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PLC Papers

Created For:

PiXL PLC 2017 Certification

Collecting like terms 1 Grade 4

Objective: Simplify algebraic expressions by collecting like terms

Question1

Simplify 7x + 4y – 4x + 3y

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

(Total 2 mark)

Question 2

Simplify 8f − f + 6f

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

(Total 1 mark)

Question 3

Simplify 8x − 3x + 5x

........................................................... (Total 1 mark)

Question4

Simplify 9x + x + 13y + y

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

(Total 2 mark)


Question 5

Simplify 13p – q + 12 – 6p + 4q - 9

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

(Total 2 mark)

Question 6

Simplify 19 + a – 2b + 15a + 8 – 2b

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

(Total 2 mark)

Total /10


Graphs of Linear Functions 1 Grade 4

Objective: Recognise, sketch and interpret graphs of linear functions.

Question 1

Sketch the graph of each function, clearly indicating the y-intercept.

a) y = 5x -3 b) y = 10 – 2x c) 2y = 4x + 8 d) y = -3x

y

a

x

(2)

y

b

x

(2)

y

c

x

(2)


y

d

x

(2)

Question 2

Which of these are linear functions? Circle your answer(s)

y = 7 – 3x

y = x 4 y = x2 + 4 2x + 3y = 5 y = x

(2)

Total /10


Substitution 1 Grade 4

Objective: Substitute numerical values into formulae and expressions, including scientific formulae Question 1.

Complete this table of values.

n 3n+2

12

47

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

(Total 2 mark)

Question 2 Calculate the value of y when x = 2 are

� = −1

3√18�2

You must show your working.

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

(Total 2 marks)


Question 3 a. Work out the value of v when u = 80, a = 10 and t = 4 b. Work out the value of v when u = 35, a = -5 and t = 12

� = � + �� a.

b.

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

(Total 2 mark)

Question 4

Work out the value of T when a. p = 5 and b. p = -1

� = 3�2 − 2�

a.

b.

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

(Total 2 marks)


Question 5

Work out the value of V to 3 sig fig when (a) π = 3.14, r = 10, and h = 15 (b) π = 3.14, r = 2.4, and h = 20

� = 1

3��2ℎ

a.

b.

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

(Total 2 marks)

Total /10


Adding and subtracting fractions 1 Grade 4 Objective: Add and subtract fractions including improper fractions and mixed numbers. Question 1 Work these out: 25 +

17 =

(2) 79 - 48 =

(2) Question 2 There are 700 counters in a bag. These are either black, blue or purple. 2/7 of the counters are blue. 1/5 are purple. a) What fraction of the counters are black?

(2)

b) How many counters are black?

(2)


Question 3 52 +

(2)

Total /10

113


Decimals1 Grade 3

Objective: Understand adding subtracting multiplying and dividing decimals.

Question 1

Three parcels weigh 7.2 kg, 15.02 kg and 3.1 kg. Find their total mass. (2)

Question 2 Work out 0.8965 ÷ 0.5

(2)

Question 3

(a) Tom gets paid £3.15 an hour. One week he worked for 26 hours. How much is his weekly pay? (2)


(b) Tom is saving for a gift. He wants to give his friend a gift that costs £100.27. How much more money does Tom need after he has worked 26 hours in one week.

(2)

Question 4

1 bottle of milk = £1.39

Lacey bought five bottles of milk. How much will it cost altogether? (2)

Total /10


Fractions and ratio problems 1 Grade 4 Objective: Identify and work with fractions in ratio problems. Question 1 Share £360 in the ratio 3:4:2. Find the value of smallest share. (2) Question 2 In similar triangles PST and PQR PQ = 6 cm , ST = 8 cm and RQ = 9 cm Find a) Side PT. Give your answer to 2 decimal places.

(2) b) Side TQ. Give your answer to 2 decimal places.

(2)

P

R Q

8cm S T

9cm

NOT TO SCALE

6cm


Question 3 Radhika wants to make a cake which weighs 560 grams. She uses sugar, flour, chocolate and eggs, in the ratio of 3:5:4:2. Calculate the weight of each ingredient: Flour: Sugar: Eggs: Chocolate: (4)

Total /10


LCM and HCF 1 Grade 4 Objective: Understand and find LCM and HCF.

Question 1

Find the HCF of 18 and 24. (2)

Question 2

Find the LCM of 14 and 3. (2)

Question 3 Express 42 as a product of its prime factors using any method.

Question 4 (2) Express 48 as a product of its prime factors using any method.

(2)


Question 5

Find the HCF of 42 and 48. (2)

Total /10


Multiplying fractions 1 Grade 4 Objective: Multiply fractions including improper fractions and mixed numbers. Question 1 Work out these questions always give your answer in the simplest form: 75 x

28 =

(2) 57 x

32 =

(2)

(2)

245 x 3

18 =


Question 2 Which is smaller?

or

(2)

Which is bigger?

or

(2)

Total /10

35 of 514 25 of 7

12

27 of 415 35 of 2

17


Rounding 1 Grade 5

Objective: Round to an appropriate degree of accuracy (e.g. to decimal places or significant figures)

Question 1.

Round the following to the given degree of accuracy

(a) 2.567889 to 1 decimal place

…………………….. (1)

(b) 409877.01233 to 2 decimal places

…………………….. (1)

(Total 2 marks)

Question 2

Round the following numbers to the degree of accuracy given

(a) 0.00654 to 1 significant figure

…………………….. (1)

(b) 7654.987 to 2 significant figures

…………………….. (1)

(Total 2 marks)


Question 3.

Round these numbers to the nearest integer

(a) 4.99987

……………. (1)

(b) 0.0098

…………………….. (1)

(Total 2 marks)

4. Tara worked out the answer to her question using a calculator. Her calculator read

34567654.123

Her teacher told her to round it to two significant figures. What number should she write?

…………………….. (2)

(Total 2 marks)


5. Round the following numbers to the degree of accuracy given

(a) 0.0098 to 3.d.p

……………. (1)

(b) 3755.4883087 to 4.d.p

…………….. (1)

(Total 2 marks)

Total /10


Compare quantities using ratio 1 Grade 4 Express a multiplicative relationship between two quantities

Question 1

A bank gives you 28 Euros when you exchange £20. How much will you get for exchanging £135?

€……………………

(Total 2 marks)

Question 2

Barry uses blue and red to make purple, in the ratio 3:5. How many tins of red will he need to mix with the 9 tins of blue?

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

(Total 2 marks)

Question 3

Louis, Steve and Ella shared some money in the ratio 2 : 3 : 5 Ella got £54.

How much money did Steve get?

£……………………

(Total 2 marks)


Question 3

Shannon and Liam share some chocolate in the ratio 4:3

Liam gets 81 grams of chocolate. Work out how many grams Shannon receives.

..........................g

(Total 2 marks)

Question 4

A shop sells freezers and cookers. The ratio of the number of freezers sold to the number of cookers sold is 5 : 2 The shop sells 140 freezers in one week.

Work out the number of cookers sold that week.

(Total 2 marks)

Total /10


Problems involving ratios 1 Grade 4 Objective: Solve problems involving ratios, e.g. conversion, comparison, scaling, mixing, concentrations Question 1

Louise and Anil share some sweets in the ratio 3 : 8

Anil gets 32 sweets. (a) How many sweets does Louise get?

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

(2) Anil also has a tin of chocolates.

There are 80 chocolates in the tin.

45% of the chocolates have toffee in the middle. (b) Work out the number of chocolates that have toffee in the middle.

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

(2) (Total 4 marks)

Question 2

Kiran is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.

Kiran wants to make 180 kg of concrete mix.

Kiran has

18 kg of cement 85 kg of sand 90 kg of gravel

Does Kiran have enough cement, sand and gravel to make the concrete mix?

(Total 3 marks)


Question 3

Jane made some almond biscuits which she sold at a fête.

She had:

5 kg of flour 3 kg of butter 2.5 kg of icing sugar 320 g of almonds

Here is the list of ingredients for making 24 almond biscuits.

Jane made as many almond biscuits as she could, using the ingredients she had.

Work out how many almond biscuits she made.

(Total 3 marks)

Total /10


Correlation 1 Grade 4

Objective: Recognise and describe correlation

Question 1.

(a) Each of these diagrams show a type of correlation. State the type of correlation shown in each diagram

(5)

(b) The heights and weights of some students were plotted on a scatter graph. Which of the diagrams above shows the relationship you would expect to see on the scatter diagram. Choose A, B, C, D or E

(1)

B

………………………………………. correlation ………………………………………. correlation

………………………………………. correlation

………………………………………. correlation

………………………………………. correlation

C

D E

A


Question 2.

Ten books were chosen and the average number of words on a page and the size of the font used to print the book were recorded.

a) Draw a diagram to show the results you would expect to see. Label the axes and plot 10 points to represent the results you would expect to see.

(3)

b) What kind of correlation does your diagram show?

(1)

Total / 10


Pie Charts 1 Grade 3

Objective: Interpret and construct pie charts for categorical data

Question 1

The pie chart shows information about the different types of pizza a group of students liked best.

(a) Which type of pizza was liked best by the least number of students?

(1)

20 of the students like Cheese and Tomato pizza best.

(b) How many of the students like Ham and Pineapple best?

(1)


Question 2

Jim carried out a survey to find out the type of TV programme people like the best.

He is going to show his results in a pie chart.

The table gives information about his results.

TV programme Number of people Angle in pie chart

News 14 35°

Sports 34

Drama 20

Soaps 48

Comedy 28

Complete the pie chart for this information.

(3)


Question 3

The table gives the number of one-bedroom, two-bedroom, three-bedroom and four-bedroom houses in Hunton.

Number of bedrooms one two three four

Number of houses 42 48 21 9

(a) In the circle, draw a pie chart to show this information.

(3)

The pie chart below gives information about the number of one-bedroom, two-bedroom, three-bedroom and four-bedroom houses in Lambton.


The total number of these houses in Lambton is 280

(b) Work out the number of one-bedroom houses.

(2)

Total Mark /10

one-bedroom

two-bedroom

three-bedroom

four-bedroom


Scatter Diagrams 1 Grade 4

Objective: Use and interpret scatter graphs

Question 1

The scatter graph shows the number of ice creams sold plotted against the midday temperature

a) How many ice creams where sold on the hottest day?

(1) b) Draw the line of best fit on the scatter graph

(1)

c) Describe the relationship between the number of ice creams sold and the midday temperature

(1) d) Predict the midday temperature if 105 ice creams were sold.

(1) e) One point has been plotted incorrectly. Draw a circle round this point.

(1)

50

0

120

80

40

160

100

60

20

140

180

52 54 58 56 60 64 62 68 66

Number of

ice creams

sold

Midday temperature (0F)


Question 2

Some runners recorded their resting pulse rates and miles run per week

a) How many runners ran 28.5 miles in a week?

(1) b) A runner who ran 53 miles in a week had a resting pulse rate of 49 beats per minute.

Plot this point on the scatter graph.

(1) c) Draw the line of best fit

(1) d) Describe the relationship between the resting pulse rate and the miles run per week.

(1) e) Use your line of best fit to predict the resting pulse rate of a runner that runs 34 miles in a week.

(1) Total / 10

45

75

65

55

70

60

50

80

20 25 30 40 35 45 55 50 65 60

Resting pulse rate

(beats per minute)

Distance run per week (miles)


Factorise single bracket 1 Grade 4

Objective: Take out common factors to factorise

Question 1

Factorise y2 + 27y

…........................................................

(Total 1 mark)

Question 2

Factorise 10x – 15

…........................................................

(Total 1 mark)

Question 3

Factorise 3f + 9

…........................................................

(Total 1 mark)

Question 4

Factorise 2x2 – 10

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

(Total 1 mark)

Question 5

Factorise 18a2 – 34

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

(Total 1 mark)

Question 6

Factorise 8y2 – 16

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

(Total 1 mark)


Question 7

Factorise 3x+6

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

(Total 1 mark)

Question 8

Factorise 8s+2t

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

(Total 1 mark)

Question 9

Factorise ac-c

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

(Total 1 mark)

Question 10

Factorise 4x2+3x

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

(Total 1 mark)

Total /10


Inequalities on number lines 1 Grade 4

Objective: Represent the solution of a linear inequality on a number line.

Question 1

Draw diagrams to represent these inequalities.

(a) x ≤ 3

(b) x > -2

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

(2)

Question 2

-3 ≤ n < 2

n is an integer

Write down all the possible values of n and represent these values on a number line.

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

(3)

Question 3

Write down the inequality that is represented by each diagram below.

(a) (b)

-2 -1 0 1 2 3 -1 0 1 2 3 4

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

(4)


(c) Which of these inequalities (a) or (b) has the most integer solutions?

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

(1)

Total /10


Linear equations one unknown 1 Grade 3

Objective: Solve linear equations with one unknown on one side

Question 1

Solve 2x = 15

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

(1)

Question 2

Solve y - 7 = 3

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

(1)

Question 3

Solve �5

= 4

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

(1)

Question 4

Solve 2�3

= −2

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

(2)

Question 5

Solve 2x – 6 = 10

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

(2)


Question 6

Solve 4r + 7 = 13

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

(2)

Question 7

Solve 3g = 0

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

(1)

Total /10


Multiplying single brackets 1 Grade 4

Objective: Multiply a single term over a bracket Question 1 Expand the following y(y+2)

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

(Total 1 mark) Question 2 Expand the following a(b+c)

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

(Total 1 mark) Question 3 Expand the following -2(m+3)

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

(Total 1 mark) Question 4 Expand the following -5(p-2)

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

(Total 1 mark) Question 5 Expand the following 3(t-1) + 5t

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

(Total 2 mark)


Question 6 Expand the following 3(d+2)+4(d-2)

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

(Total 2 mark) Question 7 Expand the following 3(y+10)-2(y+5)

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

(Total 2 mark)

Total /10


nth term of a linear sequence 1 Grade 4

Objective: Write and expression for the nth term of a linear sequence.

Question 1.

Write down the first five terms of the sequence whose nth term is given by:

(a) 3n + 4

……………………………………………………………………………………… (1)

(b) 2n – 1

……………………………………………………………………………………… (1)

(Total 2 marks)

Question 2.

Here are the first five terms of a linear sequence.

1, 5, 9, 13, 17, …

(a) Write down an expression in terms of n, for the nth tem of this sequence.

……………………………………………………………………………………… (2)

(b) Find the 10th term of this sequence.

……………………………………………………………………………………… (1)

(Total 3 marks)

Question 3.

The first five terms of a linear sequence is given below:

-3, 1, 5, 9, 13, …

Find an expression in terms of n, for the nth tem of this sequence.


………………………………………………………………………………………

(Total 2 marks)

Question 4.

The diagrams show a sequence of patterns made from square tiles.

(a) In the space below, draw pattern number 4.

(1)

(b) Write an expression in terms of n for the number of tiles in pattern n.

……………………………………………………………………………………… (1)

(c) Find the total number of tiles in pattern number 10. ……………………………………………………………………………………… (1) (Total 3 marks)

Total /10


Enlargements and negative scale factors 1 Grade 5

Objective: Identify and construct enlargements including using negative scale factors

Question 1

a) Describe the enlargement that would transform shape C onto shape D

b) Describe the enlargement that would transform shape D onto shape C

(3)

Question 2

Enlarge this shape by a scale factor of -½ from the point ( 2 , 3 )

(3)

– 5

10

5

– 5 10 5 15 x

y

– 5

10

5

– 5 10 5 15 x

y

C

D


Question 3

a) Enlarge this shape by a scale factor of -2 from the point ( 5 , 0 )

b) What happens to the interior angles of a shape when it is enlarged?

(4)

Total marks / 10

– 5

10

5

– 5 10 5

15 x

y


Enlargements – Fractional scale factors 1 Grade 4

Objective: Identify and construct enlargements using fractional scale factors Question 1. The shows a shape and its enlargement

(a) Write down the scale factor that transforms shape A onto shape B. ..........................................(1)

(b) Write down the enlargement that transforms shape B onto shape A.

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

(Total 2 marks)

Question 2.

The diagram shows some shapes, one shape has been enlarged to make one of the other shapes.

Draw a circle round these two shapes.

(Total 2 marks)

A

B


Question 3.

Enlarge this shape by a scale factor of from the origin.

(Total 3 marks)

Question 4.

Describe fully the enlargement that transforms shape K onto shape M.

(Total 3 marks)

Total /10

1 3

10

K

M

5

– 5

10

5 – 5


Calculating with fractions 1 Grade 5

Objective: Calculate exactly with fractions, including solving problems

Question 1

Work out 1 + 1 3 12

………………. (1)

Question 2

Work out 2 _ 1 3 4

………………. (1)

Question 3

Evaluate 3 + 5 . Leave you answer as a mixed number 7 6

………………. (2)

Question 4

Work out 3 x 1 4 5

………………. (1)


Question 5

Work out 2 x 3 . Give your answer as a fraction in its simplest form 9 8

………………. (2)

Question 6

Evaluate 6 ÷ 4 . Give your answer as a fraction in its simplest form 7 3

………………. (2)

Question 7

Work out 4 of 45 9

………………. (1)

Total /10


Index Laws 1 Grade 5

Objective: Calculate with roots and with integer indices

Question 1

i) a8 x a2

…………………….

