PLC Papers
Created For:
Algebraic Fractions 2
Objective:
Grade 6
Simplify algebraic fractions.
Question 1.
Find the length of the rectangle:
Area = (x+3)(x+2)2
2
Height = x -x-6
………………cm
(4)
Question 2.
The area of a square is 9x2-16. Find the perimeter of the square.
…………………………cm
(6)
Total /10
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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)
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Question 3
The rectangle and the equilateral triangle have equal perimeters.
Not drawn accurately
3(x – 1)
4(3x + 2)
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
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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)
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Question 3
Find the common factors for 3x2y –9xy and 2xy-6y
(3)
Total /10
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Cubic and Reciprocal Graphs 2
Objective:
Grade 6
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
y
1
6
6
.
x
2
3
2
4
5
1·2
6
(2)
b) Draw the graph of y =
6
on the grid below.
x
y
6
5
4
3
2
1
0
1
2
3
4
5
6
x
(2)
(c)
Use your graph to solve the equation
6
= 2·2.
x
(2)
(Total 6 marks)
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Question 2.
The table shows some values of x and y for the equation y = (x – 1)3.
–2
–1
0
1
2
3
4
x
–27
–1
0
8
y
a) Complete the table.
(2)
b) Draw the graph of y = (x – 1)3 for values of x from –2 to 4.
y
30
20
10
–2
–1
0
1
2
3
4
x
–10
–20
–30
(2)
(Total 4 marks)
Total /10
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Deduce quadratic roots algebraically 2
Objective:
Grade 6
Deduce roots algebraically.
Question 1.
y2 + 5y = 0
Solve the equation
(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)
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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
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Equation of a line 2
Objective:
Grade 5
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)
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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
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Expanding binomials 2
Objective:
Grade 5
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)
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(d) Expand and simplify (6� − 1)(7 − 3�)
……………………….
(2)
(e) Expand and simplify (2 + 5�)2
……………………….
(2)
(Total 10 marks)
Total /10
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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
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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)
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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
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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
y = 2x -2
2x - 2 = -3x + 5
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)
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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)
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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
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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)
-1
(b)
0
1
2
3
4
-3
-2
-1
0
1
2
..............................................
(4)
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(c) Which of these inequalities (a) or (b) has the most integer solutions?
...............................................
(1)
Total /10
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Inverse Functions 2
Objective:
Grade 6
Interpret the reverse process as the ‘inverse function’ including the correct notation.
Question 1.
Find � −1 (�) for each of the following functions
(a) �(�) = 7�
……………………….
(2)
(b) �(�) = 3� − 8
……………………….
(2)
(c) �(�) = 6 + 2�
……………………….
(2)
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(d) �(�) =
2�+1
5
……………………….
(2)
�
(e) �(�) = − 1
2
……………………….
(2)
(Total 10 marks)
Total /10
PiXL PLC 2017 Certification
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)
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Question 4
Solve 7(b + 2) = 4(b + 5)
..............................................
(3)
(ii) Show how you can check your solution is correct.
..............................................
(1)
Total /10
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Linear inequalities in two variables 2
Grade 6
Objective: Solve linear inequalities in two variables
Question 1.
The graph shows the region that represents the inequalities � ≤ 2� − 1, � + 2� < 10 and
� < 6 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 � + � ≤ 6, state which of the following may be true and which must be false
a) � + � < 3
….………………………(1)
c) � ≤ �
….………………………(1)
b) �� > 9
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….………………………(1)
(Total 3 marks)
Question 3.
a) Represent the inequalities � − 2� ≤ 2, 2� + � < 5 and � > 0 on the grid below by
shading the unwanted regions.
(3)
b) � and � are integer.
On your graph, mark with a cross each of the points with satisfies all three
inequalities.
(2)
(Total 5 marks)
TOTAL /10
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Parallel lines 2
Objective:
Question 1
Grade 6
Use the form � = �� + � to identify parallel lines.
(a) On the grid draw a line which has gradient -2.
(1)
(b) Write down the equation of your line.
……………………………
(1)
(Total 2 marks)
Question 2
(a) Write down the gradient of the line 3� = 4� + 1.
……………………………
(1)
(b) Write down the equation of a line parallel to � = 6�.
