Formulae and statistical tables for A‑level Mathematics and A‑level Further Mathematics AS Mathematics (7356) A‑level Mathematics (7357) AS Further Mathematics (7366) A‑level Further Mathematics (7367) v1.5 Issued February 2018 For the new specifications for first teaching from September 2017. This booklet of formulae and statistical tables is required for all AS and A‑level Further Mathematics exams. Students may also use this booklet in all AS and A‑level Mathematics exams. However, there is a smaller booklet of formulae available for use in AS and A‑level Mathematics exams with only the formulae required for those examinations included. Page Bros/E12 FMFB16
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Data booklet: Formulae and statistical tables (Further Maths)
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Formulae and statistical tables for A‑level Mathematics and A‑level Further Mathematics AS Mathematics (7356)A‑level Mathematics (7357)AS Further Mathematics (7366)A‑level Further Mathematics (7367)
v1.5 Issued February 2018
For the new specifications for first teaching from September 2017.
This booklet of formulae and statistical tables is required for all AS and A‑level Further Mathematics exams.
Students may also use this booklet in all AS and A‑level Mathematics exams. However, there is a smaller booklet of formulae available for use in AS and A‑level Mathematics exams with only the formulae required for those examinations included.
Page Bros/E12 FMFB16
2
Further copies of this booklet are available from:Telephone: 0844 209 6614 Fax: 01483 452819 or download from the AQA website www.aqa.org.uk
CopyrightAQA retains the copyright on all its publications. However, registered centres of AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within the centre.
Set and published by AQA.
AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX
3
Contents Page
Pure mathematics 4
Mechanics 10
Probability and statistics 11
Statistical tablesTable 1 Percentage points of the student’s t ‑distribution 12
Table 2 Percentage points of the χ 2 distribution 13
Table 3 Critical values of the product moment correlation coefficient 14
4
Pure mathematics
Binomial series
( )a b an
a bn
a bnr
a bn n n n n r r+ = +
+
+ … +
+ … +− − −1 2
1 2 2 bb nn ( )∈
where nr
nr n r
nr
= =−
C !! !( )
( ) ( ) ( ) ( ) ( )1 1 11.2
1 11.2
12+ = + + − + … + − … − +…
+ … < ∈x nx n n x n n n rr
x x nn r ,
Arithmetic series
Sn = 12 n (a + l) = 1
2 n [2a + (n − 1)d]
Geometric series
Sn = a rr
n( )11
−−
S∞ = ar1 −
for | r | < 1
Trigonometry: small angles
For small angle θ, measured in radians:
sin θ ≈ θ
cos θ ≈ 1 − θ2
2
tan θ ≈ θ
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 tan
1 tan tanA B
A B±
(A ± B ≠ (k + 12 )π)
sin A + sin B = 2 sin A B+2
cos A B−2
sin A − sin B = 2 cos A B+2
sin A B−2
cos A + cos B = 2 cos A B+2
cos A B−2
cos A − cos B = −2 sin A B+2
sin A B−2
5
Differentiation
f(x) f ′(x)
tan x sec2 x
cosec x −cosec x cot x
sec x sec x tan x
cot x −cosec2 x
sin−1 x 11 2− x
cos−1 x −1
1 2− x
tan−1 x 11 2+ x
tanh x sech2 x
sinh−1 x 1
1 2+ x
cosh−1 x 112x −
tanh−1 x 11 2− x
fg
( )( )xx
f ( )g( ) f ( )g
(g( ))
( )x x xx
x−2
Differentiation from first principles
f ( ) f ( )f ( )
0limh
x h xxh→
+ −=
6
Integration
∫ u dd
vx
dx = uv − ∫ v ddux
dx
∫ f ( ) d f ( )f ( )
x x x cx
= +ln
| |
f(x) ∫ f(x) dx
tan x ln | sec x | + c
cot x ln | sin x | + c
cosec x −ln | cosec x + cot x | = ln | tan ( 12
x)| + c
sec x ln | sec x + tan x | = ln | tan ( 12
x + 14π)| + c
sec2 x tan x + c
tanh x ln cosh x + c
12 2a x−
sin−1 xa
+ c (| x | < a)
12 2a x+
1a
tan−1 xa
+ c
12 2x a−
cosh−1 xa
or ln{x + x a2 2− } + c (x > a)
12 2a x+
sinh−1 xa
or ln{x + x a2 2+ } + c
12 2a x−
12a
ln | a xa x
+−
| = 1a
tanh−1 xa
+ c (| x | < a)
12 2x a−
12a
ln | x ax a
−+
| + c
Numerical solution of equations
The Newton‑Raphson iteration for solving f (x) = 0: xn+1 = xn − f ( )
f ( )
n
n
xx
Numerical integration
The trapezium rule: a
b∫ y dx ≈ 1
2h{(y0 + yn) + 2(y1 + y2 + … + yn−1)}, where h =
b an−
Complex numbers
[r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)
The roots of zn = 1 are given by z = π i
ekn2
, for k = 0, 1, 2, … , n − 1
7
