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NSW HSC 2002 : MATHEMATICS EXTENSION-2

16 pages, 56 questions, 1 questions with responses, 1 total responses,

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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 2 General Instructions Reading time 5 minutes Working time 3 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 412 Total marks 120 Attempt Questions 1 8 All questions are of equal value Total marks 120 Attempt Questions 1 8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Marks Question 1 (15 marks) Use a SEPARATE writing booklet. (a) By using the substitution u = sec x, or otherwise, find 2 3 sec x tan x dx . (b) dx By completing the square, find . 2 x + 2x + 2 2 (c) x dx Find . ( x + 3 )( x 1) 3 (d) By using two applications of integration by parts, evaluate 4 2 x e cos x dx . 0 (e) Use the substitution t = tan to find 2 4 2 d 2 + cos . 0 2 Marks Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Let z = 1 + 2 i and w = 1 + i . Find, in the form x + iy, (i) 1 (ii) (b) zw 1 . w 1 On an Argand diagram, shade in the region where the inequalities 3 0 Re z 2 and z 1 + i 2 both hold. (c) It is given that 2 + i is a root of P (z) = z3 + rz2 + sz + 20, where r and s are real numbers. (i) 1 (ii) (d) State why 2 i is also a root of P (z). Factorise P (z) over the real numbers. 2 Prove by induction that, for all integers n 1, 3 (cos i sin )n = cos(n ) i sin(n ). (e) Let z = 2(cos + i sin ) . (i) Find 1 z . (ii) Show that the real part of (iii) Express the imaginary part of 1 1 is 1 2 cos . 5 4 cos 1 z 1 in terms of . 1 z 3 2 1 Marks Question 3 (15 marks) Use a SEPARATE writing booklet. (a) y (3, 1) O (2, 0) y = f (x) x (1, 1) The diagram shows the graph of y = f (x). Draw separate one-third page sketches of the graphs of the following: 1 f ( x) (i) y= (ii) y2 = f ( x ) 2 (iii) y = f( x 2 (iv) y = ln( f ( x ) ). 2 ) 2 Question 3 continues on page 5 4 Marks Question 3 (continued) (b) y P T Q H x O c c The distinct points P cp, and Q cq, are on the same branch of the p q hyperbola H with equation xy = c2 . The tangents to H at P and Q meet at the point T. (i) Show that the equation of the tangent at P is 2 x + p2y = 2cp. (ii) (iii) 2cpq 2c . Show that T is the point , p + q p + q Suppose P and Q move so that the tangent at P intersects the x axis at (cq, 0). Show that the locus of T is a hyperbola, and state its eccentricity. End of Question 3 5 2 3 Marks Question 4 (15 marks) Use a SEPARATE writing booklet. (a) y y = 3 x2 P y = x + x2 x = 1 O x The shaded region bounded by y = 3 x2 , y = x + x2 and x = 1 is rotated about the line x = 1. The point P is the intersection of y = 3 x2 and y = x + x2 in the first quadrant. (i) Find the x coordinate of P. 1 (ii) Use the method of cylindrical shells to express the volume of the resulting solid of revolution as an integral. 3 (iii) Evaluate the integral in part (ii). 2 Question 4 continues on page 7 6 Marks Question 4 (continued) (b) R A B D S T C In the diagram, A, B, C and D are concyclic, and the points R, S, T are the feet of the perpendiculars from D to BA produced, AC and BC respectively. (i) 2 (ii) Show that DST = DCT. 2 (iii) (c) Show that DSR = DAR. Deduce that the points R, S and T are collinear. 2 From a pack of nine cards numbered 1, 2, 3, , 9, three cards are drawn at random and laid on a table from left to right. (i) What is the probability that the number formed exceeds 400? 1 (ii) What is the probability that the digits are drawn in descending order? 2 End of Question 4 7 Marks Question 5 (15 marks) Use a SEPARATE writing booklet. (a) The equation 4x3 27x + k = 0 has a double root. Find the possible values of k. (b) Let , , and be the roots of the equation x3 5x2 + 5 = 0. 2 (i) Find a polynomial equation with integer coefficients whose roots are 1, 1, and 1. 2 (ii) Find a polynomial equation with integer coefficients whose roots are 2, 2, and 2. 2 (iii) Find the value of 3 + 3 + 3 . 2 (c) y T(x0, y0) P(x1, y1) Directrix D S O Q x R E x2 + y2 = 1 , and focus S and directrix D as shown a2 b2 in the diagram. The point T (x0, y0) lies outside the ellipse and is not on the x axis. The ellipse E has equation The chord of contact PQ from T intersects D at R, as shown in the diagram. (i) Show that the equation of the tangent to the ellipse at the point P (x1, y1) is x1 x a2 (ii) y1y b2 =1. Show that the equation of the chord of contact from T is x0 x a2 (iii) + + y0 y b2 Show that TS is perpendicular to SR. 8 2 2 = 1. 