(1)

ii) x7 ÷ x3

…………………….

(1)

iii) 12�7�2�4�26

…………………….

(2)

iv) 3a2b x 4a3b

…………………….

(2)


v) (x4)6

…………………….

(1)

vi) (3x2y)3

…………………….

(3)

Total /10


Percentage Change 1 Grade 5 Objective: Solve problems involving percentage change, including original value problems.

Question 1.

Rupal buys a pair of jeans for £44 in the sale. They were originally £80.

What was the percentage discount?

………………%

(Total 2 marks)

Question 2.

Sean buys a second-hand car for £3200 and sells it for £3800.

What is Sean’s percentage profit?

………………%

(Total 2 marks)

Question 3.

Rachel’s monthly pay increased by 4% to £2236.

What was Rachel’s pay before the increase?

£………………

(Total 2 marks)


Question 4.

In a sale, normal prices are reduced by 30%.

Jane buys a road bike for £560.

What is the normal price of the bike?

£………………

(Total 2 marks)

Question 5.

During the Easter holidays, 78 Year 11 students attended a revision class.

This was 65% of all of Year 11.

How many students are there in Year 11 altogether?

………………

(Total 2 marks)

Total /10


Scale factors and similarity 1 Grade 5

Objective: Use scale factors as a link to similarity.

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. (the shaded part) (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 20 cm

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

Height = ………………………………..cm

(total 2 marks)

Total marks / 10


Histograms with equal class intervals 1 Grade 5

Objective: Construct and Interpret histograms with equal class widths for discrete as well as continuous data.

Question 1

The following data table represents the rainfall per day over a year in Mathstown

Rainfall, mm Frequency

0 < x ≤ 10 204

10 < x ≤ 20 82

20 < x ≤ 30 46

30 < x ≤ 40 24

40 < x ≤ 50 7

50 < x ≤ 60 2

Total 365

a) Draw a histogram to represent this data below


(3)

b) Estimate the total rainfall in this year.

……………………………………………..

(2)

Total for question – 5 marks

Question 2

The histogram below represents the number of eggs found in 30 batches of frogspawn.

a) Complete the table below

Eggs Frequency

0 - 49 3

50 - 99 9

100 - 149 13

150 - 199 19

200 - 249 41

250 - 299 15

(2)

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200 250 300

Freq

uen

cy

Number of eggs


b) How many batches of frogspawn were there?

……………………………………………..

(2)

c) Why does the table above not have groups labelled 0 ≤ x < 50 but rather have groups in the form 0 – 49.

……………………………………………..

(1)


Total /10


Equation of a line 1 Grade 5

Objective: Use the form y = mx + c to identify perpendicular lines.

Question 1.

The line l1 has equation 3x + 3y – 6 = 0 (a) Find the gradient of l1.

(2) The line l2 is perpendicular to l1 and passes through the point (3, 1).

(b) Find the equation of l2 in the form y = mx + c, where m and c are constants.

(3) (Total 5 marks)


Question 2.

A and B are straight lines. Line A has equation 2y = 3x + 8. Line B goes through the points (–1, 2) and (2, 8). Are lines A and B perpendicular to each other? You must show all your working.

(Total 5 marks)

Total /10


Linear inequalities in two variables 1 Grade 6 Objective: Solve linear inequalities in two variables

Question 1.

The graph shows the region that represents the inequalities � < � + 2 and 2� + � < 9 by shading the unwanted regions.

Use the graph to find the integer values of � and � that maximise the sum of � and �.

….………………………

(Total 2 marks)

Question 2.

If � + � > 100, state which of the following may be true and which must be false

a) � ≤ 10 ….………………………(1)

b) �� ≤ 100 ….………………………(1)

c) � + 2� ≤ 50 ….………………………(1)

(Total 3 marks)


Question 3.

a) Represent the inequalities � < � + 1, � + � ≤ 5 and � > −1 on the grid below by shading the unwanted regions.

(3)

b) Use your graph to find the integer values of � and � that maximise the product of � and �.

……………………… (2)

(Total 5 marks)

TOTAL /10


Pythagoras 1 Grade 5

Objective: Know and use Pythagoras's theorem for right-angled triangles

Question 1

ABC is a right angled triangle. AB = 9 cm, BC = 12 cm Calculate the length of AC.

………………………. (3) (Total 3 marks) Question 2

ABC is a right angled triangle. AB = 11 cm, AC = 18 cm Calculate the length of BC. Give your answer correct to 1 decimal place.

………………………. (Total 3 marks) (3)


Question 3

ABCD is a rectangle. AB = 19 m, AD = 13 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures.

………………………. (4) (Total 4 marks)

Total /10


Powers and Roots 1 Grade 6

Objective: Calculate and estimate powers and roots

Question 1

Evaluate

i) 53

…………………

ii) 104

…………………

iii) 18

…………………

(3)

Question 2

Work out the value of

i) √121

…………………

ii) 3√64

…………………

iii) 4√81

…………………

(3)

Question 3

The square root of 56 lies between which two integer values. Explain your answer.

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

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

(2)


Question 4

The cube root of 100 lies between which two integer values. Explain your answer.

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

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

(2)

Total /10


Cumulative Frequency 1 Grade 6

Objective: Construct and interpret cumulative frequency diagrams (for grouped discrete as well as continuous data) Question 1

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

(a) Complete the cumulative frequency table.

(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 median.

…………………………………… (1)

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

…………………………………… (2)

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

…………………………………… (2)

(Total 8 marks)

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500 600 700 800

CU

MU

LAT

IVE

FR

EQ

UE

NC

Y

WEEKLY WAGE, £


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)

(b) Use the graph to find an estimate for the lower quartile..

…………………………………… (1)

(Total 2 marks)

Total marks /10


Standard trigonometric ratios 1 Grade 7

Objective: Know and derive the exact values for Sin and Cos 0, 30, 45, 60 and 90 and Tan 0, 30, 45 and 60 degrees.

Question 1.

A right angled triangle has the dimensions as shown in the diagram.

Using the diagram, or otherwise, state the exact values of:

(a) Sin 60

(b) Cos 60

(c) Tan 60

(d) Sin 30

(e) Cos 30

(f) Tan 30

(Total 6 marks)

Question 2.

Using the triangle shown, or otherwise, find the exact values of:

(a) Sin 45

(b) Cos 45

(c) Tan 45

(d) Sin 90

(Total 4 marks)

Total /10

1

2 √3

1

1

√2


Histograms with unequal class widths 1 Grade 6

Objective: Construct and interpret a histogram with unequal class widths (for grouped discrete as well as continuous

data)

Question 1

The table shows the weight of 815 parcels handled by a sorting office in one day.

a) Draw a fully labelled histogram to show the weights of the parcels.

(4)

b) Estimate the number of parcels that weighed more than 2500 grams

…………………………………..

(2)


Question 2

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 36

30 < x ≤ 60 75

60 < x ≤ 80 24

Total 150

a) Use the information to complete the histogram

(2)

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

…………………………………

(2)


Total /10

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90

Freq

uen

cy D

ensi

ty

Time, minutes


Geometric Sequences 1 Grade 7

Objective: Recognise and use geometric sequences (rn, where n is an integer and r can be a surd)

Question 1.

Find the 5th and 6th terms of the sequences below.

(a) 81, 27, 9, 3, ____, ____, … (1)

(b) 4, 0.8, 0.16, 0.032, ____, ____, … (1)

(Total 2 marks)

Question 2.

(a) Write down the first four terms of the geometric sequence with nth term 2n.

…………………………………………………………………………………………...

(2)

(b) State the term-to-term rule of the sequence.

…………………………………………………………………………………………...

(1)

(Total 3 marks)

Question 3.

In this geometric sequence, the first term is √3 and the term-to-term rule is multiply by√3.

Continue the sequence for three more terms.

√3 , 3, 3√3 , ___, ___, ___,…

(Total 3 marks)


Question 4.

Work out the missing terms in this geometric sequence.

14, ___, 1, 2, 4, ___

(Total 2 marks)

Total /10


Represent quadratic inequalities 1 Grade 7 Objective: Represent the solution to a quadratic inequality on a number line, using set

notation and on a graph

Question 1.

a) Solve �2 + 13x + 36 ≤ 0

Represent your solution on a number line.

b) Write the integer answers for part a) in set notation.

……………………… (Total 2 marks)

Question 2.

Solve �2 − 3� ≥ 18

Display your answer on a sketch of the graph of the solution



Question 3.

For which values of x is the expression �2 − 2� − 8 greater than the expression 7 − 6� − 3�2 ? Represent the possible values of � on a number line.

………………………

(Total 3 marks)

Question 4.

