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4PM1
REVISION GUIDE
Pearson
Edexcel
IGCSE
Further Pure
Mathematics
Surds
1. √𝑎 𝑥 √𝑏 = √𝑎𝑏
2. √𝑎 ÷ √𝑏 = √
𝑎
𝑏
3. √a / √b and √a - √b are conjugate surds. The product of conjugate surds is a rational
number.
m
Indices
o.
co
1. Am x an = a m + n
2. am / an = am – n
6. 𝑎 = n√a
n
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𝑎𝑛
m
n
rlo
1
𝑛
𝑚
yr
3. (am)n = amn
4. a0 = 1
1
5. a –n =
m
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7. 𝑎 𝑛 = √a = ( √a)
ct
w
by
ed
1. log a + log b = logab
𝑎
2. log a – log b = log
w
Logarithms
𝑏
co
lle
3. a log x y = log x y a
4. log a a = 1
5.
loga𝑥
log 𝑏𝑥
=
log 𝑏𝑎
6. loga1 = 0
1
7. log a b = log 𝑏 a
Binomial theorem
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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 )
Quadratic Equation
Nature of roots
ax2 + bx + c = 0
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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
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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.
co
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3. If b2 – 4ac = 0, roots are real and equal / repeated / coincident and the curve y = ax2 + bx
+ c touches the x-axis.
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s.
4. If b2 – 4ac ≥ 0, roots are real.
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Solving Quadratic Inequality
by
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If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 > 0, range of values of 𝑥: 𝑥 < 𝛼, 𝑥 > 𝛽
If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 ≥ 0, range of values of 𝑥: 𝑥 ≤ 𝛼, 𝑥 ≥ 𝛽
If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 < 0, range of values of 𝑥: 𝛼 < 𝑥 < 𝛽
If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 ≤ 0, range of values of 𝑥: 𝛼 ≤ 𝑥 ≤ 𝛽
ed
1.
2.
3.
4.
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When α and β (α<β) are two roots of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 (a>0) and
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α, β
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1. If α and β are two roots of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0,
𝑏
a. 𝛼 + 𝛽 = −
b. 𝛼𝛽 =
𝑐
𝑎
𝑎
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, 𝑥2 − (𝑠𝑢𝑚 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠) × 𝑥 + 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠 = 0
d. Simplify the equation if needed.
3. ∝2+ 𝛽2 = (𝛼 + 𝛽)2 − 2𝛼𝛽
4. (𝛼 − 𝛽)2 = (𝛼 + 𝛽)2 − 4𝛼𝛽
5. 𝛼3 + 𝛽3 = (𝛼 + 𝛽)3 − 3𝛼𝛽(𝛼 + 𝛽)
6. 𝛼3 − 𝛽3 = (𝛼 − 𝛽)3 + 3𝛼𝛽(𝛼 − 𝛽)
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7. (𝛼 + 𝛽)3 = 𝛼3 + 𝛽3 + 3𝛼𝛽(𝛼 + 𝛽)
Set, Relation and Function
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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
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
1
A = 𝑟2 θ
2
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where r is the radius of the circle.
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Trigonometry
𝑜𝑝𝑝
1. Sin θ = ℎ𝑦𝑝
𝑎𝑑j
2. Cos θ = ℎ𝑦𝑝
𝑜𝑝𝑝
3. Tan θ = 𝑎𝑑j
1
4. Sec x = cos 𝑥
1
5. Cosec x = sin 𝑥
1
6. Cot x = tan 𝑥
cos 𝑥
7. Cot x = sin 𝑥
8. Sin2 + cos2 x = 1
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
tan 𝐴 + tan 𝐵
14. Tan (A+B) = 1 − tan 𝐴 tan 𝐵
15. Sin 2A = 2 sin A cos A
16. Cos 2A = cos2 A – sin2 A = 1 -2sin2 A = 2 cos2 A -1
2 tan 𝐴
17. Tan 2A = 1−tan2 𝐴
sin 𝐴
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18. Tan A = cos 𝐴
Arithmetic Progression (A.P)
1. Nth term = a + (n-1)d
𝑛
2. Sn = {2𝑎 + (𝑛 − 1)𝑑
2
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s.
,r<1
# -1 < r < 1 or |r| < 1.
The series is convergent. It has sum to infinity.
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a
𝑎
rd
1−𝑟
,r>1
rlo
3. Sn =
𝑟−1
𝑎 (1−𝑟𝑛)
pe
2. Sn =
o.
1. Nth term = arn-1
𝑎 (𝑟𝑛−1)
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Geometric Progression (G.P)
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1. Sα = 1−𝑟
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Otherwise the series is divergent. It has does not have sum to infinity.
Co – ordinate Geometry
1. The distance between two points A(x1 , y1) and B(x2 , y2)𝑦is
√{(𝑥2 − 𝑥1) + (𝑦2 − 𝑦1)}
2−𝑦1
2. The gradient of the line joining A(x y ) and B(x y ) is
1, 1
2, 2
𝑥2−𝑥1
3. The coordinates of the mid-point of the line joining A(x1 , y1) and B(x2 , y2) are
𝑦 +𝑦
𝑥1 + 𝑥2
) , ( 1 2) .
2
2
(
yr
ez
s.
𝑇+𝑈
pe
rlo
𝑏=
(𝑈 × 𝑥1) + (𝑇 × 𝑥2)
𝑇+𝑈
(𝑈 × 𝑦1) + (𝑇 × 𝑦2)
rd
𝑎=
o.
co
m
4. Finding coordinates when a point divides a line internally.
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5. The equation of the straight line having a gradient m and passing through the point (x1,
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.
Differentiation
1. For a curve 𝑦 = 𝑓(𝑥) represents the gradient of the tangent to the curve at any point 𝑥.
𝑑𝑦
2. If 𝑦 = 𝑎𝑥𝑛, then = 𝑎𝑛𝑥𝑛−1, where 𝑎 and n are constants.
3.
𝑑
(𝑢 + 𝑣) =
𝑑𝑥
𝑑𝑢
𝑑𝑥
𝑑𝑣
+
𝑑𝑥
𝑑𝑥
4. If 𝑦 is a function of 𝑢, and 𝑢 us a function of 𝑥, then
5. If y, 𝑢 and v are functions of 𝑥 and 𝑦 = 𝑢𝑣, then
𝑑𝑦
𝑑𝑦
=
𝑑𝑥
=𝑢
𝑑𝑥
𝑑𝑦
×
𝑑𝑢
𝑑𝑢
𝑑𝑣
𝑑𝑥
𝑑𝑢
𝑑𝑥
𝑑𝑥
+𝑣
𝑑𝑣
𝑣𝑑𝑢−
𝑢
𝑢
𝑑𝑥
(quotient rule).
𝑑𝑥
o.
yr
ez
(cos 𝑥) = − sin 𝑥
(tan 𝑥) = sec2 𝑥
s.
9.
𝑑𝑥
𝑑
(sin 𝑥) = cos 𝑥
rd
8.
𝑑
𝑑𝑥
𝑑
15.
16.
17.
18.
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
pe
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a
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(𝑎 sin 𝑏𝑥) = 𝑎𝑏 cos 𝑏𝑥
by
𝑑𝑥
𝑑
(tan𝑛 𝑥) = 𝑛 tan𝑛−1 𝑥 (sec2 𝑥)
(𝑎 cos 𝑏𝑥) = −𝑎𝑏 sin 𝑏𝑥
ed
14.
𝑑𝑥
𝑑
(cos𝑛 𝑥) = 𝑛 cos𝑛−1 𝑥 (− sin 𝑥)
(𝑎 tan 𝑏𝑥) = 𝑎𝑏 sec2 𝑏𝑥
ct
13.
𝑑𝑥
𝑑
(sin𝑛 𝑥) = 𝑛 sin𝑛−1 𝑥 (𝑐𝑜𝑠𝑥)
[sin(𝑎𝑥 + 𝑏)] = 𝑎 cos(𝑎𝑥 + 𝑏)
lle
12.
𝑑𝑥
𝑑
co
11.
𝑑
rlo
Other formulae
10.
co
The following are true only when 𝑥 is in radians:
7.
(product rule).
m
𝑑𝑦
6. If 𝑦, 𝑢 and 𝑣 are functions of 𝑥 and 𝑦 = , then
= 𝑑𝑥
𝑣2
𝑑𝑥
𝑣
(chain rule).
[cos(𝑎𝑥 + 𝑏)] = −𝑎 sin(𝑎𝑥 + 𝑏)
[tan(𝑎𝑥 + 𝑏)] = 𝑎 sec2(𝑎𝑥 + 𝑏)
Application of Differentiation
19. Stationary points or turning points of a function 𝑦 = 𝑓(𝑥) occur when
20. The second derivative (
(a) If
𝑑2𝑦
𝑑𝑥2
𝑑2𝑦
𝑑 2𝑦
𝑑𝑥 2
𝑑𝑦
𝑑𝑥
= 0.
) determines the nature of the stationery points:
is negative, the stationery point is a maximum point.
