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PLC Papers

Created For:

PiXL PLC 2017 Certification

Algebraic argument 2 Grade 5

Objective: Argue mathematically that two algebraic expressions are equivalent, and use algebra to support and construct arguments Question 1.

Show that

(3x +1)(x + 5)(2x +3) = 6x3 + 41x2 + 58x + 15

for all values of x.

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

(Total 3 mark)

Question 2. Write 3x2 + 15x + 35 in the form a(x + b)2 + c where a, b, and c are integers.

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

(Total 3 mark)


Question 3 The rectangle and the equilateral triangle have equal perimeters.

Work out an expression, in terms of x, for the length of a side of the triangle.

Give your answer in its simplest form.

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

(Total 4 mark)

Total /10

3(x – 1)

4(3x + 2)

Not drawn accurately


Algebraic terminology 2 Grade 5 Objective: Understand the meaning of the terms expression, equation, formula, inequality, term and factor (also identity). Question 1 a) How many terms does 9x2 – 5x + 6 contain? b) How many values of y will work with 5y + 4 = 24 ? c) How many values of k will work for k ( k – 4) ≡ k2 – 4k ? d) Is 4 (x – 3) equal to 4x – 12 or identical to 4x – 12? You must give a reason for your answer. (4) Question 2 Multiply out (x + 3)(x – 5) and how many terms does the final simplified expression contain? (3)


Question 3

Find the common factors for 3x2y –9xy and 2xy-6y

(3)

Total /10


Changing the subject 2 Grade 4

Objective: Change the subject of a formula Question 1. Make m the subject of

m

mT

31+=

........................................................... (Total 3 mark)

Question 2. Make x the subject of

223 xy −=

........................................................... (Total 2 mark)


Question 3. Make c the subject of

cbca 73+=−

........................................................... (Total 3 mark)

Question 4. Make T the subject of the formula

2

73 += TW

........................................................... (Total 2 mark)

Total /10


Collecting like terms 2 Grade 4

Objective: Simplify algebraic expressions by collecting like terms

Question 1 Simplify 7x + 2y – 3x + 4y

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

(Total 1 mark)

Question 2 Simplify 5f − f + 2f

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

(Total 1 mark)

Question 3 Simplify 8x − 3x + 2x + 10

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

(Total 2 marks)

Question 4 Simplify 5x + 2x + 3y + y

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

(Total 2 marks)


Question 5 Simplify 3p – q + 2 – 5p + 4q – 7

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

(Total 2 marks)

Question 6 Simplify 9 + a – 2b – 5a + 4 – 3b

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

(Total 2 marks)

Total /10


Cubic and Reciprocal Graphs 2 Grade 6

Objective: Recognise, sketch and interpret graphs of simple cubic functions and reciprocal

functions y = �� where x is not 0.

Question 1.

a) Complete the table below for y = .

x 1 2 3 4 5 6

y 6 2 1·2 (2)

b) Draw the graph of y = on the grid below.

(2)

(c) Use your graph to solve the equation = 2·2.

(2) (Total 6 marks)

x

6

x

6

1 2 3 4 5 6

4

5

6

y

x

1

0

2

3

x

6


Question 2.

The table shows some values of x and y for the equation y = (x – 1)3.

x –2 –1 0 1 2 3 4

y –27 –1 0 8

a) Complete the table. (2)

b) Draw the graph of y = (x – 1)3 for values of x from –2 to 4.

(2)

(Total 4 marks)

Total /10

–30

–20

–10

10

20

30

0 1 2 3 4 x

y

–1–2


Deduce quadratic roots algebraically 2 Grade 6

Objective: Deduce roots algebraically.

Question 1.

Solve the equation y2 + 5y = 0

(Total 3 marks) Question 2.

a) Factorise 8x2 + 8x + 2

(2) b) Hence, or otherwise, solve the equation 8x2 + 8x + 2 = 0

(1) (Total 3 marks)


Question 3.

Solve the equation 5x2 – 7x -10 = 0

Give your answers to two decimal places. You must show your working.

(Total 4 marks)

Total /10


Equation of a line 2 Grade 5

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

Question 1.

The line l1 has equation 5y - 15x + 10 = 0 (a) Find the gradient of l1.

(2) The line l2 is perpendicular to l1 and passes through the point (6, 3).

(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 7y = 9x - 3. Line B goes through the points (-5, 3) and (-14, 10). Are lines A and B perpendicular to each other? You must show all your working.

(Total 5 marks)

Total /10


Expanding binomials 2 Grade 5

Objective: Expand the product of two binomials Question 1. (a) Expand and simplify (� + 3)(4 + �)

……………………….

(2)

(b) Expand and simplify (� + 8)(� − 11)

……………………….

(2)

(c) Expand and simplify (3� − 2)(� − 4)

……………………….

(2)


(d) Expand and simplify (6� − 1)(7− 3�)

……………………….

(2)

(e) Expand and simplify (2 + 5�)2

……………………….

(2)

(Total 10 marks)

Total /10


Expressions 2 Grade 4

Objective: Use the correct algebraic notation in expressions and equations, including using brackets

Question 1

£y is shared equally between four people. How much does each person receive?

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

(Total 1 mark)

Question 2

Write an expression for the total cost of seven apples at a pence each and twenty pears at b pence each.

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

(Total 1 mark)

Question 3

Write an expression for the number that is seven times smaller than n

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

(Total 1 mark)

Question 4

Lilly buys x packs of sweets costing 55p per packet.

He pays T pence altogether.

Write a formula for the total cost of sweets

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

(Total 1 mark)


Question 5

Write down an equation for three bananas at a pence each and two grapefruit at b pence each when the total cost is £1.46?

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

(Total 1 mark)

Question 6

Amelie is x years old.

Her sister is four years older. Her brother is three times her age. The sum of their ages is 44 years.

a. Write an expression, in terms of x, for her sister’s age.

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

(Total 1 mark)

b. Form an equation in x to work out Amelie’s age.

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

(Total 2 marks)


Question 7

Two angles have a difference of 40o.

Together they form a straight line.

The smaller angle is yo

a. Write down an expression for the larger angle, in terms of y.

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

(Total 1 mark)

b. Work out the value of y.

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

(Total 1 mark)

Total /10


Factorise single bracket 2 Grade 4

Objective: Take out common factors to factorise Question 1 Factorise fully 5xy+5xt

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

(Total 1 mark)

Question 2 Factorise fully 4pq+2ps+8pt

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

(Total 1 mark)

Question 3 Factorise fully 6f2+2f3

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

(Total 1 mark)

Question 4 Factorise fully 8pqr+10prs

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

(Total 1 mark)

Question 5 Factorise 2y-2

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

(Total 1 mark)


Question 6

Factorise 9a+18b

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

(Total 1 mark)

Question 7

Factorise 6x2+9x+6

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

(Total 1 mark)

Question 8

Factorise 2h-5h2

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

(Total 1 mark)

Question 9

Factorise fully 3ad-6ac

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

(Total 1 mark)

Question 10

Factorise fully mn-kmn

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

(Total 1 mark)

Total /10


Factorising quadratic expressions 2 Grade 5

Objective: Factorise a quadratic expression of the form ax2 + bx + c including the difference of two squares Question 1.

Factorise the expression x2 + 7x + 12

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

(Total 2 mark)

Question 2. Factorise the expression x2 - 3x – 18

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

(Total 2 mark)

Question 3 Factorise the expression x2 - 24x - 25

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

(Total 2 mark)

Question 4 Factorise the expression x2 - 16

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

(Total 2 mark)

Question 5 Factorise the expression 9x2 - 49y2

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

(Total 2 mark)

Total marks / 10


Fibonacci, quadratic and simple geometric sequences 2 Grade 5

Objective: Recognise the Fibonacci sequence, quadratic sequences and simple geometric sequences (rn, where n is an integer and r is a rational number >0)

Question 1.

Here are the first five terms of a quadratic sequence.

8, 11, 16, 23, 32, ….

Write down the next two terms in the sequence.

……………… and ………………

(Total 2 marks)

Question 2.

Which of the sequences below is a geometric sequence?

Circle your answer

2, 3, 4, 5,… 2, 5, 7, 9,…

2, 6, 10, 14, 2, 4, 8, 16,…

(Total 1 mark)

Question 3.

Find the next three terms in this Fibonacci type sequence.

3, 3, 6, 9, 15,…

……………… , ………………, ………………

(Total 2 marks)


Question 4.

Write down the first five terms of the quadratic sequence with nth term 2n2 + 3.

……………………………………………………………………………………..

(Total 2 marks)

Question 5.

Write down the missing terms in this Fibonacci sequence.

1, 1, 2, 3, 5, ___, 13, ___, ….

(Total 1 mark)

Question 6.

Continue this geometric sequence for two more terms.

3, 6, 12, 24, ____, ____, …

(Total 2 marks)

Total /10


Graphical solutions to equations 2 Grade 5

Objective: Find approximate solutions to equations using a graph

Question 1

Using the graph below

(a) Find an approximate solution to the equation 2x - 2 = -3x + 5

y = 2x -2 y = -3x + 5

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

(1)

(b) What is the y coordinate of the point of intersection of the two lines y = 2x - 2 and y = -3x + 5?

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


Question 2

Here is the graph of y = 2x2 + 3x - 1

(a) Use the graph to find solutions to the equation 2x2 + 3x - 1 = 0

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

(2)

(b) Use the graph to find approximate solutions to the the equation 2x2 + 3x - 3 = 0

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

(2)


Question 3

Here is the graph of y = - x2 + 3x + 3

(a) Use the graph to find approximate solutions to the equation -x2 + 3x = -3

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

(2)

(b) Use the graph to find approximate solutions to the equation -x2 + 3x + 4 = 0

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

(2)

Total /10


Graphs of Linear Functions 2 Grade 4

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

Question 1

Match each equation to one of these sketch graphs

A y = 5x + 1 B y = 3x + 3 C y = -x + 4 D y = -3x E y = x + 3

y

3

x

y

4

x

y

3

x

y

1

x

y

x

(5)


Question 2

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

a) y = 3x + 2 b) y = 6 – x

y

a

x

(2)

y

b

x

(2)

Question 3

‘y = mx +c’ is the general form of a linear equation. What does the word linear mean?

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

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

(1)

Total /10


Graphs of quadratic functions 2 Grade 4

Objective: Recognise, sketch and interpret graphs of quadratic functions

Question 1

(a) On the grid, draw the graph for the function � = �2 − 3� − 1 for values of x from -2 to +4

(3)

(Total 3 marks)


Question 2

(a) On the grid, draw the graph for the function � = �(� − 3) for values of x from 0 to 5

(Total 3 marks)


Question 3

On the grid, draw the graph for the function � = �2 − 3 for values of x from -3 to +3 (3)

(b) Use your graph to estimate both solutions to �2 − 3 = 0 correct to 1 decimal place. (1)

……………………………………………………(Total 4 marks)

Total marks / 10


Inequalities on number lines 2 Grade 4

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

Question 1

Draw diagrams to represent these inequalities.

(a) x > -1

(b) x ≤ 4

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

(2)

Question 2

-2 < n ≤ 3

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)

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

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

(4)


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

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

(1)

Total /10


Linear Equations 2 Grade 4

Objective: Solve linear equations with one unknown on both sides and those involving brackets.

Question 1

Solve x + 31 = 5x + 7

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

(2)

Question 2

Solve 3x + 12 = 6x

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

(2)

Question 3

Solve 6(g - 3) = 12

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

(2)


Question 4

Solve 7(b + 2) = 4(b + 5)

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

(3)

(ii) Show how you can check your solution is correct.

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

(1)

Total /10


Linear equations one unknown 2 Grade 3

Objective: Solve linear equations with one unknown on one side

Question 1

Solve 5x = 3

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

(1)

Question 2

Solve y + 4 = 12

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

(1)

Question 3

Solve ℎ3

= 6

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

(1)

Question 4

Solve −5�7

= 10

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

(2)

Question 5

Solve 9d – 8 = 37

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

(2)


Question 6

Solve 6p + 2 = 38

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

(2)

Question 7

Solve 0 = 99k

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

(1)

Total /10


Multiplying single brackets 2 Grade 4

Objective: Multiply a single term over a bracket Question 1 Expand the following g(g-3)

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

(Total 2 marks) Question 2 Expand the following s(3s-4)

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

(Total 2 marks) Question 3 Expand the following -3(2x+2)

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

(Total 2 marks) Question 4 Expand the following -3q(1-q)

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

(Total 2 marks) Question 5 Expand the following 6p+3(p+2)

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

(Total 2 marks)

Total /10


Non standard real life graphs 2 Grade 4

Objective: Plot and interpret non-standard real-life graphs to find approximate solutions to kinematics

problems involving distance, speed and acceleration

Question 1

The conversion graph can be used to change between metres and feet.

(a) Use the conversion graph to change 6 metres to feet.

…………………………….feet (1)

(b) Use the conversion graph to change 8 feet to metres.

…………………………..metres (1)

Robert jumps 4 metres.

James jumps 12 feet.

(c) Who jumps furthest, Robert or James?

Explain how you got your answer.

(2)

(Total 4 marks)


Question 2

The graph represents the first part of Ian’s journey.

(a) What does the shape of the curve from A to B tell you about Ian’s speed? (1)

(b) This graph represents Sarah’s journey home.

Calculate Sarah’s speed.

Give your answer is kilometres per hour. (3)

(Total 4 marks)


Question 3

An activity centre hires out road bikes.

The graph shows the cost, C(£) of hiring a road bike for a number of days, d.

(a) Circle the correct formula connecting the cost, C and the number of days, d for hiring a road bike. (1)

C = 2d + 5 C = 5d + 10 C = 10d + 5

(b) What would it cost for me to hire a road bike for 6 days? Show how you know. (1)

(Total 2 marks)

Total marks / 10


Number Machines 2 Grade 4 Objective: Interpret simple expressions as functions with inputs and outputs Question 1 Input 2× 5+ Output

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

(Total 1 mark)

Question 2 Input 2x 3+ Output

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

.

(Total 2 marks)

Question 3 Input 2x 2× 4− Output

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

(Total 3 marks)


Question 4 Here is a number machine.

Input X3 - 4 Output

(a) Work out the output when the input is 3

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

(Total 2 marks)

(b) Work out the input when the output is 14

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

(Total 2 marks)

Total /10


Quadratic Graphs 2 Grade 6

Objective: Identify roots, intercepts and turning points of quadratic functions graphically.

Question 1.

a) Complete the table of values for y = x2 + 5

x –2 –1 0 1 2

y 9 6

(2)

b) On the grid below, draw the graph of y = 2x2 – 1 for values of x from x = –2 to x = 2

(2)



a) Complete the table of values for y = x2 – 2x – 1.

x –2 –1 0 1 2 3 4

y 7 –2 –1

(2)

b) On the grid, draw the graph of y = x2 – 2x – 1 for values of x from –2 to 4.

(2)

c) Solve x2 – 2x – 1 = x + 3

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

(Total 6 marks)


Total /10


V V

D

V V

D

D D

Reciprocal Real Life Graphs 2 Grade 5

Objective: Plot and interpret reciprocal real-life graphs

Question 1. The relationship between the volume (V) of an ambulance siren and a person’s distance (D) from the

ambulance can be illustrated by which one of the graphs below?

Graph 1 Graph 2

Graph 3 Graph 4

Graph …………

(Total 1 mark)

Question 2. A rectangle has a width of w cm, and a length l cm. The area of the rectangle is 36 cm2. The length L of the rectangle is inversely proportional to its width. L is given by � =

36�

(a) Complete the table of values below to show how the length � varies depending on the width.

� width 1 2 3 4 6 12 36 � length

(2)


�

�

(b) On the grid, draw the graph of � = 36� on the axes provided.

(3)

(c) Use your graph to estimate the value of w when L = 8

……………

(1)

(Total 6 marks)


Question 3.

(a) Some bricklayers build a wall.

It takes two people 8 hours to build a wall.

Write down an equation to show how the time T (hours) taken to build the wall, varies as the number of

people P, building the wall varies.

………………..

(2)

(b) On the axes below, sketch the graph showing the relationship between the number of bricklayers (P)

against the time (T) taken to build the wall.

(1)

(Total 3 marks)

Total /10


Simplify indices 2 Grade 5

Objective: Simplify expressions involving sums, products and powers, including using index laws 2

Question 1.

Simplify (m3)2 .........................................................

(Total 2 marks)

Question 2.

Simplify d2 × d3

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

(Total 2 marks)

Question 3.

Simplify 4y × 2y

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

(Total 2 marks)

Question 4.

Simplify 3y – 2 + 5y + 1

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

(Total 2 marks)

Question 5.

Simplify 5u2w4 × 7uw3 ............................ ..............................

(Total 2 mark)

Total /10


Simplify surds 2 Grade 5 Objective: Simplify algebraic expressions involving surds Question 1 Calculate the value of n

32 = 2n

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

(Total 1 mark)

Question 2 Calculate the value of n

n7)7( 3 =

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

(Total 1 mark)


n38132 =×

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

(Total 1 mark)


n216

1 =

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

(Total 1 mark)


Question 5 Calculate the value of k

10160 k=

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

(Total 1 mark)

Question 6 Work out the value of ( )( )

27

33133 +−

Give you answer in its simplest form

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

(Total 3 mark)

Question 7

Simplify ( )( )3839 −+ Give your answer in the form 7ba +

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

(Total 2 mark)

Total /10


Substitution 2 Grade 4 Objective: Substitute numerical values into formulae and expressions, including scientific formulae Question 1. Complete this table of values.

n 4n-1

5

39

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

(Total 2 mark)

Question 2

Calculate the value of y when x = 2 are

� = −3

5√50�2

You must show your working.

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

(Total 2 marks)


Question 3

a. Work out the value of v when u = 70, a = 5 and t = 2 b. Work out the value of v when u = 25, a = -2 and t = -5

� = � + �� a.

b.

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

(Total 2 mark)

Question 4

Work out the value of T when (a) p = 7 and (b) p = -2

� = 3�2 − 2�

a.

b.

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

(Total 2 marks)


Question 5

Work out the value of V when (a) π = 3.14, r = 5, and h = 12 (b) π = 3.14, r = 1.4, and h = 30

Give your answers to 3 significant figures

� = 1

3��2ℎ

a.

b.

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

(Total 2 marks)

Total /10


Using the equation of a straight line 2 Grade 4

Objective: Identify and interpret gradients and intercepts of linear functions, both algebraically and graphically

Question 1

(a) Write down the coordinates of the point S.

(1 mark)

The coordinates of the point T are (−3, 2).

(b) On the grid, mark this point with a cross (×). Label the point T.

(1 mark)

(c) Write down an equation of the line L.

( 2 marks)


Question 2

The straight line P has been drawn on a grid.

Find the gradient of the line P.

(2 marks)

Hence, or otherwise, find the equation of the line

(1 mark)

Question 3

A straight line, L, is parallel to the line with equation y = 1 – 3x.

The point with coordinates (6, 3) is on the line L.

Find an equation of the line L.

(3 marks)

Total marks / 10


Writing formulae and expressions 2 Grade 4

Objective: Write simple formulae or expressions from a problem

Question 1

Ben buys p packets of plain biscuits (x) and c packets of chocolate biscuits (y).

Write down an expression for the total number of packets of biscuits Ben buys.

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

(1)

Question 2

A boy is y years old.

Write expressions to represent the following statements:-

(i) How old will he be seven years from now

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

(ii) How old was he ten years ago?

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

(iii) His father is three times his age. How old is his father?

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

(1)


Question 3

Write an expression for the area of a square if the length of one side is x

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

(1)

Question 4

I think of a number, double it, add fourteen and then divide it by 4. If my number is n , write an expression to show this.

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

(2)

Question 5

A cab company charges a basic rate of £ x plus £1.50 for every kilometre travelled. Write a formulae to represent the cost of a journey of y kilometres in terms of x and y.

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

(2)

Question 6

A regular hexagon has sides of length 5h. Write an expression for the perimeter of the hexagon.

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

(1)

Total /10


Derive an equation 2 Grade 5

Objective: Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution in context.

Question 1

The diagram shows a right-angled triangle.

