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Formula sheet with algebra, trigonometry, differentiation, integrations, vectors, mechanics, probability and statistics. From university of cambridge.
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General Certificate of Education Advanced Level
General Certificate of Education Advanced Subsidiary Level
Advanced International Certificate of Education
Algebra
For the quadratic equation 0
2 ax + bx + c = :
a
b b ac x 2
2 − ±√ − =
For an arithmetic series:
un = a +( n − 1 ) d , ( ) { 2 ( 1 ) } 2
1 2
1 Sn = na + l = n a + n − d
For a geometric series:
= n −^1 un ar , ( 1 ) 1
= r r
a r S
n
∞ = r r
a S
Binomial expansion:
n n n n n n a b b
n a b
n a b
n a b a + +
− − −
1 2 2 33
1 2 3
( ) , where n is a positive integer
and !( )!
r n r
n
r
n
−
2 3
3!
( 1 ) 1 x
nn n x
nn x nx
n , where n is rational and x < 1
Trigonometry
Arc length of circle= r θ ( θ in radians)
Area of sector of circle θ
2 2
1 = r ( θ in radians)
θ
θ θ cos
sin tan ≡
cos sin 1
2 2 θ + θ≡ , θ θ
2 2 1 + tan ≡sec , θ θ
2 2 cot + 1 ≡cosec
sin( A ± B )≡sin A cos B ±cos A sin B
cos( A ± B )≡cos A cos B #sin A sin B
1 tan tan
tan tan tan( )
sin 2 A ≡ 2 sin A cos A
2 2 2 2 cos 2 ≡cos −sin ≡ 2 cos − 1 ≡ 1 − 2 sin
2 1 tan
2 tan tan 2 −
Principal values:
π π 2
1 1 2
1 − ≤sin ≤
− x
≤ ≤ π
− x
1 0 cos
π π 2
1 1 2
1 − <tan <
− x
Uniformly accelerated motion
v = u + at , s ( u v ) t 2
2
s = ut +^1 at , v^2 = u^2 + 2 as
Motion of a projectile
Equation of trajectory is:
θ
θ 2 2
2
2 cos
tan V
gx y = x −
Elastic strings and springs
l
x T
λ = , l
x E 2
2 λ =
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
r
2 ω or r
v^2
Centres of mass of uniform bodies
Triangular lamina: 3
2 along median from vertex
Solid hemisphere of radius r : r 8
3 from centre
Hemispherical shell of radius r : r 2
1 from centre
Circular arc of radius r and angle 2 α : α
r sinα from centre
Circular sector of radius r and angle 2 α : α
α
2 r sin from centre
Solid cone or pyramid of height h : h 4
3 from vertex
Summary statistics
For ungrouped data:
n
x x
= , standard deviation
2
2 2 ( ) x n
x
n
x x −
For grouped data:
f
xf x Σ
= , standard deviation
2
2 2 ( ) x f
x f
f
x x f − Σ
Discrete random variables
E( X )=Σ xp
2 2 Var( X )=Σ x p −{E( X )}
For the binomial distribution B( n , p ):
r n r r p p r
n p (^) − −
= ( 1 ) , μ = np , σ^2 = np ( 1 − p )
For the Poisson distribution Po( a ) :
e r
a p
r a r
− = , μ = a , = a
2 σ
Continuous random variables
E( X )= x f( x )d x
2 2 Var( X ) x f( x )d x {E( X )}
Sampling and testing
Unbiased estimators:
n
x x
n
x x n
s
2 2 2 ( )
1
Central Limit Theorem:
n
2 ~N ,
σ μ
Approximate distribution of sample proportion:
n
p p p
