Core 2 Key Points

HideShow resource information
  • Created by: LShan261
  • Created on: 25-04-15 18:50

Algebra and functions

  • If f(x) is a polynomial and f(a) = 0, then (x-a) is a factor of f(x)
  • If f(x) is a polynomial and f(b/a) = 0, then (ax-b) is a factor of f(x)
  • If a polynomial f(x) is / by (ax-b) then the remainder is f(b/a)
1 of 11

The sine and cosine rule

  • Sine rule
    • a/sinA = b/sinB = c/sinC
    • 2 angles & length of opposite side --> unknown side
    • 2 sides & opposite angle --> unknown angle
  • Cosine rule
    • a^2 = b^2 + c^2 - 2bccosA or b^2 = a^2 + c^2 - 2accosB or c^2 = a^2 + b^2 - 2abcosC
    • 2 sides & angle between --> unknown side
    • 3 lengths --> unknown angle
  • Rearranged form of cosine rule
    • unknown angle
    • cosA = b^2 + c^2 - a^2/2bc or cosB = a^2 + c^2 - b^2/2ac or cosC = a^2 + b^2 - c^2/2ab
  • area of a triangle
    • 1/2absinC
    • length of 2 sides (a and b) & angle C
2 of 11

Exponential and logarithms

  • function y = a^x, or f(x) = a^x --> a = constant --> exponential function
  • logan = x --> a^x = n --> a = base
  • loga1 = 0
  • logaa = 1
  • log10x --> logx
  • Laws of logarithms
    • logaxy = logax + logay --> multiplication law
    • loga(x/y) = logax - logay --> division law
    • loga(x)^k = klogax --> the power law
  • From the power law
    • loga(1/x) = -logax
  • Solve equation e.g. a^x = b by 1st taking logarithms (to base 10) of each side
  • Change of base rule
    • logax = logbx/logba
  • From the change of base rule
    • logab = 1/logba
3 of 11

Coordinate geometry in the (x,y) plane

  • Mid-point of (x1,y1) and (x2,y2)
    • (x1 + x2/2, y1 + y2/2)
  • distance d between (x1, y1) and (x2, y2)
    • d=√[(x2-x1)^2 + (y2-y1)^2]
  • Equation of the circle centre
    • (x-a)^2 + (y - b)^2 = r^2
  • Chord = line that joins two points on circumference of a circle
  • The perpindicular from the centre of  circle to chord --> bisects cord
  • Angle in a semicircle --> right angle
  • Tangent = line that meets circle @ 1 point only
  • Angle btwn tangent & radius = 90 degrees
4 of 11

The binomial expansion

  • use Pascal's Triangle to multiply out bracket
  • The binomial expansions is
5 of 11

Radian measure & its application

  • If arc AB has length, r, then angle AOB is 1 radian (1c or 1rad)
  • Radian = angle subtended @ centre of circle by an arc whose length is = to that of the radius of the circle
  • 1 radian = 180 degrees/π
  • Length of an arc
    • l = rØ   (Ø = theta)
  • Area of a sector
    • 1/2r^2Ø
  • Area of a segment
    • 1/2r^2(Ø - sinØ)
6 of 11

Geometric sequences and series

  • next term --> multiply by common ratio
  • nth term
    • ar^n-1 --> a=1st term   r=common ratio
  • Sum to n terms
    • Sn = a(1-r^n)/1-r or Sn = a(r^n - 1)/r-1
  • Sum to infinity
    • S∞ = a/1-r
7 of 11

Graphs of trigonometric functions

  • the x-y plane / into quadrants
  • for all values of Ø, definitions of sin Ø, cos Ø and tan Ø are taken to be
    • sinØ = y/r
    • cos Ø = x/r
    • tan Ø = y/x
    • x and y = coordinates     r = radius
  • 1 st quadrant --> Ø = acute
    • All functions = +ve
  • 2nd quadrant --> Ø = obtuse
    • Sine = +ve
  • 3rd quadrant --> Ø = reflex
    • 180 degrees < Ø < 270 degrees
    • Tangent = +ve
  • 4th quadrant --> Ø = reflex
    • 270 degrees < Ø < 360 degrees
    • Cosine = +ve
  • Sine & cosine have a period of 360 degrees (2π rad) --> tangent period of 180 degrees (π rad)
8 of 11

Differentiation

  • Increasing function f(x) in interval (a.b). f'(x) > 0 in the interval a≤x≤b
  • Decreasing function f(x) in the interval (a,b), f'(x) < 0 in the interval a≤x≤b
  • Maximum point = where f(x) stops increasing & begins to decrease
  • Minimum point = f(x) stops decreasing & begins to increase
  • Point of inflexion = where the m is @ a max/min value in the neighbourhood of points
  • Stationary point = 0 m --> max/min/inflexion
  • Find the coordinates of a stationary point
    • find dy/dx --> solve f'(x) = 0 --> sub into y = f(x) to find corresponding values of y
  • Determining the nature of a stationary point
    • dy/dx = 0 & d^2y/dx^2 > 0 --> minimum point
    • dy/dx = 0 & d^2y/dx^2 < 0 --> maximum point
    • dy/dx = 0 & d^2y/dx^2 = 0 --> max/min/inflexion
    • dy/dx = 0 & d^2y/dx^2 = 0 BUT d^3y/dx^3 ≠ 0 --> point of inflexion
  • need to find the max/min point
    • establish formula for y in terms of x
    • differentiate
    • derived function = 0
    • find x then y
9 of 11

Trigonometrical identities & simple equations

  • tan Ø = sin Ø/cos Ø
  • sin^2Ø + cos^2Ø = 1
  • sinx = k
    • 1st solution --> a = sin^-1(k)     (a = alpha)
    • 2nd solution --> (180 degrees - a) or (π - a), if working in radians
  • cosx = k
    • 1st solution --> a = cos^-1(k)
    • 2nd solution --> (360 degrees - a) or (2π - a)
  • tanx = k
    • 1st solution --> a = tan^-1(k)
    • 2nd solution --> (180 degrees + a) or (π + a)
10 of 11

Integration

  •  = f(b) - f(a)

  • Area beneath curve w/ equation y = f(x) & between line x=a and x=b
  • Area between a line (equation y1) and a curve (equation y2) is given by
    • Same as above BUT f(x) = (y1 - y2)
  • Trapezium rule
    • (http://upload.wikimedia.org/math/f/1/d/f1d1ce91413083417116cb908d2d5a31.png)
    • h = b-a/n and yi = f(a + ih)
11 of 11

Comments

No comments have yet been made

Similar Mathematics resources:

See all Mathematics resources »See all Core 2 resources »