Trigonometry 2 - SOLN Revision/GRADE B.pdf · Question 3 On the grid below, show by shading, the region defined by the inequalities x + y < 6 x > − 1 y > 2

PLC Papers

Created For:

year 11

GRADE B

Graphical Inequalities 1 B Grade

LO: Use inequalities to represent a region on a graph

Question 1

The lines y = x – 2 and x + y = 10 are drawn on the grid.

On the grid, mark with a cross (×) each of the points with integer coordinates that are in the region

defined by

y > x – 2

x + y < 10

x > 3 (3 marks)

Question 2

Use the grid below to show the region that satisfies the following inequalities:

X>-4 Y>2 x+y<5

(3 marks)

Question 3

On the grid below, show by shading, the region defined by the inequalities

x + y < 6 x > − 1 y > 2

Mark this region with the letter R.

(4 marks)

Total /10

Graphical Inequalities 2 B Grade


Question 1

Show the following inequalities on a grid, hence show the region that is described by all 3

inequalities.

X+y<5 2x>y y>2

(4 marks)

Question 2

A, B, C, D, E, F and G are all regions on a co-

ordiante grid.

(a) Write down the letters of all of the

regions that represent the region x�6

(1 mark)

(b) The regions D, F and G satisfy a

different inequality. Write down this

inequality.

(2 marks)

Question 3

The lines x=2 and y=x-1 are shown on

the grid.

List all of the integer co-ordinate

values that satisfy the inequalities

listed below.

x>2 y�x-1 x+y≤6

(3 marks)

Total /10

Real life graphs 1 B Grade

LO: Plot sketch and recognise graphs that represent real situations

Question 1

Two companies, Barry's Bricks and Bricks ArUs, deliver bricks.

The graph shows the delivery costs of bricks from both companies.

Prakash wants Bricks ArUs to deliver some bricks.

He lives 2 miles away from Bricks ArUs.

(a) Write down the delivery cost.

(1)

John needs to have some bricks delivered.

He lives 4 miles from Barry's Bricks.

He lives 5 miles from Bricks ArUs.

(b) Work out the difference between the two delivery costs.

(3)

(Total 4 marks)

Question 2

* You can use this graph to change between miles and kilometres.

The distance from Paris to London is 280 miles.

The distance from Paris to Amsterdam is 500 kilometres.

Is Paris further from London or further from Amsterdam?

You must show how you get your answer.

(3)

(Total 3 marks)

Question 3

The graph shows information about the speeds of two cars.

(a) Work out the gradient of the line for car A.

(2)

(b) After how many seconds is the speed of car A equal to the speed of car B?

(1)

(Total 3 marks)

Total /10

Real life graphs 2 B Grade


Question 1

The graph shows the distance travelled by two trains.

(a) Work out the gradient of the line for train A.

(2 marks)

(b) Which train is travelling at the greater speed?

You must explain your answer.

(1 mark)

(c) After how many minutes has train A gone 10 miles further than train B?

(1 mark)

Question 2

Mr Shah is thinking of having a water meter fitted for his house.

For a house with a water meter fitted, the graph shows information about the cost, in pounds (£), of

buying water.

Mr Shah does not have a water meter.

He used 50 cubic metres of water.

The cost was £80

Would the cost of the water have been cheaper if Mr Shah had a water meter?


(3 marks)

Question 3

Sophie's company pays her 80p for each mile she travels.

The graph can be used to work out how much her company pays her for travel.

Sophie travels 20 miles.

(a) Work out how much her company pays her.

(1 mark)

Sophie's company paid her £60

(b) Work out the distance Sophie travelled.

(2 marks)

Total /10

Simultaneous Equations 1 B Grade

LO: Form and/or solve linear simultaneous equations

Question 1

Solve these simultaneous equations to find the value for x and y.

x+3y=10

x+2y=7

x= y=

(2 marks)

Question 2


3x+y=9

4x+2y=14

x= y=

(2 marks)

Question 3


3x-2y=8

2x+3y=14 x= y=

(3 marks)

Question 4

At a charity event, Amy charges 10p per badge and 15p per rosette. She sells a total of 75 items and

raises £9.10. How many badges did Amy sell?

(3 marks)

Total /10

Simultaneous Equations 2 B Grade


Question 1


4x+2y=8

4x+3y=10

x= y=

(2 marks)

Question 2


6x-2y=14

2x+5y=16

x= y=

(2 marks)

Question 3


4x+2y=10

3x+5y=4

x= y=

(3 marks)

Question 4

Abi needs to transport herself AND 34 friends to a party by taxi. Black cabs can carry 5 people and

Minicabs can carry 3 people. Abi orders 9 vehicles, how many of each type does she use?

(3 marks)

Total /10

Boxplots 1 Grade B

Objective: Draw and interpret box plots for grouped data Question 1

The box plot gives information about the weights of a group of children.

(a) Write down the median.

…………………………………… kg (1)

(b) Work out the interquartile range.

…………………………………… kg (1)

There are 80 children in the group.

(c) Work out an estimate for the number of children who weigh 52 kg or more.

…………………………………… (2)

(Total 4 marks)

Question 2

Colin took a sample of 80 football players.

He recorded the total distance, in kilometres, each player ran in the first half of their matches on Saturday.

Colin drew this box plot for his results.

(a) Work out the interquartile range.

…………………………………… km (2)

There were 80 players in Colin's sample.

(b) Work out the number of players who ran a distance of more than 5.6 km.

…………………………………… (2)

Colin also recorded the total distance each player ran in the second half of their matches.

He drew the box plot below for this information.

(c) Compare the distribution of the distances run in the first half with the distribution of the distances run in the

second half.

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

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

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

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

(2)

(Total 6 marks)

Total marks /10

Boxplots 2 Grade B

Objective: Draw and interpret box plots for grouped data

Question 1

Kelly recorded the length of time 48 teachers took to travel to school on Monday.

The table shows information about these travel times in minutes.

Least time 5

Greatest time 47

Median 28

Lower quartile 18

Upper quartile 35

(a) Work out the number of teachers with a travel time of 35 minutes or more.

…………………………………… (2)

(b) On the grid, draw a box plot to show the information in the table.

(2)

Kelly then recorded the times the same 48 teachers took to travel to school on Tuesday.

The box plot shows some information about these times.

(c) Compare the travel times on Monday and on Tuesday.

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

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

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

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

(2)

(Total 6 marks)

Question 2

Jodie picks all the apples from her 56 apple trees.

For each tree she records the total weight of its apples.

The table shows some information about these total weights in kg.

Least weight 25

Greatest weight 55

Median 40

Lower Quartile 35

Upper Quartile 45

(a) Work out how many of Jodie's apple trees have a total weight of apples of less than 45 kg.

…………………………………… (2)

(b) On the grid, draw a box plot for the information in the table.

(2)

(Total 4 marks)

Total marks /10

Cumulative Frequency 1 Grade B

Objective: Draw and interpret cumulative frequency tables and diagrams Question 1

The grouped frequency table shows information about the weekly wages of 80 factory workers.

(a) Complete the cumulative frequency table.

Weekly wage (£x) Cumulative Frequency

100 < x ≤ 200

100 < x ≤ 300

100 < x ≤ 400

100 < x ≤ 500

100 < x ≤ 600

100 < x ≤ 700

(1)

(b) On the next page, draw a cumulative frequency graph for your table.

(2)

(c) Use your graph to find an estimate for the interquartile range.

…………………………………… (2)

Weekly wage (£x) Frequency

100 < x ≤ 200 8

200 < x ≤ 300 15

300 < x ≤ 400 30

400 < x ≤ 500 17

500 < x ≤ 600 7

600 < x ≤ 700 3

(d) Use your graph to find an estimate for the number of workers with a weekly wage of more than £530.

…………………………………… (2)

(Total 7 marks)

Question 2

The cumulative frequency graph shows information about the times 80 swimmers take to swim 50 metres.

(a) Use the graph to find an estimate for the median time.

…………………………………… (1)

A swimmer has to swim 50 metres in 60 seconds or less to qualify for the swimming team.

The team captain says,

"More than 25% of swimmers have qualified for the swimming team."

(b) Is the team captain right?

You must show how you got your answer.

……………………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………………… (2)

(Total 3 marks)

Total marks /10

Cum

ulative Frequency 2 Grade B


The table below shows information about the heights of 60 students.

(a) On the grid, draw a cumulative frequency graph for the information in the table.

(3)

(b) Find an estimate

(i) for the median,

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

(ii) for the interquartile range.

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

(3)

(Total 6 marks)

Question 2

The table shows information about the lengths, in seconds, of 40 TV adverts.

Time (T seconds) Frequency

10 < T ≤ 20 4

20 < T ≤ 30 7

30 < T ≤ 40 13

40 < T ≤ 50 12

50 < T ≤ 60 4

(a) Complete the cumulative frequency table for this information.

Time (T seconds) Cumulative frequency

10 < T ≤ 20 4

10 < T ≤ 30

10 < T ≤ 40

10 < T ≤ 50

10 < T ≤ 60

(1)

(b) On the grid, draw a cumulative frequency graph for your table. (2)

(c) Use your graph to find an estimate for the median length of these TV adverts.

…………………………… seconds (1)

(Total 4 marks)

Total marks /10

Tree Diagrams 1 Grade B

Objective: Use tree diagrams to represent outcomes of combined events, and calculate their probabilities

Question 1

Wendy goes to a fun fair.

She has one go at Hoopla and one go on the Coconut shy.

The probability that she wins at Hoopla is 0.4 whereas the probability that she wins on the Coconut shy is 0.3.

(a) Complete the probability tree diagram.

(2)

(b) Work out the probability that Wendy wins at Hoopla and also wins on the Coconut shy.

………………………………………… (2)

(Total 4 marks)

Question 2

Lily and Anna take a test.

The probability that Lily will pass the test is 0.6

The probability that Anna will pass the test is 0.8

(a) Work out the probability that both of these girls fail the test.

………………………………………… (3)

(b) Work out the probability that both of these girls pass the test or that both of these girls fail the test.

………………………………………… (3)

(Total 6 marks)

Total marks /10

Tree Diagrams 2 Grade B


Question 1

In a newsagent's shop, the probability that any customer buys a newspaper is 0.6.

In the same shop, the probability that any customer buys a magazine is 0.3.


(2)

(b) Work out the probability that a customer will buy either a newspaper or a magazine but not both.

……………………………………… (3)

(Total 5 marks)

Question 2

Yvonne has 10 tulip bulbs in a bag.

7 of the tulip bulbs will grow into red tulips.

3 of the tulip bulbs will grow into yellow tulips.

Yvonne takes at random two tulip bulbs from the bag.

She plants the bulbs.


(2)

(b) Work out the probability that at least one of the bulbs will grow into a yellow tulip.

……………………………………… (3)

(Total 5 marks)

Total marks /10

Compound Interest 1 Grade B

Objective: Solve problems involving compound interest

Question 1

£6000 is invested for two years at 4% per annum compound interest.

Work out the total interest earned over two years.

£ …………………………………… (2)

(Total 2 marks)

Question 2

£400 is invested for 3 years at 3% per annum compound interest.

(a) Calculate how much interest was paid in this time

£ …………………………………… (2)

(b) Calculate how much money was in the account at the end of the 3 years.

£ …………………………………… (1)

(Total 3 marks)

Question 3

A computer was purchased for £400 brand new. The computer depreciates at a rate of 18% per annum.

(a) Calculate how much the computer was worth after 2 years.

£ …………………………………… (2)

(b) After how many years was the computer worth less than £200?

…………………………………… (2)

(c) Dave said that the value after two years is the same as taking off 36%. Is this correct?

Explain your answer.

YES NO (circle your answer)

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………… (1)

(Total 5 marks)

Total marks /10

Compound Interest 2 Grade B


Question 1

£7,000 is invested for three years at 8% per annum compound interest.

Work out the total interest earned over three years.

£ …………………………………… (2)

(Total 2 marks)

Question 2

£5,000 is invested for 2 years at 7.5% per annum compound interest.

(a) Calculate how much interest was paid in this time.

£ …………………………………… (2)


£ …………………………………… (1)

(Total 3 marks)

Question 3

A car was purchase for £8,000 brand new. The car depreciates at a rate of 25% per annum.

(a) Calculate how much the car was worth after 3 years.

£ …………………………………… (2)

(b) After how many years was the car worth less than £2,000?

…………………………………… (2)




……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………………………………… (1)

(Total 5 marks)

Total marks /10

Percentage & proportion 1 Grade C SOLUTIONS

Objective: Use percentages to compare proportions

Question 1

In a class 9K, 40% of the class are girls.

�

� of the girls are left handed and

�

�� of the boys are left handed.

(4)

Question 2

Andrew scored 33 out of 60 in his Geography test.

Andrew scored 62% in his History test.

Which subject did he get a better mark in?

(3)

Question 3

Zeenat scored 53 out of 60 in her English test.

She scored 177 out of 200 in her Maths test.

Which subject did she get a better mark in?

(3)

Total /10

Percentages & Proportion 2 Grade C


Question 1

Two buses transport some pupils to school. On Monday:

Bus A has 28 pupils on it and the ratio of girls to boys is 2:5.

Bus B has 30 pupils and the ratio of girls to boys is 3:2.

On Monday, is the percentage of girls travelling to school by bus more than 50%?

Clearly show your method.

(4)

Question 2

Pupils took a test in Chemistry. The maximum mark possible is 40.

Mr Rogers, the Chemistry teacher, decides to give an A grade to any pupil able to score at least 84% on the test.

What is the lowest mark it is possible to score on the test and achieve an A grade?

(3)

Question 3

A library has some computers people can use.

At 10am there are 13 people using computers.

At 10.30am people arrive at the library and use the spare computers. All the computers are now being used.

At 11am 20% of the people using computers, leave the library. There are now 16 computers being used.

a) How many computers are there in the library?

(2)

b) What percentage of the computers in the library was being used at 10am?

(1)

Total /10

Reverse Percentages 1 Grade B

Objective: Solve problems involving reverse percentages

Question 1

In a sale, normal prices are reduced by 25%. The sale price of a saw is £12.75

Calculate the normal price of the saw.

£ …………………………………… (3)

(Total 3 marks)

Question 2

In a sale, normal prices are reduced by 12%.

The sale price of a DVD player is £242.

Work out the normal price of the DVD player.

£ …………………………………… (3)

(Total 3 marks)

Question 3

The price of all rail season tickets to London increased by 4%.

(a) The price of a rail season ticket from Cambridge to London increased by £121.60.

Work out the price before this increase.

£ …………………………………… (2)

(b) After the increase, the price of a rail season ticket from Brighton to London was £2828.80.


£ …………………………………… (2)

(Total 4 marks)

Total marks /10

Reverse Percentages 2 Grade B

Objective: Solve problems involving reverse percentage

Question 1

In a sale, the price of a shirt is reduced by 30%.

The sale price of the shirt is £28.

Work out the original price of the shirt.

£ …………………………………… (3)

(Total 3 marks)

Question 2

The price of a computer is £1292.50.

This includes an increase of VAT charged at 20%.

Work out the cost of the computer before VAT was added.

£ …………………………………… (3)

(Total 3 marks)

Question 3

In a sale, all the normal prices are reduced by 35%.

In the sale, a sofa costs £546.

(a) Calculate the normal price of the sofa

£ …………………………………… (3)

(b) How much cheaper is the sofa?

£ …………………………………… (1)

(Total 4 marks)

Total marks /10

Similar shapes 1 Grade B

Objective: Use the similarity of triangles and other 2D shapes to find the size of angles and the length of lines

Question 1

Triangles ABC and DEF are similar.

(a) Work out the length of DF (2)

DF = ………………………………………………….cm

(b) Work out the length of BC (2)

BC = ………………………………………………….cm

(total 4 marks)

Question 2

These two triangles are similar.

Not drawn accurately

(a) Work out the height of the smaller triangle (2)

Height = ……………………………………cm

The smaller triangle is placed on top of the larger triangle, as shown.

(b) Work out the area of the larger triangle that can still be seen. (2)

..................................cm2

(total 4 marks)

9 cm

10 cm

12 cm

Question 3

These two rectangles are similar. Diagrams not drawn accurately.

The dimensions of the smaller rectangle are 5cm x 8cm (height x length).

The length of the larger rectangle is 12.5 cm

What is the height of the larger rectangle? (2)

Height = ………………………………..cm

(total 2 marks)

(Total marks /10)

Similar shapes 2 Grade B


Question 1

(a) Work out the size of x

x = ………………………………………….(2)

(b) Work out the size of y

y = ………………………………………….(2)

(total 4 marks)

The two triangles are similar.

8cm

5cmy

43.2cm

x37.8cm

Question 2

AB = …………………………..

PS = …………………………..

(total 3 marks)

The diagram shows two quadrilaterals that are mathematically similar.

a) Calculate the length of AB

b) Calculate the length of PS

P

Q

RS

A

B

CD4 cm

8 cm

14 cm

21 cm

Question 3

VU = ……………………………………………..cm

(total 3 marks)

(Total marks /10)

SV is parallel to TU.RST and RVU are straight lines.RS = 9 cm, ST = 3 cm, TU = 7 cm, RV = 6 cm

Calculate the length of VU.R

S

T U

V

9 cm

3 cm

7cm

6 cm

Trigonometry 1 Grade B

Objective: Use sine, cosine and tangent ratios (SOHCAHTOA)

Question 1

Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall.

The ladder is 16 feet long.

The foot of the ladder is 4 feet from the base of the wall.

a Work out how high up the wall the ladder reaches. Give your answer to 3 significant figures. (2)

………………………………….feet

b Work out the angle the base of the ladder makes with the ground. Give your answer to 3 significant figures. (2)

………………………………….feet

(total 4 marks)

Question 2

AP is a telephone pole. The angle of elevation of the top of the pole from the point R on the ground is 42°as seen in

the diagram.

Calculate the height of the pole. Give your answer to 3 significant figures. (3)

……………………………………………metres

(total 3 marks)

Question 3

Calculate angle R to 1dp (3)

Angle R = ……………………………………….degrees

(total 3 marks)

(Total marks/10)

Trigonometry 2 Grade B


Question 1

x = ………………………°

(2 marks)

Question 2

AC = ………………………cm

(3 marks)

4.5 cm

P

Q

R11 cm

PQR is a right-angled triangle.PR = 11 cm.QR = 4.5 cmAngle PRQ = 90°

Work out the value of x.Give your answer correct to 1 decimal place.