(5x - 70)°

3x°

Work out the size of the smallest angle of the triangle.

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

(4)


Question 2

Two bunches of tulips and five bunches of daffodils costs £36. Seven bunches of tulips and four bunches of daffodils costs £72. Find the cost of a bunch of tulips.

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

(4)

Question 3

A garden in the shape of an isosceles triangle has two equal sides 8m longer than the other and the perimeter is 40m.

Derive an equation you could use to find the length of the shorter side .

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

(2)

Total /10


nth term of a linear sequence 2 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) 2n + 3

……………………………………………………………………………………… (1)

(b) 4n – 3

……………………………………………………………………………………… (1)

(Total 2 marks)

Question 2.

Here are the first five terms of a linear sequence.

5, 8, 11, 14, 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:

-5, -2, 1, 4, 7, …

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 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


Solve linear inequalities one variable 3 Grade 5

Objective: Solve linear inequalities in one variable

Question 1

Solve.

(a) 4x – 5 < 7

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

(b) Write down the largest integer that satisfies 4x – 5 < 7

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

(1)

Question 2

Solve.

(a) 4h – 5 > 2h + 2

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

(b) Write down the smallest integer that satisfies 4h – 5 > 2h + 2

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

(1)


Question 3

Solve.

2(5r – 1) < 2r + 3

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

Total /10

PLC Papers

Created For:


Algebraic Argument 2 Grade 5 Solutions

Objective: Argue mathematically that two algebraic expressions are equivalent, and use algebra to support and construct arguments Question 1.

Show that

(3x +1)(x + 5)(2x +3) = 6x3 + 41x2 + 58x + 15

for all values of x.

(3x + 1)(x + 5)(2x + 3)

(3x2+x+15x+5)(2x + 3) 6x3+9x2+2x2+3x+30 x2+45x+10x+15

6x3 + 41x2 + 58x + 15

........................ 6x3 + 41x2 + 58x + 15...................................

(Total 3 mark)

Question 2. Write 3x2 + 18x + 35 in the form a(x + b)2 + c where a, b, and c are integers.

3x2 + 18x + 35

3(x2 + 6x) + 35

3(x + 3)2 – 27 + 35

3(x + 3)2 + 8

......................... 3(x + 3)2 + 8..................................

(Total 3 mark)


Question 3 The rectangle and the equilateral triangle have equal perimeters.

Work out an expression, in terms of x, for the length of a side of the triangle.

Give your answer in its simplest form. P = 3(x-1) + 3(x-1) + 4(3x+3) + 4(3x+3) P = 6(x-1) + 8(3x+3) P = 6x-6+24x+24 P = 30x +18 Triangle side = P/3 = 10x + 6

........................................................... (Total 4 mark)

Total /10

3(x – 1)

4(3x + 3)

Not drawn accurately


Algebraic terminology 2 Grade 5 SOLUTIONS Objective: Understand the meaning of the terms expression, equation, formula, inequality, term and factor (also identity). Question 1 a) How many terms does 9x2 – 5x + 6 contain? b) How many values of y will work with 5y + 4 = 24 ? c) How many values of k will work for k ( k – 4) ≡ k2 – 4k ? d) Is 4 (x – 3) equal to 4x – 12 or identical to 4x – 12? You must give a reason for your answer. a) The expression contains 3 terms. (A1) b) 1 value of y will work. This value is when y = 4. (A1- the value of y =4 must be worked out to award mark) c) All real numbers will work for this identity. (A1) d) 4 (x – 3) is identical to 4x – 12 as when it is expanded, the answer is 4x – 12, and any value that is substituted into 4 (x – 3) and 4x – 12 will always be the same. True for all values of x. (A1) (4)


Question 2 Multiply out (x + 3)(x – 5) and how many terms does the final simplified expression contain? = x2 + 3x – 5x – 15 (M1) = x2 – 2x – 15 (M1) ∴There are 3 terms for the expansion/ simplified expression. (A1) (3)


Question 3

Find the common factors for 3x2y –9xy and 2xy-6y

3x2y –9xy = 3xy ( x – 3) (M1)

2xy-6y = 2y ( x – 3) (M1)

Common factor: y(x – 3) (A1)

(3)

Total /10


Changing the subject 2 Grade 4 Solutions

Objective: Change the subject of a formula Question 1. Make m the subject of

m

mT

31+=

3

1

1)3(

13

31

31

−==−=−

+=+=

Tm

Tm

mmT

mmTm

mT

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

(Total 3 mark)

Question 2. Make x the subject of

223 xy −=

2

3

2

3

23

23

23

23

2

22

22

22

22

2

yx

xy

xy

xy

xy

xy

−==−=−−=−

−=−=

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

(Total 2 mark)


Question 3. Make c the subject of

cbca 73+=−

b

ac

cb

a

bca

bcca

bcca

cbca

+−==+

−+=−+=−++=+=−

7

37

3

)7(3

73

73

73

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

(Total 3 mark)

Question 4. Make T the subject of the formula

2

73 += TW

3

72

3

72

372

7322

732

73

2

2

2

2

2

−==−=−+=+=

+=

WT

TW

TW

TW

TW

TW

........................................................... (Total 2 mark)

Total /10


Collecting like terms 2 Grade 4 Solutions

Objective: Simplify algebraic expressions by collecting like terms Question 1 Simplify 7x + 2y – 3x + 4y

yx 64 + ...........................................................

(Total 1 mark)

Question 2 Simplify 5f − f + 2f

f6

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

(Total 1 mark)

Question 3 Simplify 8x − 3x + 2x + 10

x7 + 10

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

(Total 2 marks)

Question 4 Simplify 5x + 2x + 3y + y

yy 47 +

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

(Total 2 marks)

Question 5 Simplify 3p – q + 2 – 5p + 4q - 7

532 −+− qp ...........................................................

(Total 2 marks)

Question 6 Simplify 9 + a – 2b – 5a + 4 – 3b

ba 5413 −− ...........................................................

(Total 2 marks)

Total /10


Cubic and Reciprocal Graphs 2 Grade 6 Solutions

Objective: Recognise, sketch and interpret graphs of simple cubic functions and reciprocal

functions y = �� where x is not 0.

Question 1.

a) Complete the table below for y = .

x 1 2 3 4 5 6

y 6 3 2 1.5 1·2 1

(M2)

b) Draw the graph of y = on the grid below.

(M2)

(c) Use your graph to solve the equation = 2·2.

Evidence of line drawn on graph (M1); x =2.7 (A1) (2)

(Total 6 marks)

x

6

x

6

1 2 3 4 5 6

4

5

6

y

x

1

0

2

3

x

6

*

*

*

*

*

*


Question 2.

The table shows some values of x and y for the equation y = (x – 1)3.

x –2 –1 0 1 2 3 4

y –27 -8 –1 0 1 8 27

a) Complete the table.

(M2)

b) Draw the graph of y = (x – 1)3 for values of x from –2 to 4.

