]
M31. (a) Equal to B1
(b) Less than B1
[2]
M32. Area of triangle = × 9 × 1 or 4.5
Length of square = √(92 + 12) M1
100 – 4 × their 4.5
√82 M1 dep
82 A1
[3]
Page 53
M33. (a) 30 × 30 M1
900 A1
(b) 50 B1
(c) 300 ÷ 30 or 180 ÷ 30
or 300 × 180 M1
their 10 × their 6
or their 54 000 ÷ their 900 M1
60 A1
[6]
M34. (a) 8 B1
(b) A&C B1
(c) Attempt to find area
Lines on diagram making triangles or rhombi; correct number of triangles/rhombi in two or more shapes: 12, 7, 8, 8 or 6,3 ½, 4, 4
M1
D&C A1
[4]
M35. (a) 122 (–) 2 × 42
oe M1
112 A1
cm2
Units mark B1
(b) 9x2
or attempt to use their 112 and 144
B1
Attempt to calculate shaded area (= 7x2)
or
(3x × 3x) (–) 2(x × x) M1
Note: score B1M1A0 (unshaded)
A1
Page 54
[6]
M36. (a) 6 B1
(b) Correct net
B1 for 4 squares in a row or column B2 for correct net for open-topped cube ( ±2 mm) SC1 for correct net in correct scale factor
B3
[4]
M37. A B and E
–1 eeoo B2
[2]
M38.
oe eg,
B2 For 2 correct and 1 in correct position
but in wrong orientation
B1 For 2 correct and 1 in wrong position
B3
[3]
M39. (a) (1) 2 2 B1
1 2 1 1
B1
(b) Correctly drawn solid
B1 Correct solid incorrect orientation
Allow with no shading
Penalise (–1) incorrect shading
B2
[4]
M40.
B1 with 1 error
SC1 90° degree rotation B2
[2]
Page 55
M41. 4 × 2 or 2 × 2 or 8 or sight of 4, 2 and 2 on diagram
M1
4 × 2 × 2 or 8 × 2 or 4 × 4 or 8 + 8
M1 dep
16 A1
[3]
M42. Number of cans in length (L) Number of cans width_ (W) Number of cans in height_ (H) (LWH = 48) For example L = 8, W = 2, H = 3 L = 4, W = 4, H = 3 L = 6, W = 4, H = 2 L = 12, W = 4, H = 1 L = 16, W = 3, H = 1 L = 12, W = 2, H = 2
Not L = 48, W = 1, H = 1 M1
Calculating dimensions from: (their) L × 74 or 75 (their) W × 74 or 75 (their) H × 108 or 110
Award this mark for two correct dimensions from × 74 (75) and × 108 (110) with L, W and H any factors of 48 apart from 1
(not 74 and 108)
Allow rounded lengths (75 and 110)
M1
For example 592 by 148 by 324 or 296 by 296 by 324 or 444 by 296 by 216
Allow rounded lengths
eg, 600 by 150 by 330 or 300 by 300 by 330 or 450 by 300 by 220
Not 3552 by 108 by 74 oe A1
[3]
M43. (a) 5 × 4 × 3
oe
eg, A “layer” of 15 × 4 A “layer” of 20 × 3 A “layer” of 12 × 5
B1
(b) 60 ÷ 8
or 7.5
or 7 × 8 = 56 and 8 × 8 = 64 M1
7 A1
[3]
Page 56
M44. (a) 30 × 12 × 4 M1
1440 A1
(b) 10 × 30
or 279 ÷ 30 (× 5) M1
279 ÷ (their 300) × 100
or (their 9.3) ÷ 10 × 100 or (their 46.5) ÷ 50 × 100
M1
93
SC2 for 7 A1
[5]
M45. Sight of 4000
B1 may be awarded later for dividing their cm3 answer by 1000
B1
Vol of cup = π × 32 × 10 M1
= 282.7 (433388)
Accept 280 ≤ vol ≤ 283 A1
(their 4000) ÷ (their 282.7(....)) DM1
= 14(.14710...)
= 14
A1
[5]
M46. 20 × 20 × 15 (6000)
or M1
Their 6000 + 1000
2.5 A1 ft
Their 7000 ÷ (20 × 20)
15 + Their 2.5 M1 dep
17.5
T & I can get all 4 marks A1
[4]
M47. (a) π × 32
3.1(4...) × 32
M1
9π
Accept 9 (×) π or π (×) 9
Do not accept fw A1
(b) π × 32 × 0.5 or 9π × 0.5 or their (a) × 0.5
3.1(4...) × 32 × 0.5
Page 57
π not needed for M1 ft M1
4.5π
Accept 4.5 (×) π or π (×) 4.5
Answer must be in terms of π A1ft
[4]
M48. (a) Fully correct rotation
B1 180° rotation with centre 0
B1 90° clockwise rotation with wrong centre
B2 90° clockwise rotation with centre 0
B2 90° anticlockwise rotation with wrong centre
B3
(b) x = –1
B1 for coordinates plotted or line shown on graph
B2
[5]
M49. (a) 6 right, 6 down
or as vector B1
(b) y = x B1
(c) Half turn or 180°
90° or 270° anti-clockwise or clockwise about (–3, –3) or (3, 3)
B1
Centre (0, 0) or origin
Check alternative fully correct for 2 marks, 2 parts correct for 1 mark.