(b) If 𝑑𝑥2 is positive, the stationary point is a minimum point.
𝑑2𝑦
(c) If 𝑑𝑥2 is zero, the nature of the stationery point depends on how the value of
𝑑𝑦
𝑑𝑥
changes
near the stationary point:
𝑑𝑦
(i) if the sign 𝑑𝑥 does not change before and after the turning point, the point is a point of
inflexion.
𝑑𝑦
(ii) if the value of 𝑑𝑥 changes from positive to negative after the turning point, the point is a
maximum point.
(iii) if the value of
𝑑𝑦
changes from negative to positive after the turning point, the point is a
𝑑𝑥
co
m
minimum point.
21. To sketch a curve, note
(i) the points where 𝑥 = 0 or 𝑦 = 0
(ii) the nature and position of the stationary points
(iii) the direction of the curve as 𝑥 and 𝑦 approach infinity.
(iv) the interval on which the gradient is positive or negative.
yr
o.
Further Applications of Differentiation
ez
22. If 𝑦 = 𝑓(𝑥) and 𝛿𝑥 and 𝛿𝑦 are small increments in 𝑥 and 𝑦 respectively, then 𝛿𝑦 =
𝑑𝑦
× 𝛿𝑥
𝑑𝑥
𝑑2𝑠
by 𝑑𝑡 and acceleration 𝑎 is given by 𝑑𝑡 or 𝑑𝑡2.
rd
𝑑𝑣
rlo
𝑑𝑠
s.
23. If the displacement of a body is given by 𝑠 = 𝑓(𝑡), where 𝑡 is the time, then velocity v is given
.p
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24. When a body is instantaneously at rest, its velocity is zero.
𝑑𝑦
25. If a curve is given by 𝑦 = 𝑓(𝑥), then the gradient of the tangent to the curve is given and the
1
w
gradient of the normal is − 𝑑𝑦 .
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𝑑𝑥
𝑑𝑥
Integration
1. J 𝑎𝑥𝑛 𝑑𝑥 =
𝑎𝑥𝑛+1
𝑛+1
2. J (𝑎𝑥 + 𝑏)𝑛𝑑𝑥 =
3.
4.
5.
6.
+𝑐
𝑛 ≠ −1
(𝑎𝑥+𝑏)^(𝑛+1)
(𝑛+1)𝑎
+𝑐
J cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐
J sin 𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐
J sec2 𝑥 𝑑𝑥 = tan 𝑥 + 𝑐
1
J cos 𝑏𝑥 𝑑𝑥 = sin 𝑏𝑥 + 𝑐
𝑏
1
7. J sin 𝑏𝑥 𝑑𝑥 = − cos 𝑏𝑥 + 𝑐
1
9. J cos(𝑎𝑥 + 𝑏) 𝑑𝑥 =
𝑎
m
𝑏
𝑏
tan 𝑏𝑥 + 𝑐
co
1
sin(𝑎𝑥 + 𝑏) + 𝑐
o.
8. J sec2 𝑏𝑥 𝑑𝑥 =
1
𝑎
ez
𝑎
1
tan(𝑎𝑥 + 𝑏) + 𝑐
s.
11. J sec2(𝑎𝑥 + 𝑏) 𝑑𝑥 =
yr
10. J sin(𝑎𝑥 + 𝑏) 𝑑𝑥 = − cos(𝑎𝑥 + 𝑏) + 𝑐
rd
12. The area bounded by the curve 𝑦 = 𝑓(𝑥), the 𝑥-axis and the lines 𝑥 = 𝑎 and 𝑥 = 𝑏 is given by
𝑏
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∫𝑎 𝑦 𝑑𝑥.
𝑏
∫𝑎 𝑥 𝑑𝑦.
co
13. The area bounded by the curve 𝑥 = 𝑓(𝑦), the 𝑦-axis and the lines 𝑦 = 𝑎 and 𝑦 = 𝑏 is given by
𝑏
14. Area between 𝑔(𝑥) and 𝑓(𝑥) = ∫𝑎 |𝑔(𝑥) − 𝑓(𝑥)| 𝑑𝑥
15. When the area bounded by 𝑦 = 𝑓(𝑥), the 𝑥-axis and the lines 𝑥 = 𝑎 and 𝑥 = 𝑏 is rotated
𝑏
through 360o about the 𝑥-axis, the volume of solid of revolution is given by 𝜋 ∫𝑎 𝑦2 𝑑𝑥.