(M2)

(Total 4 marks)

Total /10

–30

–20

–10

10

20

30

0 1 2 3 4 x

y

–1–2

*

*

* * *

*

*


Deduce quadratic roots algebraically 2 Grade 6 Solutions

Objective: Deduce roots algebraically.

Question 1.

Solve the equation y2 + 5y = 0

y (y + 5) = 0 (M1)

y = 0 or y = -5 (A2)


a) Factorise 8x2 + 8x + 2 (4x + 2) (2x + 1) = 2(2x + 1)2

(2) b) Hence, or otherwise, solve the equation 8x2 + 8x + 2 = 0

2 (2x + 1)2 = 0

x = -0.5

(1) (Total 3 marks)


Question 3.

Solve the equation 5x2 – 7x -10 = 0

Give your answers to two decimal places. You must show your working.

x = −−7 ±�(−7)2−(4×5×−10)2×5 (M1)

x = 7 ±√49+20010

x = 7 ±√24910 (M1)

x = 2.28 (A1) x = -0.88 (A1)

(Total 4 marks)

Total /10


Equation of a line 2 Grade 5 Solutions

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

Question 1.

The line l1 has equation 5y - 15x + 10 = 0 (a) Find the gradient of l1.

5y = 15x - 10 y = 3x - 2 (M1)

gradient = 3 (A1)

(2) The line l2 is perpendicular to l1 and passes through the point (6, 3).

(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 3 (M1)

y = −�3 + c

sub (6,3) 3 = -2 + c 5 = c (M1)

y = −�3 + 5 (A1)

(3)

(Total 5 marks)

Question 2.

A and B are straight lines. Line A has equation 7y = 9x - 3. Line B goes through the points (-5, 3) and (-14, 10). Are lines A and B perpendicular to each other? You must show all your working.

y = 9�7 +

37 (M1)

Product of two gradients is -1 (M1)

Gradient of B must be −7 9 (M1)

10 − 3−14− −5 =

7−9 (A1)

7−9 x

97 = -1 therefore Lines A and B are perpendicular to each other (C1)

(Total 5 marks)

Total /10


Expanding binomials 2 Grade 5 Solutions

Objective: Expand the product of two binomials Question 1. (a) Expand and simplify (� + 3)(4 + �)

(� + �)(�+ �) = ��+ �� + �� + ��= �� + �� + ��

……………………….

(2)

(b) Expand and simplify (� + 8)(� − 11)

(� + �)(� − ��) = �� − ��+ �� − ��= �� − �� − ��

……………………….

(2)

(c) Expand and simplify (3� − 2)(� − 4)

(�� − �)(� − �) = �� − �� − �� + � = �� − ��+ �

……………………….

(2)


(d) Expand and simplify (6� − 1)(7 − 3�)

(6� − 1)(7 − 3�) = �� − �� − � + �� = −�� + �� − �

……………………….

(2)

(e) Expand and simplify (2 + 5�)2

(� + ��)(� + ��) = � + ��+ ��+ �� = �� + ��+ �

……………………….

(2)

(Total 10 marks)

Total /10


Expressions 2 Grade 4 Solutions

Objective: Use the correct algebraic notation in expressions and equations, including using brackets

Question 1

£y is shared equally between four people. How much does each person receive?

4

y

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

(Total 1 mark)

Question 2

Write an expression for the total cost of seven apples at a pence each and twenty pears at b pence each.

ba 207 +

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

(Total 1 mark)

Question 3

Write an expression for the number that is seven times smaller than n

7

n

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

(Total 1 mark)

Question 4

Lilly buys x packs of sweets costing 55p per packet.

She pays T pence altogether.

Write a formula for the total cost of sweets

xT 55=

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

(Total 1 mark)


Question 5

Write down an equation for three bananas at a pence each and two grapefruit at b pence each when the total cost is £1.46?

14623 =+ ba ...........................................................

(Total 1 mark) Question 6

Amelie is x years old.

Her sister is four years older. Her brother is three times her age. The sum of their ages is 44 years.

a. Write an expression, in terms of x, for her sister’s age. 4+x

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

b. Form an equation in x to work out Amelie’s age.

4445

4434

=+=+++

x

xxx

..................................Amelie is 8 years old.........................

(Total 2 marks)

Question 7

Two angles have a difference of 40o. Together they form a straight line. The smaller angle is yo

a. Write down an expression for the larger angle, in terms of y. 40+y

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

(Total 1 mark)

b. Work out the value of y.

70

1402

401802

180402

18040

==

−==+=++

y

y

y

y

yy

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

(Total 1 mark)

Total /10


Factorise single bracket 2 Grade 4 Solutions

Objective: Take out common factors to factorise Question 1 Factorise fully 5xy+5xt = )(5 tyx +

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

(Total 1 mark)

Question 2 Factorise fully 4pq+2ps+8pt = )42(2 tsqp ++

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

(Total 1 mark)

Question 3

Factorise fully 6f2+2f3 = )3(2 2 ff +

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

(Total 1 mark)

Question 4 Factorise fully 8pqr+10prs = )54(2 sqpr +

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

(Total 1 mark)

Question 5 Factorise 2y-2 = )1(2 −y

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

(Total 1 mark)


Question 6

Factorise 9a+18b = )2(9 ba +

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

(Total 1 mark)

Question 7

Factorise 6x2+9x+6 = )232(3 2 ++ xx

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

(Total 1 mark)

Question 8

Factorise 2h-5h2 = )52( hh −

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

(Total 1 mark)

Question 9

Factorise fully 3ad-6ac = )2(3 cda −

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

(Total 1 mark)

Question 10

Factorise fully mn-kmn = )1( kmn −

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

(Total 1 mark)

Total /10


Factorising quadratic expressions 2 Grade 5 Solutions

Objective: Factorise a quadratic expression of the form ax2 + bx + c including the difference of two squares Question 1.

Factorise the expression x2 + 7x + 12

(x + 3)(x + 4)

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

(Total 2 mark)

Question 2. Factorise the expression x2 - 3x - 18

(x - 6)(x + 3)

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

(Total 2 mark)

Question 3 Factorise the expression x2 - 24x - 25

(x - 25)(x + 1)

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

(Total 2 mark)

Question 4 Factorise the expression x2 - 16

(x + 4)(x - 4)

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

(Total 2 mark)

Question 5 Factorise the expression 9x2 - 49y2

(3x + 7y)(3x - 7y)

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

(Total 2 mark)

Total marks / 10


Fibonacci, quadratic and simple geometric sequences 2 Grade 5 Solutions

Objective: Recognise the Fibonacci sequence, quadratic sequences and simple geometric sequences (rn, where n is an integer and r is a rational number >0)

Question 1.

Here are the first five terms of a quadratic sequence.

8, 11, 16, 23, 32, ….

Write down the next two terms in the sequence.

