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

NSW HSC 2001 : MATHEMATICS EXTENSION-1

12 pages, 37 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 ...

2001 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 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) Use the table of standard integrals to find the exact value of 2 0 ( dx 16 x . 2 ) d x sin 2 x . dx (b) Find (c) Evaluate 2 2 7 (2n + 3) . 1 n=4 (d) Let A be the point ( 2, 7) and let B be the point (1, 5). Find the coordinates of the point P which divides the interval AB externally in the ratio 1 : 2. 2 (e) Is x + 3 a factor of x3 5x + 12? Give reasons for your answer. 2 (f) Use the substitution u = 1 + x to evaluate 3 0 15 x 1 + x dx . 1 2 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) Let (x) = 3x2 + x . Use the definition 2 lim f ( a + h) f ( a) f ( a ) = h 0 h to find the derivative of (x) at the point x = a . (b) Find (i) x e dx 1 + ex 1 (ii) (c) 2 cos 3 x dx . 0 3 The letters A, E, I, O, and U are vowels. (i) 1 (ii) (d) How many arrangements of the letters in the word ALGEBRAIC are possible? How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th and 8th positions? 2 Find the term independent of x in the binomial expansion of 9 x2 1 . x 3 3 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. (a) The function (x) = sin x + cos x x has a zero near x = 1.2 3 Use one application of Newton s method to find a second approximation to the zero. Write your answer correct to three significant figures. (b) B C2 C1 O P A T Two circles, C1 and C2 , intersect at points A and B. Circle C1 passes through the centre O of circle C2. The point P lies on circle C2 so that the line PAT is tangent to circle C1 at point A. Let APB = . Copy or trace the diagram into your writing booklet. (i) 1 (ii) Explain why TAB = 2 . 1 (iii) (c) Find AOB in terms of . Give a reason for your answer. Deduce that PA = BA. 2 Starting from the identity sin( + 2 ) = sin cos 2 + cos sin 2 , and using the double angle formulae, prove the identity 2 (i) sin 3 = 3sin 4sin3 . (ii) 3 Hence solve the equation sin 3 = 2sin for 0 2 . 4 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. 3x 1 . x 2 3 (a) Solve (b) An aircraft flying horizontally at V m s 1 releases a bomb that hits the ground 4000 m away, measured horizontally. The bomb hits the ground at an angle of 45 to the vertical. 4 y x Assume that, t seconds after release, the position of the bomb is given by x = Vt, y = 5t 2 . Find the speed V of the aircraft. (c) A particle, whose displacement is x, moves in simple harmonic motion. Find x as a function of t if x = 4x and if x = 3 and x = 6 3 when t = 0. 5 5 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) y y = 2 cos 1 3 0 3 x 3 x The sketch shows the graph of the curve y = (x) where f ( x ) = 2 cos 1 The area under the curve for 0 x 3 is shaded. x . 3 (i) 1 (ii) Determine the inverse function y = 1(x), and write down the domain D of this inverse function. 2 (iii) (b) Find the y intercept. Calculate the area of the shaded region. 2 By using the binomial expansion, show that 3 n n 1 n n 3 3 q p + 2 q p + 1 3 ( q + p )n ( q p )n = 2 What is the last term in the expansion when n is odd? What is the last term in the expansion when n is even? (c) A fair six-sided die is randomly tossed n times. (i) Suppose 0 r n . What is the probability that exactly r sixes appear in the uppermost position? 2 (ii) By using the result of part (b), or otherwise, show that the probability that an odd number of sixes appears is 2 1 2 n 1 . 2 3 6 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) 3 Prove by induction that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for n = 1, 2, 3, (b) y x2 = 4ay R P Q O x Consider the variable point P (2at, at 2 ) on the parabola x2 = 4ay . (i) Prove that the equation of the normal at P is x + ty = at 3 + 2at . 2 (ii) Find the coordinates of the point Q on the parabola such that the normal at Q is perpendicular to the normal at P. 1 (iii) Show that the two normals of part (ii) intersect at the point R, whose coordinates are 4 1 1 x = a t , y = a t2 + 1 + . t t2 (iv) Find the equation in Cartesian form of the locus of the point R given in part (iii). 7 2 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) A particle moves in a straight line so that its acceleration is given by dv = x 1 dt where v is its velocity and x is its displacement from the origin. Initially, the particle is at the origin and has velocity v = 1. (i) Show that v 2 = ( x 1)2 . (ii) By finding an expression for 2 dt , or otherwise, find x as a function of t. dx Question 7 continues on page 9 8 2 Marks Question 7 (continued) (b) T h 4 O 3 A P Consider the diagram, which shows a vertical tower OT of height h metres, a fixed point A, and a variable point P that is constrained to move so that angle AOP is radians. The angle of elevation of T from A is radians. 3 4 Let the angle of elevation of T from P be radians and let angle ATP be radians. (i) 1 By considering triangle AOP, show that AP 2 = h2 + h2 cot2 h2 cot . (ii) By finding a second expression for AP2, deduce that cos = (iii) 1 2 sin + Sketch a graph of for 0 < < 1 22 3 cos . , identifying and classifying any 2 turning points. Discuss the behaviour of as 0 and as End of paper 9 . 2 4 BLANK PAGE 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 2001

Formatting page ...

Additional Info : New South Wales Higher School Certificate Mathematics extension-1 2001
Tags : new south wales higher school certificate mathematics (extension-1) 2001, 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: