NYJC H2 Mathematics Formula Summary Cheatsheet (very complete, old syllabus)

H2 Mathematics (9740) Chapter 1: Binomial Theorem H2 Mathematics Summarised Formulae Page 1 of 19 Last updated 19 October Chapter 1: Binomial Theorem Expansion of linear algebraic factors 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r where n is a positive integer and ! !( )!    n n r r n r Expansion of quadratic functions 22 1 2 2 2 2 ( ) [ ( )] ( ) ( ) ...12 nn n n n ax bx c c bx ax nnc c x x c x x Express the quadratic factor into the form (a + b)n where a is the constant and b would be x and x2. Expansion of (1 + x)n 2( 1) ( 1)...( 1)(1 ) 1 ... ... 1 2! ! nr n n n n n rx nx x x x r To expand (1 + kx), replace x with kx. Expansion of (a + bx)n Ascending powers of x (or descending 1/x) Descending powers of x         2 1 11 2! 1 1 ... 1... ... !                                  n n n n n r bax a bb ba bx a x a nnxxan aa xn n n r r b a Range of validity: 1b a a ax x xa b b b      (small x values) (closer to 0 gives better estimate)           2 1 11. 1 ..2! 1 ... 1 ...!                                 n n n n r nn aa bx bx bx nn abx bx aa bx bxbx n n n n x r r a b Range of validity: 1 or a a a a x x xbx b b b      (large x values) Chapter 2: Sequences and Series Simple definitions 11uS [1st term = sum of 1st term] 1;2n n nu S S n    [The difference in 2 sums gives the term] Arithmetic progression (A.P.) Geometric progression (G.P.) Definitions difference of any two successive members of the sequence is a constant each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. General term ( 1)nu a n d   where a = first term, n = no. of terms, d = common difference. 1n nu ar  where r = common ratio 1n n nT S S  Proving 1nnd u u  a non-zero constant 1 n n ur u   is a constant independent of n. Sum of first n terms  212 n nS a n d    2 n nS a l l = last term  1 1 n n ar S r     1 1 n n ar S r    1 aS r    where 0,1r exists 1Sr  Sigma Notation 12 ... b r aa a r a bu u u u u        where a and b are integers. n = (b – a + 1) i.e. upper limit – lower limit + 1. Rules of sigma notation 1 n r a an     1 G.P. 1r r a r     11 nn r r r r au a u     1 1 1 n n n rr r r r r ru v u v           1 11 n n m r m r r r r ru u v        Method of Differences Note: Watch out for method of differences when there is a small difference in the terms being subtracted. Also note that the cancelling is symmetrical, i.e. cancelling the 2nd term on the LHS will cancel the last 2nd term on the RHS. Limit of sequence 11, 0 !nn  0na  (if 0 < a < 1) 1 0na  (if a > 1) The other terms, e.g. an, an where a > 1 will tend to infinity and are not included in the limit of a sequence. Convergen ce of sequences A sequence is convergent if lim n nuL   where L is a finite number 1 , nnu L u L  SL  (Use G.C. to generate terms of sequence) Shared by Chris Chiam On owlcove.sg

