Download the transitional activity

AS Level Double Mathematics
Transition activities
The questions in this booklet should be completed, using the Head Start to AS Maths
book for guidance (‘Head Start to AS Maths’ Published by CGP (available on Amazon for
about £4) ISBN 9781841469935). The material included is all in the GCSE Maths Higher
Syllabus. Answers, with worked solutions, should be written by hand and brought to your
first Maths lesson at the College.
Additional resources that you may find interesting:
+plus magazine (http://plus.maths.org/content/) for interesting articles on application of
mathematics e.g.



The maths of infectious diseases:
Constructing our lives: the mathematics of engineering
Mathematics and the nature of reality
Enriching mathematics site (http://nrich.maths.org/public/) which has a wide range of puzzles and
articles
There are many interesting popular maths books, here are just a few examples:

‘Professor Stewart's Cabinet of Mathematical Curiosities’ by Ian Stewart ISBN-10: 1846680646

‘Fermat's Last Theorem: The story of a riddle that confounded the world's greatest minds for 358
years’ by Simon Singh ISBN-10: 1841157910

The Penguin Dictionary of Curious and Interesting Numbers (Penguin Press Science) ISBN-10:
0140261494
AS Summer Transition Work: AS Double Maths
1. Indices:
a)
What is the value of 16 - 25 + 34 - 43 + 52 - 61?
Simplify the following expressions:
b)
𝟒𝟓 × 𝟒𝟐
c)
𝟒𝟒 × 𝟒
(𝟐𝟓 )𝟐
d) 𝟓𝟎 × 𝟓𝟑 × 𝟓𝟒
𝟐𝟑
e)
𝒙𝟐 × 𝒙𝟓
(𝒙𝟑 )𝟐
Evaluate:
𝟑
f) 𝟒𝟐
k)
𝟑
g) 𝟖𝟏𝟒
h) −𝟕−𝟑
i) (
𝟖 𝟐
𝟐𝟕
)𝟑
j) 𝟏𝟎𝟎
Which of these values is the odd one out?: 26 43 85/3 163/2 326/5
2. Factorise
a) 12 + 6a
b) a2b + ab2
c) x2 – 36
d) x2 + 5x +6
e) x2 – 2x – 24
f) x3 – 6x2 + 8x
g) x³ + 5x² - 14x
h) 3x² - 10x + 3
i) 2y² + 7y + 6
j) 6x² + 11x + 4
k) 4x² + x - 3
l) 3x3 - 13x2 + 4x
3. Simultaneous equations
Solve the following pairs of simultaneous equations
a) 2x + y = 7
x+y=4
b) 3x + 4y = 23
2x + 5y = 20
c) x² + 2y = 12
y = 3x - 2
d) x² + 3xy + y² = 11
x+y=3
e)
If 5x - y = 18 and 5y - x = 12, what is the value of x - y?
f)
𝟏
𝟏
Suppose that 𝒙 − 𝒙 = 𝒚 − 𝒚 and 𝒙 ≠ 𝒚. What is the value of xy?
4. Equations
Solve the following equations
a) 5(x + 2) = 2x + 22
b) 2(x - 4) = 3x + 1
c) x² + 8x + 15 = 0
d) x² - 4x - 21 = 0
e) d² - 9d + 20 = 0
f) x² - 16 = 0
g) x² + 3x - 5 = 0
h) 2x² - 3x - 5 = 0
i) x³ + 10x² + 21x = 0
5. Rearranging Formulae
Rearrange the following to make y the subject:
a) x + y = 10
b) 4x + 2y – 6 = 0
c) 4x + y – 5 = 0
d) 3x – y – 7 = 0
e) y – 6x + 9 = 0
f) x – 2y = 10
Rearrange the following to make x the subject:
𝒙
g) 𝒚 = 𝟓 + 𝟏𝟕
𝒙
i) 𝒛 = √𝒙+𝒚
h) 𝒂 = 𝒃 − 𝒄𝒙
𝟏
𝟏
j) 𝒗𝟐 = 𝟐𝒌 (𝒙 − 𝒂)
k) 𝒅 =
𝟐(𝑺−𝒙𝒏)
𝒏(𝒏−𝟏)
6. More Equations
Solve the following equations
𝟏
a)
c)
𝟏
𝟑𝒙
𝒎
𝟐
𝟕
+ 𝟒𝒙 = 𝟐𝟎
+
𝒎
𝟑
+ 𝟑=𝟐+
b)
𝒎
𝟔
d)
𝟒
𝟑
𝒚 − 𝟓𝒚 = 𝟐
𝟕
𝒙+𝟑
𝟒
−
𝒙−𝟑
𝟓
=𝟐
7. Surds
Evaluate the following:
a) √4 × √25
b) √7 × √7
c) √(24 + √34)
f) √2 + √2 + √2 + √2 = 2x. What is the value of x?
8. Sequences:
Complete the next two terms in these sequences
a) 2 5 8 11 14 17
b) 2 4 8 16 32 64
c) 1 4 9 16 25 36
d) 2 3 5 7 11 13
e) 1 2 6 24 120 720
f) 1 1 2 3 5 8
d) √2 × √18
*
e) √6 × √10 × √15
9. Miscellaneous
a) Put the results of the following questions in order from smallest to largest
1. The number of cm in a foot
2. The number of grams in an ounce