……………………………
(1)
(Total 2 marks)
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Question 3
Here is a straight line graph.
Find the equation of the line.
……………………………
(2)
(Total 2 marks)
Question 4
Here are the equations of 4 lines:
Line L 2 :
� = 2� − 6.
Line L 3 :
3� = 2� + 2.
Line L 4 :
1
Line L 1 :
2� − � + 7 = 0
2
� = 10 + �.
Which line is NOT parallel to the other three?
……………………………
(2)
(Total 2 marks)
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Question 5
A
B
The lines A and B are parallel.
The line A passes through the point (0, -2)
The line B has equation y = -3x + 1
Write down the equation of line A.
……………………………
(2)
(Total 2 marks)
TOTAL /10
PiXL PLC 2017 Certification
Quadratic Graphs 2
Objective:
Grade 6
Identify roots, intercepts and turning points of quadratic functions graphically.
Question 1.
a) Complete the table of values for y = x2 + 5
x
–2
y
9
–1
0
1
2
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)
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(Total 4 marks)
Question 2.
a) Complete the table of values for y = x2 – 2x – 1.
x
–2
y
7
–1
0
1
2
–2
–1
3
4
(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)
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Total /10
PiXL PLC 2017 Certification
Quadratic equations (factorisation) 2
Grade 6
Objective: Solve quadratic equations by factorising.
Question 1.
Solve:
a) � 2 + 5� − 6 = 0
………………………
(1)
b) � 2 − 8� − 48 = 0
………………………
(1)
c) � 2 − 4 = 0
………………………
(1)
d) 4� 2 − 16 = 0
………………………
(1)
(Total 4 marks)
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Question 2.
Solve � 2 − 30 = �
………………………
(Total 3 marks)
Question 3.
Solve by factorising 6� 2 + 5� − 4 = 0
………………………
(Total 3 marks)
TOTAL /10
PiXL PLC 2017 Certification
Quadratic equations (graphical methods) 2
Objective:
Grade 6
Find approximate solutions to quadratic equations using a graph
Question 1.
a) Complete the table below to work out values for the graph of � = � 2 − 3� − 3 for
values of x from −3 ≤ � ≤ 5.
Plot the graph using −3 ≤ � ≤ 5 and −10 ≤ � ≤ 15.
b) Use your graph to estimate the solutions of the two roots of � 2 − 3� − 3 = 0
………………………
(Total 3 marks)
Question 2.
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a) Complete the table below to work out values for the graph of � = 2� 2 − 4� − 7 for
values of x from −2 ≤ � ≤ 4 then draw the graph.
b) Use your graph to find the y-value when � = 3.5
………………………
(Total 3 marks)
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Question 3.
a) Using suitable axes draw the graph of � = � 2 − 5� + 2 for −2 ≤ � ≤ 6
b) What is the value of � when � = 1.5
c) For what values of x does � 2 − 4� − 2 = 4
………………………
(Total 4 marks)
TOTAL /10
PiXL PLC 2017 Certification
Reciprocal Real Life Graphs 2
Objective:
Grade 5
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?
V
V
D
D
Graph 1
Graph 2
V
V
D
D
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
� length
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3
4
6
12
36
(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)
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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
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PiXL PLC 2017 Certification
Represent linear inequalities 2
Grade 6
Objective: Represent the solution of a linear inequality in two variables on a number
line, using set notation and on a graph
Question 1.
Balpreet has correctly drawn the inequalities � < 3, � + � ≥ 4, ��� � − � > −2
graphically and correctly shown the region that represents the solution by shading the
unwanted regions.
a) She then writes down the integer dataset of the solution but makes some mistakes, circle
the incorrect answers in Balpreet’s dataset.
{ (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,2), (4,3), (5,3) }
b) Write down the inequalities that would make Balpreet’s dataset the correct and complete
solution.
………………………
(Total 3 marks)
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Question 2.
The dataset shown below lists the complete integer solution set to three inequalities.
{ (2,1), (3,1), (3,2) }
Plot the points on the given axes and determine the three inequalities for which they are the
complete integer solution set.
………………………
(Total 4 marks)
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Question 3.
Represent the solution to the inequalities � + � ≥ 3, � − � < 2, ��� � ≤ 3.5
graphically on the grid below by shading the unwanted regions.
