
	Trending ▼ ICSE CBSE 10th ISC CBSE 12th CTET GATE UGC NET Vestibulares ResFinder

NSW HSC 2002 : MATHEMATICS EXTENSION-1

12 pages, 42 questions, 0 questions with responses, 0 total responses,

	nsw_hsc	+Fave Message
	Profile Timeline Uploads Q & A

Home > nsw_hsc >

Instantly get Model Answers to questions on this ResPaper. Try now!
NEW ResPaper Exclusive!

Formatting page ...

2002 H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N Mathematics Extension 1 General Instructions Reading time 5 minutes Working time 2 hours Write using black or blue pen Board-approved calculators may be used A table of standard integrals is provided at the back of this paper All necessary working should be shown in every question 411 Total marks 84 Attempt Questions 1 7 All questions are of equal value BLANK PAGE 2 Total marks 84 Attempt Questions 1 7 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Marks Question 1 (12 marks) Use a SEPARATE writing booklet. (a) Evaluate lim sin 3 x . x 0 x (b) Find (c) 6 Use the table of standard integrals to evaluate sec 2 x tan 2 x d x . 0 2 (d) x State the domain and range of the function f ( x ) = 3 sin 1 . 2 2 ( 1 ) d 3 x 2 ln x for x > 0. dx 2 (e) The variable point (3t, 2t 2 ) lies on a parabola. Find the Cartesian equation for this parabola. 2 3 (f) 2x Use the substitution u = 1 x to evaluate 1 x2 2 2 ( 3 ) 2 dx . 3 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) Solve 2x = 3. 2 Express your answer correct to two decimal places. (b) Find the general solution to 2 cos x = 3 . 2 Express your answer in terms of . (c) Suppose x 3 2x 2 + a (x + 2 ) Q (x) + 3 where Q (x) is a polynomial. 2 Find the value of a . (d) 4 Evaluate 2 sin 2 4 x d x . 0 (e) T 3 A X Y B NOT TO SCALE C In the diagram the points A, B and C lie on the circle and CB produced meets the tangent from A at the point T. The bisector of the angle ATC intersects AB and AC at X and Y respectively. Let TAB = . Copy or trace the diagram into your writing booklet. (i) Explain why ACB = . 1 (ii) Hence prove that triangle AXY is isosceles. 2 4 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (a) Seven people are to be seated at a round table. (i) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? 2 (i) Show that f (x) = ex 3x2 has a root between x = 3.7 and x = 3.8 . 1 (ii) (c) 1 (ii) (b) How many seating arrangements are possible? Starting with x = 3.8, use one application of Newton s method to find a better approximation for this root. Write your answer correct to three significant figures. 3 A household iron is cooling in a room of constant temperature 22 C. At time t minutes its temperature T decreases according to the equation dT = k (T 22) where k is a positive constant. dt The initial temperature of the iron is 80 C and it cools to 60 C after 10 minutes. (i) Verify that T = 22 + Ae kt is a solution of this equation, where A is a constant. 1 (ii) Find the values of A and k. 2 (iii) How long will it take for the temperature of the iron to cool to 30 C? Give your answer to the nearest minute. 2 5 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. (a) Lyndal hits the target on average 2 out of every 3 shots in archery competitions. During a competition she has 10 shots at the target. (i) 1 (ii) (b) What is the probability that Lyndal hits the target exactly 9 times? Leave your answer in unsimplified form. What is the probability that Lyndal hits the target fewer than 9 times? Leave your answer in unsimplified form. 2 The polynomial P(x) = x3 2x2 + kx + 24 has roots , , . (i) Find the value of + + . 1 (ii) Find the value of . 1 (iii) It is known that two of the roots are equal in magnitude but opposite in sign. 2 Find the third root and hence find the value of k. (c) A particle, whose displacement is x, moves in simple harmonic motion such that x = 16 x . At time t = 0, x = 1 and x = 4. (i) Show that, for all positions of the particle, 2 x = 4 2 x2 . (ii) What is the particle s greatest displacement? 1 (iii) Find x as a function of t. You may assume the general form for x. 2 6 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) Use the principle of mathematical induction to show that 3 2 1! + 5 2! + 10 3! + + (n2 + 1)n! = n(n + 1)! for all positive integers n. (b) 4 cm NOT TO SCALE r 12 cm h The diagram shows a conical drinking cup of height 12 cm and radius 4 cm. The cup is being filled with water at the rate of 3 cm3 per second. The height of water at time t seconds is h cm and the radius of the water s surface is r cm. (i) (ii) (c) 1 Show that r = h. 3 1 Find the rate at which the height is increasing when the height of water 1 is 9 cm. (Volume of cone = r 2 h .) 3 3 Consider the function f ( x ) = 2 sin 1 x sin 1 (2 x 1) for 0 x 1 . (i) Show that f ( x ) = 0 for 0 < x < 1. 3 (ii) Sketch the graph of y = f ( x ). 2 7 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. y (a) V NOT TO SCALE 5m x 60 m O An angler casts a fishing line so that the sinker is projected with a speed V m s 1 from a point 5 metres above a flat sea. The angle of projection to the horizontal is , as shown. Assume that the equations of motion of the sinker are x=0 and = 10 , y referred to the coordinate axes shown. (i) Let ( x , y ) be the position of the sinker at time t seconds after the cast, and before the sinker hits the water. 2 It is known that x = Vt cos . Show that (ii) y = Vt sin 5t 2 + 5 . Suppose the sinker hits the sea 60 metres away as shown in the diagram. Find the value of V if = tan 1 (iii) 3 3 . 4 For the cast described in part (ii), find the maximum height above sea level that the sinker achieved. Question 6 continues on page 9 8 2 Marks Question 6 (continued) (b) Let n be a positive integer. (i) By considering the graph of y = 1 show that x n +1 1 < n + 1 n (ii) 2 1 dx <. x n Hence deduce that 3 n 1 + 1 < e < 1 + 1 n n n +1 End of Question 6 Please turn over 9 . Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) 1 1 for all real values of x and let f ( x ) = e x + x for x 0. x e e Sketch the graph y = g(x) and explain why g(x) does not have an inverse function. Let g( x ) = e x + (i) 2 (ii) 1 (iii) (b) On a separate diagram, sketch the graph of the inverse function y = 1(x ). Find an expression for y = 1(x ) in terms of x. 3 The coefficient of x k in (1 + x)n, where n is a positive integer, is denoted by ck (so ck = nCk ). (i) Show that 3 c0 + 2c1 + 3c2 + K + (n + 1)cn = (n + 2)2 n 1. (ii) Find the sum c0 1.2 3 c1 2.3 + c2 3.4 K + ( 1)n cn (n + 1)(n + 2) . Write your answer as a simple expression in terms of n. End of paper 10 BLANK PAGE 11 STANDARD INTEGRALS n x dx = 1 x dx = ln x, x > 0 ax e dx = 1 ax e , a 0 a cos ax dx = 1 sin ax, a 0 a sin ax dx 1 = cos ax, a 0 a 2 sec ax dx = 1 tan ax, a 0 a sec ax tan ax dx = 1 sec ax, a 0 a 1 dx 2 a + x2 = 1 x tan 1 , a 0 a a 1 dx 2 a x2 x = sin 1 , a > 0, a < x < a a 1 dx 2 x a2 = ln x + x 2 a 2 , x > a > 0 1 dx 2 x + a2 = ln x + x 2 + a 2 1 n +1 x , n 1; x 0, if n < 0 n +1 ( ) ( ) NOTE : ln x = loge x, x>0 12 Board of Studies NSW 2002

Formatting page ...

Additional Info : New South Wales Higher School Certificate Mathematics Extension-1 2002
Tags : new south wales higher school certificate mathematics (extension-1) 2002, nsw hsc online mathematics extension-1, nsw hsc maths, nsw hsc mathematics extension-1 syllabus, nsw hsc maths model exam papers, mathematics sample papers, mathematics course, nsw hsc maths solved paper., australia new south wales, nsw hsc online, nsw hsc past papers, nsw hsc papers, nsw hsc syllabus, nsw board of studies, higher school certificate new south wales, nsw australia, hsc syllabus, nsw hsc exams, nsw hsc question papers, nsw hsc solved question papers, nsw hsc previous exam papers, nsw university.

nsw_hsc chat

Horizontal lines at:
Guest Horizontal lines at:
AutoRM Data:
Box geometries: