Formulas and Rules | IGCSE Edexcel Further Pure Math
Surds
1. √
√
2. √
√
√
√
3. √a / √b and √a - √b are conjugate surds. The product of conjugate surds is a rational
number.
Indices
1. Am x an = a m + n
2. am / an = am – n
3. (am)n = amn
4. a0 = 1
5. a –n =
6.
7.
= n√a
= n√am = (n√a)m
Logarithms
1. log a + log b = logab
2. log a – log b = log
3. a log x y = log x y a
4. log a a = 1
5. loga
=
6. loga1 = 0
7. log a b =
a
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Binomial theorem
1. (x + y)n = xn + (nc1 * xn-1 * y)+ (nc2 * xn-2 * y2n)+ ( nc3 xn-3 y3 ) + ( nc4 xn-4 y4 ) ………………………
2. (r+1)th term = ( ncr xn-r yr )
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Quadratic Equation
Nature of roots
#
ax2 + bx + c = 0
1. If b2 – 4ac > 0, roots are real & different / real and distinct and the curve y = ax2 + bx + c
will cut the x axis at two real and distinct points
2. If b2 – 4ac < 0, roots are not real/ imaginary / complex and the curve y = ax2 + bx + c will
lie entirely above the x axis if a > 0 and entirely below the x axis if a < 0.
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Formulas and Rules | IGCSE Edexcel Further Pure Math
3. If b2 – 4ac = 0, roots are real and equal / repeated / coincident and the curve y = ax2 + bx
+ c touches the x-axis.
4. If b2 – 4ac ≥ 0, roots are real.
Solving Quadratic Inequality
When α and β (α<β) are two roots of
1.
2.
3.
4.
If
If
If
If
, range of values of
, range of values of
, range of values of
, range of values of
(a>0) and
:
:
:
:
α, β
1. If α and β are two roots of
,
a.
b.
2. If two roots of an unknown equation is given and you want to find the equation, follow the
following steps:
a. Find the sum of the roots.
b. Find the product of the root.
(
)
c. Use the following formula,
d. Simplify the equation if needed.
(
)
3.
)
(
)
4. (
(
)
5.
(
)
(
)
6.
(
)
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Formulas and Rules | IGCSE Edexcel Further Pure Math
7. (
)
(
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)
Formulas and Rules | IGCSE Edexcel Further Pure Math
Set, Relation and Function
1. A set is any well-defined collection, list or class of objects. It may be given by either listing its
members or defining its properties clearly.
2. A relation is a set of ordered pairs.
3. A function is a relation in which every elements in the domain has a unique image in the range.
4. The range of a function f is the set of values f(x) for the given domain.
5. Two functions f and g can be combined to produce composite functions fg or gf such that
fg(x) = f(g(x)) and gf(x) = g(f(x)).
In general, fg and gf are different functions.
6. The absolute value of x, written as |x|, is defined as
x if x > 0
|x| = 0 if x = 0
-x if x < 0
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Formulas and Rules | IGCSE Edexcel Further Pure Math
Circular Measure
1. ∏ radian = 1800
2. For a sector of a circle enclosed by two radii that subtend an angle of θ radians at the
centre, the arc length s is given by
s=rθ
and the area of the sector A is given by
A=
where r is the radius of the circle.
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Formulas and Rules | IGCSE Edexcel Further Pure Math
Trigonometry
1. Sin θ =
2. Cos θ =
3. Tan θ =
4. Sec x =
5. Cosec x =
6. Cot x =
7. Cot x =
8. Sin2 + cos2 x = 1
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Formulas and Rules | IGCSE Edexcel Further Pure Math
9. sec2 x - tan2 x =1
10. cosec2 - cot2 x = 1
11. Sec2 x = 1 + tan2 x
12. Sin(A + B) = sin A cos B + cos A sin B
13. Cos(A+B) = cos A cos B – sin A sin B
14. Tan (A+B) =
15. Sin 2A = 2 sin A cos A
16. Cos 2A = cos2 A – sin2 A = 1 -2sin2 A = 2 cos2 A -1
17. Tan 2A =
18. Tan A =
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Arithmetic Progression (A.P)
1. Nth term = a + (n-1)d
2. Sn = *
(
)
Geometric Progression (G.P)
1. Nth term = arn-1
2. Sn =
3. Sn =
(
)
(
)
,r>1
,r<1
# -1 < r < 1 or |r| < 1.
The series is convergent. It has sum to infinity.
1. Sα =
Otherwise the series is divergent. It has does not have sum to infinity.
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Formulas and Rules | IGCSE Edexcel Further Pure Math
Co – ordinate Geometry
1. The distance between two points A(x1 , y1) and B(x2 , y2) is √*(
)
(
)+
2. The gradient of the line joining A(x1, y1) and B(x2 , y2) is
3. The coordinates of the mid-point of the line joining A(x1 , y1) and B(x2 , y2) are
.
/ .
/.
4. Finding coordinates when a point divides a line internally.
(
)
(
)
(
)
(
)
5. The equation of the straight line having a gradient m and passing through the point (x 1,
y2) is given by : y – y1 = m (x – x1).
6. Two lines are parallel if their gradients are equal.
7. Two lines are perpendicular to each other if the product of their gradients is -1.
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Differentiation
( ) represents the gradient of the tangent to the curve at any point .
1. For a curve
2. If
, then
(
3.
4. If
, where
and n are constants.
)
is a function of , and
us a function of , then
5. If y,
and v are functions of
and
6. If
