Circle Theorems & Vectors 1 Grade 8Question 1 

A, B, C and D are points on the circumference of a circle with centre O.

Angle ABC = 116°

Find the size of the angle marked x.

Give reasons for your answer.

(3)

Question 2 

B and C are points on the circumference of a circle, centre O. AB and AC are tangents to the circle. Angle BAC = 40°.

Find the size of angle BCO.

 (3)Question 3

OAB is a triangle

OA = a and OB = b

(a) Find the vector AB in terms of a and b

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

P is the point on AB such that AP: PB = 3:2

(b) Show that OP = 15 (2a + 3b)

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

Total /10

Diagram NOT drawn

accurately

Circle Theorems & Vectors 2 Grade 8Question 1

B, C and D are points on the circumference of a circle, centre O. ABE and ADF are tangents to the circle.

Angle DAB = 40° Angle CBE = 75°

Work out the size of angle ODC.

(4)

Question 2.

OAP is a triangle

OA = 2f + g and OB = 3h

P is the point on AB such that AP: PB = 2:1

(a) Find the vector BA in terms of f, g and h.

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

(1)

(b) Find the vector PO in terms of f, g and h

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

(2)

Diagram NOT drawn

accurately

3h

2f + g

Question 3.

OABC is a parallelogram.

X is the midpoint of OB

OA = a and OC = c

(a) Find the vector OX in terms of a and c.

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

(1)

(b) Find the vector XC in terms of a and c.

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

(2)

Total /10

Diagram NOT drawn

accurately

Circle Theorems & Vectors 3 Grade 8Question 1

ADB is the tangent at D to the circle, centre O.C is a point on the circumference.Angle OCD = 25°.

Not to scale

Calculate angle CDA.Show each step of your calculation.

[3]

Question 2

PQRS is a parallelogram.

M is the midpoint of RS

N is the midpoint of QR

PQ = 2a

PS = 2b

Use vectors to proof that the line segments SQ and MN are parallel.

(3)

Diagram NOT drawn

accurately

Question 3

B is the point on AD such that XB:BD is 1:2

A is the point on XC such that XA:XC is 1:2

XB = p and XA = q

Use vectors to explain the geometrical relationships between the line segments BA and DC.

(4)

Total / 10 Circle Theorems & Vectors 4 Grade 8

q

p

Diagram NOT drawn

accurately

Diagram NOT drawn

accurately

Question 1

ABCD is a cyclic quadrilateral within a circle centre O.

XY is the tangent to the circle at A.

CD is parallel to AB.

Angle BAD = 75o

Angle CBD = 38o

(a) Give a reason why angle BCD = 105o

(1)

(b) Work out the value of angle BAX. You must show all of your working.

Question 2.

PQRS is a parallelogram.

A is the point on PR such that PA:AR is 2:1

M is the midpoint of RS.

(a) Find PA in terms of k and j.

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

(b) Prove that Q, A and M are co-linear.

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

Total / 10

M

A Diagram NOT drawn

accuratelyk

jS

RQ

P

Conditional Probability & Histograms 1 Grade 7

Question 1

The histogram and the frequency table show some information about how much time vehicles spent in a car park.

Time, minutes Frequency

0 < x ≤ 10

10 < x ≤ 30

30 < x ≤ 60 75

60 < x ≤ 80 24

    Total 150    

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

Time, minutes

Freq

uenc

y De

nsity

a) Use the information to complete the histogram

(2)

b) Use the histogram to find the missing frequencies in the table

……… …………………………

(2)

(Total 4 marks)

Question 2.

A box contains 3 new batteries, 5 partly used batteries and 4 dead batteries.

Kelly takes two batteries at random.

Work out the probability that she picks two different types of batteries.

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

(Total 3 marks)

Question 3

Caleb either walks to school or travels by bus.

The probability that he walks to school is 0.75.

If he walks to school, the probability that he will be late is 0.3.

If he travels to school by bus, the probability that he will be late is 0.1.

Work out the probability that he will not be late.

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

(3)

(Total 3 marks)

Total /10

Conditional Probability & Histograms 2 Grade 7

Question 1.

The table shows the length of 678 phone calls made at a call centre

Time, secs Frequency

0 < x ≤ 20 20

20 < x ≤ 60 148

60 < x ≤ 120 240

120 < x ≤ 300 270

    Total 678    

a) Draw a fully labelled histogram to show the length of the phone calls.

b) Estimate the number of phone calls that lasted more than 4 minutes.

……………………………………

(2)

(Total 6 marks)

Question 2.

Laura has 9 tins of soup in her cupboard, but all the labels are missing.

She knows that there are 5 tins of tomato soup and 4 tins of vegetable soup.

She opens three tins at random.

Work out the probability that she opens more tins of vegetable soup than tomato soup.

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

(Total 4 marks)

Total /10

Conditional Probability & Histograms 3 Grade 7

Question 1.

The two way table shows the number of deaths and serious injuries caused by road traffic accidents in Great Britain in 2013.

Work out an estimate for the probability:

(a) that the accident is serious.

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

(1)

(b) that the accident is fatal given that the speed limit is 30 mph.

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

(2)

(c) that the accident happens at 20 mph given that the accident is serious.

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

(2)

Question 2.

The table and histogram show the weights of some snakes.

Weight, grams Frequency

250 < x ≤ 300 60

300 < x ≤ 325 25

325 < x ≤ 350 40

350 < x ≤ 450 35

450 < x ≤ 600 40

    Total 200    

(a) Use the information to complete the histogram

(3)

(b) Calculate an estimate for the median

…………………………………(2)

Total /10

Conditional Probability & Histograms 4 Grade 7

Question 1.

The Venn diagram shows the ice-cream flavours chosen by a group of 44 children at a party.

The choices are strawberry (S), choc-chip (C) and toffee (T).

A child is picked at random.

Work out :(a) P(S)

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

(1)

(b) P(T U C│C)

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

(2)

(c) P(C│S U T)

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

(2)

(Total 5 marks)

Question 2.

The table and histogram show the lengths of some pythons.

Length, cms Frequency

30 < x ≤ 40 20

40 < x ≤ 50 10

50 < x ≤ 70 50

70 < x ≤ 100

100 < x ≤ 150

    Total 180

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

Length, cms

Freq

uenc

y de

nsity

(a) Use the histogram to find the missing frequencies in the table

…………………………………(2)

(b) Estimate the median python length.

…………………………………(3)

(Total 5 marks)

Total /10

Equation of a circle and tangent 1 Grade 9

Question 1

(a) Write down the equation of a circle with centre (0, 0) and radius 5.

(2)

(b) Write down the centre and radius of the circle x2+ y2=20.

Simplify your answer.

Centre =

Radius =

(2)

Question 2

A graph has been drawn for you on the grid below.

Write down the equation of this graph.

(3)

Question 3

The grid below shows a circle with equation x2+ y2=25.

There are two tangents to this circle with gradient 0.

(a) Draw these tangents on the graph above.

(2)

(b) Write down the equation of these tangents.

(1)

TOTAL /10

Equation of a circle and tangent 2 Grade 9

Question 1

(a) Write down the equation of a circle with centre (0, 0) and radius 1.5.

(2)

(b) Write down the centre and radius of the circle x2+ y2=81.

Centre =

Radius =

(2)

Question 2

Here is a circle, x2+ y2=13, and a tangent to the circle.

The tangent goes through the point B(2, -3) on the circle.

Find the equation of the tangent at point B.

(4)

Question 3

The equation of a circle is x2+ y2=k .

The line y=7 is a tangent to the circle.

Work out the value of k.

(2)

TOTAL /10

Equation of a circle and tangent 3 Grade 9

Question 1

AB is the diameter of a circle.

A is (4, 3) and B is (-4, -3).

Work out the equation of the circle.

(5)

Question 2

A circle has equation x2+ y2=34.

P lies on the circle and has x-coordinate 3.

The tangent at P intersects the x-axis at point A and the y-axis at point B.

Work out the co-ordinates of A and B.

A

B

(5)

TOTAL /10

P

B

A

O

Equation of a circle and tangent 4 Grade 9

Question 1

Solve the simultaneous equation using a graphical method.

x2+ y2=16

y=2x+4

(3)

Question 2

Show that the point (5, -2) lies on the circle with (0, 0) and radius √29.

You must show your working.

……………………………

(2)

Question 3

The line l1 is a tangent to the circle x2+ y2=52 at the point P.

P is the point (4, -6).

The line l1 crosses the x–axis at the point D.

Work out the area of triangle OPD.

……………………………

(5)

TOTAL /10

D

P

Gradients and rate of change/iterative processes 1 Grades 7 - 9

Question 1

(a) Calculate the average rate of change between x = 1 and x = 3

…………………

(2)

(b) Calculate the instantaneous rate of change at x = -1

…………………(2)

Question 2

Calculate the instantaneous rate of change at x = -3

…………………

(3)

Question 3.

This iterative process can be used to find approximate solutions to the equation x3 – 3x – 1 = 0 to 1dp.

Use this iterative process to find a solution to 1 decimal place to x3 – 3x – 1 = 0.

Start with x = -1

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

(3)

Total /10

Start with a value of x

Work out the value of 3√1+3x

Is your answer to 1 decimal place the same as your value of x to 1 decimal

place?