(A3)
………………………
(Total 3 marks)
Total /10
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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
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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
( 7 )3 = 7 n
...........................................................
(Total 1 mark)
Question 3
Calculate the value of n
32 × 81 = 3n
...........................................................
(Total 1 mark)
Question 4
Calculate the value of n
1
= 2n
16
...........................................................
(Total 1 mark)
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Question 5
Calculate the value of k
160 = k 10
...........................................................
(Total 1 mark)
Question 6
(
)(
Work out the value of
3 − 3 1+ 3 3
)
27
Give you answer in its simplest form
...........................................................
(Total 3 mark)
Question 7
Simplify
(9 + 3 )(8 − 3 )
Give your answer in the form a + b 7
...........................................................
(Total 2 mark)
Total /10
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Simultaneous equations graphically & algebraically 1
Grade 6
Objective: Form and/or solve simultaneous equations
Question 1.
The straight line 2� + � = 3 has been drawn on the grid.
a) On the grid, draw the graph of � = 3� − 4
b) Use your diagram to solve the simultaneous equations
2� + � = 3
� = 3� − 4
(2)
� =.……………� =.……………
(1)
(Total 3 marks)
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Question 2.
Solve the simultaneous equations
8� − 3� = 16
6� − 7� = 31
� =.……………� =.……………
(Total 3 marks)
Question 3.
Solve, by substitution, the simultaneous equations
� + 4� = 3
2� = 3� + 5
� =.……………� =.……………
(Total 4 marks)
TOTAL /10
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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)
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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
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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)
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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
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nth term of quadratic sequences 2 Grade 6
Objective: Write and expression for the nth term of a sequence, including quadratic sequences.
Question 1.
(a)
Here are the first five terms of a sequence.
-1, 2, 7, 14, 23, …
Find the next term two terms of the sequence.
………………… and …………………
(b)
The nth term of a different sequence is 3n2
(1)
+ 4.
Work out the third term of this sequence.
…………………………… (1)
(Total 2 marks)
Question 2.
Here are the first five terms of a quadratic sequence.
0, 3, 8, 15, 24, …
(a)
Write down the next term of the sequence.
………………………… (1)
(b)
Find an expression in terms of n, for the nth term of this quadratic sequence. .
……………………………………… (2)
(Total 3 marks)
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Question 3.
Find an expression in terms of n, for the nth tem of this quadratic sequence.
-2, 7, 22, 43, 70, …
………………………………………………………………………………………
(Total 2 marks)
Question 4.
Find the nth tem of this sequence.
5, 12, 23, 38, 57, …
………………………………………………………………………………………
(Total 3 marks)
Total /10
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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)
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Question 3
Solve.
2(5r – 1) < 2r + 3
..............................................
(3)
Total /10
PiXL PLC 2017 Certification
PLC Papers
Created For:
Algebraic Fractions 2
Objective:
Grade 6
Solutions
Simplify algebraic fractions.
Question 1.
Find the length of the rectangle:
Area = (x+3)(x+2)2
2
Height = x -x-6
(�+3)(�+2)2
� 2 −�−6
(�+3)(�+2)2
(�−3)(�+2)
(�+3)(�+2)2
(�−3)(�+2)
B1 for division
M1 for factorisation
M1 for recognising common factors
(�+3)(�+2)
(�−3)
A1
………………cm
(4)
Question 2.
The area of a rectangle is 9x2-16 and one of the sides is 3x2-x-4. Find the perimeter of the rectangle.
3x2-16 = (3x+4)(3x-4) M1
3x2-x-4 = (3x-4)(x+1) M1
Missing side =
=
Perimeter = 2(
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(3x+4)(3x−4)
(3x−4)(x+1)
3�+4
�+1
3�+4
�+1
M1
A1
) +2(3x2-x-4) M1
3�+4
=2(
�+1
) +6x2-2x-8 OE A1
…………………………cm
(6)
Total /10
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
Question 3
The rectangle and the equilateral triangle have equal perimeters.
