OCR formula book
ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION
ADVANCED GENERAL CERTIFICATE OF EDUCATION
MATHEMATICS
LIST OF FORMULAE
AND
STATISTICAL TABLES
(List MF1)
Pure Mathematics
Mensuration
Surface area of sphere = 4π r2
Area of curved surface of cone = π r × slant height
Trigonometry
a2 = b2 + c2 − 2bc cos A
Arithmetic Series
un = a + (n − 1)d
Sn = 12 n(a + l) = 12 n{2a + (n − 1)d}
Geometric Series
un = arn−1 a(1 − rn )
1−r
a
S∞ = for | r | < 1
1−r
Sn =
Summations n ∑ r2 = 16 n(n + 1)(2n + 1)
r=1 n ∑ r3 = 14 n2 (n + 1)2
r=1
Binomial Series
n r +
n r+1 =
n+1 r+1 n n an−2 b2 + . . . + an−r br + . . . + bn n∈ , r 2 n! n where = n Cr = r r!(n − r)! n ( n −
1)
n(n − 1) . . . (n − r + 1) r
(1 + x)n = 1 + nx + x2 + . . . +
| x | < 1, n ∈ x + ...
1.2
1.2.3 . . . r
(a + b)n = an +
n an−1 b +
1
Logarithms and exponentials
ex ln a = ax
Complex Numbers
{r (cos θ + i sin θ )}n = r n (cos nθ + i sin nθ ) eiθ = cos θ + i sin θ
The roots of
n
= 1 are given by
=e
2π k i n , for k = 0, 1, 2, . . . , n − 1
2
Maclaurin’s Series
x2 xr f (0) + . . . + f (r) (0) + . . .
2!
r!
2
r x x ex = exp(x) = 1 + x +
+ ... +
+ . . . for all x
2!
r! x2 x3 xr ln(1 + x) = x −
+
− . . . + (−1)r+1 + . . . (−1 < x ≤ 1)
2
3 r 2r+1 x3 x5 x sin x = x −
+
− . . . + (−1)r
+ . . . for all x
3! 5!
(2r + 1)! f(x) = f(0) + xf (0) +
cos x = 1 −
x2 x4 x2r +
− . . . + (−1)r
+ ...
2! 4!
(2r )!
for all x
x3 x5 x2r+1 +
− . . . + (−1)r
+ . . . (−1 ≤ x ≤ 1)
3
5
2r + 1 x3 x5 x2r+1 sinh x = x +
+
+ ... +
+ . . . for all x
3! 5!
(2r + 1)! tan−1 x = x −
cosh x = 1 +
x2 x4 x2r +
+ ... +
+ ...
2! 4!
(2r )!
tanh−1 x = x +
x3 x5 x2r+1 +
+ ... +
+ ...
3
5
2r + 1
for all x
(−1 < x < 1)
Hyperbolic Functions
cosh2 x − sinh2 x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x
√
cosh−1 x = ln{x + (x2 − 1)}
√
sinh−1 x = ln{x + (x2 + 1)} tanh−1 x = 12 ln
1+x
1−x
(x ≥ 1)
|x| < 1
Coordinate Geometry
| ah + bk + c |
The perpendicular distance from (h, k) to ax + by + c = 0 is √ 2
(a +
ADVANCED GENERAL CERTIFICATE OF EDUCATION
MATHEMATICS
LIST OF FORMULAE
AND
STATISTICAL TABLES
(List MF1)
Pure Mathematics
Mensuration
Surface area of sphere = 4π r2
Area of curved surface of cone = π r × slant height
Trigonometry
a2 = b2 + c2 − 2bc cos A
Arithmetic Series
un = a + (n − 1)d
Sn = 12 n(a + l) = 12 n{2a + (n − 1)d}
Geometric Series
un = arn−1 a(1 − rn )
1−r
a
S∞ = for | r | < 1
1−r
Sn =
Summations n ∑ r2 = 16 n(n + 1)(2n + 1)
r=1 n ∑ r3 = 14 n2 (n + 1)2
r=1
Binomial Series
n r +
n r+1 =
n+1 r+1 n n an−2 b2 + . . . + an−r br + . . . + bn n∈ , r 2 n! n where = n Cr = r r!(n − r)! n ( n −
1)
n(n − 1) . . . (n − r + 1) r
(1 + x)n = 1 + nx + x2 + . . . +
| x | < 1, n ∈ x + ...
1.2
1.2.3 . . . r
(a + b)n = an +
n an−1 b +
1
Logarithms and exponentials
ex ln a = ax
Complex Numbers
{r (cos θ + i sin θ )}n = r n (cos nθ + i sin nθ ) eiθ = cos θ + i sin θ
The roots of
n
= 1 are given by
=e
2π k i n , for k = 0, 1, 2, . . . , n − 1
2
Maclaurin’s Series
x2 xr f (0) + . . . + f (r) (0) + . . .
2!
r!
2
r x x ex = exp(x) = 1 + x +
+ ... +
+ . . . for all x
2!
r! x2 x3 xr ln(1 + x) = x −
+
− . . . + (−1)r+1 + . . . (−1 < x ≤ 1)
2
3 r 2r+1 x3 x5 x sin x = x −
+
− . . . + (−1)r
+ . . . for all x
3! 5!
(2r + 1)! f(x) = f(0) + xf (0) +
cos x = 1 −
x2 x4 x2r +
− . . . + (−1)r
+ ...
2! 4!
(2r )!
for all x
x3 x5 x2r+1 +
− . . . + (−1)r
+ . . . (−1 ≤ x ≤ 1)
3
5
2r + 1 x3 x5 x2r+1 sinh x = x +
+
+ ... +
+ . . . for all x
3! 5!
(2r + 1)! tan−1 x = x −
cosh x = 1 +
x2 x4 x2r +
+ ... +
+ ...
2! 4!
(2r )!
tanh−1 x = x +
x3 x5 x2r+1 +
+ ... +
+ ...
3
5
2r + 1
for all x
(−1 < x < 1)
Hyperbolic Functions
cosh2 x − sinh2 x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x
√
cosh−1 x = ln{x + (x2 − 1)}
√
sinh−1 x = ln{x + (x2 + 1)} tanh−1 x = 12 ln
1+x
1−x
(x ≥ 1)
|x| < 1
Coordinate Geometry
| ah + bk + c |
The perpendicular distance from (h, k) to ax + by + c = 0 is √ 2
(a +