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Higher Maths
Question Types
You need to learn
basic movements
Exam questions
normally involve
two movements
Remember order
BODMAS
Sketching
Graphs
Steps :
1. Outside function stays
the same EXCEPT replace
x terms with a ( )
Composite
Functions
2. Put inner function in bracket
Restrictions :
Functions & Graphs
TYPE questions
(Trig , Quadratics)
1. Denominator NOT ALLOWED
to be zero
2. CANNOT take the square root
of a negative number
f ( x) 
x3  x
3x
OR y 
x3  x
3x
Function
Increasing or
Decreasing
Basic
and
Format
f’(x)
Gradient
f’(x) > 0
Differentiation
TYPE questions
(Fractions / Surds /Indices)
Stationary
Points Optimization
Steps :
1. Differentiate
2. f’(x) = 0 (statement)
3. Factorise
4. Nature Table
5. Sub x = to original
equation to find
y coordinate.
f’(x) < 0
Max / Mini
in closed intervals
Steps :
1. Find Max / Mini points
2. Find end values
3. Decide Max / Mini Points
Tangent Line
Steps :
1. Differentiate
2. Sub x = into f’(x)
to find gradient
3. Use a point on the line
and y – b = m(x – a)
Steps :
1. Setup recurrence relation
2. State if limit exists
3. Find limit
Wordy question
Steps :
Finding
Constants
Recurrence Relations
TYPE questions
(Fractions / Sim Equations)
1. Using information given
setup two equations
2. Use simultaneous
equation method to find
constants
f(x) = a(x + b)2 + c
2x2 - 8x + 9
2x2 - 8x
2(x2 - 4x)
+9
+9
2(x - 2)2 - 8 + 9
e.g. -2 1 4 5 2
use coefficients to
-2 -4 -2
factorise further
Factor Theorem
if possible !!
1 2 1 0
(x+2) is a factor
Remember to
since no remainder answer question
Completing
the square
f(x) = ( ) ( ) ( )
Factorising
cubic's polynomials
f(x) = 2(x - 2)2 + 1
Sketch
See Function
& Graphs
Discriminant
b2 – 4ac
3 scenarios
> 0
= 0 tangent !!!
<0
Quadratic
Theory questions
(Circle, Function Graphs)
Harder
discriminant
(1 - 2k)x2 - 5kx - 2k > 0
Harder
Finding
coefficients
simultaneous
equations
Steps
1. Identify a , b and c.
2. Discriminant .... = 0
and factorise.
3. Sketch and identify
solution based on
question asked.
3

x3  x
3x
-1
dx
2
0
Simple Area
under the curve
1
Basic
Find original equation given
4
Area above &
below x-axis
Do separately and
remember statement
for below x-axis
Integration
TYPE questions
Original
Equation
1
(Fractions / Surds /Indices)
AT = A1 + A2
dy
 2 x  1 and passes through (0,1)
dx
y

2 x  1 dx
y  x2  x  c
To find c sub x  0 y  1
1  02  0  c
2
c 1
y  x  x 1
Steps :
Area between
two curves
1. For limits make
equal to each other.
2. Integrate
Top – (bottom)
-2
3
4
α
3
cos2x  3cos x 1  0
5
β
Substitution
and solving
12
Steps
1. Pythagoras Theorem
Expansion
2. Expansion
With Triangles
3. SOHCAHTOA
4. Solve.
Trigonometry
TYPE questions
Sketching
y  3sin 2 x  1
f(x) = sinx
(Quadratic, Function Graphs)
Sub for cos2x
Factorise
Solve
See A3 sheet given out in Unit 1
For more solving techniques
Basic
Exact values
and radians !!!
f(2x) = sin2x
3f(2x) = 3sin2x f(2x) + 1 = 3sin2x + 1
Equation from Graph
and solving
Steps
1. Write down equation using graph
2. Using balance method to solve
See A3 sheet given out in Unit 1
For more solving techniques
radius =
(a,b)
2
2
g  f c
centre = (-g,-f)
(x - a)2 + (y - b)2 = r2
Equation
from graph
Intersection
points between
line and circle
3 possible
scenarios
x2 + y2 + 2gx + 2fy + c =0
Finding centre and radius
from circle equation
Circle TYPE
questions
(Straight Line , Quadratics)
Steps
1. 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
Does circles
touch
externally or
internally ?
externally
Dc1c2  r1  r2
internally
Dc1c2  r2  r1
Is equation
a circle ?
r>0
Equation
of tangent
(a,b)
Steps
1. Find gradient of
centre to point
2. Use m1 x m2 = -1
to find gradient of line
3. Use y – b = m(x - a)
b
Tail to tail
a  (b  c )  a  b  a  c
a b  b a
a b
cos 
a b
Vector Theory
Magnitude &
Direction
Section formula
B
A
B
m
Points A, B and C are said to be
Collinear if
A
AB  kBC
Parallel AND B is a point in common.
a
Angle between
two vectors
properties
C
θ
b
C
n
c
b
a
n
m
a
c
m n
m n
O
d
d
d
(inside)n   (outside the bracket)  (inside the bracket)
dx
dx
dx
d 3
Harder functions
( x  2 x5 )2  2( x3  2 x5 )(3x 2  10 x 4 ) Use Chain Rule
dx
d
sin ax  a cos ax
dx
Differentiations
Question Type see Basic Differentiation.
d
cos ax  a sin ax
dx
Differentiation
Further Calculus
Integration
1
 sin ax dx   a cos ax  c
1
cos
ax
dx

sin ax  c

a
Trig
Integration
Question Type see Basic Integration.
n 1
(
ax

b
)
n
(
ax

b
)
dx 
c

a(n  1)
7
(2
x

3)
6
(2
x

3)
dx 
c

14
Remember
ln e-kt = -kt
log A + log B = log AB
log A - log B = log A
B
n
log (A) = n log A
loga1 = 0
logaa = 1
Solving Exp
Equations
(half – life)
Solving Log
Equations
Logs & Exp
Question Types
Functions & Graphs
Straight Line
Graph 2
Graph 1
y = axb
y = abx
log y
log y = x log b + log a
(0,C)
x
Y = mX + C
Y = (log b) X + C
C = log a m = log b
log y = b log x + log a
Y = mX + C
Y = bX + C
C = log a
m=b
log y
(0,C)
log x
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 format
Part (a) of question
Wave Function
Type Questions
(Functions & Graphs)
Solving Equation
Normally Part (b) of question
Normally Part (b) of question
UNIT 2
3sin(x +
45o)
=1
Arrange into
x = .......
Find Max / Mini Value
Sketching
Wave Function
S
A
T
C
Normally Part (b) of question
UNIT 1