A level math paper 3 link Tutorial

1. Algebra

Modulus Functions

• The modulus function
• Graph y=|f(x)|
• Graph y=f(|x|)
  • Exam Questions
• Modulus Equations
  • Equations 1
  • Equations 2
  • Equations 3
  • Exam Questions
• Modulus Inequalities
  • Inequalities 1
  • Inequalities 2
  • Inequalities 3
  • Exam Questions

Algebraic Long Division

• Algebraic long division
  • missing terms
  • finding the remainder
  • dividing a quartic polynomial by a quadratic polynomial

Factor Theorem

• The factor theorem
  • showing that x-1 is a factor of a cubic polynomial
  • factorising a cubic polynomial
  • solving a cubic equation
  • finding constants in a polynomial given the factors

Remainder Theorem

• The remainder theorem
  • finding the remainder when a cubic polynomial is divided by x+1
  • finding the remainder when a quartic polynomial is divided by 2x-1
  • finding constants in a polynomial given the remainders

• Exam questions

Partial Fractions

• Introduction
• Denominator contains 2 linear factors
• Denominator contains 3 linear factors
• Denominator contains repeated factors
• Denominator contains repeated factors (alternative method)
• Exam Questions

Binomial Expansion

• Rational powers
• Validity
• Exam Questions

2. Logarithmic
and exponential
functions

Exponential functions

• Exponential functions: what they are and their graphs

Logarithms

• What do we mean by a log?
• Rules of logs
• Simplifying
• Equations
  • Solving equations 1
  • Solving equations 2
  • Solving equations 3
• Inequalities
  • Solving inequalities
• Exam questions

Exponential Functions

• Exponential function (e^x )
• Sketching exponential graphs based on transformations :
  • relections | translations | stretches

Natural Log Functions

• The natural logarithmic function, ln x
• Exam Questions

Modelling Curves of the form y=kxⁿ and y=ka^x

• Converting to linear form
• numerical example still to come

3. Trigonometry

Secant, Cosecant and Cotangent

• Definitions of Secant, cosecant and cotangent
  • graphs

Identities

• Pythagorean type: sin²x + cos²x ≡1 ; 1 + tan²x ≡ sec²x ; 1 + cot²x ≡ cosec²x
  • Introduction and Proof
• Examples : 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8


• Addition type: sin(A±B), cos(A±B) and tan(A±B)
  • Addition Formulae (compound angles)
  • What are they?
  • Using the formulae to get exact values:
    • Examples : 1 | 2
  • Proving identities :
    • Examples : 1 | 2

• Double angle type for sin2A, cos2A and tan2A
  • Double angle formulae
    • Examples : 1 | 2 | 3
  • Half angles
    • Half angles (1)
    • Half angles (2)

• A cos x ± B sin x and A sin x ± B cos x type
  • A cos x ± B sin x and A sin x ± B cos x

Solving Equations using:

• the identities : sin²x + cos²x ≡1 ; 1 + tan²x ≡ sec²x ; 1 + cot²x ≡ cosec²x
  • Examples: 1 | 2 | 3

• the identities for sin2A, cos2A and tan2A
  • Examples: 1 | 2 | 3

• the identities for A cos x ± B sin x and A sin x ± B cos x
  • Examples : 1 and 2
• Exam Questions

Miscellaneous Exam Practice

• Exam Questions

4. Differentiation

Differentiating the exponential function e^x

• exponential function e^x

Differentiating the natural log function, ln(x)

• The natural log function, ln(x)

Differentiating the trig. functions, sin(x), cos(x) and tan(x)

• The trig. functions, sin(x), cos(x) and tan(x)

The Chain Rule

• polynomial to a rational power types
• exponential types
• natural log types
• trigonometric types (1)
• trigonometric types (2)

The Product Rule

• The product rule

The Quotient Rule

• The quotient rule
  • example

A Special Result

• Using the result dy/dx = 1 ÷ dx/dy

Miscellaneous Exam Practice

• Methods
• Applications - tangents, normals and stationary points
• Exponential Rates of change

Parametric Functions

• Parametric functions
  • Tangents and normals
  • Stationary points
• Exam Questions

Implicit Functions

• Implicit functions
  • Example using the product rule
• Tangents and normals
• Stationary points
  • Nature of stationary points
• Exam Questions

5. Integration

(ax+b)ⁿ types

• How to do these
  • examples
• Exam Questions

The exponential functions : e^x, e^ax and e^(ax+b)

• How to do these
• Exam Questions

Reciprocal Functions : 1/x and 1/(ax+b)

• How to do these

Integrals of the form : f'(x)/f(x)

• How to do these

Integrals of the form : f'(x)e^f(x)

• How to do these : part 1
• How to do these : part 2

Trigonometric types

• sin x, cos x, sec² x
• sin(ax+b), cos(ax+b), sec² (ax+b) types
• Identity types
  • example 1
  • example 2
  • example 3
• sin² x and cos² x types
• Exam Questions

Partial fraction types

• Partial fraction types
• Exam Questions

Substitution

• Substitution 1
• Substitution 2
• Substitution 3a
• Substitution 3b
• Substitution 4 (trig types)
• Substitution 5 (exponential types)
• Substitution 6 (using limits)
• Substitution 7 (trig type with limits)
• Exam Questions

By Parts

• By parts
  • Example 1
  • Example 2
  • Example 3 (using limits)
  • Example 4 (ln types)
• Exam Questions

Mixed examples

• 1 | 2 | 3 | 4 | 5 | 6
• Mixed Exam Questions Workout

Trapezium Rule

• Numerical integration - trapezium rule
  • Type of estimate
• Exam Questions

6. Numerical
solution of
equations

Solution of Equations by:

• Graphical methods
• Change of sign
• Iteration
  • How it works
• Exam Questions

7. Vectors

Lines

• Vector equation of a line
  • example 1
  • example 2
• Angle between two lines
• Parallel lines
• Intersecting and skew lines
• Closest point to a line and shortest distance from the origin
• Exam questions

Cartesian form of a line

• Cartesian form of a line

Planes - Parametric form

• Parametric vector form
• Locating a point on a parametric form of the plane
• Equation of a plane passing through 3 points
• Equation of a plane passing through a point and parallel to two lines

Planes - Scalar product form

• Scalar product form

Planes - Cartesian form

• Cartesian form
• Finding the point of intersection between a line and a plane

Intersection of two planes

• The equation of the line of intersection between two non parallel planes

Angle between two planes

• The angle between two planes

• Equation of a plane in the form (r-a).n = 0 (still to do)
• Determine whether a line lies in a plane, is parallel or intersects (still to do)
• Find the perpendicular distance from a point to a plane, and from a
  point to a line (still to do)

8. Differential Equations

Differential equations

• Separating the variables
  • Finding a general solution and a particular solution
  • Which side should the constant C go on?
  • Working with constants in log types
  • Exponential and trig type (a little more challenging)

• Forming differential equations
  • direct proportion type
  • inverse proportion type
  • Newton's Law of Cooling

• Exam Questions

9. Complex Numbers

• Real and imaginary numbers
• Addition, Subtraction and Multiplying complex numbers and simplifying powers of i
• Complex conjugates
• Division of a complex number by a complex number
• Argand diagram
• Modulus and argument of a complex number
• Mod-Arg form of a complex number
• Solving problems with complex numbers
• Square roots of a complex number
• Solving quadratic equations with complex roots (complex conjugate pairs)
• Solving cubic equations
• Solving quartic equations
  • Exam Questions
• Modulus-argument form
• Exponential Form (Euler's relation)
• Multiplication rule for the mod and argument of two complex numbers
• Division rule for the mod and argument of two complex numbers
  • Mixed examples
• understand in simple terms the geometrical effects of conjugating a
  complex number and of adding, subtracting, multiplying and dividing
  two complex numbers

Loci in the complex plane

• The locus of a point moving in a circle : | z- z₁| = r
• The locus of a point moving along a perpendicular bisector : | z-z₁ | = | z-z₂ |
• The locus of a point moving along a half-line : arg(z-z₁) = θ
• The locus of a point moving on the arc of a circle : arg[ (z-z₁) / (z-z₂) ] = θ

• Using complex numbers to represent regions on an Argand diagram
• Exam Questions