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AM Notes By Kok YH (2011)

1. Simultaneous Equations 1. Linear & Non-Linear 2. By Inverse Matrix

2. Polynomials & Partial Fractions 1. Identities 2. Remainder & Factor Theorem 3. Cubic Equations 4. Partial Fractions

3. Indices & Surds 1. Laws of Indices / Solving Equations 2. Surds

i. Add/Sub/Multiply/Divide ii. Rationalize / Conjugate iii. Solving Equations

4. Quadratic Equations & Inequalities 1. Sum of Roots ( βα + ) and Product of

Roots (αβ ) 2. Form Quadratic Eq with Sum &

Product of Roots 3. Discriminant & Nature of Roots 4. Always +ve / Always –ve 5. Intersection Problems 6. Quadratic Inequalities

5. Modulus Functions 1. Solving Equations 2. Sketching graphs

6. Binomial Theorem 1. nCr 2. General Expansion 3. Specific Term / Trionomial

7. Exponential & Logarithmic Functions 1. Laws of Logarithms 2. Graphs of exponential & log 3. Solving Equations

8. Trigonometric Functions, Identities & Equations

1. General & Basic angle 2. 4 Quad., Negative, Complementary,

Special Angles 3. Solving Trigo Equations 4. Trigo Graphs (amplitude, period, line

of equilibrium) 5. Simple Trigo Identities

9. Further Trigonometric Identities 1. Addition / Subtraction 2. Double Angle / Half Angle 3. R Formula 4. Factor (Sum to Product / Product to

Sum)

10. Coordinate Geometry & Area of Rectilinear Figures

1. Perpendicular lines 2. Equation of straight line 3. Midpoint 4. Area of Rectilinear Figures

11. Curves & Circles 1. Circles 2. Curves

12. Linear Law 13. Differentiation

1. Basic (Product/Quotient/Chain) 2. Tangent & Normal 3. Rates of Change 4. 1st and 2nd Derivative Tests/

Stationary Point / Maxima & Minima

5. Trigonometric 6. Exponential and ln

14. Integration 1. Indefinite integral 2. Definite Integral (properties) 3. Trigo

4. Exponential & x1

15. Area of a Region 1. Area between a curve and x-axis 2. Area between a curve and y-axis 3. Area between a curve and line

16. Kinematics 1. Displacement 2. Velocity 3. Acceleration

17. Geometry Proof 1. Symmetrical & Angle Properties of

Circles 2. Midpoint Theorem 3. Intercept Theorem 4. Tangents

i. Tangent-Chord (alt. seg) ii. Intersecting Chords iii. Tangent-secant

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Simultaneous Equations

Important:

1. If there are two sets of solutions, give the answers in pairs of x and y paired correctly. 2. If question asks for coordinates, give in coordinate form (x, y)

Types of Solutions The solution will be one of the following cases: 1. A unique solution - when the gradients of the two lines are not equal 2. Infinitely many solutions - when the two lines coincide (same gradient, same y-intercept) 3. No solution - when the two lines are parallel but different y-intercept Methods to Solve

1. Substitution 2. Elimination 3. Graphical 4. Inverse Matrix (to be covered in Sec4)

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Example

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Polynomials & Partial Fractions

Polynomials

Identities

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Example

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Remainder Theorem - Gives the remainder when a polynomial f(x) is divided by a linear divisor ( )ax −

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Cases for different Divisors

Divisor Remainder

( )ax − Constant

( )cbxax ++2 ( )BAx +

( )dcxbxax +++ 23 ( )CBxAx ++2

In general, the remainder will be one degree less than the divisor.

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Factor Theorem

Example

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Cubic Equations

Solve Cubic Equations in the form 023 =+++ dcxbxax 1. Find the first factor (by trial and error)

(can use calculator to speed up process, but must show working) 2. Find the remaining two factors by:

1. equating coefficients, or 2. long division 3. Synthetic Division

3. Solve the equation Important: Difference between “Solving” and “Factorising”

• If question states )(xf , you can only factorise )(xf into its factors • If question states 0)( =xf , you can solve it (i.e. x = ?. ?, ?)

Example

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Partial Fractions

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Partial Fractions of Proper Fractions Three cases:

Case Proper Fraction Partial Fractions

1 (Linear factors) ))((

)(bxax

xP++

)()( bx

BaxA

++

+

2 (Repeated Linear Factors) 2))((

)(bxax

xP++

2)()()( bx

CbxB

axA

++

++

+

3 (Non-factorisable Quad.