Not drawn accurately
3(x – 1)
4(3x + 3)
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
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
Cubic and Reciprocal Graphs 2
Objective:
Grade 6
Solutions
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
y
1
6
6
.
x
2
3
3
2
4
1.5
5
1·2
6
1
(M2)
b) Draw the graph of y =
6
on the grid below.
x
y
6
*
5
4
3
*
2
*
*
*
1
0
1
2
3
4
5
*
6
x
(M2)
(c)
6
Use your graph to solve the equation = 2·2.
x
Evidence of line drawn on graph (M1);
x =2.7 (A1)
(2)
(Total 6 marks)
PiXL PLC 2017 Certification
Question 2.
The table shows some values of x and y for the equation y = (x – 1)3.
–2
–27
x
y
–1
-8
0
–1
1
0
2
1
3
8
4
27
a) Complete the table.
(M2)
b) Draw the graph of y = (x – 1)3 for values of x from –2 to 4.
y
30
*
20
10
*
–2
–1
*
0
*
*1
2*
3
4
x
–10
–20
*
–30
(M2)
(Total 4 marks)
Total /10
PiXL PLC 2017 Certification
Deduce quadratic roots algebraically 2
Objective:
Grade 6
Solutions
Deduce roots algebraically.
Question 1.
y2 + 5y = 0
Solve the equation
y (y + 5) = 0 (M1)
y = 0 or y = -5 (A2)
(Total 3 marks)
Question 2.
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)
PiXL PLC 2017 Certification
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)
x=
7 ±√49+200
x=
7 ±√249
2×5
(M1)
10
10
(M1)
x = 2.28 (A1)
x = -0.88 (A1)
(Total 4 marks)
Total /10
PiXL PLC 2017 Certification
Equation of a line 2
Objective:
Grade 5
Solutions
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 =
y=
−�
3
+c
−1
3
(M1)
sub (6,3) 3 = -2 + c
5 = c (M1)
−�
y=
+ 5 (A1)
3
(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
3
+ (M1)
7
Product of two gradients is -1 (M1)
Gradient of B must be
10 − 3
−14− −5
9
7
−9
=
7
−9
(A1)
−7
9
(M1)
x = -1 therefore Lines A and B are perpendicular to each other (C1)
7
(Total 5 marks)
Total /10
PiXL PLC 2017 Certification
Expanding binomials 2
Objective:
Grade 5
Solutions
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)
PiXL PLC 2017 Certification
(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
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
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
y = 2x -2
2x - 2 = -3x + 5
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)
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
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
(A1)
0
1
2
3
4
..............................................
(2)
Question 2
-2 < n ≤ 3
(A1)
-2 -1 0 1
n is an integer
2 3
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
1 ≤ x < 4
2
3
4
(M1 A1)
-3
-2
-1
0
1
2
-2 < x < 1 (M1 A1)
..............................................
(4)
PiXL PLC 2017 Certification
(c) Which of these inequalities (a) or (b) has the most integer solutions? (a) B1
...............................................
(1)
Total /10
PiXL PLC 2017 Certification
Inverse Functions 2
Objective:
Grade 6
SOLUTIONS
Interpret the reverse process as the ‘inverse function’ including the correct notation.
Question 1.
Find � −1 (�) for each of the following functions
(a) �(�) = 7�
� = ��
(M1 rearranging to make ‘x’ or ‘y’ the subject)
�
=�
�
�
(A1)
� −1 (�) =
�
……………………….
(2)
(b) �(�) = 3� − 8
� = �� − �
� + � = ��
� −1 (�) =
�+�
�
(M1 rearranging to make ‘x’ or ‘y’ the subject)
(A1)
……………………….
(2)
(c) �(�) = 6 + 2�
� = � + ��
� − � = ��
� −1 (�) =
�−�
�
(M1 rearranging to make ‘x’ or ‘y’ the subject)
(A1)
……………………….
(2)
PiXL PLC 2017 Certification
(d) �(�) =
�=
2�+1
�� + �
�
5
�� = �� + �
�
−1 (�)
=
(M1 rearranging to make ‘x’ or ‘y’ the subject)
��−�
(A1)
�
……………………….
(2)
�
(e) �(�) = − 1
�=
�
−1
2
2
�+�=
�
�
(M1 rearranging to make ‘x’ or ‘y’ the subject)
� −1 (�) = �(� + �)
(A1)
……………………….
(2)
(Total 10 marks)
Total /10
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
Linear inequalities in two variables 2
Grade 6
SOLUTIONS
Objective: Solve linear inequalities in two variables
Question 1.
