Web viewQ1. The time in minutes (T) for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters (W) working at

Q1. The time in minutes (T) for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters (W) working at that time.

It takes 20 minutes for meals to be served when 12 waiters are working.

(a) Find an equation connecting T and W.

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Answer .................................................(3)

(b) What is the minimum number of waiters that must be working for a meal to be served within 30 minutes?

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Answer .................................................(3)

(Total 6 marks)

Q2. In the diagram, O is the centre of the circle. A, B, C and D are points on the circumference. Angle AOC = 130°

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(a) Calculate the value of x. Give a reason for your answer.

Answer x = ..........................degrees

Reason ............................................................................................................

.........................................................................................................................(2)

(b) Calculate the value of y. Give a reason for your answer.

Answer y = ..........................degrees

Reason .............................................................................................................(2)

(Total 4 marks)

Q3. (a) Enlarge the shaded shape by a scale factor of 3.

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(2)

(b) How many times bigger is the area of the enlarged shape than the area of the small shape?

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Answer .................................................(2)

(Total 4 marks)

Q4. ABCD is a quadrilateral.

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AB = 7 cm, AD = 6 cm and BC = 9 cm. Angle ABC = 75° and angle ADC = 90°

Calculate the perimeter of ABCD.

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Answer ................................................. cm(Total 5 marks)

Q5. OACB is a parallelogram and M is the mid-point of BC.

= a and = b

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(a) Express the following vectors in terms of a and b

(i)

Answer .................................................(1)

(ii)

Answer .................................................(1)

(b) AM is extended to N, where .

Show that = b

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(c) What does this tell you about the position of N?

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(Total 5 marks)

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Q6. (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)

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Q7. The diagrams show a rectangle and an L shapeAll the angles are right angles.All lengths are in centimetres.The shapes are equal in area.

Calculate the value of y.

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Q8. (a) Complete the table of values for y = 3x2 – 6

x –3 –2 –1 0 1 2 3 4

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y 21 6 –3 –6 –3 21 42

(1)

(b) On the grid below, draw the graph of y = 3x2 – 6 for values of x between –3 and +4.

(2)

(c) Use your graph to write down the solutions of 3x2 – 6 = 0

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Answer ...................... and ......................(1)

(d) By drawing an appropriate linear graph, write down the solutions of

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3x2 – 5x – 6 = 0

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Answer .................................................(3)

(Total 7 marks)

Q9. The area of the screen of a television set is A square inches.The length of the diagonal of the screen is d inches.A is directly proportional to the square of d.

A television set with an area of 90 square inches has a diagonal of length 15 inches.

(a) Find an equation connecting A and d.

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Answer .................................................(3)

(b) Find the area of the screen of a television set with a diagonal of length 20 inches.

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Answer ................................................. square inches(1)

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(c) Another television set has a screen with an area of 250 square inches.

Find the length of its diagonal.

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Answer ................................................. inches(3)

(Total 7 marks)

Q10. The graph shows the function

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(a) Write down the coordinates of the point where the graph intersects with the y-axis.

Answer ( ........................ , ....................... )(1)

(b) Find the value of a.

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Answer .................................................(2)

(Total 3 marks)

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Q11. Two spheres of radius 5 cm just fit inside a tube.

Calculate the volume inside the tube not filled by the spheres.

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Answer ................................................. cm2

(Total 5 marks)

Q12. A sign maker designs a letter L.

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All arcs are quarter circles of radius 2 cm.

Not drawn accurately

Calculate the area of the L.

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Answer ................................................. cm2

(Total 4 marks)

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Q13. (a) P is inversely proportional to Q.When P = 100, Q = 32

Express P in terms of Q.

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Answer .................................................(3)

(b) P and Q are positive quantities.Sketch a graph of the relationship between P and Q on this diagram.

(1)

(c) Calculate the value of Q when P is twice as big as Q.

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Answer .................................................(2)

(Total 6 marks)

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Q14. A(1, 1) and B(–2, 4) are two points on the graph of y = x2

Here are three transformations of the graph y = x2.On each diagram the graph of y = x2 is shown dotted.The images A′ and B′ of A and B are shown.Write down the equation of the transformed graph in each case.

(a)

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y = ........................................(1)

(b)

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y = ........................................(1)

(c)

y = ........................................(1)

(Total 3 marks)

Q15. The diagram shows a solid made from a cone and a hemisphere.The radius of both shapes is r.The slant height of the cone is l.The perpendicular height of the cone is h.

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The curved surface area of the cone and the curved surface area of the hemisphere are equal.

(a) Show that l = 2r

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.........................................................................................................................(2)

(b) Find the perpendicular height, h, of the cone in terms of r.

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Answer h = ..............................................(2)

(c) Find the ratio of the volumes of the cone and the hemisphere.Give your answer in surd form.

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Answer .................................................(2)

(Total 6 marks)

Q16. (a) Show that can be written as 2x2 – 9x + 4 = 0

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.........................................................................................................................(2)

(b) Part of the graph of y = is shown on the grid below.

Draw a straight line on the grid which will enable you to solve the equation2x2– 9x + 4 = 0

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(c) Hence, or otherwise, solve the equation 2x2– 9x + 4 = 0

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Answer .................................................(2)

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(Total 7 marks)

Q17. A circle fits exactly inside a semi-circle of diameter 20 cm.

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The shaded area is a × π square centimetres.Work out the value of a.You must show your working.

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Answer a = ..............................................(Total 4 marks)

Q18. This is the graph of y = cos x for 0° ≤ x ≤ 360°

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Write the equation of each of the transformed graphs.In each case the graph of y = cos x is shown dotted to help you.

(a)

Equation y = ..............................................(1)

(b)

Equation y = ..............................................(1)

(c)

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Equation y = ..............................................(1)

(d)

Equation y = ..............................................(1)

(Total 4 marks)

Q19. ABCD is a cyclic quadrilateral.PAQ is a tangent to the circle at A.BC = CDAngle QAB = 38° and angle BAD = 76°

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Show that AD is parallel to BC.Give reasons to justify any values you write down or calculate.

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Q20. (a) A calculator displays a number in standard form as

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Which of the following numbers does the display show?Circle the correct answer.

7000 0.700 0.007 700 0.0007(1)

(b) Use your calculator to work out

cos (tan–10.45)

(i) Give all the figures in your calculator display.

Answer .................................................(1)

(ii) Write your answer to an appropriate degree of accuracy.

Answer .................................................(1)

(c) Use your calculator to work out

Answer .................................................(1)

(Total 4 marks)

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Q21. A sphere has radius r.A cone has base radius r and perpendicular height x.The volume of the sphere is double the volume of the cone.


(a) Show that x =2r

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(b) Calculate the ratio of the surface area of the sphere to the curved surface area of the cone.Give your answer in surd form.

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Answer .................................................(4)

(Total 6 marks)

Q22. The diagram shows a cuboid.AB = 3 cm, AE = 4 cm, BC = 12 cm.


(a) Find the length of BH.

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Answer ............................................ cm(2)

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(b) The angle between BH and BD is x and the angle between BH and BC is y.

Which angle is bigger, x or y?You must show your working.

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Answer .................................................(3)

(Total 5 marks)

Q23. XYZ is an isosceles triangle in which XZ = XY M and N are points on XZ and XY such that angle MYZ = angle NZY.

Prove that triangles YMZ and ZNY are congruent.

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Q24. In the diagram SR is parallel to PT.SQT and RQP are straight lines.SR = 20 cm and PT = 30 cmThe total height of the two triangles is 40 cm.


Use similar triangles to calculate the height, h cm, of triangle PQT.

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Answer h = .................................................. cm(Total 3 marks)

Q25. (a) A circle has a radius of 6 cm.A sector has an arc length of 8.4 cm.The angle at the centre of the sector is θ.


Calculate the value of θ.

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Answer ................................................. degrees(3)

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(b) A cone has base radius 6 cm and height h cm.A smaller cone of base radius 2 cm and height 3 cm is cut from the top.The remaining frustum has dimensions as shown.


Calculate the volume of the frustum.

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Answer ................................................. cm3

(5)(Total 8 marks)

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Q26. The grid below shows graphs of a curve

y = x2 + 2x – 3

and 3 straight lines

y = x + 1

y = – x – 2

and y = – x + 2

You must use the graphs to answer the following questions.

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(a) Write down a pair of simultaneous linear equations that have a solution

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x = – , y =

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Answer .................................................(1)

(b) Write down and simplify a quadratic equation whose solutions are approximately– 3.3 or 0.3.You must show clearly how you obtain your answer.

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Answer .................................................(2)

(c) Write down the approximate solutions to the equation x2 + x – 4 = 0.You must show clearly how you obtain your answer.

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Answer .................................................(2)

(Total 5 marks)

Q27. A square of side x and a quarter-circle of radius r have the same area.

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Not to scale

Express r in terms of x.Simplify your answer.

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Answer r = ............................................(Total 3 marks)

Q28. (a) ABC is a triangle.AC = 19 cm, BC = 17 cm and angle BAC = 60°

Not to scale

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Calculate the size of angle ABC.

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Answer ................................................. degrees(3)

(b) PQR is a triangle.PR = 23 cm, PQ = 22 cm and angle QPR = 48°

Not to scale

Calculate the length of QR.Give your answer to an appropriate degree of accuracy.

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Answer ................................................. cm(4)

(Total 7 marks)

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Q29. ABC is an isosceles triangle.The lengths, in cm, of the sides are

AB = 4a + 3, BC = 2b + 5 and AC = 2a + b

Not to scale

(a) AB = BC

Show that 2a – b = 1

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(b) The perimeter of the triangle is 32 cm. Find the values of a and b.

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Answer a = ..................... cm, b =....................... cm(4)

(Total 6 marks)

Q30. For a ladder to be safe it must be inclined at between 70° and 80° to the ground.

(a) The diagram shows a ladder resting against a wall.

Not to scale

Is it safe?You must show your working.

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(b) Another ladder rests against a wall.

Not to scale

Work out the closest distance that the bottom of the ladder can be from the wall so that it is safe.

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Answer ................................................. m(3)

(Total 6 marks)

Q31. A hemispherical bowl of radius 6 cm has the same volume as a cone of perpendicular height 27 cm.


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Calculate the base radius, r, of the cone.

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Q32. In the diagram below points Q and S lie on a circle centre O.SR is a tangent to the circle at S.Angle QRS = 40° and angle SOQ = 80°

Page 41


Prove that triangle QSR is isosceles.

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Q33. Match each of the sketch graphs to one of these equations.

A y = 2 – 2x B y = 2x + 2 C y = 3 – x2 D y = x3 + 4 E y =

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Graph 1 represents equation ……................



Graph 4 represents equation ……................(Total 4 marks)

Q34. The grid below shows the graph of y = x2 + 3x – 2

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(a) By drawing an appropriate straight line on the graph solve the equation

x2 + 3x – 3 = 0

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Answer .................................................(2)

(b) By drawing an appropriate straight line on the graph solve the equation

x2 + 2x – 1 = 0

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Answer .................................................(3)

(Total 5 marks)

Q35. (a) Explain why the volume of a cube increases by a factor of 8 when the side length is doubled.

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(b) June recently bought a small toy in the local shop.

ALIEN

Place inwater andit becomes

6 timesbigger!

It was originally 8 cm tall.After she placed it in water it grew to a similarly shaped alien.The height was then 14.5 cm.Is the claim on the pack justified?

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(Total 5 marks)

Q36. A marble paperweight consists of a cuboid and a hemisphere as shown in the diagram.The hemisphere has a radius of 4 cm.

Not to scale

Calculate the volume of the paperweight.

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Answer .................................................(Total 4 marks)

Q37. A circle fits inside a semicircle of diameter 10 cm as shown.


Calculate the shaded area.

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Answer ................................................. cm2

(Total 3 marks)

Q38. y is directly proportional to the square of x.When y = 5, x = 4.Find the value of y when x = 8.

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Q39. A giant paper clip is placed alongside a centimetre ruler.The curved ends are semicircles.

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Calculate the length of wire used to make the clip.

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Q40. (a) ABC is a right-angled triangle.AC = 19 cm and AB = 9 cm.

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Calculate the length of BC.

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Answer ................................................. cm(3)

(b) PQR is a right-angled triangle.PQ = 11 cm and QR = 24 cm.

Calculate the size of angle PRQ.

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Answer ................................................. degrees(3)

(Total 6 marks)

Q41. Two towns, A and B, are connected by a motorway of length 100 miles and a dual carriageway of length 80 miles as shown.

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Jack travels from A to B along the motorway at an average speed of 60 mph.Fred travels from A to B along the dual carriageway at an average speed of 50 mph.What is the difference in time between the two journeys?Give your answer in minutes.

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

Q42. A straight line has the equation y = 2x – 3

A curve has the equation y2 = 8x – 16

(a) Solve these simultaneous equations to find any points of intersection of the line and the curve.Do not use trial and improvement.You must show all your working.

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Answer .................................................(5)

(b) Here are three sketches showing the curve y2 = 8x – 16 and three possible positions of the line y = 2x – 3

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Sketch 1

Sketch 2

Sketch 3

Which is the correct sketch?

You must explain your answer.

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(2)(Total 7 marks)

Q43. The sketch shows the graph of y = sin x for 0° ≤ x ≤ 360°

You are given that sin 70° = 0.9397

(a) Write down another solution of the equation sin x = 0.9397

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Answer ................................................. degrees(1)

(b) Solve the equation sin x = –0.9397 for 0° ≤ x ≤ 360°

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Answer ................................................. degrees

................................................. degrees(2)

(c) On the axes below sketch the graph of y = sin 2x for 0° ≤ x ≤ 360°

(2)

(d) Hence write down the four solutions of the equation sin 2x = 0.9397

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Answer ................................................. degrees

................................................. degrees

................................................. degrees

................................................. degrees(3)

(Total 8 marks)

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Q44. The diagram shows the graph of the equation y = x2 + px + q

The graph crosses the x-axis at A and B (2,0).

C (–3, –5) also lies on the graph.

(a) Find the values of p and q.

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Answer p = ...................... q = ......................(4)

(b) Hence work out the coordinates of A.

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Answer ( ........................ , ........................ )(2)

(Total 6 marks)

Q45. The diagram shows a cylinder.The diameter of the cylinder is 10 cm.The height of the cylinder is 10 cm.

(a) Work out the volume of the cylinder.Give your answer in terms of π.

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Answer ................................................. cm3

(3)

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(b) Twenty of the cylinders are packed in a box of height 10 cm.The diagram shows how the cylinders are arranged inside the box.The shaded area is the space between the cylinders.

Work out the volume inside the box that is not filled by the cylinders.Give your answer in terms of π.

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Answer ................................................. cm3

(4)(Total 7 marks)

Q46. In the diagram OACD, OADB and ODEB are parallelograms.

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(a) Express, in terms of a and b, the following vectors.Give your answers in their simplest form.

(i)

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Answer .................................................(1)

(ii)

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Answer .................................................(1)

(iii)

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Answer .................................................(1)

(b) The point F is such that OCFE is a parallelogram.

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Write the vector in terms of a and b.

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Answer .................................................(2)

(c) What geometrical relationship is there between the points O, D and F? Justify your answer.

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(Total 7 marks)

Q47. A square-based pyramid with a base of side 2 cm has a volume of 2.75 cm3.

Not to scale

What is the volume of a similar square-based pyramid with a base of side 6 cm?

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Answer ................................................. cm3

(Total 2 marks)

Q48. A ruined tower is fenced off for safety reasons.To find the height of the tower Rashid stands at a point A and measures the angle of elevation as 18°.He then walks 20 metres directly towards the base of the tower to point B where the angle of elevation is 31°.

Calculate the height, h, of the tower.

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Answer ................................................. m(Total 6 marks)

Q49. The sketch below is of the graph of y = x2

On the axes provided, sketch the following graphs.The graph of y = x2 is shown dotted on each set of axes to act as a guide.

(a) y = x2 + 2

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(1)

(b) y = (x – 2)2

(1)

(c)

Page 63

(1)(Total 3 marks)

Q50. The graph of y = x2 – 4x + 8 is shown below.

Page 64

(a) (i) By drawing the graph of an appropriate straight line, solve the equation

x2 – 4x + 8 = 3x – 2

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Answer .................................................(3)

Page 65

(ii) Hence, or otherwise, solve x2 – 7x + 10 = 0

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Answer .................................................(1)

(b) The graph of y = x2 – 4x + 8 is to be used to solve the equation x2 – 5x + 4 = 0What straight line graph would need to be drawn?(You do not need to draw it, just state its equation.)

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Answer y = .................................................(2)

(Total 6 marks)

Q51. In the diagram, the sides of triangle ABC are tangents to the circle.D, E and F are the points of contact.AE = 5 cm and EC = 4 cm

Not to scale

Page 66

(a) Write down the length of CD.

Answer .................................................cm(1)

(b) The perimeter of the triangle is 32 cm. Calculate the length of DB.

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Answer .................................................cm(2)

(Total 3 marks)

Q52. Tom is investigating the equation y = x2 – x + 5

He starts to complete a table of values of y for some integer values of x.

x –2 –1 0 1 2 3

y 11 7 5 5 7 11

Tom says, "When x is an integer, y is always a prime number".Find a counter-example to show that Tom is wrong.Explain your answer.

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Page 67

Q53. A water tank is 50 cm long, 34 cm wide and 24 cm high.It contains water to a depth of 18 cm.

Four identical spheres are placed in the tank and are fully submerged.The water level rises by 4.5 cm.

Calculate the radius of the spheres.

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Q54. ABCD is a quadrilateral.AB = 7 cm, AD = 6 cm and BC = 9 cm.

Page 68

Angle ABC = 75° and angle ADC = 90°

Calculate the perimeter of ABCD.

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Answer .................................................. cm(Total 5 marks)

Q55. AB is a chord of a circle, centre O, radius 6 cm.

Page 69

AB = 7 cm

Calculate the area of the shaded segment.

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Answer ................................................. cm2

(Total 6 marks)

Q56. (a) Complete the table of values for y = (0.8)x

x 0 1 2 3 4

Page 70

y 1 0.8 0.64 0.41

(1)

(b) On the grid below, draw the graph of y = (0.8)x for values of x from 0 to 4.

(2)

(c) Use your graph to solve the equation (0.8)x = 0.76

Answer .................................................(1)

(Total 4 marks)

Page 71

Q57.

Enlarge the shaded shape by scale factor with centre of enlargement (–1, 0).(Total 2 marks)

Q58. A tin of diameter 7 cm and height 12 cm has a label around it.The label is glued together using a 1 cm overlap.There is a 1 cm gap between the label and the top and the bottom of the tin.

Find the length and the height of the label.

Page 72

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Answer Length = ......................................... cm

Height = ......................................... cm(Total 4 marks)

Q59. Dario is using trial and improvement to find a solution to the equation

Page 73

x + = 5

The table shows his first trial.

Page 74

x +

Page 75

4 4.25 Too low

Continue the table to find a solution to the equation.Give your answer to 1 decimal place.

Answer x = .................................................(Total 4 marks)

Q60. (a) Points P, Q, R and S lie on a circle.

PQ = QR

Angle PQR = 116°

Page 76

Explain why angle QSR = 32°.

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...........................................................................................................................(2)

(b) The diagram shows a circle, centre O.TA is a tangent to the circle at A.Angle BAC = 58° and angle BAT = 74°.

(i) Calculate angle BOC.

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Answer Angle BOC = ............................... degrees(1)

(ii) Calculate angle OCA.

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Page 77

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Answer Angle OCA = ............................... degrees(3)

(Total 6 marks)

Q61. Which one of the following kites is a cyclic quadrilateral? Give a reason for your answer.

Answer ......................................................

Reason .....................................................................................................................

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..................................................................................................................................(Total 2 marks)

Q62. A square-based pyramid has a base of edge 5 cm.The vertex of the pyramid is directly over the midpoint of the base.The volume of the pyramid is 100cm3.

Page 78

Find the length of the slant edge of the pyramid (marked x in the diagram).

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Answer ..................................... cm(Total 5 marks)

Page 79

Q63. A solid cube has a square hole cut through horizontally and a circular hole cut through vertically.

Both holes are cut centrally in the appropriate faces.

The dimensions of the cube and the holes are as shown in the diagram.

Calculate the volume remaining after the holes have been cut.

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Answer ............................................(Total 5 marks)

Page 80

Q64. In triangle ABC, AB = 11 cm, BC = 9 cm and CA = 10 cm.

Find the area of triangle ABC.

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Answer ..................................... cm2

(Total 5 marks)

Q65. ABCD is a rectangle with length 25 cm and width 10 cm.

Page 81

The length of the rectangle is increased by 10%.The width of the rectangle is increased by 20%.Find the percentage increase in the area of the rectangle.

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Answer ................................. %(Total 3 marks)

Q66. Solve the equation

x2 – 10x – 5 = 0

Give your answers to 2 decimal places.

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Page 82

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

Q67. (a) ABC is a right-angled triangle.AB = 5.1 cm

CAB = 48°

Find the length of BC (marked x in the diagram).Give your answer to a suitable degree of accuracy.

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Answer ..................................... cm(4)

(b) PQRS is a parallelogram.PQ = 5.1 cmPS = 6.8 cm

QPS = 48°

Page 83

Calculate the area of PQRS.

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Answer .................................... cm2

(2)(Total 6 marks)

Q68. The diagram shows a circle with centre O and radius 2.5 cm.TA is a tangent to the circle, of length 6 cm.The line from A to the centre O of the circle cuts the circumference at B.

Calculate the length of AB.

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Page 84

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Answer .................................... cm(Total 4 marks)

Q69. (a) Liquid is poured at a steady rate into the bottle shown in the diagram.

As the bottle is filled, the height, h, of the liquid in the bottle changes.

Which of the five graphs below shows this change?

Give a reason for your choice.

Page 85

Graph ..............................................................................................................

Reason ............................................................................................................

.........................................................................................................................(2)

(b) Liquid is poured at a steady rate into another container.The graph shows how the height, h, of the liquid in this container changes.

Page 86

Sketch a picture of this container.

(1)

(Total 3 marks)

Q70. The diagrams, which are not drawn to scale, show the graph of y = x2 and four other graphs A, B, C and D.

A, B, C and D represent four different transformations of y = x2.

Find the equation of each of the graphs A, B, C and D.

Page 87

(Total 4 marks)

Page 88

Q71. The diagram shows two right-angled triangles.AD = 15 cm.CD = 6 cm.

(a) Given that cos x° = , calculate the length BD.

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Answer BD = ............................................(2)

(b) Find the value of sin y°.

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Page 89

Answer sin y° = ............................................(3)

(Total 5 marks)

Q72. ABC is a triangle.ACD is a straight line.

Using a ruler and compasses only, make an accurate construction of this diagram.You must show clearly all your construction arcs.The line AD has been drawn for you.

Page 90

(Total 6 marks)

Q73. (a) The diagram shows a circle with centre O.

Work out the size of the angle marked x.

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Answer .................................. degrees(1)

(b) The diagram shows a different circle with centre O.

Page 91

Work out the size of the angle marked y.

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Answer ................................ degrees(1)

(c) A, B and C are points on the circumference of a circle with centre O.BOC is a straight line.Angle ABC = 20°

Work out the size of the angle marked z.Explain your answer.

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Answer ................................... degrees(2)

(Total 4 marks)

Q74. Two similar bottles are shown below.

Page 92

The smaller bottle is 20 cm tall and holds 480 ml of water.The larger bottle is 30 cm tall.

How much water does the larger bottle hold?

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Q75. A thin-walled glass paperweight consists of a hollow cylinder with a hollow cone on top as shown.The paperweight contains just enough sand to fill the cylinder.

The paperweight is now turned upside down.

Page 93

Calculate the depth of the sand, (marked x in the diagram).

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Q76. Two ships, A and B, leave port at 13 00 hours.Ship A travels at a constant speed of 18 km per hour on a bearing of 070°.Ship B travels at a constant speed of 25 km per hour on a bearing of 152°.

Page 94

Calculate the distance between A and B at 14 00 hours.

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Answer ................................................. km(Total 4 marks)

Q77. ABCDEFGH is a cuboid with sides of 5 cm, 5 cm and l2 cm as shown.

Calculate angle DFH.

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Page 95

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Answer ................................................. degrees(Total 5 marks)

Q78. In the diagram, the lines AC and BD intersect at E.

AB and DC are parallel and AB = DC.

Prove that triangles ABE and CDE are congruent.

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Page 96

Q79. ABC is a right-angled triangle.BC = 125 m.Angle CAB = 33°.

Find the length of AC (marked x in the diagram).Give your answer to an appropriate degree of accuracy.

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Answer ................................................. m(Total 4 marks)

Q80. Triangles ADE and ABC are similar.DE is parallel to BC.AD = 4 cm, DE = 6 cm and BC = 9 cm.

Page 97

Calculate the length of BD.

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Q81. The graph of y = sin x for 0° ≤ x ≤ 360° is shown on the grid below.The point P(90, 1) lies on the curve.

Page 98

On both of the grids that follow, sketch the graph of the transformed function.In both cases write down the coordinates of the transformed point P.

(a) y = sin (x – 45)

P (......................., ......................)(2)

Page 99

(b) y = 2sinx

P (......................., ......................)(2)

(Total 4 marks)

Q82. A firm makes cone shaped containers out of card.The card is in the shape of a sector of a circle of radius 12 cm.The angle of the sector is 270°.The straight edges are brought together to make the cone.

