HILLS 1999T - LORETO KlRRlBILLI WWW 85 CARABELLA ST...

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Unformatted text preview: LORETO KlRRlBILLI WWW 85 CARABELLA ST KIRRIBBLL! 2061 TRIAL Hsc : 1999 MATHEMATICS 3 UNIT TIME ALLOWED: 2 Hours (plus 5 minutes reading time) Teacher Responsible: Mrs B Spencer Mrs S Maxto‘n ENSTRUCTIONS: a Attempt all questions. 9 ln‘every question show all necessary working. 9 Silent calculators and approved templates may be used. a Start each question on a NEW page and hand up your paper in ONE bundle with your name marked on EVERY page. 9 All questions are of equal value. Question 1 l+\/§ l~—~/—2- a) Express ———-— + ————- inthe form of «B +bx/g 3+5 a/E—JE " b) Showthat C'="“’"+1 "C r r—l c) Find the acute angle between the lines y=4x—2 a11de +3y—9 =0 to the nearest minute. d) Find the co—ordinates of the point P which divides the interval AB externally in the ratio 3:1. A is (—4,2) and B is (6,5). e) (i) What is the maximum value of Ssin0 + 12 0056 ; and (ii) What is the first positive value of 6 for which this maximum occurs. Question 2 3) Find jx3,/(x3 —9)dx using u =x3—9 % b) Evaluate [cos2 x dx. Answer in exact value terms. 0 . 'y 1 c) Find the term independent of x in the expanswn of (x “ —— ) ‘2. x d) The remainder when x 3 +ax + b is divided by (x~2)(x+3) is 2x+1. Find a and b. Marks Question 3 a) The points P (2c1rp,ap'2 ) and Q (2aq,aq 3 lie on the parabola x2 = 4ay. The angle POQ is 90°. 0 is (0,0). Show thath = —- 4 and hence find the equation of the locus of M, the midpoint of PQ. b) A projectile is fired from O with a velocity of 20rn/s at an angle of 60°. The projectile just clears a wall 25m from the point of projection. How high is the wall? Take g = lOrn/sec2 . c) f(x)=x3—x2-—x—1. (i) Show that the equation f(x) = 0 has a root in the interval 1<x<1 (ii) Use Newton’s method once to find a better approximation for the root, taking x to be 1.5. ‘ Question 4 a) A stone is thrown into a pond and creates a circular ripple which expands so that 21.:— = 1.5m / s . Find the rate at which the area of the circle is increasing when the radius is 2m. Answer to 4 significant figures. b) A particle moving along the x-axis has a velocity given by v2=15—2x—x2. (i) Show that the centre of the motion is at x = — 1. (ii) Show that a=~n 2 x and thus the motion is simple harmonic. (iii) Find the amplitude and period of the motion. Two unequal circles touch at P. APD and BPC are straight lines. {71) Copy the diagram in a larger scale. (ii) Draw the common tangent XPY. (iii) Prove that AB is parallel to CD. Marks Question 5 Marks a). Use the expansion (If-x) " = Z ” Crx’ to pmVe that 1—"Cl+"C2~”C3+ ....... ..+(—1)"Cn=o_ 2 b) In how many ways can a committee of 4 people be selected fiom a group of 10 people if; ‘ (i) 2 particular members P and Q are included; _‘ 3 (ii) 2 particular members P and Q are excluded. Question 6 3) 2200]“ 120”? A ship is observed from the top of a 150m cliff BP with an angle of depression 9° when the ship is at the point C. Ten minutes later it is seen at D with an angle of depression of 10°. APDB = a , APCB = ,6 and ADBC = 6. BC bears 120°T. BD bears 220°T. (i) Showthat a=10°, flz9° and 6:100°. 3 (ii) Show that BD = 150 cota and BC = 150 cot ,6 and hence that 4 CD221502 (cotza + cot2 ,6 — Zeotacotflcosa). (iii) Find the speed of the ship in kin/h to three significant figures. 2 e) Solve for all x, 25in2' x = sian. Answer in radian measure. 3 Question 7 a) My loungeroom isikept at a constant temperature of 25°C. A cup of tea left standing in the robin cools at a rate proportional to the difference in temperature betWeen the tea and its surroundings so that 11—:— = k(T —— 25). After 20 minutes the temperature of the tea has dropped from 95 °c to 65°C. (i) Show that T = 25+ .42” is a solution of Lg; k(T — 25) . W t ,_ Wu (ii) Find the values of and k. (iii) Find the temperature of the tea after a further 10 minutes. b) Consider the fimctionflirj = x sin " x. (i) Show that f(x) is an even function. (ii) Find f '(x) and hence the co—ordinates of the only turning point. (iii) Determine the domain and range of f(x). (iv) Sketch f(x). Marks ...
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This note was uploaded on 11/01/2011 for the course ECON 101 taught by Professor Adam during the Two '11 term at University of Technology, Sydney.

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HILLS 1999T - LORETO KlRRlBILLI WWW 85 CARABELLA ST...

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