The graph shows the region that represents the inequalities � ≤ 2� − 1, � + 2� < 10 and
� < 6 by shading the unwanted regions.
Use the graph to find the integer values of � and � that maximise the sum of � and �.
� = 5, � = 2
so that � + � = 7
(A2)
….………………………
(Total 2 marks)
Question 2.
If � + � ≤ 6, state which of the following may be true and which must be false
a) � + � < 3
b) �� > 9
c) � ≤ �
PiXL PLC 2017 Certification
may be true (A1)….………………………(1)
must be false (A1)….………………………(1)
may be true (A1)….………………………(1)
(Total 3 marks)
Question 3.
a) Represent the inequalities � − 2� ≤ 2, 2� + � < 5 and � > 0 on the grid below by
shading the unwanted regions.
(3)
b) � and � are integer.
On your graph, mark with a cross each of the points with satisfies all three
inequalities.
Crosses on (0,1), (0,2), (1,1) and (1,2)
(A2)
If one point omitted or one error then
(A1)
(2)
(Total 5 marks)
TOTAL /10
PiXL PLC 2017 Certification
Parallel lines 2
Objective:
Question 1
Grade 6
Solutions
Use the form � = �� + � to identify parallel lines.
(a) On the grid draw a line which has gradient -2.
Any line with gradient -2
(A1)
(1)
(b) Write down the equation of your line.
Correct equation of their line
(A1)ft
(1)
(Total 2 marks)
Question 2
(a) Write down the gradient of the line 3� = 4� + 1.
m = 4/3
(A1)
(1)
(b) Write down the equation of a line parallel to � = 6�.
y = 6x + k
for any k except 0
(A1)
(1)
(Total 2 marks)
PiXL PLC 2017 Certification
Question 3
Here is a straight line graph.
Find the equation of the line.
� =
3
4
� + 1
(B1) for gradient
(B1) y-intercept
(2)
(Total 2 marks)
Question 4
Here are the equations of 4 lines:
Line L 2 :
� = 2� − 6.
Line L 3 :
3� = 2� + 2.
Line L 4 :
1
Line L 1 :
2� − � + 7 = 0
2
� = 10 + �.
Which line is NOT parallel to the other three?
Attempt to turn all lines in to y = mx + c form with not more than 1 error (M1)
Line L 3 :
3� = 2� + 2
(A1)
(2)
(Total 2 marks)
PiXL PLC 2017 Certification
Question 5
A
B
The lines A and B are parallel.
The line A passes through the point (0, -2)
The line B has equation y = -3x + 1
Write down the equation of line A.
Identify that gradient must be equal to -3 or y-intercept = -2
(M1)
y = -3x - 2
(A1)
(2)
(Total 2 marks)
TOTAL /10
PiXL PLC 2017 Certification
Quadratic Graphs 2
Objective:
Grade 6
Solutions
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)
PiXL PLC 2017 Certification
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
PiXL PLC 2017 Certification
Quadratic equations (factorisation) 2
Grade 6
Objective: Solve quadratic equations by factorising.
Question 1.
Solve:
a) � 2 + 5� − 6 = 0
(� + 6)(� − 1) = 0
� = −6 �� � = 1
(A1)
………………………
(1)
b) � 2 − 8� − 48 = 0
(� − 12)(� + 4) = 0
� = 12 �� � = −4 (A1)
………………………
(1)
c) � 2 − 4 = 0
(� − 2)(� + 2) = 0
� = 2 �� � = −2
(A1)
………………………
(1)
d) 4� 2 − 16 = 0
(2� − 4)(2� + 4) = 0
� = 2 �� � = −2
(A1)
………………………
(1)
(Total 4 marks)
PiXL PLC 2017 Certification
Question 2.
Solve � 2 − 30 = �
� 2 − � − 30 = 0
(M1)
� = −5 �� � = 6
(A1)
(� + 5)(� − 6) = 0 (M1)
………………………
(Total 3 marks)
Question 3.