Page 100

(a) Find the arc length of the card used to make the cone.Give your answer in terms of π.

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Answer ................................................. cm(2)

(b) Calculate the radius of the base of the cone.

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Answer ................................................. cm(2)

(Total 4 marks)

Q83.

OAB is a triangle where M is the mid-point of OB.

P and Q are points on AB such that AP = PQ = QB.

= a and = 2b

Page 101

(a) Find, in terms of a and b, expressions for

(i)

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Answer .................................................(1)

(ii)

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Answer .................................................(2)

(iii)

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Answer .................................................(2)

(b) What can you deduce about quadrilateral OMQP?Give a reason for your answer.

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..........................................................................................................................(2)

Page 102

(Total 7 marks)

Q84. (a)

(i) Write down the value of x.

Answer ................................................. degrees(1)

(ii) Calculate the value of y.

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Answer ................................................. degrees(1)

(b) A and C are points on the circumference of a circle centre B.AD and CD are tangents.Angle ADB = 40°.

Page 103

Explain why angle ABC is 100°.

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..........................................................................................................................(2)

(c) ABCD is a cyclic quadrilateral.PAQ is a tangent to the circle at A.BC = CD.AD is parallel to BC.Angle BAQ = 32°.

Page 104

Find the size of angle BAD.

You must show all your working.

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Answer Angle BAD = ........................... degrees(5)

(Total 9 marks)

Q85. The map below shows three boats, A, B and C, on a lake.Along one edge of the lake there is a straight path.

Treasure lies at the bottom of the lake.

The treasure is: between 150 m and 250 m from B, nearer to A than C, more than 100 m from the path.

Page 105

Using a ruler and compasses only, shade the region in which the treasure lies.

You must show clearly all your construction arcs.(Total 5 marks)

Q86. (a)

Page 106

(i) Write down the value of x.

Answer ................................................. degrees(1)

(ii) Calculate the value of y.

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Answer ................................................. degrees(1)

(b) A and C are points on the circumference of a circle centre B.AD and CD are tangents.Angle ADB = 40°.

Explain why angle ABC is 100°.

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..........................................................................................................................(2)

(c) P is a point on the circumference of a circle with centre O.PQ is a tangent of length 8 cm.The area of triangle OPQ is 24 cm2.

Page 107

Calculate the area of the circle.Give your answer in terms of π.

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Answer ................................................. cm2

(3)(Total 7 marks)

Q87. Three circles fit inside a rectangle as shown.

Two of the circles are identical and the third is larger. The circles have radii 9 cm, 9 cm and 25 cm.


Page 108

Calculate the length, l, of the rectangle.

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Answer ............................................ cm(Total 6 marks)

Q88. This diagram is made from 25 small squares and 16 large squares.

Page 109

What percentage of the diagram is shaded?

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Answer .............................................. %(Total 6 marks)

Q89. The diagram shows a regular pentagon and a regular decagon joined at side XY.

Page 110


Show that the points A, B and C lie on a straight line.

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Page 111

Q90. In the figure, AC = 9 cm, AE = 6 cm, BD = 8.5 cm, BE = 4.5 cm and DF = 5 cm, BEDF and AEC are straight lines.


(a) Show that triangles BEC and AED are similar.

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.........................................................................................................................(3)

(b) By considering triangles BEC and FEA show that AF is parallel to BC.

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Page 112

.........................................................................................................................(5)

(Total 8 marks)

Q91. Four points, A(– 4, –1), B(5, 11), C(20, 11) and D(11, –1) are joined to form a quadrilateral.


Prove by calculation that ABCD is a rhombus.

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Page 113

Q92. Find the reflection of the point A(1, 4) in the line y = 2x – 3

Use the grid below to help you.

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Answer ( ...................... , ….................... )

(Total 3 marks)

Page 114

Q93. The diagram shows four points A, B, C and D on the circumference of a circle, centre O.

PAQ is the tangent to the circle at A. PBD is a straight line.

Angle QAD = 75° Angle APB = 30°


Work out angle BCD. You must show your working.

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Page 115

Answer .................................. degrees(Total 3 marks)

Q94. A manufacturer designs a set of three similar containers to fit inside each other. The diagram shows a sketch of the containers and their oval cross-sections.


Some information about the containers is shown in this table.

Base length Height Area of card used in manufacture

Large 30 cm

Medium 20 cm 12 cm 1080 cm2

Small 10 cm

(a) Work out the height of the large container.

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Page 116

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Answer …................................... cm(3)

(b) 1080 cm2 of card is used to manufacture the medium container.

Work out the area of card used to manufacture the small container.

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Answer …................................... cm2

(2)(Total 5 marks)

Q95. The shaded shape is made from two different right-angled triangles. The area of each of the triangles is 6cm2. Each of lengths a, b, c and d are a whole number of centimetres. a + c = 10 cm a > b and c > d


Work out the perimeter of the shaded shape.

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Page 117

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Answer ........................................... cm(Total 5 marks)

Q96. The sector AOB of a circle is shown below. The length of its arc AB is 10π cm.


The sector is folded so that the straight edges meet and form a cone as shown.

(a) Calculate the radius of the base of the cone.

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Page 118

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Answer .......................................... cm(3)

(b) The volume of the cone is 80π cm3.

Work out the perpendicular height of the cone.

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Answer ......................................... cm(3)

(Total 6 marks)

Q97. OABC is a quadrilateral. D, E, F and G are midpoints of OA, AB, BC and OC respectively.

= 2a, = 2b and = 2c


Page 119

Find the following vectors in terms of a, b and c.

For example = c – a

(a)

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Answer ..................................................(1)

(b)

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Answer ..................................................(1)

(c) Use your answers to parts (a) and (b) to show that = c – a

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Page 120

(d) Explain how you can tell that DEFG is a parallelogram.

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.........................................................................................................................(1)

(Total 4 marks)

Q98. The diagrams show a trapezium and a parallelogram.


(a) Use the trapezium to explain why 2x + y = 180

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.........................................................................................................................(1)

(b) The parallelogram can be used to form another equation connecting x and y.

Tick a box to show the correct equation.

3x + y = 130 3x + y = 230

3x = y – 50 3x + y = 410(1)

Page 121

(c) Hence, or otherwise, work out the values of x and y.

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Answer x = ................ , y = …...............(3)

(Total 5 marks)

Q99. Airport runways have a two-digit number painted on them. These numbers are used to work out the direction of the runway.

To work out the three-figure bearing, multiply the runway number by 10.

Here is a diagram of a runway on a three-figure bearing of 280° and a runway on a three-figure bearing of 040°.

Page 122

(a) (i) Write down the three-figure bearing for a runway pointing due South.

Answer .............................................. °(1)

(ii) Write down the runway number for a runway pointing due South.

Answer .................................................(1)

(iii) A runway has a three-figure bearing of 060°.

Write down the runway number.

Answer .................................................(1)

(b) A runway is being painted.

By measuring the three-figure bearing, work out the runway number.

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Page 123

Answer ..................................................(2)

(c) Runways are used in both directions. This means that they have two different runway numbers, one at each end. A runway has the number 30 at one end.

What runway number is at the other end?

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Answer ..................................................(3)

(Total 8 marks)

Q100. This is the graph of y = sin x for values of x from 0° to 360°

On each of the following grids the solid line shows a transformation of the graphof y = sin x.

Write down the equation of each of the transformed graphs.

On each grid, the graph y = sin x is shown dotted to help you.

(a)

Page 124

Answer y = ......................................(1)

(b)

Answer y = ......................................(1)

(c)

Page 125

Answer y = ......................................(1)

(Total 3 marks)

Q101. In triangle ABC the length of AB is 13.2 cm. Angle BAC = 40° Angle BCA = 114°

Not drawn accuratelly

Work out the length of BC. Give your answer to an appropriate degree of accuracy.

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Answer ....................................... cm(Total 4 marks)

Page 126

Q102. Advice about wheelchair ramps is that for a ramp that rises more than 7.5 cm the

maximum gradient should be

For example


(a) Access to a Village Hall is by two steps each 16 cm high. It is proposed to build a ramp alongside the steps as shown in the diagram.


Does the proposed ramp follow the advice given? You must show your working.

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........................................................................................................................(2)

(b) For a wheelchair ramp that rises less than 7.5 cm the maximum gradient should be

Page 127

For example


A ramp is designed for a step that rises 6 cm.


Work out the length of a ramp that uses the maximum gradient.

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Answer ..................................... cm(4)

(Total 6 marks)

Q103. The diagram shows a kite ABCD. AB = x + 1 and CD = 3x – 2

Page 128


(a) Show that, in terms of x, the perimeter of the kite is 8x – 2

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.........................................................................................................................(1)

(b) The perimeter of the kite is 16 cm.

Write down and solve an equation to work out the value of x.

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Answer x = ............................... cm(3)

(Total 4 marks)

Q104. A, B and C are three points such that

= 5a – 3b and = 7.5a – 4.5b

(a) Write down a fact about the points A, B and C.

.........................................................................................................................

.........................................................................................................................

Page 129

(1)

(b) Write down the ratio of the lengths AB : BC in its simplest form.

.........................................................................................................................(1)

(Total 2 marks)

Q105. The diagram shows the graph of x2 + y2 = 25

(a) By drawing a linear graph, write down one solution to the simultaneous equations

x2 + y2 = 25 and y = x

Answer x = ................ , y = ................(2)

Page 130

(b) Explain why the x-coordinate in your answer to part (a) is an approximation of

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(Total 4 marks)

Q106. (a) The diagram shows a circle with centre O. TS is a tangent.


Work out the value of x.

.........................................................................................................................

Answer ............................... degrees(1)

(b) The diagram shows another circle.

Page 131


Write down the value of y.

.........................................................................................................................

Answer ............................... degrees(1)

(Total 2 marks)

Q107. A baker is weighing out amounts of bread dough using a machine. The machine is set to weigh 400 grams of dough. The amounts of dough are each within 10% of the weight set. 300 amounts of dough are produced by the machine.

What is the maximum total amount of bread dough that could be produced? Give your answer in kilograms.

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

Answer ......................................... kg(Total 3 marks)

Page 132

Q108. The diagram shows an accurate scale drawing of the plan view of a house and garage.

The drawing is on a centimetre grid. The front of the house is 12.5 metres long.

(a) Explain why the scale is 1 centimetre represents 2.5 metres.

.........................................................................................................................

.........................................................................................................................(1)

(b) The owner of the house is designing an extension to be joined to the back wall of the house.

The extension is rectangular and is 7.5 metres long and 5 metres wide. It must be at least 2.5 metres from the garage.

Make an accurate sketch of the possible extension on the grid.(3)

(Total 4 marks)

Page 133

Q109. The diagram shows a rhombus made of two triangles X and Y. M is the midpoint of diagonal AC.


(a) Describe fully a single transformation that maps triangle X onto triangle Y.

.........................................................................................................................

.........................................................................................................................(2)

(b) Describe fully a different single transformation that maps triangle X onto triangle Y.

.........................................................................................................................

.........................................................................................................................(3)

(Total 5 marks)

Q110. AB is a diameter of the circle, centre O. ABC is a straight line. DTC is a tangent to the circle at T. Angle BCT = 32° and angle TAB = x°

Page 134


Find the value of x. Give reasons for all angles you write down or calculate.

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Answer .................................... degrees(Total 4 marks)

Q111. This right-angled triangle has sides of lengths (x – 2) cm, (x + 5) cm and 10 cm.

Page 135


Calculate the value of x. Give your answer to an appropriate degree of accuracy.

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Answer ........................................... cm(Total 5 marks)

Q112. An old windmill is the shape of a truncated cone. The mill is 12 metres high and has 4 floors, equally spaced. The diameter of the ground floor is 8 metres and the diameter of the roof is 6 metres.

Page 136


The mill is for sale. This is the advert.

DEVELOPMENT OPPORTUNITYOLD MILL FOR SALE

Over 150 square metres of floor space

Is the claim about floor space justified? You must show your working.

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

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..................................................................................................................................

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..................................................................................................................................(Total 5 marks)

Page 137

Q113. 251 is a prime number.

(a) (i) Write down √251 Give your answer to 1 decimal place.

Answer ..............................................(1)

(ii) Explain how to test that 251 is a prime number.

................................................................................................................

................................................................................................................

................................................................................................................(2)

(b) (i) Express 2008 as a product of its prime factors.

................................................................................................................

................................................................................................................

................................................................................................................

................................................................................................................

Answer ..............................................(2)

(ii) Write down all the factors of 2008

................................................................................................................

................................................................................................................

................................................................................................................

Answer ..............................................(1)

(c) (i) Show that (x + y)(x – y) = x2 – y2

................................................................................................................

Page 138

................................................................................................................(1)

(ii) Hence find both pairs of integers x and y such that

x2 – y2 = 2008

................................................................................................................

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................................................................................................................

