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CORE 4 Summary Notes
1 Rational Expressions 1
∑ Factorise all expressions where possible
∑ Cancel any factors common to the numerator and denominator
x
2
+ 5x
x(x + 5)
x
2
x
x – 5
∑ To add or subtract - the fractions must have a common denominator
x
x x –( 4 )
6(x – 4)
x x – 4( )
x x –( 4 )
6x – 24 + 3
x x – 4( )
6x – 21
x x – 4( )
∑ To divide by a rational expression you can multiply by it’s reciprocal.
2 Rational Expressions
∑ REMAINDER THEOREM
When p(x) is divided by (ax-b) the remainder is p(
b
a
Note that x =
b
a
is the solution of the equation ax – b = 0
E.g Show that 2x –1 is a factor of 2x
3
+ 7x
2
2x – 1 =
x = 0.
3
2
Must be a factor as
the remainder is zero
∑ An Algebraic fraction is ‘PROPER’ is the degree of the polynomial that is the
numerator is less than the degree of the polynomial that is the denominator.
x + 6
x + 2
x + 2 + 4
x + 2
x + 2
x + 2
x + 2
x + 2
E.g.
x
2
+ 6x – 3
x + 2
(x + 2)(x + 4) – 11
x + 2
= x + 4 –
x + 2
x
3
2
+ x + 5
x
2
(x
2
x
2
= x – 2 +
4x – 1
x
2
∑ An identity is a statement that is true for all values of x for which the
statement is defined
∑ PARTIAL FRACTIONS
Any proper algebraic fraction with a denominator that is a product of distinct
linear factors can be written as partial fractions as the sum of proper fractions
whose denominators are linear factors.
5x + 1
(x – 1)(2x + 1)(x – 5)
can be expressed in the form
A
x – 1
B
2x + 1
C
x – 5
E.g
(x – 2)(x + 3)
A
x – 2
B
x + 3
A(x + 3) + B(x – 2)
(x – 2)(x + 3)
Looking at the numerators A(x + 3) + B(x - 2) = 5
x= 2 5A = 5 so A = 1
x= -3 -5B = 5 so B = -
REPEATED FACTOR
5x + 1
(x 1 )(2x + 1)
2
can be expressed in the form
A
x – 1
B
2 x + 1
C
(2x + 1)
2
3 Parametric Equations
∑ Two equations that separately define the x- and y- coordinates of a graph in terms
of a third variable.
∑ The third variable is called the parameter
x = t + 4 y = 1 – t
2
∑ To convert a pair of parametric equations to single Cartesian equation, eliminate the
parameter.
E.g.
x = t + 4 t = x – 4 so t
2
= x
2
y = 1 – t
2
y = 1 – (x
2
y = 8x – x
2
∑ CIRCLE and ELLIPSE
The curve x = r cos q y = r sin q is a circle with radius r and centre the origin
The curve (^) x = rcos q (^) + p y = r sin q (^) + p is a circle with radius r and centre (p,q)
The curve x = acos q y = bsin q is an ellipse, centre the origin.
Its width is 2a and its height is 2b units.
∑ For an expansion of an expression such as
x + 5
split the expression into
(3 – x)(1 + 3x)
PARTIAL FRACTIONS before attempting an expansion.
5 Trigonometric Formulae
tan(A + B) =
tanA + tanB
1 – tanA tanB
ADDITION FORMULAE
sin (A+B) = sin A cos B + sin B cos A
sin (A-B) = sin A cos B – sin B cos A
cos (A+B) = cos A cos B – sin A sin B
tan(A – B) =
tanA – tanB
1 + tanA tanB
cos (A-B) = cos A cos B + sin A sin B
DOUBLE ANGLE FORMULAE
sin 2A = 2sin A cos A
cos 2A = cos
2 A –sin
2 A = 2 cos
2 A –1 = 1 – 2sin
2 A
tan 2A =
2tanA
1 – tan
2
A
∑ asin x + bcos x can be written in the form
rsin(x + a) where a = rcos a and b = rsin a
rcos(x – a) where a = rsin a and b = rcos a
r
2
= a
2
+ b
2
e.g Find the maximum value of the expression 2 sin x + 3 cos x by expressing it in
the form
rsin(x + a)
rsin a^
3
rcos a 2
tan a =
3
2
a = 56
r
2 = a
2
2
r
2
= 2
2
2
= 13
r = 13
2sinx + 3cosx = (^) 13 sin(x + 56)
Maximum value is 13 which occurs when sin(x+56) =1 x = 34
∑ asinx – bcos x can be written in the form
rsin(x – a), where a = rcosa and b = rsina
rcos(x + a), where a = – rsina and b = – rcosa
r
2
= a
2
+ b
2
∑ Both of the above are useful in SOLVING EQUATIONS.
