Mathematics Extension 2

Transcription

1 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time 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 Total marks 0 Attempt Questions 8 All questions are of equal value 4

2 Total marks 0 Attempt Questions 8 All questions are of equal value Answer each question in a SEPARATE writing booklet Extra writing booklets are available Question (5 marks) Use a SEPARATE writing booklet e (a) Evaluate dx x ( + e ) 0 x (b) Use integration by parts to find x log e x dx (c) By completing the square and using the table of standard integrals, find x dx x+ 5 (d) (i) Find the real numbers a and b such that 5x x+ a ( )( + ) x x 4 x + bx x + 4 5x x+ (ii) Find dx x x 4 ( )( + ) (e) Use the substitution x = sinθ to evaluate 4 0 dx ( 9 x )

3 Question (5 marks) Use a SEPARATE writing booklet (a) Let z =+i and w = i Find, in the form x + iy, (i) zw 4 (ii) z (b) Let α = + i (i) Express α in modulus-argument form (ii) Show that α is a root of the equation z = 0 (iii) Hence, or otherwise, find a real quadratic factor of the polynomial z (c) Sketch the region in the complex plane where the inequalities π z i < and 0< arg( z i)< 4 hold simultaneously (d) By applying de Moivre s theorem and by also expanding (cos θ + isin θ ) 5, express cos5θ as a polynomial in cos θ π (e) Suppose that the complex number z lies on the unit circle, and 0 arg() z Prove that arg(z + ) = arg(z)

4 Question (5 marks) Use a SEPARATE writing booklet (a) The diagram shows the graph of y = f(x) y O x Draw separate one-third page sketches of the graphs of the following: (i) y = f( x) (ii) y = f( x)+ f( x) (iii) y = ( f(x)) (iv) y = e f(x) (b) Find the eccentricity, foci and the equations of the directrices of the ellipse x y + = 9 4 Question continues on page 5 4

5 Question (continued) (c) The region bounded by the curve y = (x )( x) and the x-axis is rotated about the line x = to form a solid When the region is rotated, the horizontal line segment l at height y sweeps out an annulus y l O x (i) Show that the area of the annulus at height y is given by 4π y (ii) Find the volume of the solid End of Question Please turn over 5

6 Question 4 (5 marks) Use a SEPARATE writing booklet (a) A particle P of mass m moves with constant angular velocity ω on a circle of radius r Its position at time t is given by: where θ = ωt x = r cosθ y = r sin θ, (i) Show that there is an inward radial force of magnitude mrω acting on P (ii) A telecommunications satellite, of mass m, orbits Earth with constant angular velocity ω at a distance r from the centre of Earth The Am gravitational force exerted by Earth on the satellite is, where A is a constant By considering all other forces on the satellite to be negligible, show that r A = ω r (b) (i) Derive the equation of the tangent to the hyperbola x a y = b at the point P (asec θ, btan θ) (ii) Show that the tangent intersects the asymptotes of the hyperbola at the points A acos θ b B a b sin, cosθ cosθ sin sin, cosθ and θ θ + θ + sinθ (iii) Prove that the area of the triangle OAB is ab 4 Question 4 continues on page 7 6

7 Question 4 (continued) (c) A hall has n doors Suppose that n people each choose any door at random to enter the hall (i) (ii) In how many ways can this be done? What is the probability that at least one door will not be chosen by any of the people? End of Question 4 Please turn over 7

8 Question 5 (5 marks) Use a SEPARATE writing booklet (a) Let α, β and γ be the three roots of x + px + q = 0, and define s n by s n = α n + β n + γ n for n =,,, (i) Explain why s = 0, and show that s = p and s = q (ii) Prove that for n > s n = ps n qs n (iii) Deduce that α + β + γ α + β + γ α + β + γ = 5 Question 5 continues on page 9 8