Answer x = ..................., y = ...................

x = ..................., y = ...................(4)

(Total 11 marks)

Q114. The diagram shows two right-angled triangles. AD = 40 cm CD = 7 cm

Page 139

cos x =


Find the value of sin y.

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Answer ..............................................(Total 6 marks)

Q115. OABC is a quadrilateral.

Page 140

P, Q, R and S are the mid-points of OA, AB, BC and CO respectively.

= 2a, = 2b and = 2c


(a) Write down, in terms of a and b, the vector .

Answer ..............................................(1)

(b) Write down, in terms of c and b, the vector .

Answer ..............................................(1)

(c) Show that = = b

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................(2)

(d) Using your answer to part (c) write down a geometrical fact about the line joining the mid-points of two sides of a triangle.

.........................................................................................................................

.........................................................................................................................(1)

(e) What type of quadrilateral is formed by joining the mid-points of the four sides of a quadrilateral?

Give a reason for your answer.

Page 141

Type of quadrilateral ..................................................

Reason............................................................................................................

.........................................................................................................................(2)

(Total 7 marks)

Q116. The rule for this sequence is that each term is the mean of the two previous terms.

ax y

b

(a) Find an expression for a in terms of x and y.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

Answer ..............................................(2)

(b) Find an expression for b in terms of x and y. Simplify your answer.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

Answer ..............................................(2)

(Total 4 marks)

Page 142

Q117. Four identical circular discs fit into a rectangle 10 cm long.


Ten of the same discs fit into a rectangle 22 cm long.


24 discs are placed together in the same way.

How long is the rectangle?

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Answer ..............................................cm(Total 3 marks)

Q118. Joe uses a ruler and compasses to find the centre of the circle drawn below. He starts by drawing a chord on the circle.

Page 143

Complete Joe’s construction to find the centre of the circle.

(Total 3 marks)

Q119. The rule for continuing a Fibonacci sequence is to add the last two terms to make the next term.

For example, the sequence that starts 1, 1, … continues as 1, 1, 2, 3, 5, 8, …

Two other Fibonacci sequences start a, 2a, …… and b, 4b, …

The fifth terms of these two sequences are equal.

Given that a + b = 11, work out the values of a and b.

..................................................................................................................................

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..................................................................................................................................

..................................................................................................................................

Page 144

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..................................................................................................................................

Answer a = ...................... b = .......................(Total 4 marks)

Q120. The diagram shows a cone.

The diameter of the base of the cone is x cm.

The height of the cone is also x cm.

The volume of the cone is V cm3.

Find a formula for x in terms of V and π.

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Answer .......................................................................(Total 4 marks)

Page 145

Q121. ABCDEF is a regular hexagon.

AFGH and AJKB are squares.


Show that triangle AHJ is equilateral.

..................................................................................................................................

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..................................................................................................................................(Total 4 marks)

Page 146

Q122. Solve the equation 3x2 – 5x – 7 = 0

Give your answers to 2 decimal places.

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

Q123. Triangle ABC has a right angle at B.

Angle BAC = 38°

AB = 7.21 cm


Calculate the length of BC.

Give your answer to an appropriate degree of accuracy.

Page 147

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..................................................................................................................................

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..................................................................................................................................

Answer ................................................................. cm(Total 4 marks)

Q124. The diagram shows the graph of y = sin x° for 0 ≤ x ≤ 360

(a) Write down a possible equation of the following graph.

Page 148

Answer .......................................................................(1)

(b) Write down a possible equation of the following graph.

Answer .......................................................................(1)

Page 149

(c) Write down a possible equation of the following graph.

Answer .......................................................................(1)

(Total 3 marks)

Q125. The diagram shows a hollow cylinder and a solid sphere.

The radius of the cylinder = 3 cm

The radius of the sphere = 3 cm

The height of the cylinder = 6 cm

Page 150


The sphere just fits inside the cylinder as shown.


Work out the volume of the space left inside the cylinder.

Give your answer in terms of π as simply as possible.

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Answer ................................................................ cm3

(Total 5 marks)

Page 151

Q126. (a) In the diagram O is the centre of the circle.


What is the value of x?

Answer .......................................................... degrees(1)

(b)


What is the value of y?

Answer .......................................................... degrees(1)

(Total 2 marks)

Page 152

Q127. The diagram shows two identical shapes A and B.

Describe fully the single transformation which takes shape A to shape B.

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................(Total 2 marks)

Page 153

Q128. In the diagram AB and CD are parallel.


(a) Write down the value of x.

Answer .......................................................... degrees(1)

(b) Work out the value of y.

.........................................................................................................................

.........................................................................................................................

Answer .......................................................... degrees(2)

(Total 3 marks)

Q129. This is the graph of y = x2 – 4x + 1

Page 154

By drawing an appropriate linear graph, solve the equation x2 – 5x + 3 = 0

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Answer ................................................(Total 4 marks)

Page 155

Q130. (a) A test tube is formed from a cylinder and a hemisphere as shown.

Work out the total volume of the test tube.

.........................................................................................................................

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Answer .......................................... cm3

(4)

(b) The test tube is filled with water to a depth of d cm, as shown in the next diagram.

The water occupies exactly half the full capacity of the test tube.

Page 156

Work out the value of d.

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Answer ............................................. cm(4)

(Total 8 marks)

Q131. O is the centre of the circle.

Angle PRS = 134°


Work out the size of the reflex angle POQ.

You must show your working.

..................................................................................................................................

..................................................................................................................................

Page 157


Q132. Triangle ABC has AB = 6 cm, AC = 10 cm, BC = 14 cm


Calculate the largest angle in the triangle.

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..................................................................................................................................

..................................................................................................................................


Q133. Katy is using the quadratic formula to solve a quadratic equation.

Page 158

After correctly substituting the values, she writes

(a) What is the quadratic equation Katy is trying to solve?

.........................................................................................................................

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.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

Answer ...............................................(3)

(b) Explain why Katy will not be able to find any solutions to the equation.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................(1)

(Total 4 marks)

Q134. (a) The right-angled triangle has sides shown.


Page 159

Show that x = 9 cm

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

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

Q135. Here are four equations of graphs.

Page 160

A y = 3x + 2 B 2x + 3y = 6 C y = 3x2 D y = x3

(a) Here are three sketch graphs.

Match each graph to its equation.

Equation ................... Equation ................... Equation ...................(3)

(b) On the axes below, sketch the graph of the other equation.

(1)(Total 4 marks)

Q136. A gold bar has a trapezium cross-sectional area.

The dimensions are shown in the diagram.


Page 161

(a) Calculate the cross-sectional area of the gold bar.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

Answer .......................................... cm2

(2)

(b) Gold has a density of 19.3 grams per cm3.

Work out the mass of the gold bar.

Give your answer in kilograms.

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Answer ........................................... kg(4)

(Total 6 marks)

Page 162

Q137. (a) Factorise x2 + 10x

.........................................................................................................................

.........................................................................................................................

Answer ...............................................(1)

(b) Factorise y2 – 36

.........................................................................................................................

.........................................................................................................................

Answer ...............................................(1)

(c) Solve the equation 5w + 6 = 9 – w

.........................................................................................................................

.........................................................................................................................

Answer w = .........................................(3)

(d) Solve the equation

.........................................................................................................................

.........................................................................................................................

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.........................................................................................................................

.........................................................................................................................

Answer x = .........................................(4)

Page 163

(Total 9 marks)

Q138. A restaurant serves garlic bread.

All the garlic breads are circular and the same thickness.

They can be made with different diameters as shown.

Robert is going to order a 14-inch garlic bread.

The restaurant has a special offer.

Special Offer

Get one 7-inch garlic bread and one 10-inch garlic breadfor the same price as a 14-inch garlic bread.

Robert says that if he has the special offer he will get less garlic bread.

Is Robert correct?

Page 164

You must show your working.

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................(Total 4 marks)

Q139. You have a square piece of paper which is folded in half to form a rectangle as shown.

The perimeter of the rectangle is 39 centimetres.

What is the area of the square you started with?

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

Answer ……................................... cm2

(Total 4 marks)

Page 165

Q140. Is the statement below always true, sometimes true or never true?

Tick the correct box.

The circumference of a circle of diameter 10 cm is greater than the perimeter ofa triangle with a base 10 cm.

Always true Sometimes true Never true

Explain your answer

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................

..................................................................................................................................(Total 2 marks)

Q141. Triangle T is drawn on the grid.

Page 166

(a) Draw the image of T after a rotation of 90° anticlockwise 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)

Q142. The radius of the Earth and the radius of Jupiter are in the approximate ratio 1 : 11.

The mass of the Earth and the mass of Jupiter are in the approximate ratio 1 : 320.

You will need the following information.

Page 167

• The Earth and Jupiter are spherical

Page 168

• The volume of a sphere of radius r is

(a) Show that the approximate ratio of the volume of the Earth to the volume of Jupiter is 1 : 1331.

.........................................................................................................................

.........................................................................................................................(1)

(b) You are given density =

Work out the approximate ratio of the average density of the Earth to the average density of Jupiter in the form 1 : n

.........................................................................................................................

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Answer 1:................................................(2)

(Total 3 marks)

Page 169

M1. (a) M1

2880 implies M2M1

ork = 2880 and equation seen using k

A1

(b) M1

(W = their =) their 9.79....A1 ft

so need 10 waiters.ft answer rounded up if M1 awarded.

A1 ft[6]

M2. (a) 65B1

Page 170

angle at centreB1

(b) 115ft 180 – their 65 provided reason given is not contradictory

B1 ft

Opposite angles (of cyclic quad)or other valid explanationeg x + y = 180

B1[4]

M3. (a) Correct enlargementB1 for enlargement any scale factor (not 1) Accept any orientation

B2

(b) 36 ÷ 4or 3 × 3 or 54 ÷ 6

M1

9SC1 for their (SF in (a))2

Accept ratio 1 : 9 or 9 : 1A1

[4]

M4. AC2 = 72 + 92 – 2 × 9 × 7 × cos 75M1

Page 171

AC2 = 97.3888…Accept 97.4

A1

DC2 = (their AC)2 – 62

M1

DC = 7.8(35…)or 7.84

A1 ft

29.8(35…)or 29.84 or 30 with correct working

A1 ft[5]

M5. (a) (i) – b + a or a – bB1

(ii) b – aoe

B1

(b) oe

M1

= a + b – aA1

(c) = 2or OBN a straight lineor BN = OBor B is midpoint of ON

B1[5]

Page 172

M6. (a) 360 ÷ 8or 45 seen or 6 × 180 or 1080 or (2 × 8 – 4) right angles

M1

180 – (their 45)(their 1080) ÷ 8

M1 dep

135135

A1

(b) 360 – (their 135 + 135) or 2 × 45M1

90° in XA1

Sides of X are equalor (regular) octagons so sides are equal4 lines of symmetry or rotational symmetry of order 4 scores 3 marksOther symmetry scores B1

B1[6]

M7. (3x + 2)(x + 1)Rectangle

M1

3x2 + 5x + 2Rectangle

A1

x × 3x + 5(x + y)or x × 3x + x × 5 + y × 5or x(3x + 5) + y × 5or (3x + 5)(x + y) – 3x × y

M1

Page 173

3x2 + 5x + 5yL shape

A1

5y = 2oe dependent on a previous M1 and a term in y

M1 dep

0.4oe

A1[6]

M8. (a) 6B1

(b) Plot pointsB1

Draw curveB1

(c) x = 1.4 and –1.4B1

(d) (3x2 – 6) – (3x2 – 5x – 6)Sight of (±) 5x (+ k)

M1

= 5x

Draw y = 5xB1 ft

x = 2.5, –0.8Accept 2.4 to 2.55 and –0.75 to –0.85

A1[7]

M9. (a) A ∞ d 2 or A = kd 2

Page 174

When d = 15, A = 90M1

90 = 225k

Page 175

k = 0.4 or M1

A = 0.4d 2

Page 176

Accept A = oeA1

(b) d = 20 =>

A = 0.4 × 202

A = 160160 unsupported SC1

A1

(c) A = 250

250 = 0.4d 2

Page 177

d 2 = Dep on M2 in (a)

M1

= 625

Accept M1 dep

d = 25A1

[7]

M10. (a) (0, 1)Generally marked

B1

(b) Matching any (non zero for x)Values eg, a1 = 3, a2 = 9, etc

Must show as powerM1

a = 3 A1

[3]

M11. Volume of one (or two) spheres

Allow 10 for r for M1M1

= 523.6 (1047.2){524} {1048, 1050}

oe 500π/3 or 1000π/3(523.3 to 523.7) or (1046.6 to 1047.4)

A1

Volume of cylinder

Page 178

Allow 10 for h or 10 for r for M1 (not both)M1

= 1570.8{1570, 1571}

oe 500π (1570 to 1571)A1

Volume remaining(1570.8 – 1047.2 =) 523.6, 524

Page 179

oe

Due to different values of π an answer between523.3 and 523.7 gets full marksft If both Ms awarded and one value is correct.