A

°x

AB = 23 cm.Angle ABC = 90°

Angle ACB = 21°

Calculate the length of AC.Give your answer correct to3 significant figures.

A

B C21

23 cm

°

Question 3

The diagram represents a vertical pole ACD.

AB is horizontal ground.

BC is a wire of length 8.5 metres.

The height of the pole AD is 9 metres.

For the pole to be correctly installed, the length of DC has to be at least 1 metre.

Show that the pole has been correctly installed.

(5 marks)

(Total marks / 10)

PLC Papers

Created For:

year 11

GRADE B

Graphical Inequalities 1 B Grade SOLUTIONS


Question 1

The lines y = x – 2 and x + y = 10 are drawn on the grid.

On the grid, mark with a cross (×) each of the points with integer coordinates that are in the region

defined by

y > x – 2 M1 for identifying correct region

x + y < 10 M1 for identifying correct points

x > 3 A1 for no missing or extra points (3 marks)

Question 2

Use the grid to show the region that

satisfies the following inequalities:

X>-4 Y>2

x+y<5

M1 for lines drawn correctly placed

M1 for lines dashed

A1 for correct regions shaded

(3 marks)

Question 3

On the grid below, show by shading, the region defined by the inequalities

x + y < 6 x > − 1 y > 2

Mark this region with the letter R.

M1 for EACH of 3 lines drawn correctly (M3 in total)

A1 for R in correct place

(4 marks)

Total /10

R

Inequalities 2 SOLUTIONS B Grade


Question 1

Show the following inequalities on a grid, hence show the region that is described by all 3

inequalities.

X+y<5 2x>y y>2

M1 for each line drawn correctly

A1 for correct region identified

(4 marks)

Question 2

A, B, C, D, E, F and G are all regions on a co-

ordiante grid.

(a) Write down the letters of all of the

regions that represent the region x�6

B, C and G

(1 mark)

(b) The regions D, F and G satisfy a

different inequality. Write down this

inequality.

X+y≤8 M1 for correct inequality sign and A1

for complete correct answer. (2 marks)

Question 3

The lines x=2 and y=x-1 are shown on

the grid.

List all of the integer co-ordinate

values that satisfy the inequalities

listed below.

x>2 y�x-1 x+y≤6

(3, 2 ) and (3, 3)

M1 for EACH identified

A1 for NO extras.

(3 marks)

Total /10

Real life graphs 1 B Grade SOLUTIONS


Question 1

Two companies, Barry's Bricks and Bricks ArUs, deliver bricks.

The graph shows the delivery costs of bricks from both companies.

Prakash wants Bricks ArUs to deliver some bricks.

He lives 2 miles away from Bricks ArUs.

(a) Write down the delivery cost.

£56

(1 mark)

John needs to have some bricks delivered.

He lives 4 miles from Barry's Bricks.

He lives 5 miles from Bricks ArUs.

(b) Work out the difference between the two delivery costs.

Cost from Barry’s Bricks: £50 M1

Cost from Bricks ArUs: £65 M1

Difference is £15 A1

(3 marks)

Question 2

* You can use this graph to change between miles and kilometres.

The distance from Paris to London is 280 miles.

The distance from Paris to Amsterdam is 500 kilometres.

Is Paris further from London or further from Amsterdam?

You must show how you get your answer.

Paris to Amsterdam is (50x10=500 so multiply reading on graph by 10) 310 miles M1 M1

Paris to London is only 280 so Amsterdam is further. A1

(3 marks)

Question 3

The graph shows information about the speeds of two cars.

(a) Work out the gradient of the line for car A.

Vertical distance = 5.5

Horizontal distance = 20 M1 for both distances

5.5/20 = 0.275 (accept 0.25-0.3) A1 for solution

(2 marks)

(b) After how many seconds is the speed of car A equal to the speed of car B?

35 seconds

(1 mark)

Total /10

Real life graphs 2 B Grade SOLUTIONS


Question 1

The graph shows the distance travelled by two trains.

(a) Work out the gradient of the line for train A.

60/30 = 2 M1 A1

(2 marks)

(b) Which train is travelling at the greater speed?


Train A had a greater speed because the line has a larger gradient/ is steeper in the positive

direction.

(1 mark)

(c) After how many minutes has train A gone 10 miles further than train B?

15 minutes

(1 mark)

Question 2

Mr Shah is thinking of having a water meter fitted for his house.

For a house with a water meter fitted, the graph shows information about the cost, in pounds (£), of

buying water.

Mr Shah does not have a water meter.

He used 50 cubic metres of water.

The cost was £80

Would the cost of the water have been cheaper if Mr Shah had a water meter?


40 cubic metres costs £62, so 80 costs £124 M1 M1

Water is cheaper on the meter. A1

(3 marks)

Question 3

Sophie's company pays her 80p for each mile she travels.

The graph can be used to work out how much her company pays her for travel.

Sophie travels 20 miles.

(a) Work out how much her company pays her.

£16

(1 mark)

Sophie's company paid her £60

(b) Work out the distance Sophie travelled.

£20 is for 25 miles, multiply all by 3 to find £60 M1

25 x 3 = 75 miles A1

(2 marks)

Total /10

Simultaneous Equations 1 B Grade SOLUTIONS


Question 1


x+3y=10 M1 – Subtract to eliminate x. This gives y=3

x+2y=7 M1 – Substitute y=3 in eq1 to give x+9=10, so x=1

x= 1 y= 3

(2 marks)

Question 2


3x+y=9 M1 – multiply eq1 by 2 and then subtract to give 2x=4 and x=2

4x+2y=14 M1 – Substitute x=2 in eq1 to give 8+2y=14, so 2y=6 and y=3

x= 2 y= 3

(2 marks)

Question 3


3x-2y=8 M1 – multiply eq1 by 3 and eq2 by 2 to give 9x-6y=24 and 4x+6y=28

2x+3y=14 M1 – add to give 13x=52 and x=4

A1 – substitute x=4 in eq1 to give 2y=4 and y=2 x= 4 y= 2

(3 marks)

Question 4

At a charity event, Amy charges 10p per badge and 15p per rosette. She sells a total of 75 items and

raises £9.10. How many badges did Amy sell?

