Mathematical Formulae and Statistical Tables
List MF20
List of Formulae and Statistical Tables
Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795)
For use from 2017 in all papers for the above syllabuses.
CST317
PURE MATHEMATICS
Mensuration Surface area of sphere = 4r2
Area of curved surface of cone = r ? slant height
Trigonometry a2 = b2 + c2 - 2bc cos A
Arithmetic series
un = a + (n - 1)d
Sn =
1 n(a + l) =
2
1 2
n{2a
+
(n
-
1)d}
Geometric series un = arn - 1
a(1 - rn ) Sn = 1 - r
a S = 1 - r for r < 1
Summations
n
r2 = 1 n(n +1)(2n +1) 6 r =1
n
= r3 1 n2 (n + 1)2 4 r =1
Binomial series
n r
+
r
n +
1
= nr ++11
(a
+
b)n=
a
n+
n
1
a
n-1b
+
n
2
a
n-2b 2+
...
+
n
r
a
n-rb
r+
...
+
b
n
,
(n
),
where
n = r
(1+ x) n=1+ nx + n (n -1) x2 + ... + n (n -1)...(n - r +1) xr + ... ( x < 1, n )
1.2
1.2...r
= n Cr
n! r!(n - r)!
Logarithms and exponentials ex ln a = ax
Complex numbers {r(cos + i sin )}n = rn(cos n + i sin n) ei = cos + i sin
2 ki
The roots of zn = 1 are given by z = e n , for k = 0, 1, 2, ..., n ? 1
2
Maclaurin's series
f(x) = f(0) + x f(0) + x2 f(0) + ... + xr f(r)(0) + ...
2!
r !
ex = exp(x) = 1 + x + x2 + ... + xr + ... for all x
2!
r !
ln(1 + x) = x - x2 + x3 - ... + (-1)r + 1 xr + ... (-1 < x 1)
23
r
sin x = x - x3 + x5 - ... + (-1)r x2r+1 + ... for all x
3! 5!
(2r +1)!
cos x = 1 - x2 + x4 - ... + (-1)r x2r + ... for all x
2! 4!
(2r )!
tan-1 x = x - x3 + x5 - ... + (-1)r x2r+1 + ... (-1 x 1)
35
2r +1
x3 x5
x 2 r +1
sinh x = x + + + ... +
+ ... for all x
3! 5!
(2r +1)!
x2 x4
x2r
cosh x = 1 + + + ... +
+ ... for all x
2! 4!
(2r )!
tanh-1 x = x + x3 + x5 + ... + x2r+1 + ... (-1 < x < 1)
35
2r +1
Hyperbolic functions cosh2 x - sinh2 x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2x + sinh2x
cosh-1 x = ln {x + x2 -1 } (x 1)
sinh-1 x = ln {x + x2 + 1 }
tanh-1 x =
1 2
ln
1 1
+ -
x x
(|x| < 1)
Coordinate geometry ah + bk + c
The perpendicular distance from (h, k) to ax + by + c = 0 is a2 + b2
The
acute
angle
between
lines
with
gradients
m1
and
m2
is
tan?1
m1 1+
- m2 m1m2
3
Trigonometric identities
sin(A ? B) = sin A cos B ? cos A sin B
cos(A ? B) = cos A cos B sin A sin B
tan(A ? B) = tan A ? tan B (A ? B (k + 1 ))
1 tan A tan B
2
For t = tan
1 A : sin A =
2
2t 1+ t2
, cos A =
1- t2 1+ t2
sin A + sin B = 2 sin A + B cos A - B
2
2
sin A - sin B = 2 cos A + B sin A - B
2
2
cos A + cos B = 2 cos A + B cos A - B
2
2
cos A - cos B = ?2 sin A + B sin A - B
2
2
Vectors The resolved part of a in the direction of b is a.b
b
The point dividing AB in the ratio : is ?a + b +?
i a1 b1 a2b3 - a3b2
Vector product: a ? b = |a||b| sin= n^
j
a2 = b2
a3b1
-
a1b3
k a3 b3 a1b2 - a2b1
If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given by
b = b1i + b2j + b3k, then the straight line through A with direction vector b has cartesian equation
x= - a1 y= - a2 z - a3 (= )
b1
b2
b3
The plane through A with normal vector n = n1i + n2j + n3k has cartesian equation n1x + n2y + n3z + d = 0 where d = -a.n
The plane through non-collinear points A, B and C has vector equation r = a + (b - a) + (c - a) = (1 - - )a + b + c
The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc
The perpendicular distance of (, , ) from n1x + n2y + n3z + d = 0 is n1 + n2 + n3 + d n12 + n22 + n32
Matrix transformations
Anticlockwise
rotation
through
about
O:
cos sin
-sin
cos
Reflection
in
the
line
y
=
(tan
)x:
cos 2
sin
2
sin 2
-
cos
2
4
Differentiation f(x) tan kx sin-1 x
cos-1 x
tan-1 x
f(x) k sec2 kx
1
1- x2 -1
1- x2 1 1+ x2
sec x cot x cosec x sinh x cosh x tanh x sinh-1 x
cosh?1 x
tanh-1 x
sec x tan x - cosec2 x - cosec x cot x cosh x sinh x sech2 x
1
1+ x2
1
x2 -1
1 1- x2
Integration (+ constant; a > 0 where relevant)
f(x)
f( x) dx
sec2 kx
tan x cot x cosec x
sec x
sinh x cosh x tanh x
1 tan kx k
ln|sec x|
ln|sin x|
-ln|cosec x + cot x| = ln|tan( 1 x)| 2
ln|sec x + tan x| = ln|tan( 1 x + 2
1 4
)|
cosh x
sinh x
ln cosh x
1 a2 - x2
sin?1
x a
(|x| < a)
1 a2 + x2
1 a
tan?1
x a
1 x2 - a2
cosh?1
x a
or
ln{x
+
x2 - a2 }
(x > a)
1 a2 + x2
sinh?1
x a
or
ln{x
+
x2 + a2 }
1 a2 - x2
1 ln 2a
a+x a- x
=1 a
tanh?1
x a
(|x| < a)
1 x2 - a2
1 ln x - a 2a x + a
u dv d=x uv - v du dx
dx
dx
5
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