The sequence is adding 3, 5, 7, 9, so next two term will be add 11 and 13

43 and 56 (A1, A1)

……………… and ………………

(Total 2 marks)

Question 2.

Which of the sequences below is a geometric sequence?

Circle your answer

2, 3, 4, 5,… 2, 5, 7, 9,…

2, 6, 10, 14, 2, 4, 8, 16,… (A1)

The sequence that is circled doubles

(Total 1 mark)

Question 3.

Find the next three terms in this Fibonacci type sequence.

3, 3, 6, 9, 15,… Add the two pervious terms together 9+ 15 = 24, 15 + 24 = 39, 24 + 39 = 63 24, 39, 63 (A1 for 2 correct, A2 for all 3 correct)

……………… , ………………, ………………

(Total 2 marks)


Question 4.

Write down the first five terms of the quadratic sequence with nth term 2n2 + 3.

2(12) + 3, 2(22) + 3, 2(32) + 3 etc

5, 11, 21, 35, 53, … (A1 for 3 or 4 correct, A2 for all 5 correct)

……………………………………………………………………………………..

(Total 2 marks)

Question 5.

Write down the missing terms in this Fibonacci sequence.

1, 1, 2, 3, 5, 8, 13, 21, …. (A1)

Add the two pervious terms together 3 + 5 = 8, 8 + 13 = 21

(Total 1 mark)

Question 6.

Continue this geometric sequence for two more terms.

3, 6, 12, 24, 48, 96, …

The sequence is multiplying by 2 each time (A1 for each correct one)

(Total 2 marks)

Total /10


Graphical solutions to equations 2 Grade 5 Solutions

Objective: Find approximate solutions to equations using a graph

Question 1

Using the graph below

(a) Find an approximate solution to the equation 2x - 2 = -3x + 5

y = 2x -2 y = -3x + 5

x = 1.4 (B1) (allow answers between 1.25 and 1.55)

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

(1)

(b) What is the y coordinate of the point of intersection of the two lines y = 2x - 2 and y = -3x + 5?

y = 0.8 (B1) (allow answers between 0.65 and 0.95)

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

(1)


Question 2

Here is the graph of y = 2x2 + 3x - 1

(a) Use the graph to find solutions to the equation 2x2 + 3x - 1 = 0

x = 0.3 and -1.8 (B2) (allow +/- 0.15)

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

(2)

(b) Use the graph to find approximate solutions to the the equation 2x2 + 3x - 3 = 0

x = 0.7 and -2.2 (B2) (allow +/- 0.15)

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

(2)


Question 3

Here is the graph of y = - x2 + 3x + 3

(a) Use the graph to find approximate solutions to the equation -x2 + 3x = -3

x = 3.8 and -0.8 (B2) (allow +/- 0.15)

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

(2)

(b) Use the graph to find approximate solutions to the equation -x2 + 3x + 4 = 0

x = -1 and 4 (B2) (allow +/- 0.1)

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

(2)

Total /10


Graphs of Linear Functions 2 Grade 4 Solutions

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

Question 1

Match each equation to one of these sketch graphs

A y = 5x + 1 B y = 3x + 3 C y = -x + 4 D y = -3x E y = x + 3

y B1 for each correct answer

3 E

x

y

4 C

x

y

3 B

x

y

1 A

x

y

x D

(5)


Question 2

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

a) y = 3x + 2 b) y = 6 – x

y

a 2 B1 line with negative gradient B1 intercept indicated

x

(2)

y

b 6 B1 line with negative gradient B1 intercept indicated

x

(2)

Question 3

‘y = mx +c’ is the general form of a linear equation. What does the word linear mean?

C1 linear means straight line, a linear equation forms a straight line graph.

(1)

Total /10


Graphs of quadratic functions 2 Grade 4 SOLUTIONS

Objective: Recognise, sketch and interpret graphs of quadratic functions

Question 1

(a) On the grid, draw the graph for the function � = �2 − 3� − 1 for values of x from -2 to +4 (3)

Correct coordinates (M1)

Plot points accurately (M1)

Smooth curve drawn (A1)

(Total 3 marks)


Question 2

(a) On the grid, draw the graph for the function � = �(� − 3) for values of x from 0 to 5

(Total 3 marks)





Question 3

On the grid, draw the graph for the function � = �2 − 3 for values of x from -3 to +3

x -3 -2 -1 0 1 2 3

Y 6 1 -2 -3 -2 1 6




(b) Use your graph to estimate both solutions to �2 − 3 = 0 correct to 1 decimal place.

x = ±1.7 (A1)

(Total 4 marks)

Total marks / 10


Inequalities on number lines 2 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 > -1 (A1)

(b) x ≤ 4 -2 -1 0 1 2

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

(2)

Question 2

-2 < n ≤ 3 (A1)

n is an integer

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

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

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

(3)

Question 3

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

(a) (b)

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

1 ≤ x < 4 (M1 A1) -2 < x < 1 (M1 A1)

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

(4)

0 1 2 3 4

(A1)

-2 -1 0 1 2 3


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

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

(1)

Total /10


Linear Equations 2 Grade 4 Solutions

Objective: Solve linear equations with one unknown on both sides and those involving brackets.

Question 1

Solve x + 31 = 5x + 7

24 = 4x (M1)

6 = x (A1)

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

(2)

Question 2

Solve 3x + 12 = 6x

12 = 3x (M1)

4 = x (A1)

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

(2)

Question 3

Solve 6(g - 3) = 12

6g - 18 = 12 (M1)

6g = 30

g = 5(A1)

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

(2)


Question 4

Solve 7(b + 2) = 4(b + 5)

7b + 14 = 4b + 20 (M1)

3b = 6 (M1)

b = 2 (A1)

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

(3)

(ii) Show how you can check your solution is correct.

Substitute b = 2 to get LHS = RHS 28 = 28 (M1)

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

(1)

Total /10


Linear equations one unknown 2 Grade 3 Solutions

Objective: Solve linear equations with one unknown on one side

Question 1

Solve 5x = 3

(A1) x = 0.6 or x = 35

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

(1)

Question 2

Solve y + 4 = 12

(A1) y = 8

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

(1)

Question 3

Solve ℎ3

= 6

(A1) h = 18

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

(1)

Question 4

Solve −5�7

= 10

(M1) -5c = 70 (A1) c = -14

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

(2)


Question 5

Solve 9d – 8 = 37

(M1) 9d = 45 (A1) d = 5

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

(2)

Question 6

Solve 6p + 2 = 38

(M1) 6p= 36 (A1) p = 6

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

(2)

Question 7

Solve 0 = 99k

(A1) k = 0

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

(1)

Total /10


Multiplying single brackets 2 Grade 4 Solutions

Objective: Multiply a single term over a bracket Question 1 Expand the following g(g-3) = gg 32 −

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

(Total 2 marks) Question 2 Expand the following s(3s-4) = ss 43 2 −

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

(Total 2 marks) Question 3 Expand the following -3(2x+2) = 66 −− x

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

(Total 2 marks) Question 4 Expand the following -3q(1-q) = 233 qq +−

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

(Total 2 marks) Question 5 Expand the following 6p+3(p+2) = 69636 +=++ ppp

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

(Total 2 marks)

Total /10


Non standard real life graphs 2 Grade 4 SOLUTIONS

Objective: Plot and interpret non-standard real-life graphs to find approximate solutions to kinematics

problems involving distance, speed and acceleration

Question 1

The conversion graph can be used to change between metres and feet.