B+R=75 and 10B+15R=910 M1 – Construct equations and multiple eq 1 by 10

10B+10R=750 M1 – subtract to give 5R=160 so R=32

75-32= 43 Badges sold A1 – substitute in R

(3 marks)

Total /10

Simultaneous Equations 2 B Grade SOLUTIONS


Question 1


4x+2y=8 M1 – Subtract to eliminate x. This gives y=2

4x+3y=10 M1 – Substitute y=2 in eq1 to give 4x+4=8, so x=1

x= 1 y= 2

(2 marks)

Question 2


6x-2y=14 M1 – multiply eq2 by 3 and then subtract to give 17y=34 and y=2

2x+5y=16 M1 – Substitute y=4 in eq1 to give 6x-4=14, so 6x=18 and x=3

x= 3 y= 2

(2 marks)

Question 3


4x+2y=10 M1 – multiply eq1 by 5 and eq2 by 2 to give 20x+10y=50 and 6x+10y=8

3x+5y=4 M1 – subtract to give 14x=42 so, x=3

A1 – substitute x=3 in eq1 to give 2y=-2 and y=-1 x= -1 y= 2

(3 marks)

Question 4

Abi needs to transport herself AND 34 friends to a party by taxi. Black cabs can carry 5 people and

Minicabs can carry 3 people. Abi orders 9 vehicles, how many of each type does she use?

5B+3M=35 and B+M=9 M1 – Construct equations

5B+5M=45 M1 – Multiply eq2 by 5 and subtract to give 2M=10 and M=5

9-5=4, so B=4 A1 – 5 minicabs and 4 black taxis

(3 marks)

Total /10

Boxplots 1 Grade B SOLUTIONS

Objective: Draw and interpret box plots for grouped data Question 1

The box plot gives information about the weights of a group of children.

(a) Write down the median.

48

48kg (1)

(b) Work out the interquartile range.

52 – 40

= 12

12kg (1)

There are 80 children in the group.

(c) Work out an estimate for the number of children who weigh 52 kg or more.

0.25 x 80

= 20

20 (2)

(Total 4 marks)

Question 2

Colin took a sample of 80 football players.

He recorded the total distance, in kilometres, each player ran in the first half of their matches on Saturday.

Colin drew this box plot for his results.

(a) Work out the interquartile range.

5.6 – 4.85

= 0.75

0.75km (2)

There were 80 players in Colin's sample.

(b) Work out the number of players who ran a distance of more than 5.6 km.

0.25 x 80

= 20

20 (2)

Colin also recorded the total distance each player ran in the second half of their matches.

He drew the box plot below for this information.

(c) Compare the distribution of the distances run in the first half with the distribution of the distances run in the

second half.

Median is lower in the second half compared to the first half, decreases from 5.3km to 4.75km.

Interquartile range is the same in both halves, 0.75.

Range increases from first half 2.2 (6.4 – 4.2) to second half 2.45 (6.05 – 3.6).

(2)

(Total 6 marks)

Total marks /10

Boxplots 2 Grade B SOLUTIONS

Objective: Draw and interpret box plots for grouped data

Question 1

Kelly recorded the length of time 48 teachers took to travel to school on Monday.

The table shows information about these travel times in minutes.

Least time 5

Greatest time 47

Median 28

Lower quartile 18

Upper quartile 35

(a) Work out the number of teachers with a travel time of 35 minutes or more.

0.25 x 48

=12

12 (2)

(b) On the grid, draw a box plot to show the information in the table.

(2)

Kelly then recorded the times the same 48 teachers took to travel to school on Tuesday.

The box plot shows some information about these times.

(c) Compare the travel times on Monday and on Tuesday.

Median is higher on Tuesday, 40 minutes, compared with Monday, 28 minutes.

Interquartile range is larger on Tuesday, 27 minutes (52-25) compared with on Monday, 17 minutes (35 – 18).

Range is higher on Tuesday, 49 minutes (59 – 10) compared with Monday, 42 minutes (47 – 5).

(2)

(Total 6 marks)

Question 2

Jodie picks all the apples from her 56 apple trees.

For each tree she records the total weight of its apples.

The table shows some information about these total weights in kg.

Least weight 25

Greatest weight 55

Median 40

Lower Quartile 35

Upper Quartile 45

(a) Work out how many of Jodie's apple trees have a total weight of apples of less than 45 kg.

0.75 x 56

= 42

42 (2)

(b) On the grid, draw a box plot for the information in the table.

(2)

(Total 4 marks)

Total marks /10

Cumulative Frequency 1 Grade B SOLUTIONS


The grouped frequency table shows information about the weekly wages of 80 factory workers.

(a) Complete the cumulative frequency table.

Weekly wage (£x) Cumulative Frequency

100 < x ≤ 200 8

100 < x ≤ 300 23

100 < x ≤ 400 53

100 < x ≤ 500 70

100 < x ≤ 600 77

100 < x ≤ 700 80

(B1) (1)

(b) On the grid opposite, draw a cumulative frequency graph for your table.

M1 ft from their table at least 5 points plotted correctly at the ends of intervals, condoning one arithmetic error

A1 cao for correct graph with points joined by curve or straight line segments (2)

(c) Use your graph to find an estimate for the interquartile range.

Readings at 60 and 20 (M1)

420 to 440 – 280 to 295

120 - 160 (A1) (2)

Weekly wage (£x) Frequency

100 < x ≤ 200 8

200 < x ≤ 300 15

300 < x ≤ 400 30

400 < x ≤ 500 17

500 < x ≤ 600 7

600 < x ≤ 700 3

(d) Use your graph to find an estimate for the number of workers with a weekly wage of more than £530.

80 – 71 to 74 (M1 for evidence of reading off the cf axis from £530 on the wages axis)

6 – 9 (A1) ft (2)

(Total 7 marks)

Question 2

The cumulative frequency graph shows information about the times 80 swimmers take to swim 50 metres.

(a) Use the graph to find an estimate for the median time.

68 (B1) (1)

A swimmer has to swim 50 metres in 60 seconds or less to qualify for the swimming team.

The team captain says,

"More than 25% of swimmers have qualified for the swimming team."

(b) Is the team captain right?

You must show how you got your answer.

Yes

As 28 > 20 M1 for reading from graph at time = 60 (accept 27 to 28)

OR M1 for ‘28’ ÷ 80 x 100 ( = 35) OR 25 ÷ 100 x 80 (=20)

C1 (dependent on M1) for correct decision based on decisions

35% > 25%

OR OR

53 > 60 M1 for 25 ÷ 100 x 80 ( = 20)

M1 for reading a value from the graph at cf = 20 (= 53, accept 52 to 54)

C1 (dependent on M1) for correct decision based on their figures. (2)

(Total 3 marks)

Total marks /10

Cum

ulative Frequency 2 Grade B SOLUTIONS

Objective: Draw and interpret cumulative frequency tables and diagrams

Question 1

The table below shows information about the heights of 60 students.

(a) On the grid, draw a cumulative frequency graph for the information in the table.

Cf table: 4, 9, 25, 52, 57, 60

B1 correct cumulative frequencies (may be implied by correct heights on the grid)

M1 for at least 5 of “6 points” plotted consistently within each interval

A1 for a fully correct CF graph

(3)

(b) Find an estimate

(i) for the median,

172 (B1)

(ii) for the interquartile range.

IQR = UQ – LQ M1 for readings from graph at cf = 15 or 15.25 and cf = 45 or 45.75 from a cf

graph with at least one of LQ/UQ correct from graph

12 – 14 (A1) ft

(3)

(a) O

n

(Total 6 marks)

Quest

ion 2

The table shows information about the lengths, in seconds, of 40 TV adverts.

Time (T seconds) Frequency

10 < T ≤ 20 4

20 < T ≤ 30 7

30 < T ≤ 40 13

40 < T ≤ 50 12

50 < T ≤ 60 4

(a) Complete the cumulative frequency table for this information.

Time (T seconds) Cumulative frequency

10 < T ≤ 20 4

10 < T ≤ 30 11

10 < T ≤ 40 24

10 < T ≤ 50 36

10 < T ≤ 60 40

(B1)

(1)

(b) On the grid, draw a cumulative frequency graph for your table. (2)

B1 for 4 or 5 “points” plotted correctly within ½ fully square at the end of the interval

dependent on a sensible table

B2 for points joined by curve or line segments provided no gradient is negative (ignore any

graph outside range of points

(c) Use your graph to find an estimate for the median length of these TV adverts.

36.5 to 38 seconds (1)

(Total 4 marks)

Total marks /10

Tree Diagrams 1 Grade B SOLUTIONS


Question 1

Wendy goes to a fun fair.

She has one go at Hoopla and one go on the Coconut shy.

The probability that she wins at Hoopla is 0.4 whereas the probability that she wins on the Coconut shy is 0.3.


0.7

0.3

0.6

0.7

B1 for 0.6 in correct position on tree diagram

B1 for 0.7, 0.3, 0.7 in correct positions on tree diagram (2)

(b) Work out the probability that Wendy wins at Hoopla and also wins on the Coconut shy.

0.4 x 0.3 = (M1)

0.12 (A1) (2)

(Total 4 marks)

Question 2

Lily and Anna take a test.

The probability that Lily will pass the test is 0.6

The probability that Anna will pass the test is 0.8

(a) Work out the probability that both of these girls fail the test.

1 – 0.6 = 0.4

1 – 0.8 = 0.2 B1 for 0.4 or 0.2 seen

0.4 x 0.2 M1 indication of correct branch formed on tree diagram or otherwise,

leading to 0.4 x 0.2

0.08 (3)

(b) Work out the probability that both of these girls pass the test or that both of these girls fail the test.

0.4 x 0.2 + 0.6 x 0.8 M1 for 0.6 x 0.8 or “0.4” x “0.2”

M1 0.6 x 0.8 + “0.4” x “0.2” or “0.08” + “0.48”

0.56 (A1) (3)

(Total 6 marks)

Total marks /10

Tree Diagrams 2 Grade B SOLUTIONS


Question 1

In a newsagent's shop, the probability that any customer buys a newspaper is 0.6.

In the same shop, the probability that any customer buys a magazine is 0.3.


0.3

0.6

0.7

0.3

0.4

0.7

B2 for 6 correct probabilities in the correct positions, B1 for 2, 3, 4 or 5 correct (2)

(b) Work out the probability that a customer will buy either a newspaper or a magazine but not both.

0.6 x 0.7 + 0.4 x 0.3 M1 for ‘0.6’ x ‘0.7’ or ‘0.4’ x ‘0.3’

M1 for ‘0.6’ x ‘0.7’ + ‘0.4’ x ‘0.3’

OR

M1 for ‘0.6’ x ‘0.3’ + ‘0.4’ x ‘0.7’

M1 for 1- (‘0.6’ x ‘0.3’ + ‘0.4’ x ‘0.7’)

0.54 (A1) (3)

(Total 5 marks)

Question 2

Yvonne has 10 tulip bulbs in a bag.

7 of the tulip bulbs will grow into red tulips.

3 of the tulip bulbs will grow into yellow tulips.

Yvonne takes at random two tulip bulbs from the bag.

She plants the bulbs.


�

�

�

�

�

�

�

��

�

�

B1 for 3/10 on left hand of yellow branch

B1 for rest of fractions correct on tree diagram (2)

(b) Work out the probability that at least one of the bulbs will grow into a yellow tulip.

M1 for �

�� x

�

� or

�

�� x

�

� or

�

�� x

�

�

M1 for �

�� x

�

� +

�

�� x

�

� +

�

�� x

�

�

OR

M1 for �

�� x

�

�

M1 for 1 - �

�� x

�

�

�� (A1) o.e (3)

(Total 5 marks)

Total marks /10

Compound Interest 1 Grade B SOLUTIONS


Question 1

£6000 is invested for two years at 4% per annum compound interest.

Work out the total interest earned over two years.

6000 x 1.042 = 6489.6 (M1)

Or

6000 x 4/100 = 240

6000 + 240 = 6240

6240 x 4/100 = 249.6

6240 + 249.6 = 6489.6 (M1)

6489.6 – 6000 = 489.6

£ 489.60 (A1) (2)

(Total 2 marks)

Question 2

£400 is invested for 3 years at 3% per annum compound interest.