Solve by factorising 6� 2 + 5� − 4 = 0
(3� + 4)(2� − 1) = 0
4
� = − �� � =
3
(M2)
1
2
(A1)
………………………
(Total 3 marks)
TOTAL /10
PiXL PLC 2017 Certification
Quadratic equations (graphical methods) 2
Objective:
Grade 6
SOLUTIONS
Find approximate solutions to quadratic equations using a graph
Question 1.
a) Complete the table below to work out values for the graph of � = � 2 − 3� − 5 for
values of x from −3 ≤ � ≤ 5.
Plot the graph using −3 ≤ � ≤ 5 and −10 ≤ � ≤ 15.
(A1)
(A1)
b) Use your graph to estimate the solutions of the two roots of � 2 − 3� − 5 = 0
−1.5 < � < −0.5 ��� 3.5 < � < 4.5
(A1)
………………………
(Total 3 marks)
PiXL PLC 2017 Certification
Question 2.
a) Complete the table below to work out values for the graph of � = 2� 2 − 4� − 7 for
values of x from −2 ≤ � ≤ 4 then draw the graph.
(A1)
(A1)
b) Use your graph to find the y-value when � = 3.5
0<�<1
(A1)
………………………
(Total 3 marks)
PiXL PLC 2017 Certification
Question 3.
a) Using suitable axes draw the graph of � = � 2 − 5� + 2 for −2 ≤ � ≤ 6
(A1)
(A1)
b) What is the value of � when � = 1.5
−4 < � < −3
(A1)
c) For what values of x does � 2 − 4� − 2 = 4
−1 < � < 0 ��� 5 < � < 6
(A1)
………………………
(Total 4 marks)
TOTAL /10
PiXL PLC 2017 Certification
Reciprocal Real Life Graphs 2
Objective:
Grade 5
Solutions
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?
V
V
D
D
Graph 1
Graph 2
V
V
D
D
Graph 3
B1
Graph 4
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
� length
1
2
3
4
6
12
36
36
18
12
9
6
3
1
B2- all values correct, B1- 6 values correct
(2)
PiXL PLC 2017 Certification
�
(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)
PiXL PLC 2017 Certification
�
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
PiXL PLC 2017 Certification
Represent linear inequalities 2
Grade 6
SOLUTIONS
Objective: Represent the solution of a linear inequality in two variables on a number
line, using set notation and on a graph
Question 1.
Balpreet has correctly drawn the inequalities � < 3, � + � ≥ 4, ��� � − � > −2
graphically and correctly shown the region that represents the solution by shading the
unwanted regions.
a) She then writes down the integer dataset of the solution but makes some mistakes, circle
the incorrect answers in Balpreet’s dataset.
{ (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,2), (4,3), (5,3) }
(A2)
b) Write down the inequalities that would make Balpreet’s dataset the correct and complete
solution.
� ≤ 3, � + � ≥ 4, ��� � − � ≥ −2
(A1)
………………………
(Total 3 marks)
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Question 2.
The dataset shown below lists the complete integer solution set to three inequalities.
{ (2,1), (3,1), (3,2) }
Plot the points on the given axes and determine the three inequalities for which they are the
complete integer solution set.
Plotting (2,1), (3,1) and (3,2) all correctly
(M1)
�≤3
(A1)
�≥1
� ≤�−1
(A1)
(A1)
………………………
(Total 4 marks)
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Question 3.
Represent the solution to the inequalities � + � ≥ 3, � − � < 2, ��� � ≤ 3.5
graphically on the grid below by shading the unwanted regions.
(A3)
………………………
(Total 3 marks)
Total /10
PiXL PLC 2017 Certification
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
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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)
Question 2
Calculate the value of n
( 7 )3 = 7 n
..................n = 1.5.........................................
(Total 1 mark)
Question 3
Calculate the value of n
32 × 81 = 3n
...................n = 4........................................
(Total 1 mark)
Question 4
Calculate the value of n
1
= 2n
16
.......................n = -4....................................
(Total 1 mark)
Question 5
Calculate the value of k
160 = k 10
........................k = 4...................................
(Total 1 mark)
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Question 6
(
)(
Work out the value of
3 − 3 1+ 3 3
)
27
Give you answer in its simplest form
3+9 3 − 3 −9
27
−6+8 3
27
−6+8 3
9×3
−6+8 3
3 3
−6+8 3
3 3
...........................................................