Yes No

This is an approximate solution to

x3 – 3x – 1 = 0

Use your answer as the next value of x and start again

Gradients and the rate of change/iterative processes 2 Grades 7 - 9

Question 1

(a) Calculate the average rate of change between 20 and 70 seconds

…………………(2)

(b) Calculate the instantaneous rate of change when the time is 40 seconds

…………………(2)

Question 2

Calculate the instantaneous rate of change at x = 1

…………………

(3)

Question 3.

Xn+1 = 1 + 1

Xn2

with X1 = 1.4

(a) Work out the values of X2 and X3

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

(2)

(b) Work out the solution correct to 2 decimal places.

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

(1)

Total /10

Speed (kmh-1)

Velocity (kmh-1)

Acceleration (kmh-2)

Gradients and the rate of change/iterative processes 3 Grades 7 - 9

Question 1

Match the graphs with the correct interpretation of the gradient.

(2)

Question 2

Look at the graph and find

a) A co-ordinate with a rate of change of 0

…………………

b) A co-ordinate with a negative rate of change

…………………

(2)

Question 3.

The equation x3 + 2x – 6 = 0 has one solution.

(a) Show that this solution lies in the interval 1 < x < 1.5

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

(2)

(b) Show that x3 + 2x – 6 = 0 can be written as: xn+1=3√6−2 xn

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

(2)

(c) Use the iteration xn+1=3√6−2 xn to find the solution to x3 + 2x – 6 = 0 correct to 3dp.

Use a starting value of with x0 = 1.5.

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

(2)

Total /10

Gradients and the rate of change/iterative processes 4 Grades 7 – 9

Question 1

The curve shows the height of a ball when thrown in the air. Use the curve to

a) describe what the gradient at any point would mean in context

b) draw a tangent with a gradient of 0

c) what is happening at the point when the gradient is 0?

……………………………………………………

(3)

Question 2

The distance in km travelled by a particle is given by the equation

d=3t3 +15t where t is given in hours

Find the speed of the particle at 3 hours

…………………

(3)

Question 3

This shape is a cuboid. The width, y is the same as the height. The length is 10 cm longer than the width.The volume of the cuboid is 20000 cm3.

(a) Show that y satisfies the equation y3 + 10y2 – 20000 = 0

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

(1)

(b) Use the decimal search method find the dimensions of the cuboid to the nearest mm.

y Value of y3 + 10y2 - 20000 Positive or Negative?

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

(3)

Total /10

Graphs and functions 1 Grades 7 - 8

Question 1

(a) Complete the table of values for y = 3x

x –2 –1 0 1 2

y

(1)

(b) On the grid, draw the graph of y = 3x

(2)

Question 2

The graph of y=x2−3 x+4 is drawn on the grid below.

Calculate an estimate to the gradient of the curve at the point P(3, 4).

(2)

Question 3

The graph of y=f ( x ) is shown below.

Below each sketch, write down the equation of the transformed graph

y =……………………………….

y =…………………………………..

(2)

Question 4

The graph of y = tan x for 0 ≤ x≤ 360° is shown below.

(a) Label the equations of the asymptotes shown with dotted lines.

…………………..…………

……………….……………

(2)

(b) Write down the coordinates of the three x-intercepts

(1)

TOTAL /10

Graphs and functions 2 Grades 7 - 8

Question 1

Here are the equations of six different graphs:

5 x+2 y−8=0 y=5−x y=5x

y=(x+5)2−1 y=5 x3+12x y=5x

Match one of the equations to each of the following graphs:

(3)

Question 2

The graph of y=x3+3 x2−2 x−1 is drawn on the grid below.

Calculate an estimate to the gradient of the curve at the point Q(-1, 3).

(3)

Question 3

The graph of y=f ( x ) is shown below.

Below each sketch below, write down the equation of the transformed graph

y =…………………………… y =……………………………

(2)

Question 4

Here is the graph of y = cos x for 0≤ x≤ 360°

On the axes above, sketch the graph y=cos (2 x )−2 for 0≤ x≤ 360°

(2)

TOTAL /10

Graphs and functions 3 Grades 7 - 8

Question 1

A scientific experiment on the growth of bacteria yields the following results:

x (time in hours) 1 2

y (population 1000’s) 14 112

The experiment is expected to follow the model y=abx where a and b are constants and b>0, as shown on the graph.

Calculate the values of a and b.

a =

b =

(2)

Question 2

The velocity-time graph below shows the motion of a car, travelling in a straight line at v metres per second on a motorway.

t (s)

v(ms-1 )

5 8 15

V

0

(a) Find the value of V if the car travels 125 metres in the first 5 seconds.

(1)

(b) The maximum speed the car reaches is 28 ms-1.

