Transcript Higher past paper questions, 2000

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Higher Past Papers
by Topic 2000 - 2012
Straight Line (11)
Trigonometry (18)
Composite Functions (12)
The Circle (15)
Differentiation (22)
Vectors (15)
Recurrence Relations (6)
Logs & Exponential (16)
Polynomials (12)
Wave Function (12)
Integration (16)
Wednesday, 05 April 2017
Summary of Higher
Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Higher Mindmaps
by Topic
Straight Line
Trigonometry
Composite Functions
The Circle
Differentiation
Vectors
Recurrence Relations
Logs & Exponential
Polynomials
Wave Function
Integration
Wednesday, 05 April 2017
Main Menu
Created by Mr. Lafferty Maths Dept.
Land
24 cm
30 cm
2
3
Distance between
2 points
m<0
m = undefined
 x  x y  y2 
Mid   1 2 , 1
2 
 2
Terminology
D  (x2  x1 )2  (y2  y1 )2
Median – midpoint
Bisector – midpoint
Perpendicular – Right Angled
Altitude – right angled
m=0
m>0
m1.m2 = -1
Possible values
for gradient
Straight Line
y = mx + c
Form for finding
line equation
y – b = m(x - a)
(a,b) = point on line
Parallel lines
have same
gradient
For Perpendicular lines
the following is true.
m1.m2 = -1
m = gradient
m
y 2  y1
x 2  x1
c = y intercept (0,c)
m = tan θ
Mindmaps
θ
f(x)
flip in
y-axis
+
Move vertically
up or downs
depending on k
f(x)
f(x)
-
Stretch or
compress
vertically
depending on k
y = f(x) ± k
y = f(-x)
Remember we
can combine
these together !!
y = kf(x)
Graphs &
Functions
y = -f(x)
f(x)
flip in
x-axis
y = f(kx)
y = f(x ± k)
f(x)
f(x)
-
+
Move horizontally
left or right
depending on k
Stretch or
compress
horizontally
depending on k
Mindmaps
f(x) =
x2
1
g(x) =
x
-4
x
g(f(x))
1
y
y = f(x)
Domain
But y = f(x) is x2 - 4
Restriction
Range
A complex function made
up of 2 or more simpler
functions
Similar to
composite
Area
=
1
g(x) =
x
x
Domain
f(x) =
y = g(x)
x2
x ≠2
- 4 f(g(x))
y2 - 4
Range
x2 - 4 ≠ 0
(x – 2)(x + 2) ≠ 0
Composite
Functions
+
1
x2 - 4
g(f(x)) =
x ≠ -2
1
x
But y = g(x) is
1
x
f(g(x)) =
Rearranging
1
x2
2
-4
-4
Restriction x2 ≠ 0
Mindmaps
6
x5  x

5
6

1
x
7
5
6
Format for
Differentiation
/ Integration
5
x
2
 x

2
5
3
x x
x4  x
4
7
1
2
1
4
x x  x
3
4
x
1
3
Surds
n
x x
m
x
2
5
x x  x
m
n
( mn)
Basics before
Differentiation
/ Integration
Working with
fractions
Adding
1 1 5
 
2 3 6
Subtracting
1 1 1
 
2 3 6
2

3
 x
Indices
m
n
1

x
1
3
Multiplication
1 3 3
 
2 5 10
xm
( mn )

x
xn
Division
1 4

2 5
1 5 5
 
2 4 8
Mindmaps
Nature Table
x -1 2 5
f’(x) + 0 -
Leibniz
Notation
dy
 f '( x)
dx
Max
f’(x)=0
Stationary Pts
Max. / Mini Pts
Inflection Pt
Equation of
tangent line
Gradient
at a point
Graphs
f’(x)=0
Differentiation
of Polynomials
1
2
f ( x)  2 x  x
1
2
1
2
1
f '( x)  3x  x
2
1
1
f '( x)  3x 2 
2 x
1
2
Derivative
= gradient
= rate of change
f ( x) 
f ( x)  x  2 x  1
3
2
Straight Line
Theory
f(x) = axn
then f’x) = anxn-1
2
3 4 x5
f ( x) 
9
5
4
2x
3
5
 x4
5
f '( x)  2

