A A π A A B B u

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Dortha Miles
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2 A A π A A B B u

3 u( = 1 u( = 0 A π(a) = 1 π(b) = 0 B π(a) = 0 π(b) = 1 π(a) = π(b) = 0.5 p π p p 0 b s p π (s) p 0 (s)e bπ(s), A p π (A) A e bπ p 0. π p π = p 0 p 0 p π p 0 b b > 0 p π π b < 0 p π π b = 0 p π = p 0 π e π p π (s e) }{{} p 0 (s) L(s e) }{{}}{{} e bπ(s) }{{}. p π

4 e bπ(s) s π p(s) p 0 (s) p(s e, π) p(s) L(s e) L(s π). A A

8 Z S Σ σ Σ f : S (Z)

9 A B Z = {, } S = {A, B} f(a) = f(b) = (, 0.5;, 0.5) A f(a) = f(b) = B f(a) = f(b) = F F A A F f u f f(a) = (, 1;, 0) f(b) = (, 0;, 1)

10 f f u f F u : Z R p : F (S) f g, g F g g u(g(s)) p f u(g (s)) p f. S f f u f = u f p f = p f f u π : S R π(s) = u(f(s)) s S u( ) = 1 u( ) = 0 π(a) = π(b) = 0.5u( )+0.5u( ) = 0.5 A π(a) = u( ) = 1 π(b) = u( ) = 0 A A A u p Π p : Π (S) π Σ p S g g g(s) g (s) u(g(s)) u(g (s)) S

11 p : Π (S) π π E c p π (E) = 0 p π (E) = 0 π n π p πn (E) p π (E) π = π E p π (A E) = p π (A E) A E π = π + c p π = p π E E E p p p π (s) p 0 (s)ν(s, π(s)) p 0 π ν p π (E) > 0 p π (E) > 0 π = π E p π (A) = p π (A) E E p π (A E) = p π (A E)

12 ν b ν(π(s)) = e bπ(s) s p : Π (S) p 0 b π A p π (A) e bπ p 0. A = {s} s A p π (s) p 0 (s)e bπ(s). p π (s) p π (s ) = p 0(s) )) p 0 (s ) eb(π(s) π(s. p 0 p 0 π π p 0 b = 0 b = 0 b > 0 b < 0 b = 2 p 0 (A) = p 0 (B) =

13 0.5 π(a) = π(b) = 0.5 p = p 0 u( ) = 1 u( ) = 0 A π(a) = 1 π(b) = 0 e b(π(a) π(b)) = e 2 = 2 p 0 (A)/p 0 (B) = 1 p π (A)/p π (B) = 2 A p 0 (A) = 1/2 p π (A) = 2/3 p(s π) p π (s) L(s π) = e bπ(s) p(s π) }{{} = p(s) }{{} L(s π). }{{} p 0 e bπ(s) s s p 0 p 0 (A) = p 0 (B) = 0.5 A p = 2/3 p = 1/3 p 0 (A) = 4/5 A A p 0 (A) = 4/5 p π (A) = 8/9

14 π(a) π(b) = u( ) u( ) = 1 u( ) u( ) = 2 p π (A) = 16/17 u(x) = x q r rq u(x) = x π(r) = rq r µ σ 2 p π (r r 0 ) e bπ p 0 = 1 r r 0 2πσ r 2 +µ 2 2r(µ+bqσ 2 ) 2σ 2 r = r r 0 e e bqr e (r µ) 2 2σ 2 r r 0 r r 0 e r (r µ ) 2 2σ 2 r, µ = µ+bqσ 2 p 0 (r r 0 ) N (µ, σ 2 ) p π (r r 0 ) N (µ+ bqσ 2, σ 2 ) σ 2 µ bqσ 2 σ 2 t t t + 1

15 t + 1 t t+1

16 A A π A = u A A A A U 0 (A) A p 0 U A (A) p π π = u A u A U A (A) > U 0 (A) A A A u A U A (A) U 0 (A) b e bx b > 0 b < 0 b = 0 p A (s) e bu(a(s)) p 0 (s) s p A (u A)

