# Maths c1 revision

- Created by: Anna
- Created on: 12-01-13 10:17

## Surds and Indices

a^^{m} x a^^{n} = a^^{m+n}

a^^{m}/a^^{n} = a^^{m-n}

(a^^{m})^^{n} = a^^{mn}

a^^{1/m} = ^{m}√a

a^^{-m} = 1/a^^{m}

a^^{m/n} = ^{n}√a^^{m} = (^{n}√a)^^{m}

a^^{0} = 1

(√a)^^{2} = √a √a = a

√ab = √a √b

√a/b = √a/√b

## Algebra

*Brackets/ Common factors*

a (b + c + d) = ab + ac + ad

(a + b)(c + d) = ac + ad + bc + bd

(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2

(x + y + z)(a + b + c + d) = x(a + b + c + d) + y(a + b + c + d) + z(a + b + c + d)

(y + a)^2 (x - a)^3 + (x - a)^2 = (x - a)^2((y + a)^2 (x - a) + 1)

*Algebraic fractions*

a/x + b/x + c/x = a + b + c/x

ax + ay/az = a(x + y)/az = x + y/z

## Proof

**Exhaustian**

Prove "for any integer of x, the value of f(x) = x^3 + x + 1 is an odd integer"

- even integer, 2n, substitued into equation: (2n)^3 + 2n + 1 = 4(2n^3 + n) + 1. the sum and product of any integers are also integers, multiplied by 2 is even, +1 makes it odd.
- odd integer, 2m + 1, substituted into function: (2m + 1)^3 + 2m + 1 + 1 = 2 (4m^3 + 6m^2 + 4m) +3 is and odd integer too.

**Contradiction**

Prove "if x^2 is even, then x must be even"

- if x is odd, 2k + 1, (2k + 1)^2 = 4k^2 + 4k + 1 if x is odd, so is x^2. so if x^2 is even so must x.

**Counter- example**

Disprove "for any pair of real numbers x and y, if x>y then x^2 + x > y^2 + y

- ONE CASE IS NEEDED. eg. when x = 2 and y = -4
- 2 > -4 but 6 < 12 , so the statement is untrue

## Quadratics

Quadatic formula: x = -b (+/-) **√b^2 - 4ac** / 2a

Discriminant

- > 0. two distinct real roots
- = 0. one repeated root
- < 0. no real roots

Graphs

coefficent of x^2 = positive or negative graph. u-shaped positive, n-shaped negative.

when y = 0, the x values determine where the graph crosses that axis, vice versa

the minimum/maximum co-ordinates are found from *Completing the square*

- in the form f (x) = (x - a)^2 + b
- the min/max point is (a,b)

Almost quadratics: x^4 + 6x^2 + 9 = 0 can be written as (x^2)^2 - 6(x^2) + 9 or even y^2 - 6y +9 if y=x^2

## Completing the square

QUADRATIC. (2x^2 + 8x - 5)

TAKE OUT THE COEFFIENT OF X^2. 2(x^2 + 4x) - 5

HALF THE COEFFIENT OF X AND WRITE IN SQUARE FORM. 2(x + 2)^2 - 5

EXPAND BRACKETS. (x + 2)^2 = (x^2 + 4x + 4) - 5

..

## Inequalities

Solved like equations, but need to keep inequalities correct way round

Quadratic inequalities can be shown on a graph.

-x^3 + 2x + 3 >/= (x + 1)(x - 3) so x = -1 and 3

> or = to 0 means that the quadratic is satisfied when -1 </= x </= 3

- if dividing an inequality by a negative, the sign changes direction
- also do not divide by a variable, eg x or y, as it may be negative

Simultaneous equations

- first try and get the eqautions to match, e.g. to have a shared coefficient of x or y, you can then add or take away these varibles to get 0.
- this will allow you to find the value of the other variable, and therefore the first too after substituting it into the eqaution.
- you can also substitute equations into one another or make them equal each other in order to solve an equation, especially when finding points of intersection e.g. x^2 - 4x + 5 = 2x - 3

## Co-ordinate geometry

the equation of a line between two points A (x1,y1) B (x2,y2)

Gradient

m = change in y/change in x = y1-y2 / x1-x2

parallel lines have equal gradients

perpendicular lines have negative reciprocal gradients e.g. their sum is -1

Midpoint co-ordinates

(x1+x2)/2 , (y1+y2)/2

Distance between the points uses Pythagorus

√ (x2 - x1)^2 + (y2 - y1)^2

## Circles

(x - a)^2 + (y - b)^2 = r^2

radius r and centre (a,b) or x^2 + y^2 = r^2 if centre is (0,0)

the form: x^2 + y^2 - 6x - 4y + 4 = 0 can be changed to the familiar form using *Completing the square*. this simplifies to (x - 3)^2 + (y + 2)^2 = 9. i.e. the centre = (3,-2) and r = 3

properties

- angles in a semicircle are right angles
- perpendicular from the centre to a chord bisects the chord
- a radius and tangent to the same point will meet at right angles

## Graph sketching and transformations

n^2 - u shaped graph

-n^2 - n shaped graph

## Comments

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