3
6 4 x9
Mindmaps
d
d
d
(inside)n   (outside the bracket)  (inside the bracket)
dx
dx
dx

d 2
4
x  2x
dx
 
d n
x
dx
3
2
4
3 x  2x
2 2x  8x3
Trig
Harder functions
Use Chain Rule
Rules of
Indices
n 1
n x
d
cos ( a x)
dx
Real life
d
= (velocity)
dx
= acceleration
a cos ( a x)
a sinx ( a x)
Polynomials
Differentiations
d
 (distance)
dx
= velocity
d
sin( a x)
dx
dy
0
dx
Meaning
Stationary Pts
Mini / Max Pts
Inflection Pts
Rate of
change of a
function.
Gradient at
a point.
Factorisation
Graphs
Tangent
equation
Mindmaps
Straight
line Theory
Easy to graph
functions & graphs
Completing the square
f(x) = a(x + b)2 + c
Factor Theorem
x = a is a factor of f(x)
if f(a) = 0
f(x) =2x2 + 4x + 3
f(x) =2(x + 1)2 - 2 + 3
f(x) =2(x + 1)2 + 1
-2 1 4 5 2
-2 -4 -2
1 2 1 0
(x+2) is a factor
since no remainder
If finding coefficients
Sim. Equations
Discriminant of a
quadratic is
b2 -4ac
Polynomials
Functions of the type
f(x) = 3x4 + 2x3 + 2x +x + 5
Tangency
b2 -4ac > 0
Real and
distinct roots
b2 -4ac = 0
Equal roots
Degree of a polynomial
= highest power
b2 -4ac < 0
No real
roots
Mindmaps
Limit L is equal to
L=
b
(1 - a)
Given three value in a
sequence e.g. U10 , U11 , U12
we can work out
recurrence relation
U11 = aU10 + b
U12 = aU11 + b
Use
Sim. Equations
a = sets limit
b = moves limit
Un = no effect
on limit
Recurrence Relations
next number depends on the
previous number
Un+1 = aUn + b
|a| > 1
Limit exists
when |a| < 1
|a| < 1
a > 1 then growth
a < 1 then decay
+ b = increase
- b = decrease
Mindmaps
f(x)
g(x)
Remember to
change sign to + if
area is below axis.
Remember to work
out separately the
area above and below
the x-axis .
b
A= ∫a f(x) - g(x) dx
Finding where curve
and line intersect
f(x)=g(x) gives the
limits a and b
I x
1
2
Integration is the
process of finding
the AREA under a
curve and the x-axis
Area between
2 curves
Integration
of Polynomials
2
I 
x
1
 2 x  1dx
1
 32

I    2 x  x 2 dx


4 52 2 32
I  x  x C
5
3
2
1
2
I 
IF f’(x) = axn
Then I = f(x) = ax
1
n 1
n 1

1
2
dx
x
dx
2
2
 
I  x 
 1
 2 1
1
2
Quadratic Theory
Discriminant
Graph sketching
r  g2  f 2 c
Move the circle from the origin
a units to the right
b units upwards
b2 - 4ac  0
Used for
intersection problems
between circles and lines
NO intersection
b2 - 4ac  0
Centre
(a,b)
x2 y2 2g x2f  yc0
Centre
(-g,-f)
Factorisation
The
Circle
(x a)2 (y b)2  r2
Special case
x2  y2  r2
Centre
(0,0)
2 pts of intersection
b2 - 4ac  0
line is a tangent
Pythagoras Theorem
Rotated
through 360 deg.
Two circles touch
externally if
the distance
Straight line Theory
Two circles touch
internally if
the distance
C1C2  (r2  r1)
Perpendicular
equation
m1  m2  1
C1C2  (r1  r2 )
Distance formula
2
2
CC
1 2  (y2  y1) (x2 x1)
Mindmaps
Double Angle Formulae
sin2A = 2sinAcosA
cos2A = 2cos2A - 1
= 1 - 2sin2A
= cos2A – sin2A
Addition Formulae
sin(A ± B) = sinAcosB
cos(A ± B) = cosAcosB
 cosAsinB
sinAsinB
90o
The exact
value of sinx
4
xo
4
2
o
180
Trig Formulae
and Trig equations
S
A
T
C
270o
3cos2x – 5cosx – 2 = 0
Let p = cosx 3p2 – 5p - 2 = 0
sinx = 2 (¼ + √(42 - 12) )
(3p + 1)(p -2) = 0
cosx = 2
p = cosx = 1/3
x = no soln
x = cos-1( 1/3)
sinx = ½ + 2√15)
x = 109.5o and 250.5o
sinx = 2sin(x/2)cos(x/2)
Mindmaps
0o
1.
2.
3.