H2 Mathematics (9740) Chapter 3: Mathematical Induction H2 Mathematics Summarised Formulae Page 2 of 19 Last updated 19 October Chapter 3: Mathematical Induction Format Let Pn be the statement where [copy eqn here] for all n  . LHS of P1 = n RHS of P1 = n Therefore, P1 is true. Assume that Pk is true for some k  , i.e. [replace n by k] We want to prove Pk+1, i.e. [replace k by k+1] LHS of Pk+1 = [Start proving] = RHS (Proven) Since P1 is true and Pk is true Pn is true for all n  . Conjecture Some induction questions may involve conjectures (guess) of the general formula. They will usually ask you to write the values of a sequence/sigma for n = 1, 2, 3, 4 then deduce a general equation which will then be proven by Mathematical Induction. Chapter 4: Graphing Techniques Points that must be labeled Axial intercepts (x = 0, y = 0) Turning Points (Let d 0d y x  and solve for x) Asymptotes (Express in form ()f( ) ( ) () Rxx Q x Dx Vertical asymptote: Let D(x) = 0, solve for x. Horizontal/Oblique asymptote: Q(x). Conics Circle Eclipse (oval) 2 2 2( ) ( )   x h y k r represents a circle with centre (h, k) and radius r. 22 22 ( ) ( ) 1 where    x h y k abab represents an ellipse with centre (h, k) x-intercepts at ( ,0)a and y-intercepts at (0, )b . Lines of symmetry: x = h, y = k. Hyperbola 22 22 ( ) ( ) 1 where    x h y k abab represents an horizontal hyperbola with centre (h, k). Oblique asymptotes: ()  by k x h a and ()  by k x h a . Lines of symmetry: x = h, y = k. 22 22 ( ) ( ) 1 where    x h y k abab represents a vertical hyperbola with centre (h, k). Oblique asymptotes: ()  by k x h a and ()  by k x h a . Lines of symmetry: x = h, y = k. In the process of proving, you may use what you wrote for Pk into that equation. [esp. for sigmas] Shared by Chris Chiam On owlcove.sg

H2 Mathematics (9740) Chapter 4: Graphing Techniques H2 Mathematics Summarised Formulae Page 3 of 19 Last updated 19 October y = f(x) + a y = a f(x) y = – f(x) Transformations For a > 0), the graph of y = f(x) + a is obtained by translating the graph of y = f(x) by a units in the positive y direction. This vertical translation only affects the y-values and the horizontal asymptote by the constant a. For a > 0), the graph of y = a f(x) is obtained by scaling the graph of y = f(x) through a scaling parallel to the y-axis by a scale factor of a. This vertical scaling only affects the y- values and the horizontal asymptote by the constant a. the graph of y = – f(x) is a reflection of the graph y = f(x) in the x-axis. This vertical transformation only affects the y-values and the horizontal asymptote by a negative sign. (Just negate the points/equation) y = f(x + a) y = f x a   y = f(–x) The graph of y = f(x – a) is obtained by translating the graph of y = f(x) by a units in the x direction. If a > 0, shift towards the positive x-direction, if a < 0, shift towards the negative x- direction. This horizontal translation only affects the x-values and the vertical asymptote by the constant a. The graph of f xy a   is obtained by scaling the graph of y = f(x) by a units in the x direction. If a > 0 you expand the graph, if a < 0 you compress the graph (as usual). This horizontal translation only affects the x-values and the vertical asymptote by the constant a. the graph of y = – f(x) is a reflection of the graph y = f(x) in the x-axis. This horizontal transformation only affects the x-values and the vertical asymptote by a negative sign. (Just negate the points/equation) All transformations are done in the order f    xy d c a b y = |f(x)| y = f(|x|) y2 = f(x), y =   1 f x the parts of the graph below the x-axis (<0) are reflected upwards. 1. Keep y = f(x) where f(x) 0, 2. Reflect y = f(x) where f(x) 0 above the x-axis. 2 5yx 2 5yx the x-values in the graph are modulused. 1. Discard y = f(x) where x < 0 [the part of the curve left of y-axis] 2. Keep and reflect y = f(x) where f(x) 0 to the left of the y-axis. 3 5yx 3 5yx The graph of  fyx consists of 2 parts symmetrical to the x-axis. 1. Only consider the part of the graph for which f(x) 0 (area above x-axis). 2. First sketch  fyx : When f(x) > 1,    ff xx When f(x) < 1,    ff xx y = f(x) y2 = f(x) Asymptotes and x-intercepts x-intercept at (h, 0) Vertical asy. x = h Vertical asymptote x = h x-intercept at (h, 0) Horizontal asymptotes ,0y k k 1y k 0y No horizontal asymptote If function goes to pos