Higher Maths

Question Types

Functions & Graphs

TYPE questions(Trig , Quadratics)

SketchingGraphs

CompositeFunctions

Steps :

1.Outside function staysthe same EXCEPT replacex terms with a ( )

2. Put inner function in bracket

You need to learn basic movements

Exam questionsnormally involve two movements

Remember orderBODMAS

Restrictions :

1.Denominator NOT ALLOWEDto be zero

2.CANNOT take the square rootof a negative number

DifferentiationTYPE questions

(Fractions / Surds /Indices)

Basicand

Format

3 3

3 3( ) OR

x x x xf x y

x x

FunctionIncreasing

orDecreasing

Tangent LineSteps :1.Differentiate2.f’(x) = 0 (statement)3.Factorise4.Nature Table5.Sub x = to original

equation to findy coordinate.

StationaryPoints

Gradientf’(x)

Optimization

Steps :1.Differentiate2.Sub x = into f’(x)

to find gradient3.Use a point on the line

and y – b = m(x – a)

f’(x) > 0

f’(x) < 0

Max / Mini in closed intervals

Steps :1.Find Max / Mini points2.Find end values3.Decide Max / Mini Points

Recurrence Relations

TYPE questions(Fractions / Sim

Equations)

Wordy question

Finding Constants

Steps :

1.Setup recurrence relation2.State if limit exists3.Find limit

Steps :

1.Using information givensetup two equations

2.Use simultaneousequation method to findconstants

Quadratic Theory

questions(Circle, Function Graphs)

Completing the square

Harder discriminant

Steps1.Identify a , b and c.

2.Discriminant .... = 0 and factorise.

3.Sketch and identify solution based on question asked.

2x2 - 8x + 9

f(x) = a(x + b)2 + c

2x2 - 8x + 9

2(x - 2)2 + 9

f(x) = 2(x - 2)2 + 1

- 8

2(x2 - 4x) + 9

SketchSee Function

& Graphs

Discriminantb2 – 4ac

3 scenarios

> 0

= 0 tangent !!!

< 0

(1 - 2k)x2 - 5kx - 2k > 0

Factorising cubic's polynomials

(x+2) is a factorsince no remainder

Factor Theorem

1 4 5 2-2-2 -4 -22 1 01

e.g. use coefficients to factorise further

if possible !!

Remember to answer question

f(x) = ( ) ( ) ( )

HarderFinding

coefficients

simultaneousequations

Integration TYPE questions

(Fractions / Surds /Indices)

Basic

33

31

x x

dxx

-1 2

Simple Area under the curve

0 1 4

Area above &

below x-axisDo separately and remember statement

for below x-axis

AT = A1 + A2

Area between two curves

-23

Steps :

1.For limits makeequal to each other.

2. Integrate

Top – (bottom)

OriginalEquation

Find original equation given

2 1 and passes through (0,1)dy

xdx

22 1 y x dx y x x c 2

To find sub 0 1

1 0 0 1

c x y

c c

2 1y x x

Trigonometry TYPE questions

(Quadratic, Function Graphs)

ExpansionWith Triangles

Equation from Graphand solving

Steps1.Write down equation using graph2.Using balance method to solve

Sketching

Substitutionand solving

βα3

4

12

5

Steps1.Pythagoras Theorem2.Expansion3.SOHCAHTOA 4.Solve.

cos 2 3cos 1 0x x Sub for cos2x

Factorise

Solve

See A3 sheet given out in Unit 1For more solving techniques

3sin 2 1y x f(x) = sinx f(2x) = sin2x

3f(2x) = 3sin2xf(2x) + 1 = 3sin2x + 1

See A3 sheet given out in Unit 1For more solving techniques

BasicExact values

and radians !!!

Circle TYPE questions(Straight Line ,

Quadratics)

Equationfrom graph

Finding centre and radius from circle equation Is equation

a circle ?

Intersectionpoints betweenline and circle

Steps1.Sub line equation y = ...

into circle.

2.Discriminant to establish how many points.

3.Factorise for x coordinates and sub into line equation for y coordinates

(x - a)2 + (y - b)2 = r2x2 + y2 + 2gx + 2fy + c =0

centre = (-g,-f)

radius = 2 2g f c (a,b)

r > 0

3 possiblescenarios

Equation of tangent

Steps1.Find gradient of

centre to point

2.Use m1 x m2 = -1 to find gradient of line

3.Use y – b = m(x - a)

Does circles touch

externally or internally ?

1 2 1 2

1 2 2 1

c

c

externally

internally

c

c

D r r

D r r

(a,b)

B

Vector TheoryMagnitude &

Direction

Points A, B and C are said to be Collinear if

Parallel AND B is a point in common.

Section formula

n m

b a cm n m n

properties

a b b a

( )a b c a b a c

n

m

a

c

b

NNNNNNNNNNNNNNNNNNNNNNNNNNNN

AB kBC

Angle between two

vectors

cosa ba b

Tail to tail

DifferentiationsQuestion Type see Basic Differentiation.

Harder functionsUse Chain Rule

( ) (outside the bracket) (inside the bracket)nd d dinside

dx dx dx

IntegrationQuestion Type see Basic Integration.

1( )( )

( 1)

nn ax b

ax b dx ca n

1sin cosax dx ax c

a

1cos sinax dx ax c

a

76 (2 3)

(2 3)14

xx dx c

3 5 2 3 5 2 4( 2 ) 2( 2 )(3 10 )dx x x x x x

dx sin cos

dax a ax

dx

cos sind

ax a axdx

DifferentiationFurther Calculus

Integration

Trig

Logs & Exp Question Types

Functions & GraphsStraight Line

log A + log B = log AB

log A - log B = log B

A

loga1 = 0 logaa = 1

log (A)n = n log A

log y = x log b + log a

C = log am = log b

(0,C)

log y

x

log y = b log x + log a

C = log a m = b

(0,C)

log y

log xY = bX + CY = (log b) X + C

Y = mX + CY = mX + C

y = abx Graph 1 y = axb

Graph 2

Solving Log

Equations

Solving Exp

Equations(half – life)

Remember ln e-kt = -kt

Wave Function Type Questions

(Functions & Graphs)

f(x) = a sinx + b cosx

compare to required trigonometric identities

f(x) = k sin(x + β) = k sinx cos β + k cosx sin β

Changing formatPart (a) of question

Solving EquationNormally Part (b) of question

UNIT 2

3sin(x + 45o) = 1

Sketching Wave Function

Normally Part (b) of questionUNIT 1

Find Max / Mini Value Normally Part (b) of question

AS

T C

Arrange into

x = .......