17 p 0 (u A) b > 0 p 0 (u A) b < 0 p 0 (u A) b = 0 A U A (A) U 0 (A) b A B s s p 0 e s s e B e A e u(a(s)) > u(a(s )) A B p(s)/p(s ) r r r 0 = p 0 (s)/p 0 (s ) s s r A = e bu(a(s)) /e bu(a(s )) > 1 A r e = L(s e)/l(s e) s r < r 0 < r r e < r 0 r A. A B B r A > 1 r e

18 E E A B s s A B s B u(b(s)) > u(a(s)) > u(a(s )) > u(b(s )), p 0 b 0 b > b 0 A A B r A p(s)/p(s ) r p 0 (s)/p 0 (s ) = r 0 < r A A p A (s)/p A (s ) = e b(u(a(s)) u(a(s ))) r 0 b 0 = r r 0 u(a(s)) u(a(s )). b > b 0 p A (s)/p A (s ) = e b(u(a(s)) u(a(s ))) r 0 > r B

19 p 0 (A) = p 0 (B) = 0.5 A B e L(B e)/l(a e) > 1 B b = 2 e A π(a) = 1 π(b) = 0 e p π (A) p π (B) = p 0(A) p 0 (B) ebπ(a) L(A e) ebπ(b) L(B e) = 1 e 2 e 0 L(A e) L(B e) = 2 L(A e) L(B e). B L(B e)/l(a e) > 2 1 < L(B e)/l(a e) < 2

20 t = 0 1 A λ B λ 1 A t = 0 p 0 (A)/p 0 (B) = λ 0 > 1 t = 1 e B B p 0 (B e)/p 0 (A e) = µ > 1 b λ 0 µ λ = 1 λ > 1 t = 0 p(a) λp(b) > 0 p(a) p(b) = λ 0 λ, p = p 0 t = 0 π(a) π(b) = 1 + λ A B t = 1 p π (A)/p π (B) = e b(1+λ) /µ p(a) p(b) = eb(1+λ) µ λ. λ = 1 λ 0 > 1 = λ

21 µ > e 2b λ = λ 0 e b(1+λ 0) /λ 0 µ b > 0 e b(1+λ 0) /λ 0 λ 0 λ 0 b = 1 w = 1 α A B p(a) (1 + α) + (1 p(a)) (1 α) α = 2p(A) 1 p 0 (A) = 2/3 α = 1/3 π(a) π(b) = (4/3) (2/3) = 2 b = 1 p 0 (A) = 2/3 p π (A) = 4/5 α = 2p π (A) 1 = 3/ b > 0 r 0 = p 0 (A)/p 0 (B) > 1 p(a) p(a) p(a) 1 b 1 r k = p(a)/p(b) k α k = 2p(A) 1 = 2 r k r k = 2r k (1 + r k ) r k + 1 r k+1 = r k 1 r k + 1. r k+1 r 0 = ebπ(a) e bπ(b) = eb (1+αk) e b (1 α k) = ( ) b 1 + αk = 1 α k ( ) b 2rk = r b 2 k.

22 r k = r 1+b+b2 + b k 0 b 1 p(a) = r k /(r k + 1) p(a) p(a) 1 b 1

23 A B A π A (A) > π A (B) B π B (B) > π B (A) b = 2 y + αm y m α 0 A m = 1 B m = 1

24 π(a) = 1 + α A π(b) = 1 α B π(a) π(b) = 2α b = 2 4 α y = 1 p(a) = 1/9 1 + α/9 (8/9)α = 1 7α/9 α < 9/7 p(a)/p(b) = 1/8 α (1, 9/7) p π (A) > 1/2 b = 2 U D p 0 (U)/p 0 (D) = r 0 1 r U = e 2 r 0 = r 0 /2 r r 0 < 2 r D = 2r 0 > 1

25 p 0 e bπ(s) e bπ(s)