16. When the area bounded by 𝑦 = 𝑓(𝑥), the 𝑦-axis and the lines 𝑦 = 𝑎 and 𝑦 = 𝑏 is rotated
𝑏
through 360o about the 𝑦-axis, the volume of solid of revolution is given by 𝜋 ∫𝑎 𝑥2 𝑑𝑦.
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17. If the velocity v of a particle is given as a function of time t, i.e. 𝑣 = 𝑓(𝑡), then ∫ 𝑓(𝑡) 𝑑𝑡 will
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 ∫ 𝑓(𝑡) 𝑑𝑡 will give
the expression for the velocity.
Vector
1. A vector quantity possesses both magnitude and direction. The vector −→ has a
0𝐴
magnitude or modulus |−→| and its direction is from O to A.
0𝐴
2. If a fixed point O is taken as a origin, the vector −→ is known as the position vector of A
0𝐴
m
𝐴𝐵
𝑃Q
s.
6.
ez
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o.
5.
co
3.
4.
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
𝑃Q
rlo
𝐴𝐵
rd
to b, then −→ = −→ → ℎ = m and k = n.
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a
pe
7. If the position vectors of A and B relative to an origin O are a and b respectively, the
(𝑎+𝑏)
position vector of the mid point (M) of AB is −→= 𝑚 =
.
0𝑀
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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
2
Binomial Expansion
1. 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) …
• eg: 4! = 4 * 3 * 2 * 1 = 24
𝑛!
2. (𝑛−2)! = 𝑛(𝑛−1)(𝑛−2) = 𝑛(𝑛 − 1)
(𝑛−2)
3. 𝑛𝑐1 = 𝑛
4. 𝑛𝑐2 =
5. 𝑛𝑐3 =
2!
𝑛(𝑛−1)(𝑛−2)(𝑛−3)
3!
(𝑎 + 𝑥)𝑛 = 𝑎𝑛 +
𝑛𝑐 1 * 𝑎𝑛−1 * 𝑥 + 𝑛𝑐 2 * 𝑎𝑛−2 * 𝑥2 + 𝑛𝑐 3 * 𝑎𝑛−3 * 𝑥3 + …
m
6.
𝑛(𝑛−1)
2!
yr
1
2
ez
{4(1 + )}
s.
𝑥2
1
1
➢
= 42 * (1 +
𝑥2 2
4
1
𝑥2 2
➢
= 2 (1 + )
lle
ct
ed
by
w
w
w
4
co
rd
4
)
rlo
(4 + 𝑥2 )2 =
pe
1
8.
3!
[In above formula the value of ‘a’ must be ‘1’ ]
.p
a
•
o.
co
• [Note: Above formula cannot be used when ‘n’ is negative or in fraction]
𝑛(𝑛−1) 2
𝑛(𝑛−1)(𝑛−2) 3
7. (𝑎 + 𝑥)𝑛 = 1 + 𝑛𝑥 +
𝑥 +
𝑥 + …
Graph Sketching
𝑦 = −𝑥
𝑦=𝑥+1
𝑦 = −𝑥 + 1
co
𝑦 = 𝑥2 + 1
lle
ct
ed
by
w
w
w
.p
a
pe
rlo
rd
s.
ez
yr
o.
co
m
𝑦=𝑥
𝑦 = 𝑥2 - 1
𝑦 = (𝑥 + 1)2
m
𝑦 = 𝑥2
by
w
w
w
.p
a
pe
rlo
rd
s.
ez
yr
o.
co
𝑦 = (𝑥 − 1)2
co
lle
ct
ed
𝑌 = 𝑒 𝑥 ; 𝑦 = 𝑒−𝑥; 𝑦 = − 𝑒−𝑥; 𝑦 = − 𝑒𝑥
𝑌 = 𝑒𝑥 + 1
𝑦 = − 𝑥3
by
w
w
w
.p
a
pe
rlo
rd
s.
ez
yr
o.
𝑦 = 𝑥3
co
m
𝑦 = 𝑒𝑥 - 1
co
lle
ct
ed
𝑦 = 𝑥3 + 1
𝑦 = 𝑥3 - 1
𝑦 = −𝑥3 − 1
𝑌 = (𝑥 + 1)3
𝑌 = (𝑥 - 1)3
co
lle
ct
𝑦 = (𝑥 + 1)(𝑥 + 4)(𝑥 − 2)
ed
by
w
w
w
.p
a
pe
rlo
rd
s.
ez
yr
o.
co
m
𝑦 = −𝑥3 + 1
𝑦 = (𝑥 − 1)(𝑥 − 3)(−𝑥 + 2)
ed
ct
lle
co
by
yr
ez
s.
rd
rlo
pe
.p
a
w
w
w
m
co
o.