Find the set(s) of all values for which 2�−142−�

< x + 5


………………………

(Total 3 marks)

TOTAL /10


Index Laws (Negative and Fractional) 1 Grade 7

Objective: Calculate with negative and fractional indices

Question 1

Evaluate

i) 24−

…………………….

(1)

ii) 32−

…………………….

(1)

iii)

2

1

12

−

…………………….

(2)

iv) 1

38

…………………….

(1)


v) 3

416

…………………….

(2)

vi) 2

38000−

…………………….

(3)

Total /10


Approximate solutions to equations using iteration 1 Grade 9 Objective: Find approximate solutions to equations using iteration.

Question 1.

Find the first five iterations of each iterative formulae. Start each one with �1 = 4.

a) ��+1 = 3�1 + 7

……………………………….…………………….……………….………………………

(1)

b) ��+1 = 10 − 2��

……………………………….…………………….……………….………………………

(1)

c) ��+1 =��−22

……………………………….…………………….……………….………………………

(1)

(Total 3 marks)


Question 2.

Starting with �1 = 1.6 find a root of the quadratic equation 2�2 + 5� − 13 = 0 to 2 decimal places using the iterative formula

��+1 = �13 − 5��2

………………………

(Total 3 marks)


Question 3.

a) Show that �2 − 3� + 1 = 0 can be rearranged into the iterative formula

��+1 = �3�� − 1.

(1)

b) Use the iterative formula and a starting value of �1 = 2 to obtain the solution to the equation correct to 2 decimal places.

………………………

(3)

(Total 4 marks)

TOTAL /10

PLC Papers

Created For:


Collecting like terms 1 Grade 4 Solutions

Objective: Simplify algebraic expressions by collecting like terms

Question1

Simplify 7x + 4y – 4x + 3y

yx 73 + ...........................................................

(Total 2 mark)

Question 2

Simplify 8f − f + 6f

f13

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

(Total 1 mark)

Question 3

Simplify 8x − 3x + 5x

x10

........................................................... (Total 1 mark)

Question4

Simplify 9x + x + 13y + y

yx 1410 +

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

(Total 2 mark)

Question 5

Simplify 13p – q + 12 – 6p + 4q - 9

337 ++ qp ...........................................................

(Total 2 mark)

Question 6

Simplify 19 + a – 2b + 15a + 8 – 2b

ba 41627 −+ ...........................................................

(Total 2 mark)

Total /10


Graphs of Linear Functions 1 Grade 4 Solutions

Objective: Recognise, sketch and interpret graphs of linear functions.

Question 1

Sketch the graph of each function, clearly indicating the y-intercept.

a) y = 5x -3 b) y = 10 – 2x c) 2y = 4x + 8 d) y = -3x

y B1 line with positive gradient

a B1 intercept indicated

x

-3

(2)

B1 line with negative gradient B1 intercept indicated

y

b 10

x

(2)

y B1 line with positive gradient B1 intercept indicated

c 4

x

(2)


y B1 line with negative gradient B1 intercept indicated

d

0 x

(2)

Question 2

Which of these are linear functions? Circle your answer(s)

y = 7 – 3x

y = x 4 y = x2 + 4 2x + 3y = 5 y = x B1 for any three

B2 for all four correct answers

(2)

Total /10


Substitution 1 Grade 4 Solutions

Objective: Substitute numerical values into formulae and expressions, including scientific formulae Question 1.

Complete this table of values.

n 3n+2

12 38

15 47

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

(Total 2 mark)

Question 2 Calculate the value of y when x = 2 are

� = −1

3√18�2

You must show your working.

2

63

1

363

1

183

1

−=×−=

−=

−=

y

y

y

xy

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

(Total 2 marks)


Question 3 a. Work out the value of v when u = 80, a = 10 and t = 4 b. Work out the value of v when u = 35, a = -5 and t = 12

� = � + �� a.

120

4080

41080

=+=

×+=+=

v

v

v

atuv

b.

25

6035

12535

−=−=

×−+=+=

v

v

v

atuv

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

(Total 2 mark)

Question 4

Work out the value of T when a. p = 5 and b. p = -1

� = 3�2 − 2�

a.

65

1075

)5(2)5(3

232

2

=−=

−=−=

T

T

T

ppT

b.

5

23

)1(2)1(3

232

2

=+=

−−−=−=

T

T

T

ppT

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

(Total 2 marks)


Question 5

Work out the value of V to 3 sig fig when (a) π = 3.14, r = 10, and h = 15 (b) π = 3.14, r = 2.4, and h = 20

� = 1

3��2ℎ

a.

1570

151014.33

1 2

=×××=

V

V

b.

121

576.120

204.214.33

1 2

==

×××=

V

V

V

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

(Total 2 marks)

Total /10


Adding and subtracting fractions 1 Grade 4 Solutions Objective: Add and subtract fractions including improper fractions and mixed numbers. Question 1 Work these out: 25 +

17 = (2 x 7)+(5 x 1)

(5 x 7)

=

14+535 (M1)

=

1935 (A1)

(2) 79 - 48 =

(7 x 8)−(9 x 4)

(9 x 8)

=56−3672 (M1)

=2072

=518 (A1)

(2) Question 2 There are 700 counters in a bag. These are either black, blue or purple. 2/7 of the counters are blue. 1/5 are purple. a) What fraction of the counters are black?

1- ( 27 +

15 ) = 1- 1735 (M1)

=35−1735 (M1)

=1835 (A1)

(2) b) How many counters are black?

(2) =

1835 x 700 (M1)

= 360 (A1)


Question 3

52 +

= 52 +

43 (M1) = 236 = (A1)

(2)

Total /10

113

356


Decimals1 Grade 3 Solutions

Objective: Understand adding subtracting multiplying and dividing decimals.

Question 1

Three parcels weigh 7.2 kg, 15.02 kg and 3.1 kg. Find their total mass. = 7.2kg + 15.02kg + 3.1kg (M1) = 25.32 kg (A1) (2)

Question 2 Work out 0.8965 ÷ 0.5

= 0.8965 (x 10)0.5 ( x 10)

= 8.9655 (M1)

= 1.793 (A1) (2)

Question 3

(a) Tom gets paid £3.15 an hour. One week he worked for 26 hours. How much is his weekly pay? 1 hour = £ 3.15

26 hours = (£3.15 x 26) (M1) = £ 81.90 (A1)

x 26 x 26


(2)

(b) Tom is saving for a gift. He wants to give his friend a gift that costs £100.27. How much more money does Tom need after he has worked 26 hours in one week.

=£100.27 - £81.90 (M1) =£18.37 (A1) (2)

Question 4

1 bottle of milk = £1.39

Lacey bought five bottles of milk. How much will it cost altogether? 1 bottle of milk = £ 1.39

5 bottles of milk = (£1.39 x 5) (M1) = £ 6.95 (A1) (2)

Total /10

x 5 x 5


Fractions and ratio problems 1 Grade 4 Solutions Objective: Identify and work with fractions in ratio problems. Question 1 Share £360 in the ratio 3:4:2. Find the value of smallest share. £360 / 9 = £40 2 x £40 = £80 (M1) (A1) (2) Question 2 In similar triangles PST and PQR PQ = 6 cm , ST = 8 cm and RQ = 9 cm Find a) Side PT. Give your answer to 2 decimal places. SF = 9/8 6/1.125 (M1) =5.33 cm (A1)

(2) b) Side TQ. Give your answer to 2 decimal places. 6-5.33… (M1) =0.67 cm (A1)

(2)

P

R Q

8cm S T

9cm

NOT TO SCALE

6cm


Question 3 Radhika wants to make a cake which weighs 560 grams. She uses sugar, flour, chocolate and eggs, in the ratio of 3:5:4:2. Calculate the weight of each ingredient: 560/ 14= 40 Flour: 3 x 40 = 120 grams (A1) Sugar: 5 x 40 = 200 grams (A1) Eggs: 4 x 40 = 160 grams (A1) Chocolate: 2 x 40 = 80 grams (A1) (4)

Total /10


LCM and HCF 1 Grade 4 Solutions Objective: Understand and find LCM and HCF.

Question 1

Find the HCF of 18 and 24. Factors of 18: 1,18, 2,9, 3,6 Factors of 24: 1,24, 2,12, 3, 8, 4, 6 (M1) HCF = 6 (A1) (2)

Question 2

Find the LCM of 14 and 3. Multiples of 14: 14, 28, 42 Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, (M1) LCM= 42 (A1) (2)

Question 3 Express 42 as a product of its prime factors using any method.

(M1) 2 x 3 x 7 (A1)


Question 4 (2) Express 48 as a product of its prime factors using any method.

(M1) 24 x 31(A1)

(2) Question 5

Find the HCF of 42 and 48.

