Mathematics Extension 1

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1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used. A tble of stndrd integrls is provided t the bck of this pper. Show ll necessry working in Questions 4. Section I Pges 4 0 mrks Attempt Questions 0 Allow bout 5 minutes for this section. Section II Pges mrks Attempt Questions 3 Allow bout hour 45 minutes for this section.

2 Totl mrks 0 Attempt Questions 0 All questions re of equl vlue Shde your nswers in the pproprite bo in the Multiple Choice nswer sheet provided. The curves y nd y k, where k 0, intersect t 45. How mny possible vlues of k re there? (A) One. (C) Four. (B) Two. (D) None. Consider the eqution sin bcos c 0, where 0. Let b c. Which of the following sttements is flse? (A) When 0, there re lwys ectly two solutions. (B) When 0, there re lwys no solutions. (C) When 0, there is lwys ectly one solution. (D) For ny vlue of, there re t most two solutions. 3 3 Let P p q, where p nd q re constnts, hve roots, nd. Find the vlue of. (A) 6. (B) 3. (C) 0. (D) 3.

3 4 Two prticles move in simple hrmonic motion. The mimum velocity of Prticle X is twice the mimum velocity of Prticle Y. Consider the following sttements. (I) If they hve the sme mplitude, Prticle X hs hlf the period of Prticle Y. (II) If they hve the sme mplitude, Prticle X hs twice the period of Prticle Y. (III) If they hve the sme period, Prticle X hs hlf the mplitude of Prticle Y. (IV) If they hve the sme period, Prticle X hs twice the mplitude of Prticle Y. Which of the following is correct? (A) (B) (C) (D) (I) nd (III) re true. (I) nd (IV) re true. (II) nd (III) re true. (II) nd (IV) re true. 5 Consider the following epressions. (I) (II) (III) k k k k k k 6 Which of the following re NOT generl solutions to sin, for some integer k? (A) (B) (C) (D) (I) nd (II) (I) nd (III) (II) nd (III) (III)

4 6 A rel polynomil P ( ) of degree n is divided by nother polynomil A ( ) which hs degree k, where 0 k n. Let R ( ) be the reminder term from the division. Which of the following sttements is lwys true? (A) The degree of R ( ) is greter thn k (B) The degree of R ( ) is greter thn or equl to k (C) The degree of R ( ) is less thn k (D) The degree of R ( ) is less thn or equl to k 7 Wht is the smllest number of people required in room such the probbility tht t lest two of people hve the sme birthdy is t lest 50%? Assume there re 365 dys in yer. (A) 3 (B) 4 (C) 5 (D) 6 8 Which of the following vlues of k llows the inequlity b k to hve the solution b? (A) b (B) b (C) b (D) b

5 9 Consider function f( ) for some unrestricted domin. Let f ( ) be n inverse function of f( ). Which of the following sttements is NOT necessrily true? (A) (B) f ( f ( )) f ( f ( )) (C) If f ( ) is zero t the point ( b, ) then the derivtive of point ( b, ) for some fied points ( b, ) nd ( b, ) f ( ) is undefined t the (D) If the derivtive of f ( ) is zero t the point ( b, ) then f ( ) is undefined t the point ( b, ) for some fied points ( b, ) nd ( b., ) 0 The digrm below shows polynomil with three sttionry points nd two rel roots. Newton s Method is used, with 0 s the initil vlue. Which of the following sttements bout Newton s Method is lwys flse? y (Digrm NOT to scle) O b (A) If 0, then further itertions cn pproch r. (B) If 0 b, then further itertions cnnot pproch either r or r. (C) If 0 0 b, then further itertions cn pproch r. (D) If 0 b, then further itertions cn pproch r.

6 Totl mrks 60 Attempt Questions 4 All questions re of equl vlue Answer ech question in SEPARATE writing booklet. Etr writing booklets re vilble. Question (5 mrks) Use SEPARATE writing booklet. e () Solve the inequlity e e 3 (b) Show tht sin. b b lim 0 sin (c) Suppose tht P( 0, y 0) divides the intervl A(, y ) nd B(, y ). 3 Show tht if P divides the intervl AB eternlly, then ( y y )( y y ) (d) Sketch the function f( ) cos, lbelling intercepts nd symptotes. 3 (e) (i) Use the substitution sin, where 0 nd evlute d., to 4 (ii) Hence, or otherwise, evlute d. End of Question

