Core 1 Key points

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  • Created by: LShan261
  • Created on: 24-04-15 21:37

1: Algebra and functions

  • Simplifying expressions --> Like terms & rules of indices 
    • A^m x A^n = A^m+n
    • A^m / A^n = A^m-n
    • A^-m = 1/A^m
    • A^1/m = m√A
    • A^n/m = m√A^n
    • (A^m)^n = A^mn
    • A^0 = 1
  • Expanding expressions --> Each term inside X each term outside --> Factorising = opposite
  • Quadratic expression --> ax^2 + bx + c
  • Difference of squares --> x^2 - y^2 = (x+y)(x-y)
  • Surd = √prime number
  • Manipulation surds
    • √ab = √a x √b
    • √a/b = √a / √b
  • Rationalise surds
    • 1/√a --> multiply top and bottom by √a
    • 1/(a+√b) --> multiply top and bottom by (a-√b) --> opposite for 1/(a-√b)
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2: Quadratic functions

  • General form --> y=ax^2 + bx + c
  • Solving quadratic equations
    • factorisation
    • Completing the square --> x^2 +bx = (x + b/2)^2 - (b/2)^2
    • Formula --> x= -b+-√(b^2 - 4ac) / 2a
  • QE has 2 solutions - may be =
  • To sketch a Quadratic graph 
    • Shape: a > 0 = Happy , a < 0 = Sad
    • X-axis and y-axis crossing points
    • discriminant b^2 - 4ac --> general shape
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3: Equations and Inequalities

  • Linear simultaneous equations --> elimination or substitution
  • 1. linear and 1. quadratic (simultaneous) --> substitutiom
  • X or / inequality by -ve no. --> </> = opposite
  • To solve a quadratic inequality
    • Solve corresponding quadratic equation
    • sketch graph of quadratic function
    • use sketch to find required set of values.
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4: Sketching Curves

  • Shapes of the basic curves

(http://www.met.reading.ac.uk/pplato2/h-flap/math4_4f_4.png)(http://myarnell.diymaths.com/Pictures/cubicgraph.jpg)(http://mathematicsi.com/wp-content/uploads/Sketching-Polynomials11.png)(http://www.mathscool.com/images/Graphs/curve_recip+grid.gif)

  y = x^2           y = x^3      y = (x-a)(x-b)(x-c)     y=1/x

  • Transformation
    • f(x+a) --> -a in the x-direction
    • f(x) + a --> +a in the y-direction
    • f(ax) --> stretch of 1/a in x-direction (X x-coordinates by 1/a)
    • af(x) --> stretch of a in y-direction (X y-coordinate by a)
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5: Coordinate geometry in the (x,y) plane

  • General form - y = mx + c --> m = gradient & c = y intercept
  • Gradient of line joining two points together
    • m = y2-y1 / x2 - x1
  • equation of line w/ m and passes through (x1, y1)
    • y - y1 = m(x -x1)
  • equation of line passing through two points
    • y-y1/y2-y1 = x-x1/x2-x1
  • perpendicular has gradient -1/m
  • two lines perpindicular gradient = -1.
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6: Sequences and series

  • sequence = series of numbers with a set rule
  • each number = term
  • nth term = general term
  • expressed as a formular for the nth term e.g. Un = 4n + 1
  • Arithmetic sequence
    •  Uk+1 = Uk + n
    • a + (a+d) + (a + 2d) + (a + 3d) + (a + 4d)
  • nth term of an arithmetic seqence 
    • a + (n-1)d --> a = 1st term and d = common difference
  • Sum of an arithmetic sequence
    • Sn = n/2(2a + (n-1) d)
    • Sn = n/2(a+L) --> L = last term
  • Use ∑ 
    • 10∑r=1 (5 + 2r) = 7 + 9 + ... + 25
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7: Differentiation

  • m of y = f(x) @ specific point = m of tangent to curve @ that point
  • m  of tangent @ any point = dy/dx
  • f'(x) = derived function
  • y = f(x) --> dy/dx = y = f'(x) --> dy/dx = nx^n-1
  • 2nd order derivative = d^2y/dx^2
  • Equation of tangent to curve @ point A
    • y - f(a) = f'(a)(x-a)
  • Equation of normal to curve @ point A
    • y - f(a) = -1/f'(a) (x-a)
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8: Integration

  • If dy/dx = x^n --> y = (1/n+1)(x^n+1)+c
  • If dy/dx = kx^n --> y = (kx^n+1/n+1) + c
  • ∫x^ndx = (x^n+1/n+1) +c
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