B1
[4]
M50. (a) (i) Reflection
Accept mirror image or mirrored but NOT mirror or flip
B1
in x = –1
MUST give equation of mirror line
B1
(ii) correct triangle
(b) (i) B1
(ii) (–2, –1)
1 mm tolerance on reading from graph
Page 58
B1
[5]
M51. Rotation
accept rotational symmetry B1
180 B1
(About) origin
oe B1
[3]
M52. (a) Any 90° rotation
Allow wrong length of flagpole B1
90° anticlockwise about (0,0)
B1, 90° clockwise about (0,0);
No labels award best possible mark;
no pole, correct position of flag,
–1 each time B2
(b) Fully correct B1
(c) Fully correct
ie flag drawn at (1,0), (1,–2),
(1,–3), (2,–2), (2,–3)
Reflected in x = l ,B1
Reflected in y = c, B1 B2
[6]
M53. (a) B and F B1
(b) (i) A B1
(ii) 2
Accept × 2 but not 1:2 or 2:1 B1
(c) Shape 9 squares wide or 3 squares high B1
Shape 9 squares wide and 3 squares high B1
Fully correct
SC2 SF2 or SF4 fully correct B1
[6]
M54. 12.5 ÷ 5 or 16.5 ÷ 7 or 2.5 or 2.3(....) or 2.4
oe M1
12.5 ÷ 5 and 16.5 ÷ 7 or 2.5 and 2.3(....) or 2.4
or their 2.5 × 7 or 17.5
or their 2.3(....) or 2.4 × 5 or 11.5 - 12 inc
M1
Page 59
No ticked and 12.5 ÷ 5 ≠ 16.5 ÷ 7 or 2.5 ≠ 2.3(....) or 2.4
or 17.5 ≠ 16.5
or 11.5 - 12 inc ≠ 12.5
oe eg, the lengths are different
A1
[3]
M55. Fully correct [(2, 2), (2, 4), (8, 2)]
B2 Enlargement scale factor 2 B1 Any enlargement or 2 points correct
B3
[3]
M60. (a) Fully correct rotation
B1 180° rotation with centre 0
B1 90° clockwise rotation with wrong centre
B2 90° clockwise rotation with centre 0
B2 90° anticlockwise rotation with wrong centre
B3
M61. 62 + 2.52
or 42.25 seen M1
√(their 42.25) M1dep
6.5 A1
[3]
M62. P and valid explanation.
eg, a clear indication that P has a right angle and Q does not (all angle calculations shown must be correct) and a2 + b2 = c2
(or Pythagoras’ rule) only works in a right angled triangle
B1 An incomplete/missing/incorrect explanation and
Page 60
either:
P and 90° (and 100°) correct and shown
or P (and 90°) and 100° correct and shown
or (P and) 90° and 100° correct and shown
B2
[2]
M63. 1.22 + 32 (= 10.44)
Must add two squares M1
√(Their 10.44)
Dependent on first M1 DM1
3.2(3......) A1
[3]
M64. (a) 10.15 B1
(Forms an) isosceles triangle
oe B1
(b) 9.8 ÷ 2(.0)
100 ÷ 2(.0) (× 9.8) or 50 (× 9.8)
Allow 10 ÷ 2 and 5 × 9.8
Condone attempts to change to different units by multiplying/dividing by 10, 100, ...
