List MF19
List of formulae and statistical tables
Cambridge International AS & A Level
Mathematics (9709) and Further Mathematics (9231)
For use from 2020 in all papers for the above syllabuses.
CST319
*2508709701*
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List MF
List of formulae and statistical tables
Cambridge International AS & A Level
Mathematics (9709) and Further Mathematics (9231)
For use from 2020 in all papers for the above syllabuses.
CST
PURE MATHEMATICS
Mensuration
Volume of sphere =
4 3 3 π r
Surface area of sphere =
2 4 π r
Volume of cone or pyramid =
1 3 ×^ base area^ ×height
Area of curved surface of cone = π r ×slant height
Arc length of circle = r θ ( θ in radians)
Area of sector of circle
1 2 2
= r θ ( θ in radians)
Algebra
For the quadratic equation
2 ax + bx + c = 0 :
2 4
b b ac x a
For an arithmetic series:
un = a + ( n − 1) d ,
1 1 2 2
Sn = n a ( + l ) = n {2 a + ( n −1) } d
For a geometric series:
n 1 un ar
− = ,
n
n
a r S r r
, (^) ( 1 ) 1
a S r r
∞ =^ <
Binomial series:
1 2 2 3 3 ( ) 1 2 3
n n n^ n n^ n n n n a b a a b a b a b b
K , where n is a positive integer
and
n (^) n
r r n r
n n n^ n n^ n x nx x x
- = + + + +K , where n is rational and x < 1
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f( x ) ∫
f( x ) d x
n x
1
n x
n
( n ≠ −1)
x
ln x
e
x e
x
sin x −cos x
cos x sin x
2 sec x tan x
2 2
x + a
tan
x
a a
− ^
2 2
x − a
ln 2
x a
a x a
( x > a )
2 2
a − x
ln 2
a x
a a x
( x^ < a )
d d d d d d
v u u x uv v x x x
∫ ∫
f ( ) d ln f ( ) f ( )
x x x x
∫
Vectors
If a = a 1 (^) i + a 2 (^) j + a 3 k and b = b 1 (^) i + b 2 (^) j + b 3 k then
a b. = a b 1 1 + a b 2 2 + a b 3 3 = a b cos θ
FURTHER PURE MATHEMATICS
Algebra
Summations:
1 2 1
n
r
r n n
=
∑ =^ + ,^
(^2 ) 6 1
n
r
r n n n
=
∑ =^ +^ + ,^
3 1 2 2 4 1
n
r
r n n
=
∑ =^ +
Maclaurin’s series:
2 ( ) f( ) f(0) f (0) f (0) f (0) 2!!
r x x r x x r
= + ′^ + ′′ + K + +K
2
e exp( ) 1 2!!
r x x^ x x x r
= = + + + K + +K (^) (all x )
2 3 1 ln(1 ) ( 1) 2 3
r x x (^) r x x x r
- = − + − K + − +K (–1 < x ⩽ 1)
3 5 2 1
sin ( 1) 3! 5! (2 1)!
r x x (^) r x x x r
= − + − + − +
K K (all x )
2 4 2
cos 1 ( 1) 2! 4! (2 )!
r x x (^) r x x r
= − + − K + − +K (^) (all x )
3 5 2 1 1 tan ( 1) 3 5 2 1
r x x (^) r x x x r
− = − + − + − +
K K (–1 ⩽ x ⩽ 1)
3 5 2 1
sinh 3! 5! (2 1)!
r x x x x x r
= + + + + +
K K (all x )
2 4 2
cosh 1 2! 4! (2 )!
r x x x x r
= + + + K + +K (^) (all x )
3 5 2 1 1 tanh 3 5 2 1
r x x x x x r
− = + + + + +
K K (–1 < x < 1)
Trigonometry
If
1 2
t = tan x then:
2
sin 1
t x t
and
2
2
cos 1
t x t
Hyperbolic functions
2 2 cosh x − sinh x ≡ 1 , sinh 2 x ≡ 2sinh x cosh x ,
2 2 cosh 2 x ≡ cosh x +sinh x
1 2 sinh x ln ( x x 1 )
− = + +
1 2
cosh x ln ( x x 1 )
− = + − ( x ⩾ 1)
(^1 ) 2
tanh ln (| | 1) 1
x x x x
− ^ +
MECHANICS
Uniformly accelerated motion
v = u + at ,
1 s = 2 ( u + v t ),
1 2 s = ut + 2 at ,
2 2 v = u + 2 as
FURTHER MECHANICS
Motion of a projectile
Equation of trajectory is:
2
2 2
tan 2 cos
gx y x V
Elastic strings and springs
x T l
2
x E l
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
2 ω r or
2 v
r
Centres of mass of uniform bodies
Triangular lamina:
2 3
along median from vertex
Solid hemisphere of radius r :
3 8
r from centre
Hemispherical shell of radius r :
1 2
r from centre
Circular arc of radius r and angle 2 α:
r sin α
from centre
Circular sector of radius r and angle 2 α:
2 sin
r α
from centre
Solid cone or pyramid of height h :
3 4
h from vertex
PROBABILITY & STATISTICS
Summary statistics
For ungrouped data:
x x n
= , standard deviation
2 2 ( x x ) x (^) 2 x n n
For grouped data:
xf x f
, standard deviation
2 2 ( x x ) f x f (^) 2 x f f
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
n p p p r
, μ = np ,
2
σ = np (1 − p )
For the geometric distribution Geo( p ):
1 (1 )
r pr p p
− = − ,
p
For the Poisson distribution Po( λ)
e !