3. The mean of the prime numbers between 20 and 40
4. The median of the first 10 square numbers
5. The mode of the number of days in a month
6. The circumference of a circle with radius 5 units
7. The area of a rhombus with base 8 units and height 3.8 units
8. The perimeter of a rhombus with base 8 units and height 3.8 units
9. The surface area of a cube with side length 2.3 units
10. The volume of a cube with side length 3.1 units
b) A bag contains red, yellow, green and purple marbles. When a marble is drawn from the bag it
is not replaced. At the beginning of each question there are 3 red, 3 yellow, 3 green and 3 purple
marbles in the bag.
I.
If 1 marble is drawn from the bag, what is the probability that it is red?
II.
If 2 marbles are drawn from the bag, what is the probability that they are the same colour?
III.
How many marbles should be drawn from the bag to ensure two marbles of the same colour
are drawn?
IV.
How many marbles should be drawn from the bag to ensure at least one marble of each
colour is drawn?
V.
3 marbles are drawn from the bag. What is the probability that none of them are purple?
c)
How many 2-digit numbers can be formed using the digits 1, 3, 5, 7 and 8 which are divisible
by 3?
d)
The number 114 is the sum of 4 consecutive positive integers. What are they?
e)
p and q are two positive integers. p² + q² = 170. What are the values of p and q?
10. An Introduction to Complex Numbers
You may have been told that it is not possible to find the root of a negative number. If you try to find
the value of √−𝟑 using a calculator, you will usually be presented with a “Math Error”.
However, it is possible to solve this problem by defining i to be equal to √−𝟏 . This number is known as
an imaginary number.
Using i, an expression can be written for the square root of any negative number.
Example1: Find an expression for the square root of -4 and that of -7 in terms of i
√−𝟒 = √𝟒 × √−𝟏 = 2𝒊
√−𝟕 = √𝟕 × √−𝟏 = √𝟕𝒊
A number can be made up of a real and an imaginary part, e.g. 5 + 4i. 5 is the real part of the
number and 4i is the imaginary part. These numbers are known as complex numbers.
Complex numbers result from the solution of a quadratic equation where the discriminant, (b² 4ac), is negative.
Example 2: Solve x² + 2x + 5 = 0.
Using the quadratic formula
𝒙=
−𝟐 ± √𝟐²−𝟒×𝟏 ×𝟓
𝟐 ×𝟏
𝒙=
−𝟐 ± √−𝟏𝟔
𝟐
𝒙=
−𝟐 ± 𝟒𝒊
𝟐
−𝒃±√𝒃𝟐 −𝟒𝒂𝒄
𝟐𝒂
a=1, b=2, c=5
𝒙 = −𝟏 ± 𝟐𝐢
Complex numbers can be added together by adding their real and complex parts individually.
Example 3: Add the complex numbers (4 + 5i) and (2 - i)
(4 + 5i) + (2 - i) = 6 + 4i
Complex numbers can be multiplied together. Note that since i = √−𝟏 then i × i = -1.
Example 4: Find 5i × 6i
5i × 6i = 30 × (-1)
5i × 6i = -30
Example 5: Find (3 + 2i) × (4 + 5i)
3 × (4 + 5i) = 12 + 15i
2i × (4 + 5i) = 8i - 10
(3 + 2i) × (4 + 5i) = (12 + 15i) + (8i - 10)
(3 + 2i) × (4 + 5i) = 2 + 23i
Questions:
Write the following negative roots as an expression in terms of i
a) √−𝟓
b) √−𝟗
c) √−𝟏𝟐𝟏
d) √−𝟖
Solve the following equations
e) x² + 2x + 10 = 0
Find the values of the following
h)
(2 + i) + (4 + 5i)
i)
(3 + 2i) + (5 - 10i)
j)
4 × 3i
k)
6i × -2
l)
3i × 5i
m)
3 × (4 + i)
n)
3i × (4 + i)
o)
(2 + i) × (4 + 3i)
p)
(2 - i)( (4 + 3i)
q)
(5 - 2i) × (2 - 3i)
f) x² - 3x + 4 = 0
g) 2x² - x + 3 = 0