(a) Use the conversion graph to change 6 metres to feet.

20 feet (A1) (1)

(b) Use the conversion graph to change 8 feet to metres.

2.4 metres (A1) (1)

Robert jumps 4 metres.

James jumps 12 feet.

(c) Who jumps furthest, Robert or James? Robert (A1)

Explain how you got your answer. (2)

4 metres = 13 feet

13 is bigger than 12 (M1)

(Total 4 marks)


Question 2

The graph represents the first part of Ian’s journey.

(a) What does the shape of the curve from A to B tell you about Ian’s speed? (1)

The speed is increasing oe (A1)

(b) This graph represents Sarah’s journey home.

Calculate Sarah’s speed.

Give your answer is kilometres per hour. (3)

12 minutes = 0.2 hours (M1)

Speed = distance = 18 km = 90 km/h (M1)(A1)

Time 0.2 h

(Total 4 marks)


Question 3

An activity centre hires out road bikes.

The graph shows the cost, C(£) of hiring a road bike for a number of days, d.

(a) Circle the correct formula connecting the cost, C and the number of days, d for hiring a road bike. (1)

C = 2d + 5 C = 5d + 10 C = 10d + 5 (A1)

(b) What would it cost for me to hire a road bike for 6 days? Show how you know. (1)

C = 10d + 5

C = 10(6) + 5

C = 60 + 5

Cost is £65 (A1)

(Total 2 marks)

Total marks / 10


Number Machines 2 Grade 4 Solutions Objective: Interpret simple expressions as functions with inputs and outputs Question 1 Input 2× 5+ Output

................................ 2x+5

(Total 1 mark)

Question 2 Input 2x 3+ Output

.................... x2+3

.

(Total 2 marks)

Question 3 Input 2x 2× 4− Output

............................... 2x2-4

(Total 3 marks)


Question 4 Here is a number machine.

Input X3 - 4 Output

(a) Work out the output when the input is 3

5 ...........................................................

(Total 2 marks)

(b) Work out the input when the output is 14

6

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

(Total 2 marks)

Total /10


Quadratic Graphs 2 Grade 6 Solutions

Objective: Identify roots, intercepts and turning points of quadratic functions graphically.

Question 1.

a) Complete the table of values for y = x2 + 5

x –2 –1 0 1 2

y 9 6 5 6 9

M2

b) On the grid below, draw the graph of y = x2 + 5 for values of x from x = –2 to x = 2

M2

(Total 4 marks)

*

*

*

*

*


Question 2.

a) Complete the table of values for y = x2 – 2x – 1.

x –2 –1 0 1 2 3 4

y 7 2 -1 –2 –1 2 7

M2

b) On the grid, draw the graph of y = x2 – 2x – 1 for values of x from –2 to 4.

M2

c) Solve x2 – 2x – 1 = x + 3 x2 – 3x – 4 = 0 (x – 4)( x + 1) = 0 (M1) x = 4 & x = -1 (A1)

(2)

(Total 6 marks)

Total /10

*

*

*

*

*

*

*

*

*

*

*


V V

D

V V

D

D D

Reciprocal Real Life Graphs 2 Grade 5 Solutions

Objective: Plot and interpret reciprocal real-life graphs

Question 1. The relationship between the volume (V) of an ambulance siren and a person’s distance (D) from the

ambulance can be illustrated by which one of the graphs below?

Graph 1 Graph 2

Graph 3 Graph 4 B1

Graph 2………

(Total 1 mark)

Question 2. A rectangle has a width of w cm, and a length l cm. The area of the rectangle is 36 cm2. The length L of the rectangle is inversely proportional to its width. L is given by � =

36�

(a) Complete the table of values below to show how the length � varies depending on the width.

� width 1 2 3 4 6 12 36 � length 36 18 12 9 6 3 1

B2- all values correct, B1- 6 values correct

(2)


�

(b) On the grid, draw the graph of � = 36� on the axes provided.

M2 for all points plotted correctly A1 for smooth curve (3)

(c) Use your graph to estimate the value of w when L = 8

4.5-4.7 B1

(1)

(Total 6 marks)

�


Question 3.

(a) Some bricklayers build a wall.

It takes two people 8 hours to build a wall.

Write down an equation to show how the time T (hours) taken to build the wall, varies as the number of

people P, building the wall varies.

� =��

� =��

B1 16 seen, A1 correct equation (2)

(b) On the axes below, sketch the graph showing the relationship between the number of bricklayers (P)

against the time (T) taken to build the wall.

B1

(1)

(Total 3 marks)

Total /10


Simplify indices 2 Grade 5 Solutions

Objective: Simplify expressions involving sums, products and powers, including using index laws 2

Question 1.

Simplify (m3)2 .............................m6..............................

(Total 2 marks)

Question 2.

Simplify d2 × d3

............................. d5..............................

(Total 2 marks)

Question 3.

Simplify 4y × 2y

.........................8y2..................................

(Total 2 marks)

Question 4.

Simplify 3y – 2 + 5y + 1

....................... 8y – 1...................................

(Total 2 marks)

Question 5.

Simplify 5u2w4 × 7uw3 ............................ 35u3w7...............................

(Total 2 mark)

Total /10


Simplify surds 2 Grade 5 Solutions Objective: Simplify algebraic expressions involving surds Question 1 Calculate the value of n

32 = 2n

.........................n = 5..................................

(Total 1 mark)


n7)7( 3 =

..................n = 1.5.........................................

(Total 1 mark)


n38132 =×

...................n = 4........................................

(Total 1 mark)


n216

1 =

.......................n = -4....................................

(Total 1 mark)

Question 5 Calculate the value of k

10160 k=

........................k = 4...................................

(Total 1 mark)


Question 6 Work out the value of ( )( )

27

33133 +−

Give you answer in its simplest form

27

93393 −−+

27

386+−

39

386

×+−

33

386+−

33

386+−

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

(Total 3 mark)

Question 7

Simplify ( )( )3839 −+ Give your answer in the form 7ba +

3393872 −−+

369−

369− ...........................................................

(Total 2 mark)

Total /10


Substitution 2 Grade 4 Solutions Objective: Substitute numerical values into formulae and expressions, including scientific formulae Question 1. Complete this table of values.

n 4n-1

5 19

10 39

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

(Total 2 mark)

Question 2

Calculate the value of y when x = 2 are

� = −3

5√50�2

You must show your working.

