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)
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)

 
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)
 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)
 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.
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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)
 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.
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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)

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)

Interior and Exterior Angles
Q14. (a)     Explain why the exterior angle of a regular pentagon, marked p on the diagram, is 72°.

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........................................................................................................
(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.
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.......................................................................................................
……................................................................................................
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.
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Answer .................................................
(Total 3 marks)
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)
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)

 

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°
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(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)

 
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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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.
.......................................................................................................
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Answer ......................................... cm
(2)
(Total 4 marks)
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)
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)
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)
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)
Q34. Some shapes are drawn on a 1 centimetre triangular grid.

(a) Find the perimeter of shape D.
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Answer ................................................. cm			(1)
(b) Which two shapes have the same perimeter?
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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)
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)
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)

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)

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)
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)
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)
 
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)

 

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)
 
Q50. (a)     

(i) Describe fully the single transformation that takes the shaded triangle to triangle A.
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........................................................................................................
(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)
 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)
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)

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)

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..............................
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..........................................................................          (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)
 
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)
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)

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)
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)

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.
........................................................................................................
........................................................................................................
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.
........................................................................................................
........................................................................................................
........................................................................................................
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.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Answer ...................... hours ..................... minutes
(Total 4 marks)
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)

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)

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)

 

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)

 

 

 

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)     
          
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)
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 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
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°
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]

 

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]
 
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]

 
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:
•        ∠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]

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]
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
[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]
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]

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
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
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

          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 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
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]

 

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]

 

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
          = (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
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
          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)
or 5 by 1 or 4 by 2 rectangle drawn
B2
[3]