r
pr r
− λ λ = , μ = λ,
2 σ =λ
Continuous random variables
E( X ) = x f( ) d x x ∫
2 2 Var( X ) = x f( ) d x x −{E( X )} ∫
Sampling and testing
Unbiased estimators:
x x n
2 2 2 (^ )^1 2 (^ )
1 1
x x x s x n n n
Σ − ^ Σ
Central Limit Theorem:
2
X ~ N , n
σ μ
Approximate distribution of sample proportion:
N ,
p p p n
THE NORMAL DISTRIBUTION FUNCTION
If Z has a normal distribution with mean 0 and
variance 1, then, for each value of z , the table gives
the value of Φ( z ), where
Φ( z ) = P( Z ⩽ z ).
For negative values of z , use Φ(– z ) = 1 – Φ( z ).
z 0 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
ADD
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 4 8 12 16 20 24 28 32 36
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 4 8 12 16 20 24 28 32 36
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 4 8 12 15 19 23 27 31 35
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 4 7 11 15 19 22 26 30 34
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 4 7 11 14 18 22 25 29 32
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 3 7 10 14 17 20 24 27 31
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 3 7 10 13 16 19 23 26 29
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 3 6 9 12 15 18 21 24 27
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 3 5 8 11 14 16 19 22 25
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 3 5 8 10 13 15 18 20 23
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 2 5 7 9 12 14 16 19 21
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 2 4 6 8 10 12 14 16 18
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 2 4 6 7 9 11 13 15 17
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 2 3 5 6 8 10 11 13 14
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1 3 4 6 7 8 10 11 13
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1 2 4 5 6 7 8 10 11
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1 2 3 4 5 6 7 8 9
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1 2 3 4 4 5 6 7 8
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1 1 2 3 4 4 5 6 6
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 1 1 2 2 3 4 4 5 5
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0 1 1 2 2 3 3 4 4
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0 1 1 2 2 2 3 3 4
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0 1 1 1 2 2 2 3 3
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0 1 1 1 1 2 2 2 2
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 0 0 1 1 1 1 1 2 2
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0 0 0 1 1 1 1 1 1
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0 0 0 0 1 1 1 1 1
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0 0 0 0 0 1 1 1 1
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0 0 0 0 0 0 0 1 1
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0
Critical values for the normal distribution
If Z has a normal distribution with mean 0 and
variance 1, then, for each value of p , the table
gives the value of z such that
P( Z ⩽ z ) = p.
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.
z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.
CRITICAL VALUES FOR THE t -DISTRIBUTION
If T has a t -distribution with ν degrees of freedom, then,
for each pair of values of p and ν, the table gives the value
of t such that:
WILCOXON SIGNED-RANK TEST
The sample has size n.
P is the sum of the ranks corresponding to the positive differences.
Q is the sum of the ranks corresponding to the negative differences.
T is the smaller of P and Q.
For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at
the level of significance indicated.
Critical values of T
Level of significance
One-tailed 0.05 0.025 0.01 0.
Two-tailed 0.1 0.05 0.02 0.
n = 6 2 0
For larger values of n , each of P and Q can be approximated by the normal distribution with mean
1 4
n n ( +1)
and variance
1 24
n n ( + 1)(2 n + 1).
WILCOXON RANK-SUM TEST
The two samples have sizes m and n , where m ⩽ n.
R (^) m is the sum of the ranks of the items in the sample of size m.
W is the smaller of R (^) m and m ( n + m + 1) – R (^) m.
For each pair of values of m and n , the table gives the largest value of W which will lead to rejection of the
null hypothesis at the level of significance indicated.
Critical values of W
Level of significance
One-tailed 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.
Two-tailed 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.
n m = 3 m = 4 m = 5 m = 6
Level of significance
One-tailed 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.
Two-tailed 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.
n m = 7 m = 8 m = 9 m = 10
For larger values of m and n , the normal distribution with mean
1 2
m m ( + n + 1)and variance
1 12
mn m ( + n +1)
should be used as an approximation to the distribution of R (^) m.