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HSC Extension 2 Mathematics
HSC Extension 2 Mathematics Topics     Complex Numbers      Curve Sketching      Polynomials      Integration      Conics ...
Maths Extension 2 - Complex Numbers  Complex Number Rules   z   x   iy    A complex number is represented in this form   T...
Maths Extension 2 - Complex Numbers  General Ideas of Complex Numbers Complex Numbers are written in the form of a real pa...
Maths Extension 2 - Complex Numbers  Proof by Mathematical Induction  cos     i sin      n   cos n     i sin n   Let n   1...
Maths Extension 2 - Complex Numbers  Expressing Complex Numbers in Mod-Arg Form    3   3i Mod z    x2   y 2  9 9   18  3 2...
Maths Extension 2 - Complex Numbers  Theory  a   ib     c   id    a   ib       c   id   Example Addition  3   4i     2   5...
Maths Extension 2 - Complex Numbers  Finding Square Roots of Complex Numbers x   iy x   iy    a   ib    a   ib 2   a2     ...
Maths Extension 2 - Complex Numbers  Complex Numbers on the Argand Diagram  Z2  Z1  Subtraction of Vectors Flip the tail, ...
Maths Extension 2 - Complex Numbers  LOCUS  z      a   ib      z      x   iy    z         4   3i       z         2     5i ...
Maths Extension 2 - Graphs  Graphing y   f x  y   f  x  y 10  2   x-2  -3   x-2  2 -3  5 x 10  5  5  10  5  x  C 10  5  10...
Maths Extension 2 - Graphs  y   C. f  x  y    C   1, the function is    steeper      0   C   1, the function is    shallow...
Maths Extension 2 - Graphs  y       f  x  y    Flip about the x-axis   All positive become negative   All negative become ...
Maths Extension 2 - Graphs  y   f  x  y 10    All negative y values become positive  5 x 10  5  5  10  5  10  y   f  x   y...
Maths Extension 2 - Graphs  y   f  x   y    If y value is   1, y value is smaller but still   1   If y value is   1, y val...
Maths Extension 2 - Graphs  y   1 f   x   y 5  or y     f   x        1  x    Asymptotes where y   0   For f x , y values  ...
Maths Extension 2 - Graphs  Graphing y   f x , y   g x  y   g  x  y   sin x  y 10 5 x 10  5  5  10  5 10  y   f   x    g  ...
Maths Extension 2 - Graphs  y   f   x .g   x  y    y co-ordinates are multiplied  10  y  5 5 x x 5  10  5  5  5  10  5 10 ...
Maths Extension 2 - Graphs  y   f  g  x     Composite functions y   f  u   , u   g  x   y 10  Take  f   x    e x g   x    ...
Maths Extension 2 - Polynomials  Polynomials   Definitions and properties of polynomials   Factors   Roots   Fields   Q Ra...
Maths Extension 2 - Polynomials  Definitions and properties of polynomials Polynomial Expression P x    p0xn   p1xn-1   p2...
Maths Extension 2 - Polynomials  Factors and Roots Factor A polynomial that divides into another and has remainder 0  x   ...
Maths Extension 2 - Polynomials  Finding zeros over the complex field 1. If one root is complex, then one of the other roo...
Maths Extension 2 - Polynomials  Factorization and Division of Polynomials Factorizing polynomials 1. Simple factorizing. ...
Maths Extension 2 - Polynomials  Division of polynomials P x  Dividend    A x    Divisor     Q x     Quotient    R x    Re...
Maths Extension 2 - Polynomials  Factor and Remainder Theorems Remainder Theorem   If a polynomial P x  is divided by  x  ...
Maths Extension 2 - Polynomials  Rational Roots Let P x  have degree n with integer coefficients. p Suppose P x  has a rat...
Maths Extension 2 - Polynomials  Multiplicity     Repeated Roots _x   a is a repeated root of P x  or multiplicity r. so  ...
