PURE MATHEMATICS
Mensuration
Surface area of sphere = 4πr 2
Area of curved surface of cone = πr × slant height
Arithmetic series
S n a l n[ a n d] u a n d n n
( ) 2 ( 1)
= ( 1)
2
1
2
= 1 + = + −
+ −
Geometric series
1 for 1
1
(1 )
1
= − <
−
= −
=
∞
−
r r
S a r S a r u ar n n n n
Summations
( )1 2
1
1
+ = Σ= r n n n r
( 1)(2 1) 6
1
1
2 + + = Σ= r n n n n r
2 2
4
1
1
3 ) 1 ( + = Σ= r n n n r
Trigonometry – the Cosine rule a2 = b2 + c2 − 2bc cos A
Binomial Series
∈ + + ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
+ + ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
+ ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
+ = + − − a − b b n r n a b n a b n a b n an n n n r r n (
2
1
( ) 1 2 2 … … ) where !( )!
C ! r n r n r n r n −
= = ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
+ = + + − + + − − + x + x < n∈ r x n nx n n x n n n r r ( 1,
1.2
( 1) ( 1)
1.2
(1 ) 1 ( 1) 2 …
…
… … )
Logarithms and exponentials ax = exln a
Complex numbers
{r(cosθ + i sinθ )}n = r n (cos nθ + i sin nθ ) eiθ = cosθ + i sinθ
The roots of z n = 1 are given by n k z
2π i
= e , for k = 0, 1, 2, … , n −1
N
R klj 5
Maclaurin’s series f( ) f(0) f (0) 2! f (0) … ! f ( ) (0) …
2
= + ′ + ′′ + + r + r r x x x x r x x x x x x r e exp( ) 1 2! ! for all
2
= = + + +…+ +… r x x x x x x r + = − + − + (−1)r+ + (−1 < ln(1 ) 2 3 2 3 1
… … 1) x r x x x x x r r for all
(2 1)!
( 1)
3! 5! sin 3 5 2 1
… +…
+
= − + − + −
+
x r x x x x r r for all
(2 )!
( 1)
2! 4! cos 1
2 4 2
= − + −…+ − +…
Hyperbolic functions cosh2 x − sinh2 x = 1 sinh 2x = 2sinh x cosh x cosh 2x = cosh2 x + sinh 2 x cosh−1 x = ln{x + x2 −1} (x 1) sinh−1 x = ln{x + x2 +1}
( 1)
1
tanh ln 1 2
1 1 < ⎟⎠
⎞
⎜⎝
⎛
−
− = + x x x x
Conics
Ellipse Parabola Hyperbola Rectangular hyperbola Standard form 1 2
2
2
2
+ = b y a x ax y 2 = 4 1 2
2
2
2
− = b y a x 2 xy = c
Asymptotes none none b y a x = ± x = 0, y = 0
Trigonometric identities sin(A ± B) = sin Acos B ± cos Asin B cos(A ± B) = cos Acos B ∓ sin Asin B
( ( )π)
1 tan tan tan( ) tan tan 2
± = ± A ± B ≠ k + 1
A B
A B A B
∓
2 cos 2 sin A + sin B = 2sin A + B A − B
2
sin
2
sin A − sin B = 2cos A + B A − B
2
cos
2
cos A + cos B = 2 cos A + B A − B
2
sin
2
cos A − cos B = −2sin A + B A − B
6 klm
Vectors
The resolved part of a in the direction of b is b a.b
The position vector of the point dividing AB in the ratio λ :μ is λ μ μ λ
+
a + b
Vector product:
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
−
−
−
× = = =
1 2 2 1
3 1 1 3
2 3 3 2
3 3
2 2
1 1 sin ˆ a b a b a b a b a b a b a b a b a b k j i a b a b θ n
If A is the point with position vector a i j k 1 2 3 = a + a + a and the direction vector b is given by b i j k 1 2 3 = b + b + b , then the straight line through A with direction vector b has cartesian equation
− = − = − =λ
3
3
2
2
1
1 b z a b y a b x a
The plane through A with normal vector n i j k 1 2 3 = n + n + n has cartesian equation n1x + n2 y + n3 z = d where d = a.n
The plane through non-collinear points A, B and C has vector equation r = a +λ (b − a) + μ (c − a) = (1−λ −μ )a +λb + μc