Year 12 Summer Task
The Maths Department is very pleased to see that you have decided to opt to do A-Level Mathematics.
You will no doubt already realise that this is not an easy option and the course is very challenging, even for
students who achieved a grade A* at GCSE. In order to help you in your transition from GCSE to A-Level you
need to complete this task over the summer, prior to starting the course in September.
The techniques used to complete the Summer Task are things that you should have already met and are
readily used at A-Level. Completing the questions will help refresh your memory and prepare you for the
challenge of A-Level Maths. You must attempt ALL questions.
Core 1 is a non-calculator module and so this work must be completed without the using a calculator. Leave
answers as exact fractions if you need to, but try to always cancel these to their lowest form. For the
questions involving surds and/or indices, it is vital that you include sufficient working out to demonstrate that
you have not used a calculator.
The task needs to be completed and handed in as part of the enrolment process.
Please ensure your name is clearly visible on the work.
The Mathematics Department
Section
1
2
Marks
Target Setting and Review of Learning
________________________________________________________________
________________________________________________________________
3
________________________________________________________________
4
________________________________________________________________
5
________________________________________________________________
6
7
________________________________________________________________
________________________________________________________________
8
9
Total
Marks
________________________________________________________________
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Section 1: Collecting Like Terms and Removing Brackets
Simplify the following expressions as much as possible:
1. x + 1 + 3x + 5
__________________ (1)
2. xy + 4xy
__________________ (1)
3. 12a + b – 3a + 8b
__________________ (1)
4. x2 + 3x2 – x
__________________ (1)
5. x + y + xy + 4xy
__________________ (1)
6. x + x + 1 – 2x + 3 – 3x
__________________ (1)
7. 6(x + 2)
__________________ (1)
8. 3y(2x – 7)
__________________ (1)
9. 4x(3 – 9x)
__________________ (1)
10. 2y + 5(y + 2)
__________________
→
__________________ (1)
11. 2(x + 3) + 3(x – 2)
__________________
→
__________________ (1)
12. 10(2x + 3) – 3(4x -1)
__________________
→
__________________ (1)
13. (x + 7)(x + 4)
__________________
→
__________________ (1)
14. (x + 2)(x – 9)
__________________
→
__________________ (1)
15. (x – 13)(x + 2)
__________________
→
__________________ (1)
16. (x + 11)2
__________________
→
__________________ (1)
17. (x – 5)2
__________________
→
__________________ (1)
18. (3x – 5)(2x + 1)
__________________
→
__________________ (1)
19. (2x + 1)(2x – 1)
__________________
→
__________________ (1)
20. (x + 2)(x + 3) – x(x + 5)
__________________
→
__________________ (1)
21. (x - 2)(x – 7) – x(x – 9)
__________________
→
__________________ (1)
21
Section 2: Solving Linear Equations
Solve the following equations:
1. 2x – 7 = 21
X = ________ (2)
2. 6y – 11 = 4y + 31
y = ________ (2)
3. 3(x – 1) = 2(2x + 5)
X = ________ (2)
4. 8(x + 3) = 6 – 2(2x – 1)
X = ________ (2)
5. (x + 2)(x + 3) – x(x + 2) = 0
X = ________ (2)
6.
2𝑥𝑥−6
3
𝑥𝑥−3
=
5
X = ________ (2)
7.
𝑥𝑥−10
2
−
2𝑥𝑥+3
3
=2
X = ________ (2)
8.
3𝑥𝑥+8
4
= 1+
2𝑥𝑥−2
5
X = ________ (2)
9. 3 −
2𝑥𝑥−6
10
= 𝑥𝑥 + 4
X = ________ (2)
18
Section 3: Factorising Quadratic Expressions
Factorise fully the following expressions
1. 5x – x2
__________________ (1)
2. 2x2 – 10x
__________________ (1)
3. x2 + 5x + 6
__________________ (1)
4. x2 – 8x + 12
__________________ (1)
5. x2 – 10x + 9
__________________ (1)
6. x2 + 2x – 8
__________________ (1)
7. x2 – 5x – 36
__________________ (1)
8. x2 – 9x
__________________ (1)
9. x2 – 9
__________________ (1)
10. x2 – 144
__________________ (1)
11. x2y – 9xy
__________________ (1)
12. x2 + 3xy + 2y2
__________________ (1)
13. 4x2 – 49
__________________ (1)
14. 25y2 – 100x2
__________________ (1)
15. 36x4 – 9x2
__________________ (1)
16. 2x2 + 8x + 6
__________________ (1)
17. 4x2 + 14x +12
__________________ (1)
18. 3x2 + 7x – 6
__________________ (1)
19. 5x2 + 20x + 15
__________________ (1)
20. 6x2 + 11x – 7
__________________ (1)
20
Section 4: Completing the square
Re-write the following in the form: a(bx + c)2 +d
1. x2 + 12x
2. x2 - 16x
3. 5x2 + 20x
4. 2x2 - 10x
8
Section 5: Solving Quadratic Equations
Solve the following quadratic equations by factorising
1. x2 + 6x + 5 = 0
__________________
x = ________ x = ________ (2)
2. x2 - 8x + 7 = 0
__________________
x = ________ x = ________ (2)
3. x2 - 10x + 24 = 0
__________________
x = ________ x = ________ (2)
4. x2 + 9x - 36 = 0
__________________
x = ________ x = ________ (2)
5. x2 - 5x - 36 = 0
__________________
x = ________ x = ________ (2)
6. x2 - 3x - 40 = 0
__________________
x = ________ x = ________ (2)
7. x2 + x - 42 = 0
__________________
x = ________ x = ________ (2)
8. 3x2 + 10x + 3 = 0
__________________
x = ________ x = ________ (2)
9. 5x2 - 9x - 2 = 0
__________________
x = ________ x = ________ (2)
Solve the following quadratic equations using the quadratic formula. Remember you cannot use a calculator,
instead you should leave your answers in surd form (with roots).