Factors of 42= 1, 42, 2, 21,3, 14, 6, 7 Factors of 48= 24, 2, 16, 3, 2, 4, 8, 6 (M1) HCF: 6 (A1) (2)

Total /10


Multiplying fractions 1 Grade 4 Solutions Objective: Multiply fractions including improper fractions and mixed numbers. Question 1 Work out these questions always give your answer in the simplest form: 75 x

28 = 1440 =

720 (A2)

(2) 57 x

32 = 1514 (A2)

(2)

=145 x

258 (M1)

=354 (A1)

(2)

245 x 3

18 =


Question 2 Which is smaller?

or

= 6320 > 3 (A2)

(2)

Which is bigger?

or

=65 <

97 (A2)

(2)

Total /10

35 of 514 25 of 7

12

27 of 415 35 of 2

17


Rounding 1 SOLUTIONS Grade 5

Objective: Round to an appropriate degree of accuracy (e.g. to decimal places or significant figures)

Question 1.

Round the following to the given degree of accuracy

(a) 2.567889 to 1 decimal place

(B1)……2.6……………….. (1)

(b) 409877.01233 to 2 decimal places

(B1……………409877.01……….. (1)

(Total 2 marks)

Question 2

Round the following numbers to the degree of accuracy given

(a) 0.00654 to 1 significant figure

(B1……0.007……………….. (1)

(b) 7654.987 to 2 significant figures

(B1……7700……………….. (1)

(Total 2 marks)


Question 3.

Round these numbers to the nearest integer

(a) 4.99987

(B1 …5…………. (1)

(b) 0.0098

(B1……0……………….. (1)

(Total 2 marks)

4. Tara worked out the answer to her question using a calculator. Her calculator read

34567654.123

Her teacher told her to round it to two significant figures. What number should she write?

……(B2)…35000000…………….. (2)

(Total 2 marks)


5. Round the following numbers to the degree of accuracy given

(a) 0.0098 to 3.d.p

(B1 …0.010…………. (1)

(b) 3755.4883087 to 4.d.p

(B1…3755.4883………….. (1)

(Total 2 marks)

Total /10


Compare quantities using ratio 1 Grade 4 Solutions Express a multiplicative relationship between two quantities

Question 1

A bank gives you 28 Euros when you exchange £20. How much will you get for exchanging £135? 135 ÷ 20 = 6.75 M1 6.75 × 28 = €189 A1

€……………………

(Total 2 marks)

Question 2

Barry uses blue and red to make purple, in the ratio 3:5. How many tins of red will he need to mix with the 9 tins of blue? 9 ÷ 3 = 3 M1 3 × 5 = 15 A1

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

(Total 2 marks)

Question 3

Louis, Steve and Ella shared some money in the ratio 2 : 3 : 5 Ella got £54.

How much money did Steve get?

54 ÷ 5 = 10.8 M1 3 × 10.8 = £32.40 A1

£……………………

(Total 2 marks)


Question 3

Shannon and Liam share some chocolate in the ratio 4:3

Liam gets 81 grams of chocolate. Work out how many grams Shannon receives.

81 ÷ 3 = 27 M1 4 × 27 = 108 A1

..........................g

(Total 2 marks)

Question 4

A shop sells freezers and cookers. The ratio of the number of freezers sold to the number of cookers sold is 5 : 2 The shop sells 140 freezers in one week.

Work out the number of cookers sold that week.

140 ÷ 5 = 28 M1 2 × 28 = 56 A1

(Total 2 marks)

Total /10


Problems involving ratios 1 Grade 4 Solutions Objective: Solve problems involving ratios, e.g. conversion, comparison, scaling, mixing, concentrations

Question 1

Louise and Anil share some sweets in the ratio 3 : 8

Anil gets 32 sweets. (a) How many sweets does Louise get?

32 ÷ 8 = 4

3 × 4 M1

12 sweets A1 ...........................................................

(2) Anil also has a tin of chocolates.

There are 80 chocolates in the tin.

45% of the chocolates have toffee in the middle. (b) Work out the number of chocolates that have toffee in the middle.

× 800 M1

36 chocolates A1 ...........................................................

(2) (Total 4 marks)

Question 2

Kiran is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.

Kiran wants to make 180 kg of concrete mix.

Kiran has

18 kg of cement 85 kg of sand 90 kg of gravel

Does Kiran have enough cement, sand and gravel to make the concrete mix?

1 + 3 + 5 = 9 1 × 20 = 20 kg Cement ( not enough)

180 ÷ 9 = 20 M1 3 × 20 = 60 kg of sand (enough) M1

5 × 20 = 100 kg of gravel

He has enough sand and gravel, but is 2kg short of cement. C1


(Total 3 marks)

Question 3

Jane made some almond biscuits which she sold at a fête.

She had:

5 kg of flour 3 kg of butter 2.5 kg of icing sugar 320 g of almonds

Here is the list of ingredients for making 24 almond biscuits.

Jane made as many almond biscuits as she could, using the ingredients she had.

Work out how many almond biscuits she made.

500g of flour 5000 ÷ 150 = 33.3(3) 3000g of butter 3000 ÷ 150 = 30 M1 2500g of icing sugar 2500 ÷ 75 = 33.3(3) 320g of almonds 320 ÷ 10 = 32 3 × 24 M1 720 biscuits A1

(Total 3 marks)

Total /10


Correlation 1 Grade 4 SOLUTIONS

Objective: Understand and use the vocabulary of correlation (weak, strong, positive, negative, no correlation)

Question 1

(a) Each of these diagrams show a type of correlation. State the type of correlation shown in each diagram

(5)

(b) The heights and weights of some students were plotted on a scatter graph. Which of the diagrams above shows the relationship you would expect to see on the scatter diagram. Choose A, B, C, D or E

C (Strong positive correlation) 1M

(1)

E

Perfect positive correlation Strong negative correlation

no correlation

Strong positive correlation

Weak negative correlation

B

C

D

A


Question 2

Ten books were chosen and the average number of words on a page and the size of the font used to print the book were recorded.

a) Draw a diagram to show the results you would expect to see. Label the axes and plot 10 points to represent the results you would expect to see.

10 points, none of the points should be touching the axes 1M

Showing negative correlation 1M Axes labelled 1M

(3)

b) What kind of correlation does your diagram show? Negative correlation 1M

(Allow positive correlation if part (a) is incorrect) (1)

Total / 10

Average number

of word on a page

Font size


Pie Charts 1 Grade 3 SOLUTIONS

Objective: Interpret and construct pie charts for categorical data

Question 1

The pie chart shows information about the different types of pizza a group of students liked best.

(a) Which type of pizza was liked best by the least number of students?

Pepperoni B1

(1)

20 of the students like Cheese and Tomato pizza best.

(b) How many of the students like Ham and Pineapple best?

10 B1

(1)


Question 2

Jim carried out a survey to find out the type of TV programme people like the best.

He is going to show his results in a pie chart.

The table gives information about his results.

TV programme Number of people Angle in pie chart

News 14 35°

Sports 34 34144 x 360° = 85°

Drama 20 20144 x 360° = 50°

Soaps 48 48144 x 360° =120 °

Comedy 28 28144 x 360° = 70°

144

Complete the pie chart for this information.

(3)

M1 for correct method 34/144 x 360 or 34/14 x 35 or 35/14 x 34

A1 for at least 2 correctly calculated angles or at least 1 correct sector drawn

A1 for correct pie chart including labels


Question 3

The table gives the number of one-bedroom, two-bedroom, three-bedroom and four-bedroom houses in Hunton.

Number of bedrooms one two three four

Number of houses 42 48 21 9

�� x 360° =

126° �� x 360° =

144° �� x 360° =

63° �� x 360° =

27°

(a) In the circle, draw a pie chart to show this information.

(3)

M1 for correct method

A1 for at least 2 correctly calculated angles or at least 1 correct sector drawn

A1 for correct pie chart including labels


The pie chart below gives information about the number of one-bedroom, two-bedroom, three-bedroom and four-bedroom houses in Lambton.

The total number of these houses in Lambton is 280

(b) Work out the number of one-bedroom houses.

¼ x 280 = 70 one-bedroom houses M1 for ¼ of 280 oe

A1 for 70

(2)

Total Mark /10

one-bedroom

two-bedroom

three-bedroom

four-bedroom


Scatter Diagrams 1 Grade 4 SOLUTIONS

Objective: Use and interpret scatter graphs

Question 1

The scatter graph shows the number of ice creams sold plotted against the midday temperature

a) How many ice creams where sold on the hottest day? 186 ice creams 1M

(1) b) Draw the line of best fit on the scatter graph

Any line between the two lines drawn on the diagram 1M (1)

c) Describe the relationship between the number of ice creams sold and the midday temperature Positive correlation 1M (As the temperature increases the number of ice creams sold increases)

(1) d) Predict the midday temperature if 105 ice creams were sold.