400 x 1.033 = 437.09 (M1)

437.09 – 400 = 37.09

£ 37.09 (A1) (2)


£ 437.09 (A1) (1)

(Total 3 marks)

Question 3

A computer was purchased for £400 brand new. The computer depreciates at a rate of 18% per annum.

(a) Calculate how much the computer was worth after 2 years.

400 x 0.822 = £268.96 M1

£ 268.96 (A1) (2)

(b) After how many years was the computer worth less than £200?

400 x 0.823 = £220.55

400 x 0.824 = £180.85 (M1)

4 years (A1) (2)




You don’t double the 18% that comes off, the 18% in the first year will be different to 18% in the second

Examples of calculations could show this. (C1) (1)

(Total 5 marks)

Total marks /10

Compound Interest 2 Grade B SOLUTIONS


Question 1

£7,000 is invested for three years at 8% per annum compound interest.

Work out the total interest earned over three years.

7000 x 1.083 = 8817.98 (M1)

Or

7000 x 8/100 = 560

7000 + 560 = 7560

7560 x 8/100 =604.8

7560 + 604.8 = 8164.8

8164.8 x 8/100 = 653.18

8164.8 + 653.18 = 8817.98 (M1)

£ 1817.98 (A1) (2)

(Total 2 marks)

Question 2

£5,000 is invested for 2 years at 7.5% per annum compound interest.


5000 x 1.0752 = 5778.13 (M1)

5778.13 – 5000 =

£ 778.13 (A1) (2)


£ 5778.13 (A1) (1)

(Total 3 marks)

Question 3

A car was purchased for £8,000 brand new. The car depreciates at a rate of 25% per annum.

(a) Calculate how much the car was worth after 3 years.

8000 x 0.753 = 3375 (M1)

£ 3375 (A1) (2)

(b) After how many years was the car worth less than £2,000?

8,000 x 0.754 = 2531.25

8,000 x 0.755 = 1898.44 (M1)

5 years (A1) (2)




You don’t double the 25% that comes off, the 25% in the first year will be different to 25% in the second.

Examples of calculations could show this. (C1) (1)

(Total 5 marks)

Total marks /10

Percentage & proportion 1 Grade C SOLUTIONS


Question 1

In a class 9K, 40% of the class are girls.

�

� of the girls are left handed and

�

�� of the boys are left handed.

What percentage of the class 9K is left handed?

A1 60% of class is girls

M1 Suitable method to find 1/5 of 40% and 1/10 of 60%

A1 at least one of 8% and 6%

A1 14%

(4)

Question 2

Andrew scored 33 out of 60 in his Geography test.

Andrew scored 62% in his History test.

Which subject did he get a better mark in?

M1 attempt to convert 34/60 to a percentage eg 33/60 = 11/20 = 55%

A1 Geography = 55%

C1 History

(3)

Question 3

Zeenat scored 53 out of 60 in her English test.

She scored 177 out of 200 in her Maths test.

Which subject did she get a better mark in?

M1 attempt to convert 53/60 to a percentage = (88.3%)

M1 attempt to convert 177/200 to a percentage = (88.5%)

C1 Maths

(3)

Total /10

Percentages & Proportion 2 Grade C SOLUTIONS


Question 1

Two buses transport some pupils to school. On Monday:

Bus A has 28 pupils on it and the ratio of girls to boys is 2:5.

Bus B has 30 pupils and the ratio of girls to boys is 3:2.

On Monday, is the percentage of girls travelling to school by bus more than 50%?

Clearly show your method.

M1: No of girls on Bus A 28÷7 × 2 on Bus B 30÷5×3 oe

A1: 26

M1 Either ��

�� or 29 calculated

A1: cao No with working above

(4)

Question 2

Pupils took a test in Chemistry. The maximum mark possible is 40.

Mr Rogers, the Chemistry teacher, decides to give an A grade to any pupil able to score at least 84% on the test.

What is the lowest mark it is possible to score on the test and achieve an A grade?

M1: 0.84 × 40

A1: 33.6

A1 34 marks

(3)

Question 3

A library has some computers people can use.

At 10am there are 13 people using computers.

At 10.30am people arrive at the library and use the spare computers. All the computers are now being used.

At 11am 20% of the people using computers, leave the library. There are now 16 computers being used.

a) How many computers are there in the library?

M1: 80% = 16 computers and other % calculated which is a factor of 100 eg 20% = 4 or × 1.25 seen

A1: 20 computers

(2)

b) What percentage of the computers in the library was being used at 10am?

A1: 65%

(1)

Total /10

Reverse Percentages 1 Grade B SOLUTIONS

Objective: Solve problems involving reverse percentages

Question 1

In a sale, normal prices are reduced by 25%. The sale price of a saw is £12.75

Calculate the normal price of the saw.

1 – 0.25 = 0.75

12.75 / 0.75

= 17.00

£17 (3)

(Total 3 marks)

Question 2

In a sale, normal prices are reduced by 12%.

The sale price of a DVD player is £242.

Work out the normal price of the DVD player.

1 – 0.12 = 0.88

242 / 0.88 =

275

£275 (3)

(Total 3 marks)

Question 3

The price of all rail season tickets to London increased by 4%.

(a) The price of a rail season ticket from Cambridge to London increased by £121.60.


4% = 121.60 £��.��

� x 100=

3040

£3040 (2)

(b) After the increase, the price of a rail season ticket from Brighton to London was £2828.80.


2828.80 / 1.04 = 2720

£2720 (2)

(Total 4 marks)

Total marks /10

Reverse Percentages 2 Grade B SOLUTIONS

Objective: Solve problems involving reverse percentage

Question 1

In a sale, the price of a shirt is reduced by 30%.

The sale price of the shirt is £28.

Work out the original price of the shirt.

1 – 0.30 = 0.70

28 / 0.70

= 40

£40 (3)

(Total 3 marks)

Question 2

The price of a computer is £1292.50.

This includes an increase of VAT charged at 20%.

Work out the cost of the computer before VAT was added.