65

30

105

3

1005

3

2505

3

−=−=

×−=

−=×−=

y

y

y

y

y

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

(Total 2 marks)


Question 3

a. Work out the value of v when u = 70, a = 5 and t = 2 b. Work out the value of v when u = 25, a = -2 and t = -5

� = � + �� a.

80

2570

=×+=

+=v

v

atuv

b.

35

)5()2(25

=−×−+=

+=v

v

atuv

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

(Total 2 mark)

Question 4

Work out the value of T when (a) p = 7 and (b) p = -2

� = 3�2 − 2�

a.

130

14147

)7(2)7(3

232

2

=−=−=−=

T

T

T

ppT

b.

16

412

)2(2)2(3

232

2

=+=

−−−=−=

T

T

T

ppT

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

(Total 2 marks)


Question 5

Work out the value of V when (a) π = 3.14, r = 5, and h = 12 (b) π = 3.14, r = 1.4, and h = 30

Give your answers to 3 significant figures

� = 1

3��2ℎ

a.

314

12514.33

1 2

=×××=

V

V

b.

5.61

544.61

304.114.33

1 2

==

×××=

V

V

V

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

(Total 2 marks)

Total /10


Using the equation of a straight line 2 Grade 4 Solutions

Objective: Identify and interpret gradients and intercepts of linear functions, both algebraically and graphically

Question 1

(a) Write down the coordinates of the point S.

(1,4) (1 mark)

The coordinates of the point T are (−3, 2).

(b) On the grid, mark this point with a cross (×).

Label the point T.

See grid (1 mark)

(c) Write down an equation of the line L.

X=3

( 2 marks)

T


Question 2

The straight line P has been drawn on a grid.

Find the gradient of the line P.

1 up / 2 across = 0.5 (positive) M1 A1

(2 marks)

Hence, or otherwise, find the equation of the line

Y=0.5x+1

(1 mark)

Question 3

A straight line, L, is parallel to the line with equation y = 1 – 3x.

The point with coordinates (6, 3) is on the line L.

Find an equation of the line L.

Substitute (6,3) into y=a-3x to find intersect

3=a-(3x6)

3=a-18 M1

21=a M1

So y=-3x+21 or y=21-3x A1 (3 marks)

Total marks / 10


Writing formulae and expressions 2 Grade 4 Solutions

Objective: Write simple formulae or expressions from a problem

Question 1

Ben buys p packets of plain biscuits (x) and c packets of chocolate biscuits (y).

Write down an expression for the total number of packets of biscuits Ben buys.

px + cy (B1)

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

(1)

Question 2

A boy is y years old.

Write expressions to represent the following statements:-

(i) How old will he be seven years from now

x + 7 (B1) ..............................................

(1)

(ii) How old was he ten years ago?

x - 10 (B1)

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

(iii) His father is three times his age. How old is his father?

3x (B1)

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

(1)


Question 3

Write an expression for the area of a square if the length of one side is x

x2 (B1)

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

(1)

Question 4

I think of a number, double it, add fourteen and then divide it by 4. If my number is n , write an expression to show this.

Sight of ‘(2x + 14)’ (M1) ¼ (2x + 14) oe (A1)

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

(2)

Question 5

A cab company charges a basic rate of £ x plus £1.50 for every kilometre travelled. Write a formulae to represent the cost of a journey of y kilometres in terms of x and y.

£( x + 1.5y) oe (B1)

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

(2)

Question 6

A regular hexagon has sides of length 5h. Write an expression for the perimeter of the hexagon.

6 x 5h = 30h (B1)

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

(1)

Total /10


Derive an equation 2 Grade 5 Solutions

Objective: Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution in context.

Question 1

The diagram shows a right-angled triangle.

(5x - 70)°

3x°

Work out the size of the smallest angle of the triangle.

8x – 70 = 90 (or 8x – 70 + 90 = 180) (M1)

8x = 160 (M1)

x = 20 (A1)

smallest angle = (5 x 20) - 70 = 30° (B1)

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

(4)


Question 2

Two bunches of tulips and five bunches of daffodils costs £36. Seven bunches of tulips and four bunches of daffodils costs £72. Find the cost of a bunch of tulips.

2t + 5d = 36 7t + 4d = 72 (M1)

→ 8t + 20d = 144 35t + 20d = 360

27t = 216 (M1)

t = 8 (A1)

a bunch of tulips costs £8 (C1)

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

(4)

Question 3

A garden in the shape of an isosceles triangle has two equal sides 8m longer than the other and the perimeter is 40m.

Derive an equation you could use to find the length of the shorter side .

3x + 16 = 40 (M1 A1)

(award M1 A0 for a correct diagram only)

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

(2)

Total /10


nth term of a linear sequence 2 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) 2n + 3

2 × 1 + 3, 2 × 2 + 3, etc 5, 7, 9. 11, 13, … (A1)

……………………………………………………………………………………… (1)

(b) 4n – 3

4 × 1 - 3, 4 × 2 - 3, etc 1, 5, 9, 13, 17, …(A1)

……………………………………………………………………………………… (1)

(Total 2 marks)

Question 2.

Here are the first five terms of a linear sequence.

5, 8, 11, 14, 17, …

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

The sequence is adding 3 and it is 2 more than the 3 times table 3n + 2 (A1)

……………………………………………………………………………………… (2)

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

3 × 10 + 2 32 (A1)

……………………………………………………………………………………… (1)

(Total 3 marks)


Question 3. The first five terms of a linear sequence is given below:

-5, -2, 1, 4, 7, …

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

The sequence is adding 3 and it is 8 less than the 3 times table 3n – 8 (A1)

………………………………………………………………………………………

(Total 2 marks)

Question 4.

The diagrams show a sequence of patterns made from 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.

n + 2 (A1)

……………………………………………………………………………………… (1)

(c) Find the total number of tiles in pattern number 10.

12 (A1) ……………………………………………………………………………………… (1) (Total 3 marks)

Total /10


Solve linear inequalities one variable 2 Grade 5 Solutions

Objective: Solve linear inequalities in one variable

Question 1

Solve.

(a) 4x – 5 < 7

4x < 12 (M1) x < 3 (A1)

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

(b) Write down the largest integer that satisfies 4x – 5 < 7

2 (B1)

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

(1)

Question 2

Solve.

(a) 4h – 5 > 2h + 2

2h - 5 > 2 (M1) 2h > 7 (M1)

h > 3.5 (oe) (A1)

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

(b) Write down the smallest integer that satisfies 4h – 5 > 2h + 2

4 (B1)

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

(1)


Question 3

Solve.

2(5r – 1) < 2r + 3

10r - 2 < 2r + 3(M1) 8r < 5 (M1)

x < 5/8 (oe) (A1)

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

Total /10

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