M1
4.9
or 490 A1
5 B1 ft
[5]
M65. tan 38 = BC/7.21
or BC ⁄ sin 38 = 7.21 ⁄ sin 52
Page 61
M1
7.21 × tan 38
or 7.21 × sin 38 ÷ sin 52 M1 dep
5.6 (3 …) A1
5.6 or 5.63 B1 ft
[4]
M66. (a) x2 = 412 – 402
M1
x2 = 81 or x = (= 9) A1
(b) (n + 1)2 – n2 = m2
M1
n2 + 2n + 1 – n2 = m2
M1 dep
m2 = 2n + 1 A1
m2 is odd since 2n + 1 is odd
A1
m is odd since odd × odd = odd A1
[7]
M67. (a) Between 2.4 and 2.6 exclusive B1
(b) 1.7 to 1.8 inclusive B1
(c) 4 × value at 2 pints
oe 8 × value at 1 pint; 8 ÷ their value in (b); continuation of upward scale M0
M1
4.3 to 4.8 A1
[4]
Page 62
M68. 10 × 50 or 500
or 50 × 7 or 350 M1
(their 500) × 7 or 3500
or (their 350) × 10 M1
(their 3500) ÷ 1000
or correct conversion (their g) → kg
M1
3.5
oe A1
[4]
M69. (a) 76
± 0.2 B1
(b) 340
± 2 B1
(c) 87
± 0.5 B1
[3]
M70. (a) mm or cm
Accept equivalent Imperial units throughout, but penalise the first occurrence ( ≡ inches or in)
B1
(b) Litres
Accept l ( ≡ gallons or gal) B1
(c) m2 or hectares
( ≡ square yards or acres) B1
(d) kg or kilos or tonnes or Newtons
Accept tons ( ≡ pounds or lb or stone or cwt)
B1
[4]
Page 63
M71. (a) 72 ×
72 × 0.625, 72 ÷ 1.6 M1
45 A1
(b) 3.5
oe or 210 B1
× 60 M1
57.1(4) A1
Round their answer to 1dp or 0dp
57 or 57.1, or 60 with working B1 ft
[6]
M72. (a) 80 × 1.75
accept 80 × 1.45 and 80 × 105
M1
140 A1
(b) {190 – (their 140)} ÷ (2.25 – 1.75)
Or (their 50) ÷ 0.
Allow (their 50) in 30 minutes M1
100
ft from their (a) A1 ft
[4]
M73. Time is M1
= 5.2 hours
5 hours 20 minutes M1 A1
A1
Page 64
= (5 hours +) 0.2 × 60 min M1
= 5 hours 12 min A1
[4]
M74. (a) AC or C B1
(b) Scaling method used or 3 ×
eg, 1 mile in 20 minutes 3 × 40 ÷ 60
Do not accept 3 miles in 1 hour
M1
2 A1
(c) (0) 55
Tolerance ± 2° B1
[4]
M75. (a) AC or C B1
(b) Scaling method used or 3 ×
eg, 1 mile in 20 minutes 3 × 40 ÷ 60
Do not accept 3 miles in 1 hour
M1
2 A1
(c) (0) 55
Tolerance ± 2° B1
[4]
M75. Circle or part circle centred on both A and B
M1
Radii within ± 1 mm of 4 and 6 cm and large enough arcs to intersect
Check horizontally or vertically with printed grid
A1
Page 65
Correct region indicated
ft if one of circles within tolerance
A1 ft
[3]
M76. Allow embedded solutions, but if contradicted M marks only
(a) Bearing 037° ± 2 B1
290° ± 2 B1
Correct intersection of lines
Ignore any x. Within sq of grid intersection
B1
(b) 6 to 6.2 cm B1
Their 6.1 × 5 M1
30 to 31 A1
[6]
M77. C marked within limits of loci
B1 bearing from A ± 2°
B1 bearing from B ± 2° B3
[3]
M78. (a) (i) (0)25
±2° B1
(ii) 295
±2° B1
(b) Correct course (±2°) drawn with all construction arcs shown
B1 Incorrect or no course shown with all construction arcs attempted
or Correct course (±2°) drawn with some construction arcs shown, not arcs from A and/or B
B2
[4]
M79. (a) Equal arcs from L and M
Arcs greater than 0.5LM within 2 mm
Must have two intersections M1
Page 66
Perpendicular drawn A1
(b) Equidistant from 2 fixed points
oe B1
[3]
M80. Arcs on PQ and QR and equal intersecting arcs
Allow if arcs are drawn from points P and R
M1
Bisector accurate to ± 2°
59.5 – 63.5 A1
[2]
M81. Angle of 43° drawn (± 2°) or line 6.5cm drawn (± 2 mm) and ruled
B1
Complete correct triangle drawn within the tolerance shown on the overlay
B1
[2]
M82. (a) line and arc any radius B1
2nd arc same radius and 2nd line
±2° accuracy B1
(b) Both arcs intersecting
correct radius and region shaded or indicated
B1 for either arc, correct radius ± 2mm
B2
[4]
M32. 39 ÷ 3 or 39 ÷ 6 or 19.5 ÷ 3 or 19.5 ÷ 6
oe M1
13 or 6.5 seen A1
13 × 13 M1
169 A1
[4]
M29. (a) 20
B1
(b) 3 by 3 square drawn
B1 12 ÷ 4 (= 3)
Page 67
or 5 by 1 or 4 by 2 rectangle drawn
B2
[3]