For
10. x2 - 2x - 5 = 0
ax2 + bx + c = 0
𝑥𝑥 =
−𝑏𝑏±√𝑏𝑏 2 −4𝑎𝑎𝑎𝑎
2𝑎𝑎
x = ________ x = ________ (2)
11. x2 + 6x + 4 = 0
x = ________ x = ________ (2)
12. x2 + 8x + 5 = 0
x = ________ x = ________ (2)
Solve the following quadratic equations using a method of completing the square.
13. x2 + 8x + 10 = 0
x = ________ x = ________ (2)
14. 2x2 - 8x + 7 = 0
x = ________ x = ________ (2)
15. 15 – 6x -2x2 = 0
x = ________ x = ________ (2)
30
Section 6: Solving Linear Simultaneous Equations
Solve the following linear simultaneous equations
1. 2x + y =1 7
x+y=9
x = ________ y = ________ (2)
2. x + 2y = 7
3x – y = 14
x = ________ y = ________ (2)
3. 2x + y = 7
x – 3y = 0
x = ________ y = ________ (2)
4. x + 2y = 5
3x + y = 5
x = ________ y = ________ (2)
5. 4x + 2y = 8
3x – 3y = -3
x = ________ y = ________ (2)
6. 2x - 3y = -4
3x + 2y = 7
x = ________ y = ________ (2)
7. x + 2y = 8.5
3x + y = 10.5
x = ________ y = ________ (2)
14
Section 7: Rearranging Formulae
Rearrange the following to make x the subject of the formula
1. y = mx + c
2. 𝑎𝑎 =
𝑥𝑥
+ 𝑑𝑑
3. 𝑗𝑗 =
𝑘𝑘
+ 𝑚𝑚
𝑐𝑐
𝑥𝑥
4. 𝑣𝑣 = 𝑥𝑥𝑣𝑣 3
5. 𝑎𝑎 = 𝜋𝜋𝑥𝑥 2
6. 𝑣𝑣 = 𝜋𝜋𝑟𝑟 2 𝑥𝑥
7. 𝑥𝑥 2 + 𝑦𝑦 2 = 𝑧𝑧 2
8. 𝑎𝑎 = 2𝜋𝜋𝜋𝜋 + 2𝜋𝜋𝜋𝜋𝜋𝜋
8
Section 8: Basic Indices
Simplify the following leaving your answer in index form
1. 𝑎𝑎3 × 𝑎𝑎2
2. 𝑎𝑎14 ÷ 𝑎𝑎5
3. 4𝑥𝑥 2 × 5𝑥𝑥 4
4.
5.
_________________________ (1)
_________________________ (1)
_________________________ (1)
𝑎𝑎7 ×𝑎𝑎3
𝑎𝑎5
_________________________ (1)
10𝑦𝑦 2 ×2𝑦𝑦4
4𝑦𝑦 3
_________________________ (1)
6. 𝑏𝑏 3 × 𝑏𝑏 −3
_________________________ (1)
7. 2𝑥𝑥 2 𝑦𝑦 × 3𝑥𝑥 2 𝑦𝑦 2
_________________________ (1)
8. 8𝑦𝑦 2 𝑧𝑧 5 × 4𝑦𝑦 7 𝑧𝑧 3
_________________________ (1)
9.
10𝑎𝑎2 𝑏𝑏 3
5𝑎𝑎5 𝑏𝑏
10. 24𝑐𝑐 0
_________________________ (1)
_________________________ (1)
4𝑗𝑗 2 𝑘𝑘 4 𝑖𝑖 6
11. 20𝑗𝑗5 𝑘𝑘 2 𝑖𝑖 5
_________________________ (1)
12.
10𝑥𝑥𝑥𝑥 2 𝑧𝑧 3 ×2𝑥𝑥2 𝑦𝑦 4 𝑧𝑧 2
4𝑥𝑥 5 𝑦𝑦 4 𝑧𝑧 6
_________________________ (1)
13. (𝑥𝑥 2 )3
_________________________ (1)
14. (2𝑥𝑥 2 )3
_________________________ (1)
15. (3𝑥𝑥 2 𝑦𝑦 3 )2
_________________________ (1)
Work out the value of the following and it is vital that you include sufficient working out to demonstrate that
you have not used a calculator.
16. 212 ÷ 28
_________________________ (1)
17. 37 ÷ 34
_________________________ (1)
18. (29 )0
_________________________ (1)
1
19. 812
_________________________ (1)
1
20. 1692
_________________________ (1)
1
21. 164
_________________________ (1)
2
22. 273
_________________________ (1)
3
23. 325
_________________________ (1)
2 2
24. �5�
_________________________ (1)
1 3
25. �3�
_________________________ (1)
3 0
26. �10�
_________________________ (1)
27. 5−1
_________________________ (1)
28. 2−3
_________________________ (1)
1 −2
29. �3�
_________________________ (1)
3
30. 16−4
_________________________ (1)
1
31. 1000−3
32.
_________________________ (1)
1
1 −2
�49�
_________________________ (1)
32
Section 9: Surds
Simplify each of the following, rationalising the denominator if necessary and include all working out
1. √28
2. √72
3. 5√27
4. √20 + √80
5. √12 + 3√48 + √75
6.
7.
6
√3
10√2
√5
8. �3 + √2�
2
P
8