What is the acceleration between t = 5 and t = 8 ?

(2)

(c) If the final deceleration is 4 ms-2 how long did it take the car to stop ?

(1)

Question 3

The graph of y=f ( x ) is transformed to give the graph y=−2 f (x+1 ).

(a) The point P on the graph y= f ( x ) is transformed to the point Q on the graph y=−2 f (x+1 )..

The coordinates of P are (4, -2). Find the coordinates of the point Q.

……………………………

(1)

(b) The point R on the graph y= f ( x ) is transformed to the point S on the graph y=−2 f (x+1 )..

The coordinates of S are (0, -10). Find the coordinates of the point R.

……………………………

(1)

Question 4

Here is the graph of y = cos x for 0≤ x≤ 360°

(a) Use the graph to find the value of cos 45 °………………………

(1)

(b) Hence, or otherwise, find the solutions in the range 0 ≤ x≤ 360°of cos ( x )=−√32

………………………… (1)

TOTAL /10

Graphs and functions 4 Grades 7 - 8

Question 1

The price of a new car varies according to the following formula

C=15000(2−t10 )

where C is the price (£’s) and t is the age in years from brand new.

(a) State the value when the car is brand new

(1)

(b) Calculate an estimate for the value of the car after 10 years.

(1)

Question 2

The velocity-time graph below shows the motion of a car, travelling in a straight line at v metres per second, in the 20 seconds after it starts from rest.

(a) Calculate the acceleration of the car in the first 5 seconds of the journey.

(1)

(b) Was the car’s change of speed greatest between 5 and 10 seconds, or between 15 and 20 seconds.

Explain your answer.

(1)

(c) Between 10 and 15 seconds, how far did the car travel?

(2)

Question 3

Describe the transformation that maps the curve with equation y=tan (x ) on to the curve with equation:

(a) y=tan (2 x )+5

(1)

(b) y=3 tan (x−4)

(1)

Question 4

Here is the graph of y = sin x for 0 ≤ x≤ 360°

Explain why there is only 1 solution to the equation sin ( x )=−1, for x-values in the range 0 ≤ x≤ 360°.

(2)

TOTAL /10

Number 1 Grades 7-8

Question 1

…………………….

(2)

Question 2

How many numbers having 3 distinct digits can be formed with the digits 1, 2, 3, 4?

……………………

(2)

Question 3

Expand and simplify

…………………

(2)

Question 4 . .

Change 0.47 into a fraction.

(2)

Question 5.

The following number has been rounded to two significant figures. Find the upper and lower

bounds

38

Upper Bound …….…………………

Lower Bound ……….………………

(2)

Total /10

Number 2 Grades 7-8Question 1

…………………….

(2)

Question 2

A train can travel between two stations A and B through any of 6 routes. How many routes can the train take between stations A and B and back again if, to return from B to A it can take any route

……………………

(2)

Question 3

Rationalise the denominator

…………………

(2)

Question 4

. . .Convert the recurring decimal 0.178 in to a fraction.

(2)

Question 5.

The following number has been rounded to two significant figures. Find the upper and lower

bounds

0.035

Upper Bound…………….………….……...

Lower Bound………………….………

(2)

Total /10

Number 3 Grades 7 - 8Question 1

(a) Write 25 as a power of 125

………………………………..

(1)

(b) Write 4 as a power of 32

………………………………..

(1)

Question 2

Mr Green has blue, white, and pink shirts and grey and black ties. Does he have enough different coloured shirts and ties to wear and different combination of shirt and tie for every day of week? You must give a reason for your answer.

(2)

Question 3

Write in the form , where a and b are integers.

……………………….

(2)

Question 4 . .

Change 0.23 into a fraction.

(2)

Question 5

The height of five children is recorded to the nearest centimetre

110cm 131cm 115cm 104cm

Find the greatest and least possible mean for the data.

Greatest possible mean………………………..

(1)

Least possible mean……… …………………

(1)

Total /10

Number 4 Grades 7 - 8Question 1

(a) Write 81 as a power of 27

………………………………..

(1)

(b) Write 16 as a power of 14

………………………………..

(1)

Question 2

How many four digit numbers are there that only include the digits 3, 4 and 5?

……………………

(2)

Question 3

Grace has made several mistakes in her ‘simplifying surds homework. In each case explain her error and give the correct answer.

(a)

(1)

(b)

(1)

Question 4

. .Convert the recurring decimal 0.257 into a fraction.

(2)

Question 5

An apple weighs 64grams to the nearest gram. A bag contains 10 apples. The bag weighs 12grams to the nearest gram. Find the upper and lower bounds of the weight of the apples and the bag altogether.

Upper Bound …………………………....(1)

Lower Bound ……………………………(1)

Total /10