6 Differential Equations
∑ Key points from core 3
The derivative of e
ax
is ae
ax
Ú
e
ax
dx =
e
ax
+ c
a
Ú
ax
dx =
lnÁax + bÁ + c
+ b a
Ú
f '(x)
dx = lnÁf(x)Á + c
f (x )
Ú
a
2
+ x
2
dx =
a
tan
Ë
Á
x ˆ
a
+ c
Ú
a
2
2
dx = sin
Ë
Á
a
+ c
∑ An equation that involves a derivative is called a Differential Equation. They are
used to model problems involving rates of change.
e.g The rate of growth of a population is proportional to the size of the population.
Let the population at time t to be P
dP
= kP where k is a constant
dt
∑ SEPARATING THE VARIABLES – a method of solving differential equations.
Find the general solution of dy^ = 2x(y + 4) y > 0
- Separate the variables
dx
dy
y + 4
= 2x dx
- Integrate both sides
Ú
1
y + 4
dy =
Ú
2x dx
ln(y + 4) = x
2
y + 4 = e
x
2
x
2
e
c = Ae
x
2
where A = e
c
y = Ae
x 2
∑ USING TRIGONOMETRICAL IDENTITIES
Ú
sin x cos x dx =
Ú
sin 2 x
cos 2 x + c
Using the identity
2sinx cosx = sin 2x
Using the identity Ú
cos
2
Ú
x dx = (c
os 2 x + 1) dx
cos 2x = 2cos
2 x -
Replacing x by ½ x in the identity
Ú
si n
2
Ú
cos 2
x dx =
x = 1 – 2sin
2
x
9 Vectors
∑ A vector has two properties :
Magnitude (or size)
and Direction
∑ Vectors with the same magnitude and direction are equal.
∑ The modulus of a vector is its magnitude.
The modulus of the vector a is written |a|
∑ Any vector parallel to the vector a may be written as (^) l a where (^) l is a non-zero
real number and is sometimes called a scalar multiple of a.
- a has the same magnitude but is in the opposite direction to a.
∑ Vectors can be added and subtracted using the ‘triangle law’.
∑ Vectors can be written in column vector form such as
Ë
Ê
Á
3
- 7
∑ A unit vector is a vector with a magnitude of 1.
The vectors i, j and k are unit vectors in the direction of the x-, y- and z- axes
respectively.
As column vectors i =
Ë
, j = , k =
Ê
Á
1 ˆ
0
0 ¯
Ë
Ê
Á
0 ˆ
1
0 ¯
Ë
Ê
Á
0 ˆ
0
1 ¯
Vectors can be written as linear combinations of these unit vectors,
e.g.
Ë
= 3 i – 7 j + k
Ê
Á
3 ˆ
The magnitude (or modulus) of the vector Ë
= 3 i – 7 j + k is
Ê
Á
3 ˆ
1 ¯
2
2
2
The distance between two points (x1, y1, z 1 ) and (x2, y2, z 2 ) is
( x 2
2
2
2
∑ For every point P there is a unique vector
OP
(where O is a fixed origin) which is
called the position vector of the point P.
The point with coordinates (x, y, z) has position vector Ë
Ê
Á
x ˆ
y
z ¯
For two points A and B with position vectors OA and OB the vector AB is
given by
∑ The general form of a vector equation of a line is
r (^) = p (^) + l d
r is the position vector of any point on the line,
p is the position vector of a particular point on the line,
l is a scalar parameter,
d is any vector parallel to the line (called a direction vector )
N.B. Since p and d are not unique then your equation might not
look identical to the one given in the back of the book!