26 b

28 p : Π (S) π A B C p π (A) p π (B) p π (C) S Σ S S = {A, B} π p π (A) = (π(a) π(b))2 + 1 (π(a) π(b)) p π (B) = 1 (π(a) π(b)) A B π(a) π(b)

29 p π = p π π π S Σ S π p 0 h : Π S R + π A S p π (A) p 0 (A)h π (A). p 0 µ : S X R + π A S p π (A) p 0 (A)µ A (π(a)). p 0 ν : X R + π A S p π (A) p 0 (A)ν(π(A)). p 0 (S) b R π A S p π (A) p 0 (A)e bπ(a).

30 a p 0 = p a S = {A S : p 0 (A) > 0} h π (A) = p π (A)/p 0 (A) A S h π (A) = 0 A / S A S h π p 0 (A) = 0 p π (A) = 0 A / S A S x X π(a, x) A x A a E 1,..., E n S S A π(a, x) a E i E j i j E = E i E j p π(a,x) (E i )/p π(a,x) (E j ) = p 0 (E i )/p 0 (E j ) 1 p π(a,x) (A) = i µ A (x) = p π(a,x) (E i ) = i p π(a,x) (E j ) p 0 (E j ) ( )( ) 1 p0 (A) pπ(a,x) (A) p 0 (A) 1 p π(a,x) (A) p 0 (E i ) = p π(a,x)(e j ) (1 p 0 (A)) p 0 (E j ) p π(a,π(a)) (A) p 0 (A)µ A (π(a)) = (1 p 0 (A)) 1 p π(a,π(a)) (A) = p 0(E j ) pπ(a,π(a))(a) p π(a,π(a)) (E j ) π A B S π π A B a C S E j = C p π (A) p π (B) = p π (A) p π (B) = p π (A)/p π (C) p π (B)/p π (C) = p 0(C) p 0 (C) pπ(a,π(a))(a)/p π(a,π(a)) (C) p π(b,π(b)) (B)/p π(b,π(b)) (C) = p 0(A)µ A (π(a)) p 0 (B)µ B (π(b)) A, B S A S A / S µ A (x) = 1 p π (A) = p 0 (A) = 0 A / S µ A π A S C

31 A S ν : X R + ν(x) = µ A (x) x X x x π = x B = A A S x X p x (A) p x (A ) = p 0(A) p 0 (A ) µa(x) ν(x) x p x = p a = p µ A (x) = ν(x) p π (A) p 0 (A) ν(π(a)) A S A / S p π (A) = p 0 (A) = 0 A / S A, B S x y x, y x + y X p x π x,y p x (s) = x s A p x (s) = 0 s / A π x,y = p x + y p π x,y = p px p π x,y (A)/p π x,y (B) = p px (A)/p px (B) ν(x + y)/ν(y) = ν(x)/ν(0) σ(x) = (ν(x)/ν(0)) σ x y σ(x + y) = σ(x) + σ(y) m N y = mx σ(mx) = mσ(x) n N y = x/n σ(x) = σ(ny) = nσ(y) σ(x/n) = σ(x)/n y = x σ( x) = σ(x) b = σ(1) q X σ(q) = bq ν(q) = ν(0)e bq x X {q n } n N x p πqn p πx ν(q n ) ν(x) ν(q n ) = ν(0)e bqn ν(q n ) ν(0)e bx ν(x) ν(0)e bx ν(x) = ν(0)e bx ν x R