(Total 3 mark)
Question 7
Simplify
(9 + 3 )(8 − 3 )
Give your answer in the form a + b 7
72 + 8 3 − 9 3 − 3
69 − 3
69 − 3
...........................................................
(Total 2 mark)
Total /10
PiXL PLC 2017 Certification
Simultaneous equations (linear) 2
Grade 6
SOLUTIONS
Objective: Form and/or solve simultaneous equations
Question 1.
The straight line 2� + � = 3 has been drawn on the grid.
(M1)
(A1)
a) On the grid, draw the graph of � = 3� − 4
(2)
b) Use your diagram to solve the simultaneous equations
2� + � = 3
� = 3� − 4
� = 2.……………� = 2.……………(A1)
(1)
(Total 3 marks)
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Question 2.
Solve the simultaneous equations
8� − 3� = 16
6� − 7� = 31
24� − 9� = 48
24� − 28� = 124
(M1)
19� = −76 ⇒ � = −4
(A1)
8� + 12 = 16
8� = 4 ⇒ � =
1
2
(A1)
� =.……………� =.……………
(Total 3 marks)
Question 3.
Solve, by substitution, the simultaneous equations
� + 4� = 3
2� = 3� + 5
� + 2(3� + 5) = 3
(M1)
7� + 10 = 3
(M1)
7� = −7 ⇒ � = −1
(A1)
4� = 4 ⇒ � = 1
(A1)
−1 + 4� = 3
� =.……………� =.……………
(Total 4 marks)
TOTAL /10
PiXL PLC 2017 Certification
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)
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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
PiXL PLC 2017 Certification
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)
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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
→ 8t + 20d = 144
27t = 216
t=8
(M1)
35t + 20d = 360
(M1)
(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
PiXL PLC 2017 Certification
nth term of quadratic sequences 2 Grade 6
Solutions
Objective: Write and expression for the nth term of a sequence, including quadratic sequences.
Question 1.
(a)
Here are the first five terms of a sequence.
-1, 2, 7, 14, 23, …
Find the next term two terms of the sequence.
Sequence is going up in 3, 5, 7, 9, so next two terms will be add 11 then add 13.
34 and 47 (A1)
………………… and …………………
(b)
The nth term of a different sequence is 3n2
(1)
+ 4.
Work out the third term of this sequence.
= 3(32) + 4
=3x9+4
= 31 (A1)
…………………………… (1)
(Total 2 marks)
Question 2.
Here are the first five terms of a quadratic sequence.
0, 3, 8, 15, 24, …
(a)
Write down the next term of the sequence.
Sequence is going up in 3, 5, 7, 9, so nest time will be add 11
35 (A1)
………………………… (1)
(b)
Find an expression in terms of n, for the nth term of this quadratic sequence. .
first difference = 3, 5, 7, 9
Second difference = 2 so n2
Sequence = 0, 3, 8, 15, 24
n2 = 1, 4, 9, 16, 25,
difference = -1, -1, -1, -1, -1,
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nth term is n2 - 1 (A1)
……………………………………… (2)
(Total 3 marks)
Question 3.
Find an expression in terms of n, for the nth tem of this quadratic sequence.
-2, 7, 22, 43, 70, …
First difference =
9, 15, 21, 27
Second difference = 6, 6, 6, 6, so 3n2
Sequence = -2, 7, 22, 43, 70, …
3n2 = 3, 12, 27, 48, 75
Difference = -5, -5, -5, -5, -5
nth term is 3n2 – 5 (A1 for each term)
………………………………………………………………………………………
(Total 2 marks)
Question 4.
Find the nth tem of this sequence.
5, 12, 23, 38, 57, …
First difference = 7, 11, 15, 19
Second difference = 4, 4, 4, 4, so 2n2
Sequence = 5, 12, 23, 38, 57
2n2 = 2, 8, 18, 32, 50,
Difference = 3, 4, 5, 6, 7 nth term is n + 2
nth term is 2n2 + n + 2 (A1 for each term)
………………………………………………………………………………………
(Total 3 marks)
PiXL PLC 2017 Certification
Total /10
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
Question 3
Solve.
2(5r – 1) < 2r + 3
10r - 2 < 2r + 3(M1)
8r < 5 (M1)
x < 5/8 (oe) (A1)
..............................................
(3)
Total /10
PiXL PLC 2017 Certification