9 Question 5 (continued) (b) A particle of mass m is thrown from the top, O, of a very tall building with an initial velocity u at an angle α to the horizontal The particle experiences the effect of gravity, and a resistance proportional to its velocity in both the horizontal and vertical directions The equations of motion in the horizontal and vertical directions are given respectively by x = kx and y = ky g, where k is a constant and the acceleration due to gravity is g (You are NOT required to show these) y u O α x ue kt (i) Derive the result x = cosα from the relevant equation of motion ( ) (ii) Verify that kt y = ( kusinα + g) e g satisfies the appropriate equation k of motion and initial condition (iii) (iv) Find the value of t when the particle reaches its maximum height What is the limiting value of the horizontal displacement of the particle? End of Question 5 Please turn over 9

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11 Question 6 (5 marks) Use a SEPARATE writing booklet (a) (i) Prove the identity cos a+ b x cos a b x cosax cosbx (ii) Hence find cosx cosxdx ( ) + ( ) = (b) A sequence s n is defined by s =, s = and, for n >, ( ) s = s + n s n n n (i) Find s and s 4 ( ) (ii) Prove that x + x x x+ for all real numbers x 0 (iii) Prove by induction that sn n! for all integers n (c) (i) Let x and y be real numbers such that x 0 and y 0 (ii) (iii) x+ y Prove that xy Suppose that a, b, c are real numbers Prove that a 4 + b 4 + c 4 a b + a c + b c Show that a b + a c + b c a bc + b ac + c ab (iv) Deduce that if a + b + c = d, then a 4 + b 4 + c 4 abcd

12 Question 7 (5 marks) Use a SEPARATE writing booklet ( ) (a) The region bounded by 0 x, 0 y x x is rotated about the y-axis to form a solid Use the method of cylindrical shells to find the volume of the solid (b) S A B P T Two circles and intersect at the points A and B Let P be a point on AB produced and let PS and PT be tangents to and respectively, as shown in the diagram Copy or trace the diagram into your writing booklet (i) Prove that ASP SBP (ii) Hence, prove that SP = AP BP and deduce that PT = PS (iii) The perpendicular to SP drawn from S meets the bisector of SPT at D Prove that DT passes through the centre of Question 7 continues on page

13 Question 7 (continued) (c) Suppose that α is a real number with 0 < α < π Let P n = cos α cos α cos α Kcos α n 4 8 (i) Show that Pn sin α α n Pn sin n = sinα (ii) Deduce that P n = n α sin n (iii) Given that sinx < x for x >0,show that sinα cos α α α α cos cos Kcos n 4 8 < α End of Question 7 Please turn over

14 Question 8 (5 marks) Use a SEPARATE writing booklet (a) Suppose that ω =, ω, and k is a positive integer (i) Find the two possible values of + ω k + ω k (ii) Use the binomial theorem to expand ( + ω) n and ( + ω ) n,where n is a positive integer (iii) Let l be the largest integer such that l n Deduce that n n n n n n n K = + ( + ) + ( + ) ω ω l (iv) If n is a multiple of 6, prove that n n n n n K + n = + ( ) Question 8 continues on page 5 4

15 Question 8 (continued) p (b) Suppose that π could be written in the form,where p and q are positive integers q Define the family of integrals I n for n = 0,,, by I n π n q π = x n! π 4 n cos xdx You are given that I 0 = and I = 4q (Do NOT prove this) (i) Use integration by parts twice to show that, for n, I n = π π n n n n q π n! π 4 ( ) x cos xdx q 4 π x x cos xdx n! π 4 ( ) (ii) By writing x π π as where appropriate, deduce that 4 x 4 ( ) n n n I = 4n q I p q I,for n (iii) Explain briefly why I n is an integer for n = 0,,, (iv) Prove that p 0 < < p I n q n n! for n = 0,,, n p (v) Given that p <, if n is sufficiently large, deduce that π is q n! irrational End of paper 5

16 STANDARD INTEGRALS n x dx n+ = x, n ; x 0, if n< 0 n + x dx = ln x, x > 0 e ax dx a e ax =, a 0 cosax dx = sin ax, a 0 a sin ax dx = cos ax, a 0 a sec ax dx = tan ax, a 0 a sec ax tan ax dx = sec ax, a a 0 a x dx x = a tan, 0 + a a a x dx x = sin, a> 0, a< x < a a ( ) > > dx = ln x + x a, x a x a ( ) dx = ln x + x + a x + a NOTE : ln x = log x, x > 0 e 0 6 Board of Studies NSW 00