32 p : Π (S) p 0 b R π A B π A B p 0 (B) > 0 p π (A) p π (B) = p 0(A) p 0 (B) ebπ(a) e bπ(b) a Π p = p a S p π (A) > 0 π Π A S Σ(S) Σ S Π(S) Π Σ(S) p S (S, Σ(S)) b S R π Π(S) A S a Π(S) A S p 0 (A) = p a (A) p S (A)e ba p 0 (A) = p S A S A Σ(S) b = b S π Σ(S) A S p π (A) p 0 (A)e bπ(a) A B p 0 (B) > 0 π δ π (A, B) = p π (A)/p π (B) p 0 (A)/p 0 (B) δ π (A, B) = b(π(a) π(b)) E 1, E 2,... E n S A E 1 π Π π = π(a) A E 1 π = π(b) π Π(S) h = π(a) E 1 h = π(b) δ π (A, B) = δ π (A E 1, B) = δ π (A E 1, B) = δ π (A E 1, B E 2 ) = δ π (A E 1, E 2 ) = δ π (A E 1, E 2 ) = δ π (E 1, E 2 ) = b(π(a) π(b)) π Π(S) p π(a) = p π(b) = p 0 p = p a Σ Σ(S) π π(s)

33 π Π π Π E π E p π (E) = 1 π π {E 1,..., E n } π g p π ( i E i ) = 1 p π ( i E i ) = 1 p π (A i E i ) = p π (A) p π (A) = i p π(a E i ) p π (A) i p 0(A E i )e bπ(a E i) p 0 (S \ i E i ) = 0 A ebf p = i p 0(A E i )e bπ(a E i) p π (A) A ebπ p 0 {A n } n N b 0 π n p π (A n )/p π (A 1 ) = p π (A 1 ) = 0 b 0 p : Π π M R x e bx M b = 0 b 0 p 0 b π π {A n } n N π E E π π E y π E p 0 (s E : π(s) y) = p 0 (E)/2 E π E(y) E \ E(y) n E 2 n

34 M R e bx M x X X {x n } n N n p 0 (A n )e bxn p 0 (A 1 )e bx 1 π π(a n) = x n n N π(s) = x 1 n A n n N π n π n (A n ) = x n π n (s) = x 1 s / A n π π n A 1 A n N N 1 n N = p π (A 1 ) n N p π (A n ) = p π (A 1 ) n N p 0 (A n )e bxn p 0 (A 1 )e bx 1 p π (A n ) p π (A 1 ) = p π(a 1 ) p πn (A n ) p πn (A 1 ) n N p π(a 1 ) n N 1 = Np π (A 1 ) N p π (A 1 ) = 0 A 1 e bx p : Π (S) p 0 b R A B p 0 (B) > 0 π m > 0 π(s) m s A B p π (A) p π (B) = A ebπ p B ebπ p p 0 b π π M e bx M x X b 0 b = 0 n N [m, M] 2 n (M m)/2 n s I n (s) e bf I min n π n π n (s) = I min n (s) (s)/b π n f

35 e bπ(s) (M m)/2 n e bπn(s) e bπ(s) s e bπn(s) e bπ(s) s p πn (A) p πn (B) = A ebπn(s) p n B ebπn(s) p p π (A) p π (B) = n p πn (A) n p πn (B) = n = n A ebπn(s) p n B ebπn(s) p = A ebπ(s) p B ebπ(s) p p 0 (B) > 0 p πn (s) [m, M] A B A A n = {s A : e bπ(s) 2 n } π p 0 b A B p 0 (B) > 0 π m > 0 e bπ(s) m A B m π A B p 0 (B) > 0 π M R e bx M x X n N A n = {s A : e bπ(s) 2 n } B n n A \ A n = n B \ B n = p 0 (B) > 0 n 0 N n n 0 p 0 (B n ) > 0 A n B n n n 0 p π (A) p π (B) = n = p π (A n ) p π (B n ) = n A ebπ(s) p B ebπ(s) p e bπ(s) p A n B n e bπ(s) p = n e bπ(s) p A n n B n e bπ(s) p e bx M x X M

36 p 0 (B n0 ) > 0 π(s) 2 n 0 B n 0 m > 0 n n 0 B n e bπ(s) p B n0 e bπ(s) p 2 n 0 p 0 (B n0 ) m

37 π : S R π(s) = u(o(s)) p π (s) e bπ(s) c = c C p π (s i )u(c(s i )) i π α = β = 2 p 0 b(π( ) π( )) = 4

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is