A1 ft[5]

M12. Breaks down into areas of rectangles and areas of (quarter) circlesAny combination of rectangles and circles

or 12.56... or is enough evidence for area of circlesNB 12.56 from 2 × π × 2, if seen is M0NB 3.14 on its own is not evidence of the area of a quarter circle as it is π

M1

Uses an ‘addition’ method (method 1) and finds

Area of one (or 5) ‘external’ quadrants

or

or Uses a ‘subtraction’ method (methods 2 and 3) and finds5 × area one quadrant

or M1 dep

= 0.8584..., {0.9, 0.86, 0.858}

or = (× 5) 4.292... {4.3, 4.29}15.71, 15.7

A1

52.3 or 52.29... 68 – 5π

Page 180

Allow 52 if 52.3 or 52.29... or a full method seenA1

[4]

M13. (a) or or M1

Page 181

k = 3200 or M1

P = 3200/Q or PQ = 3200 or Q = 3200/PA1

(b) Correct sketch graphB1

(c) (Their 3200)or 2Q = (Their 3200) ÷ Qor Q = (Their 3200) ÷ 2Q

M1

(Q =) 40ft Their value of k

A1 ft[6]

M14. (a) B1

(b) B1

(c) oe

B1[3]

M15. (a)

Page 182

Or M1

l = 2rClearly shown since answer given

A1

(b) h2 = 4r2 – r2

Attempt to use Pythagoras’ theorem correctlyM1

h = √3rh2 = 3r2 is sufficient or h = √(3r2)

A1

(c) √3r :

ft with Their h if 1st M1 earnedM1 dep

√3 : 2 A1

[6]

M16. (a) Multiply through by x, 4 = x(9 – 2x) is enough

M1

Expanding and rearranging must be seen (answer given)A1

(b) Attempt at

identified as being required lineM1

Points worked out

Page 183

eg, table of values (2 points minimum)M1 dep

Correct line plottedWith ruler, must intersect the curve twice

A1

(c) Solutions can come from factorising

ie, B1

x = 4No ft from incorrect factors

B1[7]

M17. or Condone use of π = 3.(14...)

M1

Page 184

and M1

(Their 50π) – (Their 25π)M1 dep

25 A1

[4]

M18. (a) y = cosx + 1y = 1 + cosx

B1

(b) y = 2cosxB1

(c) y = cos2xB1

(d) y = cos(90 – x), y = cos (x +270)y = cos (x – 90) or y = sin x

B1[4]

M19. ABD = 66 (Alt segment)or angles in triangle if ADB found first

B1

DCB = 104 (opposite in cyclic)In all alternatives, for first 3 B marks do not award B1the first time no reason or wrong reason given, otherwiseaccept angles identified in answer or on diagram.NB Mark ‘positively’ ie, ignore wrong values or reasons unless totally contradictory.

B1

DBC = 38 (isosceles)

Page 185

CBA = 104 B1

CBA + BAD = 180 (interior)In all alternatives, reason must be given for final B1Accept ‘allied’ or ‘angles between parallel lines’.Dependent on correct angles.

B1

Alt. 1

ADB = 38 (Alt segment)B1

DCB = 104 (opposite in cyclic)B1

CBD = 38 (isosceles)B1

CBD = ADB (alternate)

Use of ‘Z angles’ is not acceptableDependent on correct angles

B1

Alt. 2

ADB = 38 (Alt segment)B1


BDC = 38 (isosceles)

ADC = 76B1

BDC + BCD = 180 (interior)Dependent on correct angles

B1

Alt. 3

ADB = 38 and ABD = 66 (Alt segment)B1


Page 186

CBD = CBD = 38 (isosceles)B1

DCB = CBA and CDA and BAD = (isosceles trapezium)B1

[4]

M20. (a) 0.007B1

(b) (i) 0.9119215(052)B1

(ii) 0.9, 0.91, 0.912, 9 or 9.1 or 9.12 × 10-1

ft their answer for (b)(i) to 1, 2 or 3sfeg Gradians (b)(i) 0.02221673729 B0 (b)(ii) 0.02, 2 × 10–2, etc B1 ft

B1 ft

(c) 0.00805 or 8.05 × 10-3

B1[4]

M21. (a) π r 3 = 2 × π r 2 x

Must include the factor of 2Allow use of h instead of x

M1

Simplified to give x = 2rAlternatively

Page 187

Allow substitution of 2r for height of cone and verification of result

Page 188

ie 2 × Vol cone = 2 × × π × r 2 × 2r M1

Page 189

= π r 3 (must be seen) A1A1

(b) (l)2 = r 2 + 4r 2

(l)2 = r 2 + (2r)2 is M1(l)2 = r 2 + 2r 2 is M0

M1

(l) = √5 rA1

Surface area cone = π × r × √5 rUsing their l if from an attempt at Pythagoras

M1

4 : √5 Allow √5 : 4SC2 for a complete numerical solution

A1[6]

M22. (a) Correct Pythagoras in two appropriate right-angled trianglesor simply BH ² = 12² + 3² + 4²

M1

13 A1

(b) HB = 13, HC = 5 or DB = √153with attempt at trig. Ratio

Explanations may not involve any calculationseg BC < BD or HC > HD together with some comparison such as BH is common(diagrams drawn, to illustrate, are appropriate)

M1

Two correct, comparable trig. ratios

Page 190

eg sin x = and sin y =

For example: BH is common and triangles BHD and BHC are right-angled, so y must be bigger because the height is greater

A1

y Good explanation and correct conclusion … this earns all 3 marks

A1[5]

M23. YZ = ZYB1

Angle MZY = angle NYZbase angles of (Isosceles) ∆ XYZ

Note Reason necessaryeg you might see If XZ = XY then angle XZY = angle XYZ

B1

Angle MYZ = angle NZYB1

Triangles congruent, ASANote Dependent on earning first 3 marksMust give correct reason for congruence (ASA)Only allow AAS if complete argument stating‘third angles equal’

B1 dep[4]

M24. Sight of correct ratio or scale factor ie 20 : 30, 2 : 3, , 1

Page 191

oe sight of or earns this markM1

× 40

Page 192

oe eg might work out then subtractM1

24 Note 2 : 3 ratio might be scaled up to give ratio of 16 : 24 (M1, M1)Must state h = 24 for A1Alternatively h/30 = (40 – h)/20 M1 20h = 30(40 – h) 20h = 1200 – 30h M1 50h = 1200 h = 24 A1

A1[3]

M25. (a) M1

A1

80.2(1...)r= 12 giving 40.1 is Ml, Al, A0r = 3 giving 160.4 is Ml, Al, A0

A1

(b) M1

h = 9 (cm)h = 12 gives Ml, A0

A1

Ml for difference of two cone volumesAl if all correct

M1, A1

Page 193

(V) = 327 or 326.7 ............................(cm3)Accept 330 if working seen, ft their h if bothM's awarded.

A1 ft[8]

ALTERNATIVE

linear scale factor 1:3Must be used. Just writing it down does notqualify as a method unless progress is made.

M1

Volume scale factor 1:27A1

Page 194

Volume small cone M1

Volume large cone 27 × (their 12.566)339.292...

DM1

(V) = 327 or 326.7 ............................(cm3)

Accept 330 if working seen.A1

Scs

12.566 onlyB1

339.29 onlyM1, A1, M1

M1,A1[8]

M26. (a) y = x+ 1, y = – x – 2x + 1= –x – 2

B1

–x – 2 = x2 + 2x – 3M1

(b) x2 + 3x – 1 = 0Simplified to 3 terms in x2, x and constante.g. x2 = 1 – 3x

A1

x2 + 2x – 3 = × + 1M1

(c) 1.6, –2.6

Page 195

Accept 1.5 to 1.6, – 2.5 to – 2.6A1

[5]

M27. M1

A1

Page 196

oe e.g. A1

[3]

M28. (a)

Page 197

Accept M1

Sin B = 0.9679(1...)A1

B = 75.4(...)A1

(b) x2 = 222 + 232 – 2 × 22 × 23 × cos 48M1

x2 = 335.8(...)A1

x = 18.32(....)ft only if an error made in calculation of x2

but not on (222 + 232 – 2 × 22 × 23 ( = 1)) cos 48(= 0.669 = 0.818)

A1 ft

18 or 18.3Independent mark. Award if value > 3sf seenor calculation seen.

B1 ft[7]

M29. (a) 4a + 3 = 2b + 5M1

(b) 4a – 2b = 2 (-2)Must indicate division by 2

A1

4a+3+2b+5+2a+b=32

6a + 3b = 24

2a + b = 8Bl for any version

M1

Page 198

(1) × 3: 6a – 3b = 3M1

12a = 27For attempt to eliminateAB or 4a + 3 =12 and BC or 2b + 5 = 12

M1

a = 2.25A1

[6]

M30. (a) Sight of tan unless alternative method usedM1

Tan–1(5.59/1.5)90° – tan–1(l.5/5.59), 1.5tan70 and 1.5tan80

DM1

74.(98) or 75° so safe4.1(2) and 8.5(1)

A1

(b) Sight of cosM1

4 × cos80DM1

0.690.7 with working

A1[6]

M31. Vol Hemisphere =

Ml, M1

Page 199

M1

144π(3 × 4 × 63 × π) ÷ (2 × 3 × 27 × π) = r2 or...

A1

their 144π...(4 × 63) ÷ (2 × 27) = r2 A1

M1

(r =) 4(r =) 4

A1[4]

M32. angle OSQ = angle OQS = 50°Isosceles triangle OQSPenalise ‘no reason’

B1

angle OSR = 90° → angle QSR = 40°Tangent-radius property first time only

B1

angle QSR = angle QRS (Isosceles)B1

[3]

M33. (Graph 1) DB1

(Graph 2) AB1

(Graph 3) EB1

(Graph 4) CB1

[4]

Page 200

M34. (a) Line y= 1 drawn or points on curveAccept y = 1 written in body of script.

M1

0.8, –3.8 (±0.1)A1

(b) Attempt to split equation intox2 + 3x – 2 = ax + b

Or x2 + 3x – 2 -(x2 + 2x – 1)Or x2 + 3x – 2 + ax + b = x2 + 2x – 1

M1

Line (y = x – 1) drawnA1

0.4, –2.4 (±0.1)f.t. on their line if Ml awarded,e.g. y = x + 1(1, –3), y = 1 – x(0.6 (0.7),–4.6 (–4.7)),y = –1 –x(0.2, –4.2)

A1 ft[5]

M35. (a) Linear scale factor is 2Allow numerical examples but must be complete23 = 8, 43 = 64, 64 = 8 × 8, but the increase bya factor of 8 must be shown and not assumed B2

B1

Volume scale factor is lsf3

Allow algebra (2x)3 = 8x3

B1

(b) (14.5 ÷ 8)3 or 1.81253

Page 201

3√6 = 1.817 83 × 6, 14.53 ÷ 6M1

=5.95(4)8 × 1.817 ≈ 14.5 ≈ 14.53 ≈ 83

A1

Volume increases by about 6 × so claim justified.Allow ‘Almost but not quite’

A1[5]

M36. Use of Must use 4 or 8 as radius.

M1

(Volume hemisphere =) 133.9to 134.1 (inclusive)

133.97 if π = 3.14 used.A1

(Volume paperweight =)500+(their 134) (=634)

If Ml awarded.A1 ft

cm3

This mark is independentB1

[4]

M37. Area semicircle – area circleAccept π × 102 ÷ 2 and/or π × 52 for M1

M1

π52 ÷ 2 – π(2.5)2 (= 12.5π – 6.25π = 39.27 – 19.63)Accept fractions, decimals or in terms of π

A1

Page 202

19.6(....)(= 6.25π)ft on one error only, e.g.Accept fractions, decimals or in terms of π.Use of π as 3.14 gives 19.625 AlCommon errors e.g.π × 102 ÷ 2 – π × 52 = 157.1 – 78.5 = 78.6= Ml,A0,A0ft.π × l02 ÷ 2 – π × 2.52 = 157.1 – 19.6 = 137.5= M1,AO,A1 ftπ × 52 ÷ 2 – π × 52 = 39.3 – 78.5 = –39.3Ml, AO, AO (non-sensible answer)

A1 ft[3]

M38. y = kx2 or y a x2

oe 5 = k × 16M1

k = 0.3125 oe

A1

20A1

[3]

M39. Breaks problem down into sum of lines and (semi-)circlesM1

Length of lines 4.1 + 5.9 + 4.7 + 2.9 (= 17.6)Sc 17.6 only B1

A1

Use of 2 πr ÷ 2or πd ÷ 2 but must use with numbers.

DM1

Length of semi-circles0.9π + 0.6π + 0.7π (= 6.9(11..))

