maths revision, c1 as maths

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  • Created by: Dharshini
  • Created on: 03-12-12 22:02

INDICES

  • am x an = am+n
  • am/an = am-n
  • a-m = 1/am
  • a0 = 1
  • am/n = n√(a)m
  • (am)n = am*n
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SURDS

  • √(a) * √(a) = a
  • √(a) * √(b) = √(ab)
  • √(a/b) = √(a) /√(b)
  • -√(a)2  = a
  • -√(a)3 = -√(a) * -√(a) * -√(a) = -a√(a)
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DISCRIMINANT

  • Discriminant tells you how many solutions an equation has
  • Discriminant → the b2 – 4ac part of a quadratic equation
  • Roots → another name of solutions of an equation
  • b2 – 4ac > 0 → 2 solutions/roots
  • b2 – 4ac = 0 → 1 solution/roots/repeated roots
  • b2 – 4ac < 0 → no solutions/roots
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SOLVING QUADRATIC INEQUALITIES

EG (2x – 4) (x+3) > 0

  • Step 1: solve quadratic-

x = (4/2) = 2  or  x = -3

  • Step 2: sketch graph 
  • Step 3: check regions for positive and negative

Therefore:

x < -3  and  x > 2

because within this region y in below 0

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COORDINATE GEOMETRY

  • (difference in y)/(difference in x) → (dy/dx)
  • y = mx + c
  • 2 lines are parallel if they have the same gradient
  • 2 lines are perpendicular if their gradients multiply to give -1
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GRAPH TRANSFORMATION

·         y = f(x)

f(x) + c

Moves up or down by c

f(x + c)

Moves left or right by –c

af(x)

Stretched in y direction by scale factor of a

f(ax)

Stretches in x direction by scale factor of (1/a)

-f(x)

Reflects in x axis

f(-x)

Reflects in y axis

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DIFFERENTIATION

  • Used to find gradients in a curve (eg for rate of change)
  •  (dy/dx) is the equation for differentiation
  • If y = axn; then (dy/dx) = naxn-1
  • Multiply by the power; take 1 away from the power
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INTERGRATION

  • The reverse process of differentiation
  •  Add 1 to the power
  • Divide by the new power
  • y = axn; axn dx = (axn+1)/(n+1)
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SEQUENCES

Position

0

1

2

3

Term

-2

2

6

10

1st difference

 

+4

+4

+4

 

To work out the formula for this sequence first find out what the 0th term is.

Un = 4n – 2

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ARITHMETIC PROGRESSION

The nth term for an arithmetic progression is : a + (n − 1)d; where a is the first term, n is the number of terms and d is the common difference.

 Sn = a + (a + d) + (a + 2d) + . . . + (L − 2d) + (L − d) + L

Sn is the sum of the arithmetic series, a is the first term, d is the common difference and L is the last term.

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PROOF OF Sn

Sn = a + (a + d) + (a + 2d) + . . . + (L − 2d) + (L − d) + L

Sn = L + (L − d) + (L − 2d) + . . . + (a + 2d) + (a + d) + a

Copy out the formula but reverse it. Then add the two equations together:

2Sn = (a + L) + (a + L) + (a +L) + . . . + (a + L) + (a +L) + (a + L)

2Sn = n (a+L)

Sn = (n/2) (a+L)

L = a + (n − 1)d

Sn = (n/2) (2a+(n − 1)d)

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SIGMA NOTATION

n

Un

r

  • n is the end point of the sequence
  • r is the starting point of the sequence
  • Un is the nth term
  • is the symbol for sum of.
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