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ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION ADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252 January 2007 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 ) Sn = 1−r a S∞ = for | r | < 1 1−r Summations n ∑ r2 = 16 n(n + 1)(2n + 1) r=1 n ∑ r3 = 14 n2 (n + 1)2 r=1 Binomial Series n n n+1  + =  r r+1 r+1 n n n (a + b)n = an +   an−1 b +   an−2 b2 + . . . +   an−r br + . . . + bn n ∈ , 1 2 r n n ! where   = n Cr = r r!(n − r)! n(n − 1) 2 n(n − 1) . . . (n − r + 1) r (1 + x)n = 1 + nx + x + ... + x + ... | x | < 1, n ∈  1.2 1.2.3 . . . r 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 2π k i The roots of n = 1 are given by  = e n , for k = 0, 1, 2, . . . , n − 1 2 Maclaurin’s Series x2  xr f(x) = f(0) + xf  (0) + 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)! x2 x4 x2r cos x = 1 − + − . . . + (−1)r + ... for all x 2! 4! (2r )! x3 x5 x2r+1 tan−1 x = x − + − . . . + (−1)r + . . . (−1 ≤ x ≤ 1) 3 5 2r + 1 x3 x5 x2r+1 sinh x = x + + + ... + + . . . for all x 3! 5! (2r + 1)! x2 x4 x2r cosh x = 1 + + + ... + + ... for all x 2! 4! (2r )! x3 x5 x2r+1 tanh−1 x = x + + + ... + + ... (−1 < x < 1) 3 5 2r + 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)} (x ≥ 1) √ sinh−1 x = ln{x + (x2 + 1)} 1+x tanh−1 x = 12 ln   | x | < 1 1−x Coordinate Geometry | ah + bk + c | The perpendicular distance from (h, k) to ax + by + c = 0 is √ 2 (a + b2 )  m − m2  The acute angle between lines with gradients m1 and m2 is tan−1  1    1 + m1 m2  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 ± tan B tan(A ± B) = A ± B ≠ k + 12 π  1 ∓ tan A tan B 2t 1 − t2

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