Page 203

2.8, 1.9,2.2A1

Total = 24.5(...)ft on 1 arithmetical or ‘reading from scale’error and both M's awarded.4.1 = 2.9 + 0.6 + 0.6, 5.9 = 0.6 + 0.6 + 2.9 + 1.8,4.7 = 2.9 + 1.8, 2.9 = 2.9

A1 ft[5]

M40. (a) BC2 = 192 – 92(= 280)x2 + 92 = 192

M1

BC = √280For squaring, subtracting and evidence of square rooting

DM1

BC = 17 or 16.7(....)17 with no working gets 3

A1

(b) Sight of tangentM1

orAngle = tan–l (1 ÷ 24)

tan –1 (0.458)M2 for any complete correct methodSin = 11/√697 or 11/26.4Cos = 24/√697 or 24/26.4

DM1

25 or 24.6(....)25 with no working gets 3Radians 0.43 gradians 27.35Penalise on first occurrence only.

A1[6]

Page 204

M41. 100 ÷ 60 or 80 – 501.66 or 1.6

M1

100/60 × 60 or 80/50 × 60100 min or 96 min

DM1

Their(100 – 96) or reversedDM1

4A1

[4]

M42. (a) (2x – 3)2 = 4x2 – 6x – 6x + 9condone one error 4x2+ 9 is two errors

M1

4x2 – 12x + 9 = 8x – 16or 4x2 – 20x +25 (= 0)

for equating expressions and/or simplifyingthis must lead to a quadratic equation

M1

(2x – 5)(2x – 5) (– 0)ft from their quadratic equation (if 'formula' used,substitution must be completely correct)

M1

x = 2.5A1

y = 2A1

(b) Only one solution so straight line must be a tangent to the curveHence sketch 2

ft from their solution(s) to (a)clear solution(s) in (a) ‘correct’ sketch in (b) can earnB1 (no explanation) or B2 (with explanation)

Page 205

B2 ft

ALTERNATIVE

(a) oe for setting up attempt to eliminate x

M1

y2 – 4y + 4 (= 0)oe condone one errorthis must lead to a quadratic equation

M1

(y – 2)(y – 2)(–0)ft from their quadratic equation (if ‘formula’ used,substitution must be completely correct)

M1

y = 2A1

x = 2.5A1

(b) Only one solution so straight linemust be a tangent to the curveHence sketch 2

ft from their solution(s) to (a)clear solution(s) in (a) ‘correct’ sketch in (b) can earnB1 (no explanation) or B2 (with explanation)

B2 ft[7]

M43. (a) x = 110430° scores B1

B1

(b) x = 250, x = 290B1 for each

B2

(c) correct sketch of ‘double’ cycleB1 for sketch only as far as 180 or slight inaccuracy

B2

Page 206

(d) x = 35, x = 55must have 35 and ½ of their (a)

B1

x = 215B1 ft

x = 235B1 for each of (their 35 + 180), (their 55 + 180)

B1 ft[8]

M44. (a) 0 = 4 + 2p + q–5 = 9 – 3p + q

for substitution of both sets of coordinatesallow one error

M1

5 = –5 + 5poe for correct attempt at elimination of p or q

DM1

p = 2A1

q = –8p = 2 and q = –8 from no obvious working scores 4

A1

(b) Solving their x2 + px + q = 0if ‘formula’ used substitution must be completely correct

M1

(–4, 0)A1

[6]

Page 207

M45. (a) π(×)52

condone 3.1... × 52

M1

π (×) 52 × 10 or (their area) × 10π(×) 52 × 10 or (their area) × 10

condone 3.1... × 10 × 52

their area must contain πM1

250π or 250 × π or π × 250775 to 790 scores M2 A0do not accept π250ignore fw250π can be recovered in (b)

A1

(b) 40 × 5010 × 10 × 10 40 × 50

M1

their 2000 × 10their 1000 – their 250π 20 × their (π × 52)

M1

20 × their 250π20 × their (1000 – 250π) their 2000 – their 500π

M1

20000 – 5000π20(1000 – 250) 10(2000 – 500π)4290 to 4500 scores M3 A0 ignore fw except 15000π

A1[7]

M46. (a) (i) a + bb + a

B1

(ii) 2a + bb + 2a

Page 208

B1

(iii) b – a– a + b

B1

(b) CF = OEM1

a + 2ba + b + b oe

A1

(c) Straight line because OD = a + b

B1

3 times bigger because OF = 3a + 3b

B1[7]

M47. 2.75 × 27M1 for 33

M1

74.25.....Accept 74 or 74.3Height first pyramid is 2.0625Height second is 6.1875

Volume is × 62 × 6.1875 = 74.25 This line M1, A1.

A1[2]

Page 209

M48. Angle ATB = 13°B1

M1 for use of sine rule, A1 for correct substitution.

Page 210

= M1, A1

BT = 27.47 (41539..)AT = 45.79112344

A1

H = BT × sin31M1

H=14.2 or 14.15(........)Ft only if both Ms awarded. NB 14.2 can come from BT = 27.5 or AT = 46 Deduct 1 for pa if seen.

A1 ft[6]

M49. (a) Parallel curve translated up y axis‘2’ need not be marked, needs to look symmetrical

B1

(b) Parallel curve translated in positive direction along x axisMust 'sit on' x axis and look symmetrical

B1

(c) Curve through (0,0) nearer to x axis than originalMust look symmetrical

B1[3]

M50. (a) (i) y = 3x – 2 plottedmust draw correct line

M1

Page 211

x = 2, x = 5A1 for each, must be correct answers...no ft.coordinates given ... lose 1 mark

A2

(ii) x = 2, x = 5must have both solutions (ft answers from part (a) earns 1 mark)

B1

(b) x2 – 4x +8 = x + 4allow one slip in manipulation

M1

y = x + 4Straight line to be clearly stated

A1[6]

M51. (a) 4B1

(b) (32 – 4 – 4 – 5 – 5) (÷ 2) or 14 or 16 – 4 – 5

or equivalentM1

7A1

[3]

M52. Any correctly evaluated counter example with non-prime conclusion.

Page 212

Examples–4 and 5 => 25 and not prime–5 and 6 => 35 and not prime–8 and 9 => 77 and not primeaccept any indication of "not prime"

B1 any correctly evaluated trial with no conclusionExamples – 3 and 4 => 17 –4 and 5 => 25 –5 and 6 => 35 –6 and 7 => 47 –7 and 8 => 61 –8 and 9 => 77or incorrectly evaluated trial that gives a counterexample with non-prime conclusion

B2[2]

M53. Extra volume = 50 × 34 × 4.5= 7650

M1

1912.5A1

Page 213

r3 = their 1912.5Dependent on ‘their 1912.5’ coming from a volume calculation.

M1

r3 = (3 × their 1912.5) ÷ 4πAllow (3 × 7650) ÷ 4π

DM1

R = 7.7, 7.70, 7.700…A1

[5]

M54. AC2 = 72 + 92 – 2 × 7 × 9 × cos75M1

AC2 = 97……. , AC= 9.9, 9.86……A1

Their AC2 – 62

AC2 must be > 36 = 61.38888 if correctM1

DC = 7.8(3…)Answer must be accurate to 2 sf or better

A1 ft

Perimeter = 29.8(.…)ft their DC + 22 but both Ms must be awarded.

A1 ft[5]

M55. Angle at centre = 2 × sin–1( )

Half angle sin–1( ) gets M1M1

= 71.(...)A1

Page 214

Area sector = ‘their 71’ ÷ 360 × π × 62

M for use of area sector formula not for πr2 ÷ 4 for example.M1

Area sector = 22.4(….)_A1

Their sector – their triangle areaMust make a valid attempt at calculating the area of the triangle. (17.06…) and at least one of the previous M marks must be awarded.

DM1

Area segment = 5.3…..A1

[6]

M56. (a) 0.51(2)B1

(b) Correct plots B1 ft

Smooth curve±0.5 square Use of ruler or ”double‘ lines or discontinuities B0

B1 ft

(c) 1.2ft their graph. If ”double‘ line at y = 0.76 then B0. Within tolerance of their graph

B1 ft[4]

M57.

Page 215

B2 fully correctB1 for any translation of correct answer. Alternative scheme.M1 for ”rays‘ from at least 3 corners through (–1, 0) andattempt at drawing a reduced shape in 3rd quadrant.A1 if correct shape

B2[2]

M58. C = π × 7C = 2π × 3.5 Must substitute numbers.C = πd or 2πr is M0 until used.NB π × 3.5 is M0 as wrong method (πr)

M1

= 21.98 – 22

3.14 × 7 = 21.98, × = 22A1

Length = 22.98 to 23ft their 21.99 + 1 if M1 awarded.

A1 ft

Height = 10 cmAllow answers transposed.

B1[4]

Page 216

M59. Trial for x > 4All trials correctly evaluated to at least 1 d.p.,rounded or truncated. NB Condone odd erroras this may be ”recovered‘ later.

B1

Trial for 4 < x ≤ 55 → 5.2, 4.5 → 4.72, 4.6 → 4.81, 4.7 → 4.91

B1

Trials for 4.7 = ≤ x ≤ 4.85 and answer 4.84.75 → 4.96, 4.76 → 4.97, 4.77 → 4.979…, 4.78 → 4.989…,4.79 → 4.998…, 4.8 → 5.008..or 5 4.85 → 5.056

B1

Trial for 4.75 ≤ x < 4.8 and answer 4.8NB. Minimum for full marks. e.g. test 4.75, test 4.8,state 4.8 as answer.

B1[4]

M60. (a) angle QPR = 32°Base angle of isosceles triangle

B1

angle QSR = 32°equal to angle QPR, angles in same segment oeprecise explanations for 2 marks

B1

(b) (i) 116°B1

(ii) Line from O to A, creating 90°Join OA using alt seg thm

angle OAB = 16°angle CA(X) = 48° M1angle BCA = 74° M1

M1

angle OAC = 42°angle OAC = 42° M1angle BCO = 32° M1

Page 217

M1

angle OCA = 42°angle OCA = 42° A1angle OCA = 42° A1

A1[6]

M61. Kite 2 and reasone.g. 110 + 70 = 180sum of (opposite) angles = 180

B1 Kite 2B2

[2]

M62. Height = 100 × 3 ÷ 25Alternative: B1 for length of median of side

M1

= 12(= 12.25)

A1

Diagonal base = √50 = 7.07 (or half base = 3.535...)M1 for pythagoras on half of side triangle.

B1

x = √(122 + 3.5352)x = √(12.252 + 2.52)

M1

= 12.5A1

[5]

Page 218

M63. Volume cube = 203 = 8000B1

Square hole = 102 × 20 = 2000B1

Circular holes = 2π × 42 × 5 = 502.7B1

Volume left = 5500ft 8000 – 2000 – their cylinders. Only ft if attempt madeto find volume circular holes not using 20 cm as height(e.g. 5 or 4 or 10)

B1 ft

cm3 Units mark independent

B1[5]

M64. cos A =

If other angles found answers are (to 1d.p)C = 70.5, B = 59.0

M1

cos A = 0.6363..Mark as scheme with different values

A1

A = 50.5°A1

Area = 0.5 × 10 × 11 × sin 50.5DM1

= 42.43 cm2

A1 ft[5]

Page 219

M65. 27.5 × 12 – 250 (=80)330/250 or 330/2.5 get M1

M1

cv/250 (×100)For completion of method

DM1

32% increase32% must be stated.Special cases all get M1, DM1, A0Misreads both as 10% => 21%Misreads both as 20% => 44%Misread both as decreases => 28%

A1

ALTERNATIVE

1.10 × 1.20M1 for 110% × 120%

M1

= (their 1.32) – 1A1 for 132% or equivalent

DM1

32% increaseA1 stating answerSpecial cases above with equivalent values.

A1[3]

M66. (x – 5)2 – 30 = 0For attempt at (x – 5)2,

M1

x = ±√30 + 5For –5 and –30

A1

Page 220

x = 10.48, –0.48Both answers (Accept 10.5, –0.477)

A1

ALTERNATIVE

Page 221

x =

For substitution into formula (allow one error)M1

x = Correct substitution

A1

x = 10.48, –0.48Both answers (accept 10.5, –0.477)

A1[3]

M67. (a) Sight of sine 48 or cos 42M1

x = 5.1 × sin 485.1 × cos 42. √(5.12 – (5.1 cos48)2) M2

DM1

x = 3.79003861 cmA1

x = 3.8, 3.79Rounding mark is independent for a value or calculation that is 4sf or greater

B1

(b) Area = 6.8 × (their x)5.1 × 6.8 × sin 48

M1

= 25.77 cm2

f.t. their value for x.A1 ft

[6]

Page 222

M68. 62 + 2.52 ( = 42.25)Squaring and adding M1 sight of tan

M1

√42.25Square root DM1 For 6 ÷ cos22.6 or 2.5 ÷ sin22.6

DM1

6.5A1

AB = 4Ft their 6.5 – 2.5s.c. 2.95 from incorrect pythagoras B1

A1 ft[4]

M69. (a) Graph DB1

Slow, steady, faster steadySteady rate and an indication why quicker at top.

