TAILIEUCHUNG - MEI STRUCTURED MATHEMATICS EXAMINATION FORMULAE AND TABLES

{ } In order to discuss all the subsets of a given set, let us introduce the following terminology. We shall call the original set the universal set, one-element subsets will be called unit sets, and the set which contains no members the empty set. We do not introduce special names for other kinds of subsets of the universal set. As an example, let the universal set U consist of the three elements {a, b, c}. The proper subsets of U are those sets containing some but not all of the elements of U. The proper subsets consist of three two-element sets namely, {a, b}, {a, c}, and {b,. | MEI STRUCTURED MATHEMATICS EXAMINATION FORMULAE AND TABLES 1 Arithmetic series General kth term last nth term l Sum to n terms Geometric series General kth term Sum to n terms Uk a k - 1 d un a n - l d Sn - n a l - n 2a n - 1 d 1 Uk a rk 1 a 1 - rn a rn - 1 Sn - - - - n 1 - r r - 1 Infinite series x2 xr f x f 0 xf 0 -- f 0 . -- f r 0 . 2 r f x f a x-a f a 2X2-7 22 f a . -x--af- -a . 2 r 2r r f a x f a xf a -- ĩ a . x f r a . 2 r 2r ex exp x 1 x -- . -- . all x 2 r Sum to infinity - - - 1 r 1 1 - r S c Binomial expansions When n is a positive integer 2 a b n a n a-1 b 2 an2 b2 . n an rbr . bn n e N where n nc _n n Í n n r Cr r n- r r r 1 r 1 General case n n - 1 n n- 1 . n-r 1 1 x n 1 nx - 2 --- x2 . i2f- xr . x 1 n e IR Logarithms and exponentials ____ log x exln a ax loga x --- Numerical solution of equations Newton-Raphson iterative formula for solving f x 0 xn 1 xn Complex Numbers f xn ----- r cos d j sin d n rn cos nd j sin nd ejQ cos d j sin d The roots of zn 1 are given by z exp n--j for k 0 1 2 . n-1 .2 .3 r xx x ln 1 x x - -- -- - . -1 r . - 1 x 1 2 3 r r3 r5 r2r 1 sinx x - -- -- - . -1 r -2 ---7 . all x 3 5 v 2r 1 .2 .4 2r r cosx 1 - -- -- - . -1 r . all x 2 4 7 2r r3 5 Y2r 1 arctanx x - -- -- - . -1 r -- . - 1 x 1 3 5 v 2r 1 sinh x x -- -5 24 cosh x 1 -- -- cos x -2- -4- artanh x x -- 5 x2r 1 Tã7TnT . a x 2r 1 x 2r ----- . all x 2r x2r 1 2r T . - 1 x 1 ALGEBRA Hyperbolic functions cosh2x - sinh2x 1 sinh2x 2sinhx coshx cosh2x cosh2x sinh2x arsinh x ln x Vx2 1 arcosh x ln x x x2 -1 x 1 1 _L _x artanh x - ln I 1 - x x 1 Matrices A . . z. . XX cos d -sin d Anticlockwise rotation through angle d centre O sin q cos q Finite series Yr2 - n n 1 2n 1 r 1 6 ir3 - n2 n 1 2 r 1 4 Reflection in the line y x tan d cos 2d sin 2d sin 2d -cos 2d Cosine rule cos A -- -- etc. 2hc a2 h2 c2 -2hc cos A etc. A BC a Trigonometry sin 0 0 sin 0 cos 0 cos 0 sin 0 cos 0 0 cos 0 cos 0 sin 0 sin 0 tan 0 tan 0 tan 0 0 iitfet--- 0 00 k -nl For t tan - 0 sin 0 - -- cos 0 1 - -- 2 1

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