Rearrange into sin =
Find solution in Basic Quads
Remember Multiple solutions
o
90
2
S
180o
A
C
T

0
30
o
0
Period
360o
-1
sin x
1 Amplitude
-1
cos x
30o
6
45o
radians
then x 180 ÷ π
360o
90o
sin
cos
tan
0
1
0
1
2
1
2
3
2
1
2
1
3
3
2
1
1
2
0
1
3
undefined
Trigonometry
sin, cos , tan
Amplitude
Period
o
60o
Basic Strategy
for Solving
Trig Equations
0
0

÷180 then X π
degrees
3
o
270
2
1
o
Exact Value Table
Complex Graph
3
Basic Graphs
2
1
Period
0
1
0
-1
Amplitude
-1
o 180o
90
Period
tan x
90o
180o
270o
360o
y = 2sin(4x + 45o) + 1
Max. Value =2+1= 3 Period = 360 ÷4 = 90o
Mini. Value = -2+1 -1 Amplitude = 2 Mindmaps
 u1   a1   u1  a1 
    

PQ  a   u2    a2    u2  a2 
u   a  u  a 
3
 3  3  3
Addition
Scalar product
 ku1 


k a   ku2 
 ku 
 3
Basic
properties
Q
B
P
A
a  b  a1b1  a2b2  a3b3  0
2 vectors
perpendicular if
a  b  a1b1  a2b2  a3b3
Component form
a  a1  a2  a3
Magnitude
PQ
same for
subtraction
a
Vectors are equal if
they have the same
magnitude &
direction
a  b  a b cos
scalar product
Vector Theory
Magnitude &
Direction
Notation
 u1 
 
Component form PQ   u2 
u 
 3
Unit vector form
a  a1i  a2 j  a3k
Mindmaps
b
Tail to tail
a  (b  c )  a  b  a  c
a b
cos 
a b
a b  b a
a
Angle between
two vectors
properties
C
θ
Vector Theory
Magnitude &
Direction
Section formula
B
B
A
m
Points A, B and C are said to
be Collinear if
A
AB  kBC
B is a point in common.
b
Mindmaps
C
n
c
b
a
n
m
a
c
m n
m n
O
y
y = logax
(a,1)
To undo log take exponential
loga1 = 0
logaa = 1
(1,0) x
log A + log B = log AB
To undo exponential take log
Basic
log graph
Basic
exponential
graph
y = axb
abx
Can be transformed into
a graph of the form
Can be transformed into
a graph of the form
log y = x log b + log a
(0,C)
x
a0 = 1
a1 = a
(1,a)
Logs &
Exponentials
Basic log rules
log y
(0,1)
x
log A - log B = log A
B
n
log (A) = n log A
y=
y
y = ax
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
Mindmaps
f(x) = a sinx + b cosx
Compare coefficients
compare to required
trigonometric identity
a = k cos β
f(x) = k sin(x + β)
= k sinx cos β + k cosx sin β
b = k sin β
Square and add then
square root gives
k  a 2  b2
Process
example
Divide and inverse tan gives
  tan 1
Wave Function
b
a
a and b values
decide which
quadrant
transforms
f(x)= a sinx + b cosx
Write out required form
into the form
f ( x)  k sin( x   )
f ( x)  k sin( x   )
OR
f ( x)  k cos( x   )
Related topic
Solving trig equations
Mindmaps