Factor) ))(()(2 bxaxxP++

)()( 2 bxCBx

axA

++

++

To find the unknown constants and coefficients of the partial fractions, 2 methods:

1. Comparing coefficients 2. Substitution

Example

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Partial Fractions for Improper Fraction One extra step:

Note: can use long division to break up the improper fraction into sum of a polynomial and proper fraction

Example

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Indices & Surds

Laws of Indices

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Solving Equation involving Indices Two methods: Method 1

1. Change to same base 2. Equate the powers

Method 2

1. Use substitution (to change to quadratic e.g.) 2. Solve the equation 3. Change back to original index form 4. Solve for the unknown

Tip: We normally use Method 2 if there are + or – in the equation. Example (Method 1)

Example (Method 2)

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Surds

Multiplication & Divison

aaa

ba

ba

ba

abba

=×

==÷

=×

baba

baba

−≠−

+≠+

Addition & Subtraction

anmanam

anmanam

)(

)(

−=−

+=+

Example

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Example

Example

Solution

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Rationalizing the Denominator (Conjugate)

• To rationalize the denominator, multiple it by its conjugate • The table below shows the conjugate for the different forms of surds

Surd Conjugate

a a

an an

anm + anm −

bnam + bnam −

Example

Solving Equations involving Surds

• Square both sides to remove the square root • Because of this, we need to check the final answer(s) whether they are valid. Check by substituting

into the original surd equation. For two equal surd expression, we may use the following property to form equations:

If ndcnba +=+ ,

where a, b, c and d are rational numbers and n is a surd,

then ca = and db =

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Quadratic Equations Main Topics:

1. Sum of Roots ( βα + ) and Product of Roots (αβ )

2. Form Quadratic Equation with Sum and Product of Roots

3. Discriminant & Nature of Roots

4. Always +ve and always –ve

5. Intersection Problems

6. Quadratic Inequalities

Sum of Roots ( βα + ) and Product of Roots (αβ ) To evaluate other expressions of βα , , some useful properties: ( ) αββαβα 2222 −+=+ (must remember!)

)(3)( 333 βααββαβα +−+=+ (can be derived by expanding & simplifying 3)( βα +

or ( )[ ]αββαβαβα 3)( 233 −++=+

( )[ ]αββαβαβα −+−=− 233 )(

( )( ) 2222

22222222244

)(2

))((2)()(

αββα

βαβαβαβα

−+=

−+=+=+ (can be derived)

( )( )( )βαβαβαβα −++=− 2244

Given a quadratic equation 02 =++ cbxax with roots βα ,

• Sum of roots, ab−

=+ βα

• Product of roots, ac

=αβ

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Form Equations with Sum & Product of Roots Example

Solution

Two ways to form quadratic equations: 1. Given roots βα , ,

Equation: 0))(( =−− βα xx 2. Given sum of roots βα + and product of roots αβ

Equation: 0)(2 =++− αββα xx Or (if the sum and product of roots are of other expressions of βα , )

0)()(2 =+− rootsofproductxrootsofsumx

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Discriminant & Nature of Roots

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Always +ve / Always –ve Conditions for y to be always +ve or –ve (important! Must remember)

Condition Illustration a acb 42 −

For y to be always positive

0>a

(so that it will be a “smiley

face”)

042 <− acb

(so that the curve does not intersect the x-axis)

For y to be always negative

0<a

(so that it will be a “sad”

face)

042 <− acb

(so that the curve does not intersect the x-axis)

Example

Solution

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Intersection Problems Given a straight line and a quadratic curve,

1. We can “combine” the two into a new equation. 2. By looking at the D of this new equation, we can find out if the line and the curve:

a. Meet at two points b. Meet at one point (in which case, the line is called a tangent) c. Do not meet at all

Example

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Quadratic Inequalities Steps to solve a quadratic inequality 02 >++ cbxax ,

1. Factorize to find the x-intercepts 2. Sketch & read answers from the graph

Example

Solution

)2)(1(22 −+=−− xxxx Hence, x-intercepts are (-1, 0) and (2, 0)

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Modulus Function Overview:

1. Definition 2. Properties 3. Graphs of Absolute Function 4. Solving Equations

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When sketching an absolute function graphs, you must include the following critical points:

1. Min or Max point 2. X and Y intercepts 3. Ending points based on the range of x given

Example Sketch the graph of 521 +−= xy for 34 ≤≤− x . Example

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Tip: Order of Graph to Sketch If you are given a graph of the general form, hbaxy ++±=

Sketch in the following order:

1. Sketch the absolute function first baxy +=

2. “Flip” the absolute function if necessary (e.g. if baxy +−=

3. Shift the graph up or down depending on h Solving Equations Example

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Binominal Theorem

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Binomial Theorem

In the expansion of nba )( + ,

the general term, or the ( )thr 1+ term, is rrn

r bar

nT −

+

=1

where ( )!!!rnr

nC

r

nr

n

−==

Note: Formula for Binomial Theorem is provided on formula sheet in exams. You just need to apply.

General Term

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Example

Example

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General Term Formula

Example