Matrix transformations
Anticlockwise rotation through θ about O: θ θθ θ
−
cos sinsin cos
Reflection in the line y = (tan θ) x: θ θθ θ
−
cos2 sin2sin2 cos2
The matrices for rotations (in three dimensions) through an angle θ about one of the axes are:
θ θθ θ
−
1 0 00 cos sin0 sin cos
for the x‑axis
θ θ
θ θ
−
cos 0 sin0 1 0sin 0 cos
for the y‑axis
θ θθ θ
−
cos sin 0sin cos 00 0 1
for the z‑axis
Summations
rr
n2
1=∑ = 1
6n(n + 1)(2n + 1)
rr
n3
1=∑ = 1
4n2(n + 1)2
Maclaurin’s series
f (x) = f (0) + x f ′(0) + x2
2!f ″(0) + … + x
r
r
!f (r)(0) + …
ex = exp(x) = 1 + x + x2
2! + … + x
r
r
! + … for all x
ln(1 + x) = x − x2
2 + x3
3 − … + (−1)r+1 x
r
r + … (−1 < x 1)
sin x = x − x3
3! + x5
5! − … + (−1)r x
r
r2 1
2 1
+
+( )! + … for all x
cos x = 1 − x2
2! + x4
4! − … + (−1)r x
r
r2
2( )! + … for all x
8
Vectors
The resolved part of a in the direction of b is a.bb
Vector product: a × b = | a || b | sin θ n̂ = ijk
a ba ba b
a b a ba b a ba b a b
1 1
2 2
3 3
2 3 3 2
3 1 1 3
1 2 2 1
=
−−−
If A is the point with position vector a = a1i + a2 j + a3k, then
• the straight line through A with direction vector b = b1i + b2j + b3k has equationx a
b− 1
1 =
y ab− 2
2 =
z ab− 3
3 = λ (Cartesian form)
or
(r − a) × b = 0 (vector product form)
• the plane through A and parallel to b and c has vector equation r = a + sb + tc
Area of a sector
A = 12
∫ r 2 dθ (polar coordinates)
Hyperbolic functions
cosh2 x − sinh2 x = 1
sinh 2 x = 2 sinh x cosh x
cosh 2 x = cosh2 x + sinh2 x
cosh−1 x = ln{x + x2 1− } (x 1)
sinh−1 x = ln{x + x2 1+ }
tanh−1 x = 12
ln11
+−
xx
(| x | < 1)
Conics
Ellipse Parabola Hyperbola
Standard form xa
2
2 + yb
2
2 = 1 y2 = 4ax xa
2
2 − yb
2
2 = 1
Parametric form x = a cos θy = b sin θ
x = at2y = 2at
x = a sec θy = b tan θ
Asymptotes none none xa
= ± yb
9
Further numerical integration
The mid‑ordinate rule: y x h y y y yn na
bd » ( )1
232
32
12
+ + + +− −∫
where h = b a
n−
Simpson’s rule: a
b∫ y dx » 1
3 h y y y y y y y yn n n0 1 3 1 2 4 24 2+( ) + + + +( ) + + + +( ){ }− −
where h = b a
n−
and n is even
Numerical solution of differential equations
For dd
yx
= f (x) and small h, recurrence relations are:
Euler’s method: yn +1 = yn + hf (xn), xn +1 = xn + h
For dd
yx
= f (x, y):
Euler’s method: yr +1 = yr + hf (xr, yr), xr +1 = xr + h
Improved Euler method: yr +1 = yr–1 + 2hf (xr, yr), xr +1 = xr + h
Arc length
s = 12
+
∫ d
ddy
xx (Cartesian coordinates)
s = dd
dd
dxt
yt
t
+
∫
2 2 (parametric form)
Surface area of revolution
Sx = 2π y yx
x12
+
∫ d
dd (Cartesian coordinates)
Sx = 2π y xt
yt
tdd
dd
d
+
∫
2 2 (parametric form)
10
Mechanics
Constant acceleration
s = ut + 12
at2 s = ut + 12
at2
s = vt − 12
at2 s = vt − 12
at2
v = u + at v = u + at
s = 12
(u + v)t s = 12
(u + v)t
v2 = u2 + 2as
Centres of mass
For uniform bodies:
Triangular lamina: 23
along median from vertex
Solid hemisphere, radius r: 38
r from centre
Hemispherical shell, radius r: 12
r from centre
Circular arc, radius r, angle at centre 2α: sinr αα
from centre
Sector of circle, radius r, angle at centre 2α: r αα
2 sin3
from centre
Solid cone or pyramid of height h: 14
h above the base on the line from centre of base to vertex
Conical shell of height h: 13
h above the base on the line from centre of base to vertex
11
Probability and statistics
Probability
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A) × P(B | A)
Standard deviation
Σ( ) Σx x x xn n− = −
2 22
Discrete distributions
Distribution of X P(X = x) Mean Variance
Binomial B(n, p)nx
p x(1 − p)n − x np np(1 − p)
Poisson Po(λ) e−λxλx!
λ λ
Sampling distributions
For a random sample X1, X2, …, Xn of n independent observations from a distribution having mean μ and variance σ 2:
X is an unbiased estimator of μ, with Var (X ) = σn
2
S 2 is an unbiased estimator of σ 2, where S 2 = X Xn
i −( )−
∑ 2
1
For a random sample of n observations from N(μ, σ 2):
X μσn
− ~ N(0, 1)
X μSn
− ~ tn−1
Distribution-free (non-parametric) tests
Contingency tables: O EE
i i
i
−( )∑2
is approximately distributed as χ 2
12
TABLE 1 Percentage points of the student’s t-distribution
The table gives the values of x satisfying P(X x) = p, where X is a random variable having the student’s t‑distribution with v degrees of freedom.