3 Marks Question 6 (15 marks) Use a SEPARATE writing booklet. (a) A particle of mass m is suspended by a string of length l from a point directly above the vertex of a smooth cone, which has a vertical axis. The particle remains in contact with the cone and rotates as a conical pendulum with angular velocity . The angle of the cone at its vertex is 2 , where > , and the string 4 makes an angle of with the horizontal as shown in the diagram. The forces acting on the particle are the tension in the string T, the normal reaction to the cone N and the gravitational force mg. NOT TO SCALE l T N mg (i) (ii) (iii) Show, with the aid of a diagram, that the vertical component of N is N sin . mg Show that T + N = , and find an expression for T N in terms of sin m, l and . The angular velocity is increased until N = 0, that is, when the particle is about to lose contact with the cone. Find an expression for this value of in terms of , l and g. Question 6 continues on page 10 9 1 3 2 Marks Question 6 (continued) (b) For n = 0, 1, 2, let 4 In = tan n d . 0 1 ln 2 . 2 (i) Show that I1 = (ii) Show that, for n 2, 1 3 In + In 2 = (iii) 1 . n 1 For n 2, explain why In < In 2 , and deduce that 3 1 1 . < In < 2(n + 1) 2(n 1) (iv) By using the recurrence relation of part (ii), find I5 and deduce that 2 3 < ln 2 < . 3 4 End of Question 6 10 2 Marks Question 7 (15 marks) Use a SEPARATE writing booklet. (a) y0 y Draining water The diagram represents a vertical cylindrical water cooler of constant cross-sectional area A. Water drains through a hole at the bottom of the cooler. From physical principles, it is known that the volume V of water decreases at a rate given by dV dt = k y , where k is a positive constant and y is the depth of water. Initially the cooler is full and it takes T seconds to drain. Thus y = y0 when t = 0, and y = 0 when t = T . dy = k y. A 1 (i) Show that (ii) By considering the equation for dt dt , or otherwise, show that dy 4 2 1 t for 0 t T . y = y0 T (iii) Suppose it takes 10 seconds for half the water to drain out. How long does it take to empty the full cooler? Question 7 continues on page 12 11 2 Marks Question 7 (continued) (b) Suppose 0 < , < and define complex numbers zn by 2 zn = cos( + n ) + i sin( + n ) for n = 0, 1, 2, 3, 4. The points P0, P1, P2 and P3 are the points in the Argand diagram that correspond to the complex numbers z0, z0 + z1 , z0 + z1 + z2 and z0 + z1 + z2 + z3 respectively. The angles 0, 1 and 2 are the external angles at P0, P1 and P2 as shown in the diagram below. P3 2 P2 1 y P1 0 P0 O (i) x Using vector addition, explain why 2 0 = 1 = 2 = . (ii) Show that P0OP1 = P0 P2P1, and explain why OP0 P1P2 is a cyclic quadrilateral. 2 (iii) Show that P0 P1P2P3 is a cyclic quadrilateral, and explain why the points O, P0, P1, P2 and P3 are concyclic. 2 (iv) Suppose that z0 + z1 + z2 + z3 + z4 = 0. Show that 2 = 2 . 5 End of Question 7 12 Marks Question 8 (15 marks) Use a SEPARATE writing booklet. (a) Let m be a positive integer. (i) By using De Moivre s theorem, show that 2 2 m + 1 2 m + 1 2m 2m 2 sin(2 m + 1) = sin 3 + cos sin cos 1 3 K + ( 1)m sin 2 m +1 . (ii) 3 Deduce that the polynomial 2 m + 1 m 2 m + 1 m 1 p( x ) + K + ( 1)m x x 1 3 has m distinct roots k k = cot 2 2 m + 1 (iii) where k = 1, 2, K, m . Prove that 2 2 m m(2 m 1) . cot 2 + cot 2 + K + cot 2 = 2 m + 1 2 m + 1 2 m + 1 3 (iv) You are given that cot < 1 for 0 < < . 2 Deduce that 2 1 1 1 (2 m + 1)2 < 2 + 2 +K+ 2 . 6 1 2 m 2 m(2 m 1) Question 8 continues on page 14 13 2 Marks Question 8 (continued) (b) a C a E D K N L M x P B a F a A In the diagram, AB and CD are line segments of length 2a in horizontal planes at a distance 2a apart. The midpoint E of CD is vertically above the midpoint F of AB, and AB lies in the South North direction, while CD lies in the West East direction. The rectangle KLMN is the horizontal cross-section of the tetrahedron ABCD at distance x from the midpoint P of EF (so PE = PF = a). (i) By considering the triangle ABE, deduce that KL = a x , and find the area of the rectangle KLMN. 4 (ii) Find the volume of the tetrahedron ABCD. 2 End of paper 14 BLANK PAGE 15 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 16 Board of Studies NSW 2002

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Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2002
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