A LEVEL MATHEMATICS

Why study A Level Mathematics? Pythagoras said “All things are number”

And some thoughts

University opportunities It’s enjoyable Variety of jobs Happier person It’s logical University opportunities More Money Salary is a number and numbers make us happy It’s stimulating Better wages It’s rewarding

A Level Maths is made up of 4 modules: 2 Pure Modules (Algebra,Geometry,Calculus,etc) 2 Applied Module (Statistics and Mechanics) You have no choice which applied module you do for Maths although there is some choice for Further Maths. Your options are: 2 Pure Modules Choices from Pure Modules Further Statistics Further Mechanics Decision Maths

Why study A Level Mathematics Pythagoras said “All things are number”

Algebra Inequalities

Inequalities < means “is less than” What are Inequalities? Inequalities are just another time-saving device invented by lazy mathematicians. They are a way of representing massive groups of numbers with just a couple of numbers and a fancy looking symbol. < means “is less than”  means “is less than or equal to” > means “is greater than”  means ”is greater than or equal to”

Representing Inequalities on a Number line x < 3 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 x > -4 and x ≤ 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

Solving Linear Inequalities If you multiply or divide by a NEGATIVE, the inequality sign swaps around! Why on earth does the sign swap around?

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Solving Linear Inequalities Graphically

For questions like this, just deal with each inequality in turn, shading as you go! Note: When you have got more than one inequality like this, it’s normally best to shade the region you DON’T WANT, so you can leave the region you do want blank! √ Putting it all together leaves us the blank region in the middle that satisfies all the inequalities!

5. Solving Quadratic Inequalities ! 1. Do the same to both sides to make the quadratic inequality as simple as possible 2. Sketch the simplified quadratic inequality 3. Use the sketch to find the values which satisfy the inequality Example or

Solve the inequality x2 + x – 6 > 0. Example 1 Solve the inequality x2 + x – 6 > 0. Example 2 Solve the inequality x(x + 2) ≤ 1.

(a) x2 + 3x – 10 < 0 -5 < x < 2 Solve these inequalities using whichever method you prefer. (a) x2 + 3x – 10 < 0 -5 < x < 2 (b) x2 + 11x + 18 ≥ 0 x ≤ -9 or x ≥ -2 (c) x2 – 9x + 18 > 0 x < 3 or x > 6 (d) 2p2 + 5p – 3 ≤ 0 -3 ≤ p ≤ 1/2 (e) 3a2 – a < 0 0 < a < 1/3 (f) 6t2 + 5t – 6 ≥ 0 t ≤ -3/2 or t ≥ 2/3

Quadratic Inequalities: Problem Solving Harry is digging a rectangular flower bed. The perimeter of the bed is to be 60m and the area is to be at least 200m2. Work out the range of possible values for the width of the flower bed.

Quadratic Inequalities: Problem Solving Example 4 The diagrams show the first three pentagonal numbers, 1, 5, 12 ... The nth pentagonal number is Work out the first pentagonal number that is greater than 100. n (3n – 1) 2

Match up the question to the answer.! Inequalities Match up the question to the answer.! 1. 2. 3. 4. 5. 6. 7. 8.

answers 1g 2a 3e 4d 5c 6 7b 8f