Value from the dotted line drawn on the graph using their line of best fit 1M (1)

e) One point has been plotted incorrectly. Draw a circle round this point. Circle drawn round the point indicated on the graph 1M

50

0

120

80

40

160

100

60

20

140

180

52 54 58 56 60 64 62 68 66

Number of

ice creams

sold

Midday temperature (0F)


(1)

Question 2 Some runners recorded their resting pulse rates and miles run per week

a) How many runners ran 28.5 miles in a week?

3 runners 1M (1)

b) A runner who ran 53 miles in a week had a resting pulse rate of 49 beats per minute. Plot this point on the scatter graph. Plot point in position shown 1M

(1) c) Draw the line of best fit

Line drawn between the red lines shown on the diagram 1M

(1) d) Describe the relationship between the resting pulse rate and the miles run per week.

Negative correlation 1M (The resting pulse rate decreases as the number of miles run per week increases)

(1) e) Use your line of best fit to predict the resting pulse rate of a runner that runs 34 miles in a week.

Value from the dotted line drawn on the graph using their line of best fit 1M

(1) Total / 10

45

75

65

55

70

60

50

80

20 25 30 40 35 45 55 50 65 60

Resting pulse rate

(beats per minute)

Distance run per week (miles)


Factorise single bracket 1 Grade 4 Solutions

Objective: Take out common factors to factorise

Question 1

Factorise y2 + 27y = )27( +yy

…........................................................

(Total 1 mark)

Question 2

Factorise 10x – 15 = )32(5 −x

…........................................................

(Total 1 mark)

Question 3

Factorise 3f + 9 = )3(3 +f

…........................................................

(Total 1 mark)

Question 4

Factorise 2x2 – 10 = )5(2 2 −x

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

(Total 1 mark)

Question 5

Factorise 18a2 – 34 = )179(2 2 −a

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

(Total 1 mark)

Question 6

Factorise 8y2 – 16 = )2(8 2 −y

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

(Total 1 mark)


Question 7

Factorise 3x+6 = )2(3 +x

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

(Total 1 mark)

Question 8

Factorise 8s+2t = )4(2 ts +

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

(Total 1 mark)

Question 9

Factorise ac-c = )1( −ac

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

(Total 1 mark)

Question 10

Factorise 4x2+3x = )34( +xx

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

(Total 1 mark)

Total /10


Inequalities on number lines 1 Grade 4 Solutions

Objective: Represent the solution of a linear inequality on a number line.

Question 1

Draw diagrams to represent these inequalities.

(a) x ≤ 3 (A1)

(b) x > -2 -1 0 1 2 3

(A1)

-3 -2 -1 0 1 ..............................................

(2)

Question 2

-3 ≤ n < 2 (A1)

n is an integer -3 -2 -1 0 1 2

Write down all the possible values of n and represent these values on a number line.

-3, -2, -1, 0, 1 (M1 A1)

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

(3)

Question 3

Write down the inequality that is represented by each diagram below.

(a) (b)

-2 -1 0 1 2 3 -1 0 1 2 3 4

-2 ≤ x < 3 (M1 A1) 0 < x ≤ 3 (M1 A1)

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

(4)


(c) Which of these inequalities (a) or (b) has the most integer solutions? (a) B1

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

(1)

Total /10


Linear equations one unknown 1 Grade 3 Solutions

Objective: Solve linear equations with one unknown on one side

Question 1

Solve 2x = 15

(A1) x = 7.5 oe

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

(1)

Question 2

Solve y - 7 = 3

(A1) y = 10

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

(1)

Question 3

Solve �5

= 4

(A1) t = 20

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

(1)

Question 4

Solve 2�3

= −2

(M1) 2b = -6 (A1) b = -3

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

(2)


Question 5

Solve 2x – 6 = 10

(M1) 2x = 16 (A1) x = 8

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

(2)

Question 6

Solve 4r + 7 = 13

(M1) 4r = 20 (A1) r = 5

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

(2)

Question 7

Solve 3g = 0

(A1) g = 0

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

(1)

Total /10


Multiplying single brackets 1 Grade 4 Solutions

Objective: Multiply a single term over a bracket Question 1 Expand the following y(y+2) = yy 22 +

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

(Total 1 mark) Question 2 Expand the following a(b+c) = acab +

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

(Total 1 mark) Question 3 Expand the following -2(m+3) = 62 −− m

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

(Total 1 mark) Question 4 Expand the following -5(p-2) = 105 +− p

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

(Total 1 mark) Question 5 Expand the following 3(t-1) + 5t = 38533 −=+− ttt

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

(Total 2 mark)


Question 6 Expand the following 3(d+2)+4(d-2) = 278463 −=−++ ddd

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

(Total 2 mark) Question 7 Expand the following 3(y+10)-2(y+5) = 20102303 +=−−+ yyy

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

(Total 2 mark)

Total /10


nth term of a linear sequence 1 Grade 4 Solutions

Objective: Write and expression for the nth term of a linear sequence.

Question 1.

Write down the first five terms of the sequence whose nth term is given by:

(a) 3n + 4

3 × 1 + 4, 3 × 2 + 4, etc 7, 10, 13, 16, 19, … (A1) ……………………………………………………………………………………… (1)

(b) 2n – 1

2 × 1 – 1, 2 × 2 – 1, 1, 3, 5, 7, 9, … (A1) ……………………………………………………………………………………… (1)

(Total 2 marks)

Question 2.

Here are the first five terms of a linear sequence.

1, 5, 9, 13, 17, …

(a) Write down an expression in terms of n, for the nth tem of this sequence.

The sequence is adding 4 and it is 3 less than the 4 times table 4n – 3 (B1 for 4n and B1 for -3) ……………………………………………………………………………………… (2)

(b) Find the 10th term of this sequence.

4 × 10 – 3, 37 (A1)

……………………………………………………………………………………… (1)

(Total 3 marks)


Question 3.

The first five terms of a linear sequence is given below:

-3, 1, 5, 9, 13, …

Find an expression in terms of n, for the nth tem of this sequence.

The sequence is adding 4 and it is 7 less than the 4 times table 4n – 7 (B1 for 4n and B1 for -7)

………………………………………………………………………………………

(Total 2 marks)

Question 4.

The diagrams show a sequence of patterns made from square tiles.

(a) In the space below, draw pattern number 4.

(A1)


(1)

(b) Write an expression in terms of n for the number of tiles in pattern n.

2n + 2 (A1)

……………………………………………………………………………………… (1)

(c) Find the total number of tiles in pattern number 10. 22 ( A1) ……………………………………………………………………………………… (1) (Total 3 marks)

Total /10


Enlargements and negative scale factors 1 Grade 5 Solutions

Objective: Identify and construct enlargements including using negative scale factors

(2)

Question 1

a) Describe the enlargement that would transform shape C onto shape D

Enlargement scale factor 3 1M

Centre ( 15 , 3 ) 1M

b) Describe the enlargement that would transform shape D onto shape C

Enlargement scale factor 1/3 1M

Centre ( 15 , 3 )

(3)

Question 2

Enlarge this shape by a scale factor of -½ from the point ( 2 , 3 )

1M construction lines

(at least one in correct position)

1M using the correct point

1M correct size

(3)

– 5

10

5

– 5 10 5 15 x

y

– 5

10

5

– 5 10 5 15 x

y

C

D


Question 3

1M construction lines

(at least one in correct position)

1M using the correct point

1M correct size

a) Enlarge this shape by a scale factor of -2 from the point ( 5 , 0 )

b) What happens to the interior angles of a shape when it is enlarged?

1M The angles stay the same

(4)

Total marks / 10

– 5

10

5

– 5 10 5

15 x

y


Enlargements – Fractional scale factors 1 SOLN Grade 4 Solutions

Objective: Identify and construct enlargements using fractional scale factors Question 1. The shows a shape and its enlargement

(a) Write down the scale factor that transforms shape A onto shape B. ...................2 1 mark..............(1)

(b) Write down the enlargement that transforms shape B onto shape A.

............... ½ (or 0.5) 1 mark..........(1)

(Total 2 marks)

Question 2.

The diagram shows some shapes, one shape has been enlarged to make one of the other shapes.

Draw a circle round these two shapes. 1 mark each shape

(Total 2 marks)

A

B


Question 3.

Enlarge this shape by a scale factor of from the origin.

(Total 3 marks)

Question 4.

Describe fully the enlargement that transforms shape K onto shape M.

