This document contains multiple examples and problems related to trigonometry, Pythagorean theorem, algebra, number work, formulas, and probability. It includes examples of finding hypotenuses using Pythagorean theorem, calculating trigonometric ratios, factorizing algebraic expressions, solving simultaneous equations, expanding brackets, converting between fractions and decimals, calculating percentages, identifying sequences, writing formulas, finding probabilities, calculating volumes of shapes, and describing regions with inequalities.

1. Trig & Pythag - 1
What is Pythagoras’ theorem?
a² + b² = h²
(where h is the hypotenuse)

2. Trig & Pythag - 2
How long is the hypotenuse?
h = 18cm
10cm
15cm

3. Trig & Pythag - 3
What are the trigonometric formulae?
sin A=opp
hyp
cos A=adj
hyp
tan A=opp
adj

4. Trig & Pythag - 4
Find angle A
A
A = 53°
6cm
B
8cm
C

5. Trig & Pythag - 5
A ladder leans against a vertical
wall so that is makes an angle of
35° with the wall.
The top of the ladder is 2m up the wall.
How far out is the foot of the ladder?
2.9 m

6. Algebra - 1
Factorise the following expression:
12z² – 6z
= 6z(2z – 1)

7. Algebra - 2
Expand the following expression:
(x + 3)(x + 6)
= x² + 9x + 18

8. Algebra - 3
Factorise the following expression:
x² + 5x + 6
= (x + 3)(x + 2)

9. Algebra - 4
Solve the following pair of simultaneous equations:
x+y=5
3x + y = 11
x=3
y=2

10. Algebra - 5
Expand the following expression:
(5x – 2)(5x + 2)
= 25x² – 4

11. Number work - 1
Write the following fraction as a mixed number:
25 = 8⅓
3

12. Number work - 2
Without a calculator work out:
1.23 – 0.67 = 0.56

13. Number work - 3
Fill in the blank:
3.7 x 4.2 = 15.54

14. Number work - 4
Express 12mm as a percentage of 6cm.
20%

15. Number work - 5
Write the following number as an ordinary
Number:
3.45 x 10³ - 9 x 10-¹
3450 – 0.9 = 3499.1

16. Formulas - 1
What are the next 3 terms of this sequence?
2, 5, 10, 17, 26, 37, 40…

17. Formulas - 2
If R = pq – w
R=2
Find R when p = 2, q = 3 and w = – 4

18. Formulas - 3
A taxi firm charges 50p per mile
plus a fixed charge of £2.00. Write
down a formula for the cost (C) of
hiring this taxi to travel ‘n’ miles.
C = 2 + 0.5n

19. Formulas - 4
If y = zx – 4w
w=3
Find w when y = 3, z = 3 and x = 5

20. Formulas - 5
Use an appropriate formula to find the
radius of the following circle.
Give your answer 3 sf
r = 3.18cm
Circumference = 20cm

21. Pot Luck - 1
Give the inequality which describes the shaded
region.
x≤2

22. Pot Luck - 2
Sally writes the letters of the word ‘MATHEMATICS'
on separate cards and places them in a bag. She then
draws a card at random.
What is the probability that Sally chooses the
letter 'A'?
P(A) = 2
11

23. Pot Luck - 3
You are about to leave for holiday, but you forgot socks!
You race back to your room, but all the lights are off, so
you can't see the colour of the socks.
Never mind, because you remember that in your drawer
there are ten pairs of white socks, ten pairs of black
socks, and eleven pairs of blue socks, but they are all
mixed up.
How many of your socks do you need to take before
you can be sure to have at last one matching pair?
4 socks

24. Pot Luck - 4
What is the volume of this prism?
Volume = 40cm³

25. Pot Luck - 5
Give the set of inequalities which describe the
unshaded region.
x≤3
y≥x
y≥1