Maths Extension 2 - Polynomials  For polynomials which have  Double roots Differentiate one time Triple roots Differentiat...
Maths Extension 2 - Polynomials  Relationship between the roots and coefficients of a polynomial equation Quadratic   ax2 ...
Maths Extension 2 - Polynomials  Example Roots are    ,    ,    for 2x3     4x2     3x     1 1.         1          1      ...
Maths Extension 2 - Polynomials  5.      i  4  P x  is a cubic 2x3     4x2     3x     1 2x4     4x3     3x2     x x.P x  i...
Maths Extension 2 - Polynomials  Relationship   Transformation methods Example If    ,    ,    are the roots of 2x3     5x...
Maths Extension 2 - Polynomials  Example If    ,    ,    are the roots of 2x3   3x2   x     5   0, form the equation whose...
Maths Extension 2 - Polynomials  E  2x3   3x2   x     5   0 x  y 3  2    2    y 2       3    y 2           y 2         5  ...
Maths Extension 2     Conics  Conics   General Equations of Conics   Parabola, Circle, Ellipse, Hyperbola, Rectangular Hyp...
Maths Extension 2     Conics  Shapes     From Phoenix Senior Maths Notepad Study Guides by George Fisher  Eccentricity of ...
Maths Extension 2     Conics  General Equation of Conics Circle Basic  x2   y 2   r 2   x     h  2     y     k  2   a 2 x ...
Maths Extension 2     Conics  Ellipse      P  M  Foci    ae, 0  Directrices x      a e       x2 y2    1 a2 b2  Major Axis ...
Maths Extension 2     Conics  Hyperbola      D     Foci    ae, 0  Directrices x      a e     x2 y2      1 a 2 b2  Asymptot...
Maths Extension 2     Conics  Conics in Detail Circle Equation  x2   y 2   r 2   x     h  2     y     k  2   a 2 Eccentric...
Maths Extension 2     Conics  Circle     Tangent x2   y2   a2  P x1 , y1   OR P a cos   , a sin       2x   2 y  dy dx dy d...
Maths Extension 2     Conics  Circle     Normal x2   y2   a2 dy dx y1 x1 y1 x     x1 y1 xy1 xy1     yx1   0  y x y     y1 ...
Maths Extension 2     Conics  Ellipse Equation  x2 y2    1 a2 b2 b 2   a 2  1     e   x       ae,0   a x    e xx1 yy1   2 ...
Maths Extension 2     Conics  Area of the Ellipse by integration y2 b2  a  A       y dx    a  a  b2 x 2 b     2 a    2    ...
Maths Extension 2     Conics  Chords in an Ellipse A chord through the centre is a diameter Eg. PR  C P  A chord through a...
Maths Extension 2     Conics  Ellipse     Tangent dy dx        Gradient of Tangent  xb 2 ya 2  At  x1, y1   x1b 2 y1a 2 b ...
Maths Extension 2     Conics  Hyperbola Equation  x2 y2      1 a 2 b2 b2   a 2 e2     1 x       ae,0   a x    e xx1 yy1   ...
Maths Extension 2     Conics  Hyperbola     Tangent dy b2 x   2 dx a y  Gradient of Hyperbola At  x1, y1   b 2 x1 a 2 y1 b...
Maths Extension 2     Conics  Rectangular Hyperbola Equation Eccentricity Foci  e  2 a 2       a,   a   a x    2   x   y  ...
Maths Extension 2     Conics  Gradient of a Rectangular Hyperbola xy   c 2 y c2   x dy c2      2 dx x             c Chord ...
Maths Extension 2     Conics  Rectangular Hyperbola     Tangent dy c2      2 dx x 2 y1      2 x1 1      2 p      1 p2     ...
Maths Extension 2 - Integration           Standard Integrals Function   Derivative Product     Integrate by Parts Partial ...
Maths Extension 2 - Integration  INTEGRALS  c is a constant  x n  1   c _________ n        1   n  1         x n dx        ...