(Total 3 marks)

Total /10

1 3

1 mark attempt to draw construction lines

1 mark correct centre used

1 mark correct size

1 mark attempt to draw at least 2 construction lines

1 mark correct centre ( – 4 , – 4) 1 mark correct SF 1/5 or 0.2

K

M


Calculating with fractions 1 Grade 5 Solutions

Objective: Calculate exactly with fractions, including solving problems

Question 1

Work out 1 + 1 3 12 12 + 3 = 15 (A1)

36 36 36 ……………….

(1)

Question 2

Work out 2 _ 1 3 4 12 + 3 = 15 (A1)

36 36 36

………………. (1)

Question 3

Evaluate 3 + 5 . Leave you answer as a mixed number 7 6 18 + 35 = 53 (M1)

42 42 42

= 1 11 (A1) 42

………………. (2)

Question 4

Work out 3 x 1 4 5

= 3 (A1) 20

………………. (1)


Question 5

Work out 2 x 3 . Give your answer as a fraction in its simplest form 9 8 = 6 (M1) 72 = 1 (A1) 12

………………. (2)

Question 6

Evaluate 6 ÷ 4 . Give your answer as a fraction in its simplest form 7 3 = 18 (M1) 28 = 9 (A1) 14

………………. (2)

Question 7

Work out 4 of 45 9 = 20 (A1)

………………. (1)

Total /10


Index Laws 1 Grade 5 Solutions

Objective: Calculate with roots and with integer indices

Question 1

i) a8 x a2

a10 (A1)

…………………….

(1)

ii) x7 ÷ x3

x4 (A1)

…………………….

(1)

iii) 12�7�2�4�26

6b3c4 (M1 for any two correct terms, A1 for correct answer)

…………………….

(2)

iv) 3a2b x 4a3b

12a5b2 (M1 for any two correct terms, A1 for correct answer)

…………………….

(2)


v) (x4)6

x24 (A1)

…………………….

(1)

vi) (3x2y)3

27x6y3 (M1 for each correct term, A1 for fully correct answer)

…………………….

(3)

Total /10


Percentage Change 1 Grade 5 SOLUTIONS Objective: Solve problems involving percentage change, including original value problems.

Question 1.

Rupal buys a pair of jeans for £44 in the sale. They were originally £80.

What was the percentage discount? 80−4480 × 100 (M1)

45 (A1)………………%

(Total 2 marks)

Question 2.

Sean buys a second-hand car for £3200 and sells it for £3800.

What is Sean’s percentage profit? 3800−32003200 × 100 (M1)

18.75 (A1)………………%

(Total 2 marks)

Question 3.

Rachel’s monthly pay increased by 4% to £2236.

What was Rachel’s pay before the increase?

£2236 = 104% (M1)

£21.50 = 1%

£2150 = 100%

£2150 (A1)………………

(Total 2 marks)


Question 4.

In a sale, normal prices are reduced by 30%.

Jane buys a road bike for £560.

What is the normal price of the bike?

£560 = 70% (M1)

£8 = 1%

£800 = 100%

£800 (A1)………………

(Total 2 marks)

Question 5.

During the Easter holidays, 78 Year 11 students attended a revision class.

This was 65% of all of Year 11.

How many srudents are there in Year 11 altogether?

78 = 65% (M1)

1.2 = 1%

120 = 100%

120 (A1)………………

(Total 2 marks)

Total /10


Scale factors and similarity 1 Grade 5 SOLUTIONS

Objective: Use scale factors as a link to similarity.

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

(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 20 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


Histograms with equal class intervals 1 Grade 5 SOLUTIONS

Objective: Construct and Interpret histograms with equal class widths for discrete as well as continuous data.

Question 1

The following data table represents the rainfall per day over a year in Mathstown

Rainfall, mm Frequency

0 < x ≤ 10 204

10 < x ≤ 20 82

20 < x ≤ 30 46

30 < x ≤ 40 24

40 < x ≤ 50 7

50 < x ≤ 60 2

Total 365

a) Draw a histogram to represent this data below

(3)

0

50

100

150

200

250

0 10 20 30 40 50 60 70

FR

EQ

UE

NC

Y

Rainfall, mm


b) Estimate the total rainfall in this year.

5 x 204 + 15 X 82 + 25 x 46 + 35 x 24 + 45 x 7 + 55 x 2 M1

1020 + 1230 + 1150 + 840 + 315 + 110 = 4665

……4665mm……………A1…………………………..

(2)


Question 2

The histogram below represents the number of eggs found in 30 batches of frogspawn.

a) Complete the table below

Eggs Frequency

0 - 49 3

50 - 99 9

100 - 149 13

150 - 199 19

200 - 249 41

250 - 299 15

A2

(2)

b) How many batches of frogspawn were there?

………100………B1……………………………..

(2)

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200 250 300

Freq

uen

cy

Number of eggs


c) Why does the table above not have groups labelled 0 ≤ x < 50 but rather have groups in the form 0 – 49.

……Discrete data as opposed to continuous………………C1………………………..

(1)


Total /10


Equation of a line 1 Grade 5 Solutions

Objective: Use the form y = mx + c to identify perpendicular lines.

Question 1.

The line l1 has equation 3x + 3y – 6 = 0 (a) Find the gradient of l1.

3y = -3x + 6 y = -x + 2 (M1)

gradient = -1 (A1)

(2) The line l2 is perpendicular to l1 and passes through the point (3, 1).

(b) Find the equation of l2 in the form y = mx + c, where m and c are constants. Product of two gradients = -1 therefore gradient of l2 = 1 (M1)

y = x + c sub (3,1) 1 = 3 + c -2 = c (M1) y = x – 2 (A1)

(3)

(Total 5 marks)

Question 2.

A and B are straight lines. Line A has equation 2y = 3x + 8. Line B goes through the points (–1, 2) and (2, 8). Are lines A and B perpendicular to each other? You must show all your working.

y = 1.5x + 4 (M1) Product of two gradients is -1 (M1)

Gradient of B must be −2 3 (M1)

8 − 22− −1 =

63 = 2 (A1)

2 x 1.5 = 3 therefore Lines A and B are not perpendicular to each other (C1) (Total 5 marks)

Total /10


Linear inequalities in two variables 1 Grade 6 SOLUTIONS Objective: Solve linear inequalities in two variables

Question 1.

The graph shows the region that represents the inequalities � < � + 2 and 2� + � < 9 by shading the unwanted regions.

Use the graph to find the integer values of � and � that maximise the sum of � and �. � = 2,� = 3 or � = 3,� = 2 (A2) so that � + � = 5

….………………………

(Total 2 marks)

Question 2.

If � + � > 100, state which of the following may be true and which must be false

a) � ≤ 10 may be true (A1)….………………………(1)

b) �� ≤ 100 may be true (A1)….………………………(1)

c) � + 2� ≤ 50 must be false (A1)….………………………(1)

(Total 3 marks)


Question 3.

a) Represent the inequalities � < � + 1, � + � ≤ 5 and � > −1 on the grid below by shading the unwanted regions.

(3)

b) Use your graph to find the integer values of � and � that maximise the product of � and �. � = 3,� = 2 (A2)

so that �� = 6

……………………… (2)

(Total 5 marks)

TOTAL /10


Pythagoras 1 Grade 5 Solutions

Objective: Know and use Pythagoras's theorem for right-angled triangles

Question 1

ABC is a right angled triangle. AB = 9 cm, BC = 12 cm Calculate the length of AC. AC2 = 92 + 122 (M2 square, add) = 81 + 144 = 225 AC = 15 (A1)

…………15cm…………. (3) (Total 3 marks) Question 2

ABC is a right angled triangle. AB = 11 cm, AC = 18 cm Calculate the length of BC. Give your answer correct to 1 decimal place. BC2 = 182 - 112 (M2 square, subtract)

= 324 - 121 = 203 BC = 14.2 (A1)

………14.2 cm……. (Total 3 marks) (3)


Question 3

ABCD is a rectangle. AB = 19 m, AD = 13 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. BD2 = 192 + 132 (M2 square, add)

= 361 + 169 = 530 BD = 23.0 (A2 correct, correct to 3sf)

……23.0 m…………. (4) (Total 4 marks) Total /10

Total marks / 10


Powers and Roots 1 Grade 6 Solutions

Objective: Calculate and estimate powers and roots

Question 1

Evaluate

i) 53 125 (A1)

…………………

ii) 104 10000 (A1)

…………………

iii) 18 1 (A1)

…………………

(3)

Question 2

Work out the value of

i) √121 11 (A1)

…………………

ii) 3√64 4 (A1)

…………………

iii) 4√81 3 (A1)

…………………

(3)

Question 3

The square root of 56 lies between which two integer values. Explain your answer.

The root of 56 lies between 7 and 8 (A1)

because 7 x 7 = 49 and 8 x 8 = 64 (C1)

(2)


Question 4

The cube root of 100 lies between which two integer values. Explain your answer.