DB1

(b) Any container with uniform vertical cross-sectionAllow 2-D, e.g. Rectangle, and/or substantially uniform X-section

B1[3]

M70. Graph A is y = (x – 3)2

B1

Page 223

Graph B is y (x + 3)2

B1

Graph C is y = –x2

B1

Graph D is y = 3 – x2

B1[4]

M71. (a) BD/15 = 2/3M1

BD =10Or equivalent eg. BD = 2/3 × 15

A1

(b) Correct use of PythagorasM1

BC = 8A1 ft

sin y = 8/10f.t. with their BD iff BD > 6 for this M mark

A1 ft[5]

M72. 1st arc AD, capable of being used in 60° constructionM1

2nd arc from ‘correct’ point crossing 1st arcM1 dep

Angle of 60° (± 2°)A1

‘B’ correctly markedNeed not be labelled ... only award if first M1 earned

B1

Page 224

Complete construction for perp. from ‘B’ to AD (i.e. all arcs)M1

‘C’ correct (±2°)Need not be labelled

A1[6]

M73. (a) 100B1

(b) 130B1

(c) 70 seenB1

Full explanation. E.g 180 – (90 + 20)Minimum requirement 90 – 20

B1[4]

M74. 1.53 (× 480)M1 for s.f.3 e.g. (1.5)3 or (0.666)3

M1

= 1620A1

ml or cm3

NB 1.62 1 gets M1, A1, B1.This is an independent units mark.So 720 ml gets M0, A0, B1cl and I are acceptable iff an attempt made to convert answer.

B1[3]

Page 225

M75. volume cylinder = 113.(...) cm3

Accept 36πB1

volume cone = 18.8(...)cm3

Accept 6πB1

Volume (their cylinder – their cone) ÷ 9πAccept 30π ÷ 9π

M1

3.3(3)A1

= 5.3(3...)Accept fraction. (5 1/3)f.t. iff M1 awarded.Consistent use of diameter for radius gives 144π (= 452.39) for cylinder and 24π (=75.40) for cone.Volume = 120π (376.99). Volume ÷ 36π = 3.333.. + 2 = 5.333 Give B0, B1, M1, A1, A1 f.t.Hence do not give full marks if answer seen on answer line. Check working before awarding full marks.Do not accept 5 as a answer.

A1 ft[5]

M76. Angle APB = 82°B1

x2 = 182 + 252 – 2 × 18 × 25 × cos 82Use of cosine rule 324 + 625 – 900 cos (their angle)( = 949 – 125.25 = 823.7....)f.t their angle for M1.

M1

Page 226

x2 = 823.7(..........)x = 49 cos 82 or 6.81948.....gets M1, A0, A0

A1 ft

x = 28.7(.....)Follow through on an incorrect angle only

A1 ft[4]

M77. HF2 = 52 + 122

DF2 = 52 + 52 + 122 = 194DF = 13.9(2....) gets M1,A1.

M1

HF = 13B2 if HF = 13 stated

A1

Correct Right angled triangle DFHFollow through their HF if first M1 awardedDo not accept lines on diagram joining DF and FH as evidence unless right angled marked or something done with lengths

M1

tan-1(5/13)DM1 if both previous M’s awarded.DM1 for appropriate ratio if other lengths used.

DM1

=21(.0...)°Ft on HF or DF only.

A1 ft[5]

Page 227

M78. AB = DCB1

Angle EAB = Angle ECDAlternate angles or Z angles must be stated

Angle EBA = angle EDCAlternate angles or Z angles must be stated

Angle AEB = Angle DECOpposite must be statedNB only maximum of two can be scored here.

B1, B1

Therefore congruent because ASA, AAS etcDependent on first B1(AB=DC) but can be awarded if angles stated as equal but not justified.This final step is needed for full marks. Can be given in words e.g.Angle, Side, Angle

DB1

This is a proof and the explanation must be rigid.Other versions of the proof are (for example)

Stating that ABCD a parallelogramB1

AE = EC

DE = EB

AB = CD

Any angles as above (with justification)NB B marks for length are dependent on the first B1 but angles are not.NB 3 statements need to be made here to get both marks. Sides equal do not need justifying but angles do.NB if only 2 length statements are made and no other statements then give DB1

B2

Conclusion SAS or SSS.Dependent on first B1

DB1[4]

Page 228

M79. Sight of sineM1

125 ÷ sin 33Accept 125 ÷ 33sin

DM1

229(.5..........)A1

230 or 229Follow through any value ≥ 4 s.f. or calculation seen,e.g. 125 × sin 33 = 68 or 68.1

B1[4]

M80. Scale factor = 9 ÷6 = 1.5Accept 9/6 or 6/9 or 4/6 or 6/4 or ratios

Page 229

or M1

AB = 4 × 1.5 (= 6)Correct use of their scale factor36 = 24 + 6x or equiv

M1

BD = their AB – 4 = 2A1 cao

[3]

M81. (a) Attempt at translation of 45° to the rightM1

P = (135,1)A1

(b) Attempt at sine curve of twice the amplitude of the originalM1

P = (90, 2)A1

[4]

M82. (a) × (2 × π × 12)M1

18πNot π18, unless notation previously penalisedπ × 18 is acceptable

A1

Page 230

(b) 2 × π × r = their 18πOr their 18π ÷ 2π

M1

r = 9

r = of 12 = 9 scores 2 marksA1 ft

[4]

M83. (a) (i) BA = a – 2bor equivalent

B1

(ii) MQ = MB + BA = b + (a – 2b)Attempt to set up a route, must include substitution of a and b, condone one error

M1

MQ = a + b or (a + b)Need not be simplified

A1

(iii) OP = OA + AB = a + (2b – a)or OP = OB + BA = 2b + (a – 2b)

M1

OP = a + bM1 as above, need not be simplified for A1

A1

(b) OP = 2 × MQB1

Trapeziumonly if accompanied by a sound reason

B1[7]

Page 231

M84. (a) (i) 40B1

(ii) 140 or 180 – their xDo not ft if answer = 140 in (a)(i)

B1 ft

(b) Logical and precise explanation(Either written or as a calculation)

One angle labelled or stated correctly, no reason, B1B2

(c) angle ADB = 32°Reasons not needed in any partAlt seg.thm.In all parts accept angles

B1

angle DBC = 32°Alternate anglesmarked on the diagram

B1 dep

angle BDC = 32°Base angles Isos. Triangleas ‘evidence’

B1 ft

angle BCD = 116°Angle sum of triangle

B1 ft

angle BAD = 64°Opp. angles cyclic quad.

B1 ft[9]

##

Page 232

All loci must be within lake

Arcs centre B radius 3 cm (±2mm)and 5cm (±2mm)

Consistent use of scale 1 cm to 100mB3 max(-1 MR)

B1

Construction arcs for perp.Bisector

Two sets of intersecting arcs centres A and CB1

Perp. Bisector of AC (±2mm)B1

Line 2cm (±2mm) from pathB1

Correct region shadedB1

[5]

M86. (a) (i) 40B1

(ii) 140 or 180 – (their x)Do not ft if answer = 140 in (a)(i)

B1 ft

(b) Logical and precise explanation(either written or as calculation)

B1 for 1 angle labelled or stated correctly, no reasonB2

(c) 24 ÷ 8 × 2or OP = 6

M1

Page 233

π(or 3.14) × (their 6)2

M1

36π or 36 × π or π × 36allow π36SC2 108 to 114 or π × 9 oeSC1 27 to 28.5

A1[7]

M87. Side of 34B1

50 – 18 or 25 – 9M1

Side of 32 or 16A1

162 + x2 = 342

M1

x = 302 × ‘8, 15, 17’ triangle get M1, A1

A1

64A1

[6]

M88. Outer of total

Page 234

Outer shaded area = of totalB1

Inner squares of totalB1

Outer shaded area = 16%B1

Page 235

Inner shaded area = M1

Page 236

11.52% A1

Total = 27.52%A1

Alt

Some indication that 3L = 5WB1

L side 15 and S side 9B1

Diagram is 75 × 75 (= 5 625)B1

M1 For using either part of numeratorA1 For both and adding

M1, A1

Total = 27.25%A1

[6]

M89. 360 ÷ 10 or 360 ÷ 5or 36 or 72 or 144 or 108NB Angles may be marked on diagram

M1

144 and 108or 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

Page 237

∠XBC = ∠XCB = (180 – 108) ÷ 2or ∠XBC = 36

M1

∠ABX + ∠XBC = 144 + 36 ( = 180)oeeg, ∠CBX calculated from ΔBXC equals exterior angle of decagon

A1[5]

M90. (a) Lengths CE = 3, ED = 4These will probably be marked on diagram

B1

Angle BEC = angle AEDB1

3 ÷ 4.5 = 4 ÷ 6 ( or vice versa)oe B1 For any comparison of value from both triangles eg, ratio3 ÷ 4 = 4.5 ÷ 6 ( or vice versa)

B1

(b) EF = 9B1

9 ÷ 6 = 4.5 ÷ 3 or 9 ÷ 4.5 = 6 ÷ 3 (or vice versa) BCE and AEF have twosides in same ratio.

Angle BEC = angle AED so similarAngle AFE = angle EBC (alternate)Hence AF parallel to BC

E4 Full explanationE3 Partial explanation with 3 relevant factsE2 Partial explanation with 2 relevant factsE1 Partial explanation with 1 relevant fact

E4[8]

Page 238

M91. BC = AD = 15B1

92 + 122 (= 225)Reference to 3, 4, 5 triangle

M1

Both AB and CD (√225) = 15Scaling up to 9, 12, 15

A1

ABCD is a rhombus as all sides same lengthA1

Alt 1

Gradient BD –2 and

gradient AC = M1

Hence BD perpendicular to ACA1

Showing any two sides = 12M1

ABCD is a rhombus as diagonal perpendicular and two sides sameA1

Alt 2

Gradient BD = –2 and

gradient AC = M1

Hence BD perpendicular to ACA1

Midpoint AC andmidpoint BD = (8, 5)

Method for finding midpoint (calculation or drawing)must be shown

M1

Page 239

ABCD is a rhombus as diagonal perpendicular and intersect at midpointA1

Alt 3

Vector BD = 12i andvector AD = 12i

Vector AB = 9i + 12j andvector DC = 9i + 12j

M1, A1

Showing either of AB or DC = 12A1

ABCD is a rhombus as opposite sides of equal lengths parallel and twoadjacent sides same

A1[4]

M92. y = 2x – 3 drawnB1

(1, 4) marked and perpendicular to line drawnPerpendicular can be implied.

B1

(5, 2)Accept coordinates on graph

B1[3]

M93. ∠ABD = 75or ∠PDA (= 75 – 30) = 45 and ∠PAB = 45Allow on diagram

B1

∠BAD = 60

Page 240

Allow on diagramB1

120ft (their) 60 not ∠BAD = 90Allow on diagram

B1 ft[3]

M94. (a) or or 1.5

or or or 0.666(6 …)M1

(their) 1.5 × 12or 12 ÷ (their) 0.66

M1

18A1

(b) 1080 ÷ or 1080 ×

or

Page 241

1080 ÷ or 1080 × or 1080 ÷ 4 or 1080 × 0.25

M1

270A1

[5]

M95. a, b, c and d must be integers (including ft)

(a =) 6 and (b =) 2 or (c =) 4 and (d =) 3

or (a or c =) 12 and (b or d =) 1Any indication eg, on diagram

B1

(their) 62 + (their) 22

or(their) 42 + (their) 32

ft (their) a and b or (their) c and dM1

√((their) 40 or 6.3(...)or√((their) 25 or 5

M1

6.3( …) or √40 or 2√10and5

A1

22.3( …)or 16 + √40 or 16 + 2√10Allow 22 with method

A1[5]

Page 242

M96. (a) 10π = 2πror 10π = πd

M1

10π ÷ 2πd = 10 or 10 ÷ 2

M1 dep

5A1

(b) 80π = π52hoe

M1

Page 243

oeeg, 240 = 25h (no fraction and no π)

M1 dep

Page 244

9.6 or or or or

oe ft 240 ÷ their radius2

Ignore fwA1 ft

[6]

M97. (a) 2b – 2aoe b + b – a – a

B1

(b) 2c – 2boe

B1

(c) (2b – 2a) + (2c – 2b) or b – a + c – bB1

(d) DG and EF are parallel and equal in length

or DG and EF are equal (vectors)

or GF and DE are equal (vectors)oeBoth c – aBoth are same vectorsDo not need vector arrowsDo not accept properties of parallelogramNot enough to say they are parallel

B1[4]

M98. (a) Valid explanation

Page 245

eg allied angles (add up to 180) inside parallel lines (add up to 180) y + y + 2x + 2x = 360 so y + 2x = 180 2y + 4x = 360 In a C add up to 180Condone interior angles (add up to 180)

B1

(b) 3x + y = 230oe

B1

(c) Attempt to eliminate a variable(with 2x + y = 180)

eg 6x + 2y = 460 and 6x + 3y = 540 and subtractionNote: Full marks can be awarded for this part on follow through

M1

x = 50A1 ft

y = 80A1 ft

3x + y = 130 → x = –50, y = 2803x = y –50 → x = 26, y = 1283x + y = 410 → x = 230, y = –280

[4]

M99. (a) (i) 180B1

(ii) 18Their (a)(i) ÷ 10Not ft 280 or 040 or > 360

B1 ft

(iii) 06Do not allow 6

B1

(b) 210

Page 246

Allow 208 – 212B1

21SC1 15Not 21.0

B1 ft

(c) 300 – 180180 ÷ 10

M1

12030 – 18

A1

12A1 ft

[8]

M100. (a) y = sin x + 2B1

(b) y = 3 sin xB1

(c) y = sin (x – 90)B1

[3]

M101. oe

M1

Page 247

13.2 × M1 dep

9.287( ...) or 9.29A1

9.29 or 9.3ft Their BC to 1 dp or 3 sf

B1ft[4]

M102. (a) No and showing

oe No and …eg 0.125 > 0.083 ...