7 Question (5 mrks) Use SEPARATE writing booklet. () The region bounded by the grph 0 nd 4 4 y sin cos is rotted bout the is to form solid., the is nd the lines 3 Find the volume of the solid formed. (b) (i) Prove tht sin sin tn, cos cos nd hence write down similr epression for tn. (ii) The digrm shows ABC with ngles nd where, nd corresponding sides nd b respectively. b Prove tht tn b. b tn Question continues on the net pge

8 (c) The ccelertion of prticle is given by k k, where is the displcement of the prticle from the origin, in metres, nd k is positive constnt. The prticle is initilly t the origin, nd hs initil velocity. (i) Find the displcement-time eqution of the prticle. 3 (ii) Hence, find the limiting vlue of s t. (d) Two points P p, p nd, Q q q lie on the prbol 4y, where 0 such tht PQ is focl chord. Tngents drwn t P nd Q intersect t T p q, pq (Do NOT prove this). y (Digrm NOT to scle) O (i) Show tht the re of PQT is 3 p p. 3 (ii) The vlue of P moves rte of unit per second. For wht vlues of p is the re of the tringle incresing? End of Question

9 Question 3 (5 mrks) Use SEPARATE writing booklet. () (i) Show tht k k k 3... k n is divisible by n!, where n nd k re positive integers. (ii) Let m be n rbitrry positive integer. 3 Use mthemticl induction on n to prove tht mn! is divisible by m! n for ll positive integers m nd n. (b) An n-sided die is rolled m times, where m n, nd the number fcing upwrds is recorded. (i) Show tht the probbility of cquiring prticulr number ectly k times is given by P k m k n m k n m (ii) Deduce tht if m is not divisible by n, then the chosen number m is most likely to pper pproimtely times. n (iii) Eplin why if m is divisible by n, then the chosen number is m mn most likely to pper ectly or times. n n Question 3 continues on the net pge

10 (c) In ABC, circle is inscribed such tht it is tngentil to ll three sides. Let 4 the centre of this circle be O. Another circle is drwn such tht the vertices of ABC lie on the circumference of the circle. From verte A, line is drwn to O. From O, line is drwn to meet the midpoint M of rc BC. A O B C M Prove tht OCM is isosceles. (d) Prove tht for some positive integer 0 k n. 3 n n n n n n k k k 3 k n k n k k k k k k n... 0 n n k End of Question 3

11 Question 4 (5 mrks) Use SEPARATE writing booklet. () An n digit pssword is mde using digits 0,,,, 9, with repetition llowed. 3 The pssword is entered by pressing series of buttons in the correct order. An observer notices tht Alice uses k 0 distinct numbers for her pssword. How mny possible combintions of numbers re there, in terms of k nd n? Justify your nswer. Question 4 continues on the net pge

12 (b) A prticle moves in simple hrmonic motion h metres in the ir long horizontl rod. The prticle s displcement is given by Asin nt, where A, n 0 nd 0 nt. At some moment t T, the prticle is relesed from the horizontl rod nd undergoes projectile motion. The prticle lnds metres wy from the origin. Let t 0 denote the time between when the prticle is relesed, to when the prticle hits the ground. y O (i) Prove tht when tn nt with respect to T., the horizontl rnge is mimised 4 nt 0 (ii) Hence, or otherwise, show tht when is mimised, then A. Question 4 continues on the net pge

13 (c) Suppose tht the growth t time t of popultion of P (in millions) cn be modeled by the differentil eqution dp P P dt k, where is positive integer. k (i) Suppose tht the initil popultion is million. k Show tht P stisfies the bove differentil eqution. t e (ii) Now consider the rte of chnge of nother popultion N (in millions) which cn be modeled by the differentil eqution b N, dt where b is positive integer. The initil popultion of N is lso k million nd the limiting popultion of N is the sme s the limiting popultion of P. Show tht k bt N k e stisfies the differentil eqution bove. (iii) Prove tht if b b, then there eists time t 0 where the two 4 popultions P nd N will be equl to ech other. End of Em

14 STANDARD INTEGRALS n n n d, n ; 0, if n 0 d ln, 0 e d e, 0 cos d sin, 0 sin d cos, 0 d 0 sec tn, sec tn d sec, 0 d tn, 0 d sin, 0, d ln, 0 d ln NOTE: ln log, 0 e Bored of Studies NSW 04