The cube root of 100 lies between 4 and 5 (A1)

because 4 x 4 x 4 = 64 and 5 x 5 x 5 = 125 (C1)

(2)

Total /10


Cumulative Frequency 1 Grade 6 SOLUTIONS

Objective: Construct and interpret cumulative frequency diagrams (for grouped discrete as well as continuous data) Question 1

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

(a) Complete the cumulative frequency table.

(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 median.

…………360…………B1……… (1)

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

440 – 285 = (155) M1

……………155…………A1………… (2)

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

Read off graph 73 M1

80 – 73 = (7)

……………7…………B1…………… (2)

(Total 8 marks)

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500 600 700 800

CU

MU

LAT

IVE

FR

EQ

UE

NC

Y

WEEKLY WAGE, £


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…………A1…………… (1)

(b) Use the graph to find an estimate for the lower quartile.

………53……………A1…………… (1)

(Total 2 marks)

Total marks /10


Standard trigonometric ratios 1 Grade 7 Solutions

Objective: Know and derive the exact values for Sin and Cos 0, 30, 45, 60 and 90 and Tan 0, 30, 45 and 60 degrees.

Question 1.

A right angled triangle has the dimensions as shown in the diagram.

Using the diagram, or otherwise, state the exact values of:

(a) Sin 60 = ��ℎ�� = √32

(b) Cos 60= ��ℎ�� = 12

(c) Tan 60= �� = √31

(d) Sin 30= ��ℎ�� = 12

(e) Cos 30= ��ℎ�� = √32

(f) Tan 30= �� = 1√3

(Total 6 marks)

Question 2.

Using the triangle shown, or otherwise, find the exact values of:

(a) Sin 45= ��ℎ�� = 1√2

(b) Cos 45= ��ℎ�� = 1√2

(c) Tan 45= �� = 1

(d) Sin 90= ��ℎ�� = 1

(Total 4 marks)

Total /10

1

2 √3

1

1

√2


Histograms with unequal class widths 1 Grade 6 SOLUTIONS

Objective: Construct and interpret a histogram with unequal class widths (for grouped discrete as well as continuous

data)

Question 1

The table shows the weight of 815 parcels handled by a sorting office in one day.

a) Draw a fully labelled histogram to show the weights of the parcels.

(4)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Freq

uen

cy D

ensi

ty

Weight w, grams


b) Estimate the number of parcels that weighed more than 2500 grams

0.12 x 1500 = (180) M1

………180……………A1…………..

(2)

Question 2

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

Time, minutes Frequency Class Width

Freq. Density

0 < x ≤ 10 15 10 1.5

10 < x ≤ 30 36 20 1.8

30 < x ≤ 60 75 30 2.5

60 < x ≤ 80 24 20 1.2

Total 150

a) Use the information to complete the histogram

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90

Freq

uen

cy D

ensi

ty

Time, mins


(2)

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

1.5 x 10 = 15 B1

1.8 x 20 = 36 B1

………15 and 36 …………………………

(2)


Total /10


Geometric Sequences 1 Grade 7 Solutions

Objective: Recognise and use geometric sequences (rn, where n is an integer and r can be a surd)

Question 1.

Find the 5th and 6th terms of the sequences below.

(a) 81, 27, 9, 3, ____, ____, … 1, 13 (A1) (1)

Common ratio is 13 .

3 × 13 = 1 and 1 ×

13 = 13

(b) 4, 0.8, 0.16, 0.032, ____, ____, … 0.0064, 0.00128 (A1) (1)

Common ratio is 0.2. 0.0032 × 0.2 = 0.0064 and 0.0064 × 0.2 = 0.00128

(Total 2 marks)

Question 2.

(a) Write down the first four terms of the geometric sequence with nth term 2n.

21, 22, 23, 24, (M1)

2, 4, 8, 16, (A1)

…………………………………………………………………………………………...

(2)

(b) State the term-to-term rule of the sequence.

The term-to-term rule is multiply the previous term by 2.

…………………………………………………………………………………………...

(1)

(Total 3 marks)


Question 3.

In this geometric sequence, the first term is √3 and the term-to-term rule is multiply by√3.

Continue the sequence for three more terms.

√3 , 3, 3√3 , ___, ___, ___,… 9, 9√3, 27

3√3 × √3 = 3× √3 × √3 = 3 × 3 = 9 (M1)

9 × √3 = 9√3 (A1)

9√3 × √3 = 9× √3 × √3 = 9 × 3 = 27 (A1)

(Total 3 marks)

Question 4.

Work out the missing terms in this geometric sequence.

14 ,12 , 1, 2, 4, 8

Common ratio is 2 . (M1) 14 × 2 = 12 and 4 × 2 = 8 12 and 8 (A1)

(Total 2 marks)

Total /10


Represent quadratic inequalities 1 Grade 7 Solutions Objective: Represent the solution to a quadratic inequality on a number line, using set

notation and on a graph

Question 1.

a) Solve �2 + 13x + 36 ≤ 0

Represent your solution on a number line.

(� + 9)(� + 4) ≤ 0 −9 ≤ � ≤ −4

(A1)

b) Write the integer answers for part a) in set notation.

{ -9, -8, -7, -6, -5, -4 } (A1)


Question 2.

Solve �2 − 3� ≥ 18


�2 − 3� − 18 ≥ 0

(� − 6)(� + 3) ≥ 0 (M1)

(A1)



Question 3.

For which values of x is the expression �2 − 2� − 8 greater than the expression 7 − 6� − 3�2 ? Represent the possible values of � on a number line.

�2 − 2� − 8 > 7 − 6� − 3�2 4�2 + 4� − 15 > 0 (M1)

(2� + 5)(2� − 3) > 0 (M1)

� < − 52 , � > 32

(A1)


Question 4.

Find the set(s) of all values for which 2�−142−�

< x + 5


2� − 14 < (� + 5)(2 − �) 2� − 14 < 10 − �2 − 3� (M1) �2 + 5� − 24 < 0 (� + 8)(� − 3) < 0 (M1)

(A1)


TOTAL /10


Index Laws (Negative and Fractional) 1 Grade 7 Solutions

Objective: Calculate with negative and fractional indices

Question 1

Evaluate

i) 24−

1 (A1) 16

…………………….

(1)

ii) 32−

1 (A1) 8

…………………….

(1)

iii)

2

1

12

−

= 122 (M1) 144 (A1)

…………………….

(2)

iv) 1

38

2 (A1)

…………………….

(1)


v) 3

416

4 16= 2 (M1)

8 (A1)

…………………….

(2)

vi) 2

38000−

3 8000= 20 (M1)

20-2 = 2

1

20 (M1)

1 (A1) 400

…………………….

(3)

Total /10


Approximate solutions to equations using iteration 1 Grade 9 SOLUTIONS Objective: Find approximate solutions to equations using iteration.

Question 1.

Find the first five iterations of each iterative formulae. Start each one with �1 = 4.

a) ��+1 = 3�1 + 7 �2 = 19, �3 = 64, �4 = 199, �5 = 604, �6 = 1819 (A1)

(1)

b) ��+1 = 10 − 2�� 2 = 2, �3 = 6, �4 = −2, �5 = 14, �6 = −18 (A1)

(1)

c) ��+1 =��−22 �2 = 1, �3 = − 12 , �4 = − 54 , �5 = − 138 , �6 = − 2916 (A1)

(1)

(Total 3 marks)

Question 2.

Starting with �1 = 1.6 find a root of the quadratic equation 2�2 + 5� − 13 = 0 to 2 decimal places using the iterative formula

��+1 = �13 − 5��2

�2 = 1.58113 … (M1) �3 = 1.59598 … �4 = 1.58431 … �2 = �4 to 2dp (C1) � = 1.58 (A1)

………………………

(Total 3 marks)


Question 3.

a) Show that �2 − 3� + 1 = 0 can be rearranged into the iterative formula

��+1 = �3�� − 1. �2 = 3� − 1 � = √3� − 1 ��+1 = �3�� − 1 (M1)

(1)

b) Use the iterative formula and a starting value of �1 = 2 to obtain the solution to the equation correct to 2 decimal places. �2 = 2.23606 … (M1) �3 = 2.38918 … �4 = 2.48345 … �5 = 2.53975 … �6 = 2.57279 … �7 = 2.59198 … �8 = 2.60306 … �9 = 2.60944 … �10 = 2.61310 … �9 = �10 to 2dp (C1) � = 2.61 (A1)

………………………

(3)

(Total 4 marks)

TOTAL /10

plc papers - wordpress.com · 2016-10-20 · three parcels weigh 7.2 kg, 15.02 kg and 3.1 kg. find...

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