It Takes 20 Minutes for Meals to Be Served When 12 Waiters Are Working

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.

......

Answer ......

(3)

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

......

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°

(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.

(2)

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

......

Answer ......

(2)

(Total 4 marks)

Q4.ABCD is a quadrilateral.
AB = 7 cm, AD = 6 cm and BC = 9 cm.
Angle ABC = 75° and angle ADC = 90°

Calculate the perimeter of ABCD.

......

Answer ...... cm

(Total 5 marks)

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

(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

......

(2)

......

(1)

(Total 5 marks)

Q6. (a) Calculate the size of an interior angle of a regular octagon.

......

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.

......

(3)

(Total 6 marks)

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

Calculate the value of y.

......

Answer ...... cm

(Total 6 marks)

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

x / –3 / –2 / –1 / 0 / 1 / 2 / 3 / 4
y / 21 / 6 / –3 / –6 / –3 / 21 / 42

(1)

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

(2)

......

Answer ...... and ......

(1)

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

3x2– 5x – 6 = 0

......

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.

......

Answer ......

(3)

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

......

Answer ...... square inches

(1)

Find the length of its diagonal.

......

Answer ...... inches

(3)

(Total 7 marks)

Q10. The graph shows the function

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

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

(1)

(b) Find the value of a.

......

Answer ......

(2)

(Total 3 marks)

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

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

......

Answer ...... cm2

(Total 5 marks)

Q12. A sign maker designs a letter L.

All arcs are quarter circles of radius 2 cm.

Not drawn accurately

Calculate the area of the L.

......

Answer ...... cm2

(Total 4 marks)

Q13. (a)P is inversely proportional to Q.

When P = 100, Q = 32

Express P in terms of Q.

......

Answer ......

(3)

(b)P and Q are positive quantities.

Sketch a graph of the relationship between P and Q on this diagram.

(1)

......

Answer ......

(2)

(Total 6 marks)

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)

y = ......

(1)

(b)

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.

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

(a) Show thatl = 2r

......

(2)

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

......

Answer h = ......

(2)

......

Answer ......

(2)

(Total 6 marks)

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

......

(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 equation
2x2– 9x + 4 = 0

......

(3)

......

Answer ......

(2)

(Total 7 marks)

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

Not drawn accurately

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

......

Answer a = ......

(Total 4 marks)

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

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)

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 = CD
Angle QAB = 38° and angle BAD = 76°

Not drawn accurately

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

......

(Total 4 marks)

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

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)

Answer ......

(1)

(Total 4 marks)

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.

Not drawn accurately

(a) Show that x =2r

......

(2)

(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.

......

Answer ......

(4)

(Total 6 marks)

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

Not drawn accurately

(a) Find the length of BH.

......

Answer ...... cm

(2)

(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.

......

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.

......

(Total 4 marks)

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

Not drawn accurately

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

......

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.

Not drawn accurately

Calculate the value of θ.

......

Answer ...... degrees

(3)

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

Not drawn accurately

Calculate the volume of the frustum.

......

Answer ...... cm3

(5)

(Total 8 marks)

Q26. The grid below shows graphs of a curve

y = x2 + 2x –3

and 3 straight lines

y = x + 1

y = – x –2

andy = – x + 2

You must use the graphs to answer the following questions.

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

x = –,y =

......

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.

......

Answer ......

(2)

......

Answer ......

(2)

(Total 5 marks)

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

Not to scale

Express r in terms of x.
Simplify your answer.

......

Answer r = ......

(Total 3 marks)

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

Not to scale

Calculate the size of angle ABC.

......

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.

......

Answer ...... cm

(4)

(Total 7 marks)

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

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

Not to scale

(a)AB = BC

Show that 2a – b = 1

......

(2)

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

......

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.

......

(3)

(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 itissafe.

......

Answer ...... m

(3)

(Total 6 marks)

Q31. A hemispherical bowl of radius 6 cm has the same volume as a cone of perpendicular height27cm.