and
and
are functions of
The following are true only when
7.
(
)
8.
(
)
9.
(
)
(chain rule).
, then
(product rule).
, then
(quotient rule).
is in radians:
Other formulae
10.
(
)
11.
(
)
(
12.
(
)
(
)
13.
(
)
14.
(
)
15.
(
)
16.
,
(
)-
(
)
17.
,
(
)-
(
)
18.
,
(
)-
(
)
(
)
)
Application of Differentiation
19. Stationary points or turning points of a function
20. The second derivative (
(a) If
( ) occur when
.
) determines the nature of the stationery points:
is negative, the stationery point is a maximum point.
(b) If
is positive, the stationary point is a minimum point.
(c) If
is zero, the nature of the stationery point depends on how the value of
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changes
Formulas and Rules | IGCSE Edexcel Further Pure Math
near the stationary point:
(i) if the sign
does not change before and after the turning point, the point is a point of
inflexion.
(ii) if the value of
changes from positive to negative after the turning point, the point is a
maximum point.
(iii) if the value of
changes from negative to positive after the turning point, the point is a
minimum point.
21. To sketch a curve, note
(i) the points where
or
(ii) the nature and position of the stationary points
(iii) the direction of the curve as and approach infinity.
(iv) the interval on which the gradient is positive or negative.
Further Applications of Differentiation
22. If
( ) and
and
are small increments in
23. If the displacement of a body is given by
by
and acceleration
is given by
or
and
respectively, then
( ), where is the time, then velocity v is given
.
24. When a body is instantaneously at rest, its velocity is zero.
25. If a curve is given by
gradient of the normal is
( ), then the gradient of the tangent to the curve is given
.
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Formulas and Rules | IGCSE Edexcel Further Pure Math
Integration
1.
2.
(
(
)
(
) (
)
)
3.
4.
5.
6.
7.
8.
9.
(
)
10.
(
)
(
11.
(
)
)
(
)
(
)
12. The area bounded by the curve
∫
and
is given by
( ), the -axis and the lines
and
is given by
.
13. The area bounded by the curve
∫
( ), the -axis and the lines
.
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14. Area between ( ) and ( ) = ∫ | ( )
15. When the area bounded by
( )|
( ), the -axis and the lines
and
through 360o about the -axis, the volume of solid of revolution is given by
16. When the area bounded by
o
( ), the -axis and the lines
is rotated
∫
and
through 360 about the -axis, the volume of solid of revolution is given by
is rotated
∫
17. If the velocity v of a particle is given as a function of time t, i.e.
( ), then ∫ ( )
give the expression for the distance covered.
18. If the acceleration α of a particle is given as function of time t, i.e. α=f(t), then ∫ ( )
the expression for the velocity.
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.
will
will give
Formulas and Rules | IGCSE Edexcel Further Pure Math
Vector
1. A vector quantity possesses both magnitude and direction. The vector → has a
magnitude or modulus |→ | and its direction is from O to A.
2. If a fixed point O is taken as a origin, the vector → is known as the position vector of A
3.
4.
5.
6.
with references to O.
Two vectors are equal if they have the same magnitude and direction.
Vector addition: If two vectors acting at a point are represented both in magnitude and
direction by the adjacent sides of a parallelogram, then their sum is represented both in
magnitude and direction by the diagonal of the parallelogram through the point.
a = b  |a| = |b| and a is parallel to b.
ha = kb  |ha| =|hb| and a is parallel to b.
or, h = k = 0 if a is not parallel to b.
If → = ha + kb and → = ma + nb, where h, k, m and n are constants, and a is not parallel
to b, then →
→
= m and k = n.
7. If the position vectors of A and B relative to an origin O are a and b respectively, the
position vector of the mid point (M) of AB is →
8. The scalar product of two vectors a and b is given by
a-b = |a||b| cos θ
where θ is the angle between the two vectors.
9. Commutative law: a x b = b x a
10. Distributive law: a(b+c) = a x b + a x c
11. If a = x1i + y1j and b = x2i + y2j, then
a x b = x1x2 + y1y2
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(
)
.
Formulas and Rules | IGCSE Edexcel Further Pure Math
Binomial Expansion
1.
2.
(
(
 eg:
)(
(
)(
)
(
)(
)
)
(
)
)
3.
4.
5.
6. (
)
(
)(
)(
)
)

7. (
[Note: Above formula cannot be used when ‘n’ is negative or in fraction]
(
)

8. (
(
(
)
)(
)
[In above formula the value of ‘a’ must be ‘1’ ]
)
{ (
)}


.
.
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/
/
Formulas and Rules | IGCSE Edexcel Further Pure Math
Graph Sketching
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(
(
)
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Formulas and Rules | IGCSE Edexcel Further Pure Math
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(
(
)
)(
(
)(
)
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(
)
)(
)(
)
Formulas and Rules | IGCSE Edexcel Further Pure Math
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