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Algebra ‘C/D’
Borderline Revision
Document
Algebra content that could be on Exam Paper 1 or Exam Paper 2 but:
If you need to use a calculator for *** shown (Q8 and Q9) then do
so.
Questions are numbered and presented in exam style format
MUST BE ABLE TO DO
SHOULD TRY
FANCY A CHALLENGE!
Topic: I can……
Acid test
factorise, expand and simplify
expressions
(grade C/D)
1a) factorise a single bracket
b) Use the rules of indices with a bracket
c) Use the rules of indices with a fraction
2a) expand two single brackets and then simplify
b) Use rules of indices with harder expressions
expand the product of two brackets
(grade C)
factorise a complex quadratic
(grade B)
use algebra within a proof
(grade A)
solve a quadratic inequality
(grade A)
spot errors in solving equations
(grade B/C)
form equations and make substitutions
into formulae (grade B/C) ***
3. Expand two brackets and then simplify
recognise when to and know how to
calculate the equation of a straight line.
(grade B/C)
change the subject of a simple formula
(grade C)
solve simultaneous equations and
inequalities (grade A/B)
form algebraic rules and use them for
arithmetic and diagram sequences
(grade B/C)
4. Take out a common factor then factorise the remaining quadratic
using the difference of two squares
5. Use expanding and factorising techniques to form an algebraic
proof for an identity.
6. Solve a quadratic inequality by factorisation method, modelled on
quadratic equations.
7a) Spot an error in a solution for a linear equation
b) spot an error in a solution for a quadratic equation
8. Form and solve a linear equation in order to find an area.
9.a) Use substitution to find values in a linear formula
b) Use a quadratic formula and form an inequality
10 and 11. Form equations of a line using the properties of
perpendicular gradients.
12 Recognise parallel lines from their equations
13 Use the method applied to equations to change the subject in a
linear formula with three unknowns.
14 and 15a Solve simultaneous linear equations to find x and y
15b Solve a pair of simultaneous linear inequations
16 Solve simultaneous equations where one is not linear
17a) Find a rule for the nth term of an arithmetic sequence of
numbers
b) Prove if a given number is a term in an arithmetic sequence
18. Solve problems for a diagram sequence by forming a rule
R A G
Q1.
(a) Factorise
y2 + 27y
...........................................................
(1)
(b) Simplify
(t3)2
...........................................................
(1)
(c) Simplify
...........................................................
(1)
Q2. (a) Expand and simplify
3(y – 2) + 5(2y + 1)
...........................................................
(2)
(b) Simplify
5u2w4 × 7uw3
...........................................................
(2)
Q3. Expand and simplify (m + 7)(m + 3)
...........................................................
Q4. Factorise fully 20x2 − 5
...........................................................
Q5. Prove algebraically that
(2n + 1)2 − (2n + 1) is an even number for all positive integer values of n.
(3)
Q6.
Solve x2 > 3x + 4
...........................................................(3)
Q7. Steve is asked to solve the equation 5(x + 2) = 47.
Here is his working.
5(x + 2) = 47
5x + 2 = 47
5x = 45
x=9
Steve's answer is wrong.
(a) What mistake did he make?
.............................................................................................................................................
.............................................................................................................................................
(1)
Liz is asked to solve the equation 3x2 + 8 = 83
Here is her working.
3x2 + 8 = 83
3x2 = 75
x2 = 25
x=5
(b) Explain what is wrong with Liz's answer.
.............................................................................................................................................
.............................................................................................................................................
(1)
Q8***. ABCD is a rectangle.
EFGH is a trapezium.
All measurements are in centimetres. The perimeters of these two shapes are the same.
Work out the area of the rectangle.
........................................................... cm2 (5)
Q9***. A school has a biathlon competition.
Each athlete has to throw a javelin and run 200 metres.
(a) The points scored for throwing a javelin are worked out using the formula
P1 = 16(D − 3.8)
where P1 is the number of points scored when the javelin is thrown a distance D metres.
(i)
Lottie throws the javelin a distance of 42 metres.
How many points does Lottie score?
(4)
The points scored for running 200 metres are worked out using the formula
P2 = 5(42.5 − T)2
where P2 is the number of points scored when the time taken to run 200 metres is T seconds.
Suha scores 1280 points in the 200 metres.
(b) (i) Work out the time, in seconds, it took Suha to run 200 metres.
The formula for the number of points scored in the 200 metres should not be used for T > n.
(ii) State the value of n.
Give a reason for your answer.
(4)
Q10.
A(−2, 1), B(6, 5) and C(4, k) are the vertices of a right-angled triangle ABC.
Angle ABC is the right angle.
Find an equation of the line that passes through A and C.
Give your answer in the form ay + bx = c where a, b and c are integers.
...........................................................(5)
Q11. Here is a circle, centre O, and the tangent to the circle at the point P(4, 3) on the circle.
Find an equation of the tangent at the point P
...........................................................(3)
Q12. Here are the equations of four straight lines.
Line A
Line B
Line C
Line D
y = 2x + 4
2y = x + 4
2x + 2y = 4
2x − y = 4
Two of these lines are parallel.
Write down the two parallel lines?
Line ................................ and line ................................(1)
Q13.
Make t the subject of the formula
...........................................................(2)
Q14. Solve the simultaneous equations
2x – 4y = 19
3x + 5y = 1
x = ........................................
y =.....................................(4)
Q15. (a) Solve the simultaneous equations
3x + 5y = 4
2x − y = 7
(3)
(b) Find the integer value of x that satisfies both the inequalities
x+5>8
and
2x − 3 < 7
(3)
Q16. Solve algebraically the simultaneous equations
x2 + y2 = 25
y − 2x = 5
...........................................................(5)
Q17. Here are the first four terms of an arithmetic sequence.
6
10
14
18
(a) Write an expression, in terms of n, for the nth term of this sequence.
...........................................................(2)
The nth term of a different arithmetic sequence is 3n + 5
(b) Is 108 a term of this sequence?
Show how you get your answer.
(2)
Q18. The diagrams show a sequence of patterns made from grey tiles and white tiles.
The number of grey tiles in each pattern forms an arithmetic sequence.
(a) Find an expression, in terms of n, for the number of grey tiles in Pattern n.
(2)
The total number of grey tiles and white tiles in each pattern is always the sum of the squares of two
consecutive whole numbers.
(b) Find an expression, in terms of n, for the total number of grey tiles and white tiles in Pattern n.
Give your answer in its simplest form.
(3)
(c) Is there a pattern for which the total number of grey tiles and white tiles is 231?
Give a reason for your answer.
(2)
The total number of grey tiles and white tiles in any pattern of this sequence is always an odd number.
(d) Explain why.
(2)
Mark Scheme
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Q8.
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Q10.
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Q14.
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Q18.