A Level Further Mathematics A (H245) Formulae …
Oxford Cambridge and RSA
A Level Further Mathematics A (H245)
Formulae Booklet
A Level Further Mathematics A
*9990860740*
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? OCR 2022 [603/1325/0] DC (SLM) 320372/1
A Level Further Mathematics A
CST322
2
Pure Mathematics
Arithmetic series
Sn
=
1 2
n (a + l)
=
1 2
n
"2a
+
(n - 1) d,
Geometric series
a (1 - rn) Sn = 1 - r
S3
=
a 1-r
for
r
11
Binomial series
(a + b) where
n n
= Cr
an =
+n CnrC=1 aJLKKnrnNPOO-
1b =
+ n C2an-2b2
n! r! (n - r) !
+
f
+
n
Cr
an-rbr
+
f
+
bn
(n ! N),
(1
+
x) n
=
1
+
nx
+
n
(n 2!
1)
x2
+
f
+
n
(n
-
1)
f (n r!
-
r
+
1)
xr
+
f
^ x 1 1, n ! Rh
Series
/ / n r2
r=1
=
1 6
n (n + 1) (2n + 1),
n
r3
r=1
=
1 4
n2
(n
+
1)
2
Maclaurin series
f
(x)
=
f
(0)
+
f
l(0)
x
+
f
m (0) 2!
x2
+
f
+
f
(r) (0) r!
xr
+
f
ex
=
exp (x)
=
1
+
x
+
x2 2!
+
f
+
xr r!
+
f
for
all
x
ln (l
+
x)
=
x
-
x2 2
+
x3 3
-
f
+
(-1) r+1
xr r
+
f (-1
1
x
#
1)
sin
x
=
x
-
x3 3!
+
x5 5!
-
f
+
(-1)
r
x 2r + 1 (2r + 1)
!
+
f
for
all
x
cos
x
=
1
-
x2 2!
+
x4 4!
-
f
+
(-1)
r
x2r (2r)
!
+
f
for
all
x
(1
+
x) n
=
1
+
nx
+
n
(n 2!
1)
x2
+
f
+
n
(n
-
1)
f (n r!
-
r
+
1)
xr
+
f
^ x 1 1, n ! Rh
Matrix transformations
Reflection
in
the
line
y
=
!x
:
JLKK!
0 1
! 10NPOO
Anticlockwise rotation through i about O : JLKKcsoinsii -cosisniiNPOO
? OCR 2022
A Level Further Mathematics A
3
Rotations through i about the coordinate axes. The direction of positive rotation is taken to be anticlockwise when looking towards the origin from the positive side of the axis of rotation.
10 0
Rx = > 0 cos i -sin i H
0 sin i cos i
cos i 0 sin i
Ry = > 0 1 0 H
-sin i 0 cos i
cos i -sin i 0
Rz = > sin i cos i 0 H
0 01
Differentiation
f (x) tan kx sec x cot x cosec x
arcsin x or sin-1x
arccos x or cos-1x arctan x or tan-1x
Quotient rule
y = uv,
dy dx
=
v
du dx
-u v2
dv dx
Differentiation from first principles
f
l(x)
=
f (x lim h"0
+
h) h
- f (x)
Integration
y
f l(x) f (x)
dx
=
ln
f (x)
+
c
yf
l(x)^f (x)hn
dx
=
n
1 +
1^f (x)hn+1
+
c
Integration
by
parts
y
u
dv dx
dx
=
uv -
y
v
du dx
d
x
f l(x) k sec2kx sec x tan x - cosec 2 x -cosec x cot x
1 1 - x2 -1 1 - x2
1 1 + x2
? OCR 2022
A Level Further Mathematics A
Turn over
4
The mean value of f (x) on the interval [a, b] is 1 y bf (x) dx b-a a
Area
of
sector
enclosed
by
polar
curve
is
1 2
y
r
2di
f (x)
1 a2 -x2
1 a2 +x2
1 a2 +x2
1 x2 -a2
y f(x)dx
sin-1
JKK L
x a
NOO P
^ x 1 ah
1 a
tan
-1
JKK L
x a
NOO P
sinh
-1
JKK L
x a
NOO P
or
ln (x +
cosh
-1
JKK L
x a
NOO P
or
ln (x +
x2 + a2) x2 - a2)
(x 2 a)
Numerical methods
y Trapezium rule:
b a
ydx
.
1 2
h {(y0
+ yn)
+ 2 (y1
+ y2
+ f + yn-1)
},
where
h
=
b-a n
The Newton-Raphson iteration for solving
f
(x)
=
0
:
xn+1
=
xn
-
f (xn) f l(xn)
Complex numbers
Circles: z - a = k
Half lines: arg (z - a) = a
Lines: z - a = z - b
De Moivre's theorem: {r (cos i + i sin i)} n = rn (cos ni + i sin ni)
Roots
of
unity:
The
roots
of
zn
=
1
are
given
by
z
=
exp JLKK2rn k
iNOO P
for
k
=
0, 1,
2, f,
n-1
Vectors and 3-D coordinate geometry
Cartesian equation of the line through the point A with position vector a = a1i + a2 j + a3k in direction
u
=
u1i + u2 j + u3k
is
x - a1 u1
=
y - a2 u2
=
z
- a3 u3
^=
mh
Cartesian equation of a plane n1 x + n2 y + n3 z + d = 0
JKa1 NO JKb1 NO i a1 b1 JKa2 b3 - a3 b2 NO
Vector
product:
a
#
b
=
K K L
a2 a3
O O P
#
K K L
b2 b3
O O P
=
j k
a2 a3
b2 b3
=
K K L
a3 a1
b1 b2
-
a1 a2
b3 b1
O O P
? OCR 2022
A Level Further Mathematics A
5
The distance between skew lines is D = ^b - ah. n , where a and b are position vectors of points on each line and n
n is a mutual perpendicular to both lines
The
distance between a
point
and
a line
is
D
=
ax1 + by1 - c a2 +b2
,
where
the coordinates
of
the point
are (x1,
y1) and
the equation of the line is given by ax + by = c
The distance between a point and a plane is D = b . n - p , where b is the position vector of the point and the n
equation of the plane is given by r . n = p
Small angle approximations
sin i
.
i,
cos i
.
1-
1 2
i
2,
tan i
.
i
where
i
is
small
and
measured
in
radians
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 1 " tan A tan B
(A ! B ! (k + 21) r)
Hyperbolic functions cosh2x - sinh2x = 1 sinh-1x = ln [x + (x2 + 1)] cosh-1x = ln [x + (x2 - 1)], x $ 1
tanh-1x
=
1 2
ln JLKK11 -+ xxNPOO,
-1
1
x
1
1
Simple harmonic motion
x = A cos (~t) + B sin (~t)
x = R sin (~t + {)
Statistics
Probability
P (A , B) = P (A) + P (B) - P (A + B)
P (A + B) = P (A) P (B A) = P (B) P (A B)
or
P(A
B)
=
P (A + B) P (B)
? OCR 2022
A Level Further Mathematics A
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