Maths Extension 2 - Integration  1 sin ax   c a         cos ax dx            sin ax dx  1       cos ax   c a         sec 2...
Maths Extension 2 - Integration  Trigonometric Identities sin2     cos2     1    sin2     1     cos2     cos2     1     si...
Maths Extension 2 - Integration  STANDARD INTEGRALS Basic Form n  1     x  5  dx  1     x  dx or     x    1 dx       ax   ...
Maths Extension 2 - Integration  Trigonometric Functions      sin ax  dx      sin 3x  dx      sin  3  x dx  1       cos ax...
Maths Extension 2 - Integration      sec  2  ax dx      sec  2  3x dx      tan x dx     tan  2      tan  3  1 tan ax   c a...
Maths Extension 2 - Integration  u   f  x  du   f     x dx  Function and Derivative        f   x    . f     x   n       dx...
Maths Extension 2 - Integration      uv  dx   uv         vu   dx  Integrating by parts      x cos x dx  Let u   x du  1 dx...
Maths Extension 2 - Integration  1. 2. 3. 4.  Split to Standard Integrals A, B A, Bx   C Completing the Square x  1 1     ...
Maths Extension 2 - Integration  Finding x in terms of u Let u   du       dx     dx    Substitution  1  12       4   x    ...
Maths Extension 2 - Integration  Trigonometry Substitution a2     x2 x    a 2  Let x   a sin    Let x   a sec   Let x   a ...
Maths Extension 2 - Integration  t   tan       ________                  2  T-Formula sin       2t 1  t2  1    t2 cos     ...
Maths Extension 2 - Integration  Reduction Formula Show      2  In      0 cos n x dx_______n     2  In    n     1 In-2    ...
Maths Extension 2  Volumes  Volumes    Slicing    Volume of Pyramid    Relationships   Ratios    Slicing    Revolution    ...
Maths Extension 2  Volumes  Slicing Volume of a Pyramid  h h  h  k  k  k                 b a  b  a     k      Find the vol...
Maths Extension 2  Volumes  Relationship 4 h 5 l  3  4 h 5  12  w l  3 h 10 5  w  12  10  Volume of One Slice y 4 h 0 y   ...
Maths Extension 2  Slicing Shape The base of a certain solid is x 2   y 2   9 . Plane sections are perpendicular to the y ...
Maths Extension 2  Shells y   f  x    Volumes  R r  Area of one strip A   f   x  dx     x  f  x    Volume of One Shell V  ...
Maths Extension 2  Volumes  Find the volume of the solid obtained by revolving the region x   y   2 , x   0 , y   0 about ...
Maths Extension 2  Find the volume of the solid obtained by revolving the ellipse  V    2       xy.dx a  y2 b2  a    2    ...
Maths Extension 2  Washer Method Area of one slice A z       R 2       r 2      R2     r 2       R   r   R     r         x...
4 Unit Maths     Mechanics Newton   s Second Law of Motion.                                      Fnet is defined as the su...
4 Unit Maths     Mechanics  Polar                                                                                         ...
4 Unit Maths     Mechanics Going slow     When an object moves slowly, it moves down the bank.                            ...
Maths Extension 2     Harder Maths Extension 1  3U   Harder Maths Extension 1   Harder 3U   Inequalities   Induction  http...
Maths Extension 2     Harder Maths Extension 1  3U   Inequalities Proof a     2ab   b 2     0    0 a2   b2 2     a 2   b 2...
Maths Extension 2     Harder Maths Extension 1  3U   Example 4 1 a b 1 a b  Are they equal   ab  a   2ab   b     ab a 2   ...
Maths Extension 2     Harder Maths Extension 1  3U   Induction Example 1 f  x     xe x f    x     xe x   e x f      x     ...
Maths Extension 2     Harder Maths Extension 1  3U   Example 2 Recurrence Induction A sequence of terms U n , n   1, 2, 3,...