Page 248

or

144 ÷ 12 < 16 16 × 12 or 192 > 144 The length should be 192

B1 or or 0.111 …seen

192 seen oeDeduct 1 for Yes

B2

(b) 48B1

62 + (their 48)2

or 2340M1

√(their) 2340M1dep

48.3(7 ...) or 48.4A1

[6]

M103. (a) x + 1 + x + 1 + 3x – 2 + 3x – 2oe eg, (4x – 1) × 2

B1

(b) 8x – 2 = 16or 8x = 18

B1

8x = 18oe

M1

Page 249

(x =) A1

[4]

M104. (a) Collinearoe

B1

(b) 2 : 1B1

[2]

M105. (a) y = x drawnM1

(x =) 3.5 and (y =) 3.5± 0.1Readings from their y = x

A1

(b) 2x2 = 25x2 + x2 = 25

M1

x2 = 12.5Must see method

A1[4]

Page 250

M106. (a) 23B1

(b) 44B1

[2]

M107. 300 × 440Note 300 × 440 or 300 × 400 scores M0

M1

132 000A1

132 kgB1ft

[3]

M108. (a) eg, 5 squares = 12.5or 5 cm = 12.5

12.5 ÷ 5 (= 2.5)12.5 ÷ 2.5 = 5

B1

(b) Rectangle drawn 3 cm by 2 cmB1

Distance of rectangle from garage ≥1 cmB1

Attached to back wallB1

[4]

Page 251

M109. (a) ReflectionNote parts (a) and (b) are marked as a pair witheither answer in either part

B1

(In the line) ACB1

(b) RotationB1

180°B1

About MB1

[5]

M110. Angle ATB = 90°(angle on diameter)

B1

Angle BTC = x°(alternate segment)

B1

32 + x + x + 90 = 180oe

M1

29A1

Alt

Angle OTC = 90°(radius perp to tangent)

B1

Page 252

Angle TOC = 58°B1

2x = 58oe

M1

29A1

[4]

M111. (x + 5)2 + (x – 2)2 = 102

x2 + 25 + x2 ± 4 = 100 implies M1M1

x2 + 10x + 25 + x2 – 4x + 4 = 100SC1 x2 + 10 x + 25 + x2 – 4 x + 4 = 10

A1

2x2 + 6x – 71 = 0For rearranging into a 3 term quadratic = 0or going to cts straight away.

M1 dep

Use of quadratic formula or completing the square to solveAny evidence of formula or cts gets M1Allow use of graphical calculator

M1 dep

4.6, 4.64, 4.654.64(4..) for T & I

A1[5]

M112. Diameters = , 7,

Page 253

M1 for attempt to find radii or diameterseg, 2 ÷ 4 or 1 ÷ 4

M1, A1

Page 254

π × 42 + π × 3 + π × 3 + π × 3

M1 for Σ π r2 for at least 3 floors with any different radii50.26, 44.18, 38.48, 33.18, (28.27) values rounded ortruncated

M1, A1

Yes as 165 or 166...Must reach conclusion

A1[5]

M113. (a) (i) (±) 15.8B1

(ii) Divide by numbers and reference to answers being whole numbersand/or decimals

E1

Reference to dividing minimum 3 primesE1

Full explanation E2, partial explanation E1

(b) (i) Evidence of dividing 2008 by primes or splitting into a ‘tree’Must do at least 2 divisions or 2 ‘branches’

M1

(1) × 2 × 2 × 2 × 25123 × 251

A1

(ii) 1, 2, 4, 8, 251, 502, 1004, 2008B1

(c) (i) x2 + xy – xy (yx) – y2

B1

(ii) (x + y)(x – y) = 2008 × 1 or1004 × 2 or 502 × 4 or

Page 255

251 × 8M1

x + y = 2008 or 1004 or 502 or 251x – y = 1 or 2 or 4 or 8

M1

x = 503, y = 501x = 253, y = 249

A1 for any correct solution includingx = 1004.5, (y = 1003.5)x = 129.5, (y = 121.5)

A2[11]

M114. BD = × 40M1

BD = 25A1

BC = √(their 252 – 72)cos y = 7/their 25 M1

M1

√ their 576y = 73.7 A1

M1

24Sin their 73.7 M1

A1

or 0.96Ignore y = 73.7° after answer seen

A1[6]

Page 256

M115. (a) 2b – 2aoe

B1

(b) 2b – 2cft 2b – 4c if 2b – 4a given in (a)

B1 ft

(c) oe ft their vectors

M1

Page 257

c + (2b –2c) or a + (2b – 2a)NB Answer given so could be fiddled

A1

(d) Parallel to third side (other side, base) or half lengthoe

B1

(e) ParallelogramB1

Opposite sides same vectoroe

B1[7]

M116. (a) oe

M1

(a = ) 2y – xA1

(b) M1

Page 258

(b = ) oe

A1[4]

M117. Evidence of searching for a patternor r = 2 or d = 4or 6 extra discs gives extra 12 cm

eg, 4 → 10, 10 → 22 or 5 → 10, 11 → 22 or markings on diagram or diagram of 24 discs (2 rows)6 extra discs gives extra 12 cm

M1

Page 259

2n + 2 or 2(n + )

or 14 extra discs gives 28 cmor 20 extra discs gives 40 cm

12d + r or 25r22 + 2810 + 40

M1

50A1

[3]

M118. Attempt at perpendicular bisector of given chord (intersecting arcscentred on end points both sides)

Two 90° angles constructed at each end ofchord and extended to meet circle

M1

A perpendicular bisector within tolerance (± 1 mm)Must cut both sides of circle unless anotherperpendicular bisector drawnTwo diagonals of rectangle drawn

A1

Centre within ± 1 mmA1

[3]

M119. 8a8a = 14b or 4a = 7b M1

B1

14b4a + 4b = 44 or 7a + 7b = 77 M1

B1

Page 260

a = 7 answers only with no working is zero marks11b = 44 or 11a = 77 A1

B1

b = 4 allow answers reverseda = 7 and b = 4 A1

B1[4]

M120. (V) = πCondone missing bracket

M1

A correct step moving towards (their) formula for x3

eg, 3V ⁄ π = x3 ⁄ (their 4)

Page 261

or V = π × x3 oeM1dep

x3 = (their 12)V ⁄πRearranging to get (their) formula for x3

M1dep

x = 3√(12V ⁄ π)SC3 x = 3√(6V ⁄ π) oeSC2 x = 3√(3V ⁄ π) oe

A1[4]

M121. BAF = 120°This can just be statedor exterior angle of hexagon = 60°or reflex FAB = 240°

B1

360 – (120 + 90 + 90) = 60°oe

HAJ must be shown to be 60° by calculationB1

AH = AJThis can just be stated or shown on diagram

B1

AJH = AHJ = (180 – 60) ÷ 2Dep on first B2

B1dep[4]

Page 262

M122. Allow one erroror [5 ± √59] ÷ 6

M1

(5 ± √ 109) ÷ 6A1

2.57 and –0.91SC1 2.3 and –0.45

A1[3]

M123. tan 38 = BC/7.21or BC ⁄ sin 38 = 7.21 ⁄ sin 52

M1

7.21 × tan 38or 7.21 × sin 38 ÷ sin 52

M1 dep

5.6 (3 …)A1

5.6 or 5.63B1 ft

[4]

M124. (a) y = 2 sin xoe y = 2 f( x ) ‘y =’ or ‘x =’ missing penalize first time only

B1

(b) y = 1 + sin xoe y = 1 + f( x )

Page 263

B1

(c) y = – sin xoe y = – f( x ) y = sin (x + 180) y = sin (x – 180)

B1[3]

M125. π × 3 × 3 × 6Cylinder

M1

54π169-170

A1

4 × π ×3 × 3 × 3 ÷ 3

Page 264

Sphere × π ×3 × 3 × 3M1

36π113 - 114

A1

18πTheir 54π – their 36πFollow through dependent on award of at least oneM1

A1 ft[5]

M126. (a) 53B1

(b) 85B1

[2]

M127. ReflectionB1

y = –xoe

B1[2]

Page 265

M128. (a) 41B1

(b) 180 – 6771 + their 41 oe 360 – 41 – 67 – 139

M1

113A1

[3]

M129. Attempt to rewritex2 – 5x + 3 = 0 as x2 – 4x + 1 = x – 2

M1

Identify (y =) x – 2A1

Accurately draws line y = x – 2M1 dep

(x =) 0.7 and 4.3ft their line if both M1s earned

A1[4]

M130. (a) (height of cylinder =) 9B1

π × 32 × their 9 or 81πM1

Page 266

× π × 33 or 18πoe

M1

99πA1

(b) Their 49.5π – their 18π or 31.5π

Page 267

Must see use of ‘ of their 99π’M1

Their 31.5π = π × 32 × hdep on previous M1

M1 dep

h = their 31.5π ÷ (π × 32) or 3.5M1

d = 6.5Strand (ii)For correct answer supported by logical working showingkey steps

Q1[8]

M131. Angle PRQ = 180° – 134° or 46°M1

Angle POQ = 2 × their 46° or 92°M1

Reflex angle POQ = 268°A1

[3]

M132. Cos A = (102 + 62 – 142) ÷ (2 × 6 × 10)M1

Page 268

–A1

120°A1

[3]

M133. (a) 2x2 – 7x + 9 = 0oe –1 eeooB2 fully correct expression

B3

(b) Reference to square root of negative numberB1

[4]

M134. (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 + 1A1

m2 is odd since 2n + 1 is oddA1

m is odd since odd × odd = oddA1

[7]

Page 269

M135. (a) DB1

AB1

CB1

(b) Negative gradient and through point on positive y-axis‘2’ need not be marked

B1[4]

M136. (a) × (7 + 11) × 5M1

45A1

(b) Their 45 × 16or 720

M1

19.3 × their 720M1

13896A1

13.896ft if both Ms awarded

A1 ft[6]

Page 270

M137. (a) x(x + 10)B1

(b) (y + 6)(y – 6)B1

(c) 5w + w = 9 – 6Allow one sign error

M1

6w = 3For collecting like termsft their first line

M1

Page 271

oe Accept A1

(d) LCM of 12 used correctly or attempt at LHS multiplied by 12M1

6x + 9 + 4x – 20Allow one error

M1

10x – 11 = 1810x – 11 = 3 scores A0

A1

2.9ft from one arithmetic error but not from 10x – 11 = 3

A1 ft[9]

M138. π × 3.5 × 3.5 or π × 5 × 5 orπ × 7 × 7

12.25π or 25π or 49πM1

π × 3.5 × 3.5 + π × 5 × 5M1

47.25π and 49πA1

He is correctft if both Ms awarded

A1 ft[4]

M139. 39 ÷ 3 or 39 ÷ 6

Page 272

or 19.5 ÷ 3 or 19.5 ÷ 6oe

M1

13 or 6.5 seenA1

13 × 13M1

169A1

[4]

M140. Sometimes trueB1

Valid explanationeg, height of triangle can vary

B1[2]

M141. (a) Fully correct rotationB1 180° rotation with centre 0B1 90° clockwise rotation with wrong centreB2 90° clockwise rotation with centre 0B2 90° anticlockwise rotation with wrong centre

B3

(b) x = –1B1 for coordinates plotted or line shown on graph

B2[5]

Page 273

M142. (a) Ratio is 1:113 and 113 = 1331B1

(b) : oe

M1

1 : 0.24A1

[3]

Page 274

Web viewQ1. The time in minutes (T) for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters (W) working at

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