Not drawn accurately

Calculate the base radius, r, of the cone.

......

Answer ...... cm

(Total 4 marks)

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°

Not drawn accurately

Prove that triangle QSR is isosceles.

......

(Total 3 marks)

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

Ay = 2 – 2xB y = 2x + 2Cy = 3 –x2Dy = x3 + 4E y =

Graph 1 represents equation ……......

Graph 2 represents equation ……......

Graph 3 represents equation ……......

Graph 4 represents equation ……......

(Total 4 marks)

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

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

x2 + 3x – 3 = 0

......

Answer ......

(2)

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

x2 + 2x– 1 = 0

......

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.

......

(2)

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

ALIEN
Place in
water and
it becomes
6 times
bigger! /

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?

......

(3)

(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.

......

Answer ......

(Total 4 marks)

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

Not drawn accurately

Calculate the shaded area.

......

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.

......

Answer ......

(Total 3 marks)

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

Calculate the length of wire used to make the clip.

......

Answer ...... cm

(Total 5 marks)

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

Calculate the length of BC.

......

Answer ...... cm

(3)

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

Calculate the size of angle PRQ.

......

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.

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.

......

Answer ...... minutes

(Total 4 marks)

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

A curve has the equationy2 = 8x– 16

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

......

Answer ......

(5)

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

Sketch 1

Sketch 2

Sketch 3

Which is the correct sketch?

You must explain your answer.

......

(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

......

Answer ...... degrees

(1)

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

......

Answer ...... degrees

...... degrees

(2)

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

......

Answer ...... degrees

...... degrees

(3)

(Total 8 marks)

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.

......

Answer p = ...... q = ......

(4)

(b) Hence work out the coordinates of A.

......

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.

......

Answer ...... cm3

(3)

(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.

......

Answer ...... cm3

(4)

(Total 7 marks)

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

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

(i)

......

Answer ......

(1)

(ii)

......

Answer ......

(1)

(iii)

......

Answer ......

(1)

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

Write the vector in terms of a and b.

......

Answer ......

(2)

......

(2)

(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?

......

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.

......

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

(1)

(b)y = (x – 2)2

(1)

(c)

(1)

(Total 3 marks)

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

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

x2– 4x + 8 = 3x– 2

......

Answer ......

(3)

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

......

Answer ......

(1)

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

......

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

(a) Write down the length of CD.

Answer ...... cm

(1)

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

......

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.

......

Answer ......

(Total 2 marks)

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.

......

Answer ...... cm

(Total 5 marks)

Q54.ABCD is a quadrilateral.
AB = 7 cm, AD = 6 cm and BC = 9 cm.
Angle ABC = 75° and angle ADC = 90°

Calculate the perimeter of ABCD.

......

Answer ...... cm

(Total 5 marks)

Q55.AB is a chord of a circle, centre O, radius 6 cm.
AB = 7 cm

Calculate the area of the shaded segment.

......

Answer ...... cm2

(Total 6 marks)

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

x / 0 / 1 / 2 / 3 / 4
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)

Answer ......

(1)

(Total 4 marks)

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.

......

Answer Length = ...... cm

Height = ...... cm

(Total 4 marks)

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

x + = 5

The table shows his first trial.

x / x + / Comment
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°

Explain why angle QSR = 32°.

......

(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.

......

Answer Angle BOC = ...... degrees

(1)

(ii) Calculate angle OCA.

......

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 ......

......

(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.

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

......

Answer ...... cm

(Total 5 marks)

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.

......

Answer ......

(Total 5 marks)

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

Find the area of triangle ABC.

......

Answer ...... cm2

(Total 5 marks)

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

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.

......

Answer ...... %

(Total 3 marks)

Q66. Solve the equation

x2– 10x – 5 = 0

Give your answers to 2 decimal places.

......

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.

......

Answer ...... cm

(4)

(b)PQRS is a parallelogram.
PQ = 5.1 cm
PS = 6.8 cm
QPS = 48°