1292.50 / 1.20

= 1077.0833333

= 1077.08

£1077.08 (3)

(Total 3 marks)

Question 3

In a sale, all the normal prices are reduced by 35%.

In the sale, a sofa costs £546.

(a) Calculate the normal price of the sofa

1 - 0.35 = 0.65

546 / 0.65

= 840

£840 (3)

(b) How much cheaper is the sofa?

840 - 546

= 294

£294 (1)

(Total 4 marks)

Total marks /10

Similar shapes 1 Grade B SOLUTIONS


Question 1

Triangles ABC and DEF are similar.

(a) Work out the length of DF (2)

Scale factor = 1.5 (M1)

9 x 1.5 = 13.5 cm (A1)

DF = ………………………………………………….cm

(b) Work out the length of BC (2)

Uses correct division or multiplication (M1)

10.5 / 1.5

Or

10.5 x 2/3

= 7 cm (A1)

BC = ………………………………………………….cm

(total 4 marks)

Question 2

These two triangles are similar.

Not drawn accurately

(a) Work out the height of the smaller triangle (2)

Scale factor = ¾ (M1) 10 x ¾ = 7.5 cm (A1)

Height = ……………………………………cm

The smaller triangle is placed on top of the larger triangle, as shown.

(b) Work out the area of the larger triangle that can still be seen. (the shaded part) (2)

Area larger triangle = 0.5 x 12 x 10 = 60

attempts to find areas and subtract with at least one area correct (M1)

Area of smaller triangle = 0.5 x 9 x 7.5 = 33.75

60 – 33.75 = 26.25 cm2 (A1)

..................................cm2

9 cm

10 cm

12 cm

(total 4 marks)

Question 3

These two rectangles are similar. Diagrams not drawn accurately.

The dimensions of the smaller rectangle are 5cm x 8cm (height x length).

The length of the larger rectangle is 12.5 cm

What is the height of the larger rectangle? (2)

Scale factor = 2.5 (M1)

5 x 2.5 = 12.5 cm (A1)

Height = ………………………………..cm

(total 2 marks)

(Total marks /10)

Similar shapes 2 Grade B SOLUTIONS


Question 1

(a) Work out the size of x

Scale factor = 43.2/ 8 = 5.4 (M1)

5 x 5.4 = 27 cm (A1)

x = ………………………………………….(2)

(b) Work out the size of y

37.8 / 5.4 (M1)

= 7 cm (A1)

y = ………………………………………….(2)

(total 4 marks)

The two triangles are similar.

8cm

5cmy

43.2cm

x37.8cm

Question 2

Scale factor 14/4 = 3.5 (M1)

8 x 3.5 = 28 cm (A1)

21 / 3.5 = 6 cm (A1)

AB = …………………………..

PS = …………………………..

(total 3 marks)

The diagram shows two quadrilaterals that are mathematically similar.

a) Calculate the length of AB

b) Calculate the length of PS

P

Q

RS

A

B

CD4 cm

8 cm

14 cm

21 cm

Question 3

Scale factor = �

� (M1)

RU = 6 x �

� = 8 (M1)

VU = 2 cm (A1)

VU = ……………………………………………..cm

(total 3 marks)

(Total marks /10)

SV is parallel to TU.RST and RVU are straight lines.RS = 9 cm, ST = 3 cm, TU = 7 cm, RV = 6 cm

Calculate the length of VU.R

S

T U

V

9 cm

3 cm

7cm

6 cm

Trigonometry 1 Grade B SOLUTIONS


Question 1

Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall.

The ladder is 16 feet long.

The foot of the ladder is 4 feet from the base of the wall.

a Work out how high up the wall the ladder reaches. Give your answer to 3 significant figures. (2)

√162 - 42

√256 -16 correct substitution (M1)

√240

15.49

15.5 (3sf) (A1)

………………………………….feet

b Work out the angle the base of the ladder makes with the ground. Give your answer to 3 significant figures. (2)

cos x = �

�� correct ratio and substitution (M1)

cos x = 0.25

x = 75.522

x = 75.5° (A1)

………………………………….feet

(total 4 mark

Question 2

AP is a telephone pole. The angle of elevation of the top of the pole from the point R on the ground is 42°as seen in

the diagram.

Calculate the height of the pole. Give your answer to 3 significant figures. (3)

Tan 42° = ��

�� (M1)

0.9004 = ��

�� (M1)

13.5(06) m = height of pole (A1)

……………………………………………metres

(total 3 marks)

Question 3

Calculate angle R to 1dp (3)

Cos R = ��

�� =

�

�

Cos R = 0.42857

R = 64 .6° (1dp)

Angle R = ……………………………………….degrees

(total 3 marks)

(Total marks/10)

Trigonometry 2 Grade B SOLUTIONS


Question 1

Tan x = opp/adj

Tan x = 4.5/11

Tan x = 0.4090 (M1)

x = 22.249

x = 22.2 (1dp) (A1)

x = ………………………°

(2 marks)

Question 2

Sin 21 = 23/hyp

0.35836 = 23/hyp

Hyp = 23/0.35836 (M1)

Hyp 64.1798 (A1)

64.2 (3sf) (A1)

AC = ………………………cm

(3 marks)

4.5 cm

P

Q

R11 cm

PQR is a right-angled triangle.PR = 11 cm.QR = 4.5 cmAngle PRQ = 90°

Work out the value of x.Give your answer correct to 1 decimal place.

A

°x

AB = 23 cm.Angle ABC = 90°

Angle ACB = 21°

Calculate the length of AC.Give your answer correct to3 significant figures.

A

B C21

23 cm

°

Question 3

The diagram represents a vertical pole ACD.

AB is horizontal ground. BC is a wire of length 8.5 metres.

The height of the pole AD is 9 metres.

For the pole to be correctly installed, the length of DC has to be at least 1 metre.

Show that the pole has been correctly installed.

Sin 68 = opp/hyp

0.9271 = opp/8.5

Opp = 7.88106 metres

9 - 7.88106 = 1.1189 m for length DC

So YES it is installed correctly because DC is over 1m long.

(5 marks)

(Total marks / 10)

Documents

Trigonometry 2 - SOLN Revision/GRADE B.pdf · Question 3 On the grid below, show by shading, the region defined by the inequalities x + y < 6 x > − 1 y > 2