0 HIGHER SCHL CERTIFICATE EXAMINATIN Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be used A table of standard integrals is provided at the back of this paper In Questions 6, show relevant mathematical reasoning and/or calculations Total marks 00 Section I Pages 8 0 marks Attempt Questions 0 Allow about 5 minutes for this section Section II Pages 9 9 90 marks Attempt Questions 6 Allow about hours and 5 minutes for this section 80
Section I 0 marks Attempt Questions 0 Allow about 5 minutes for this section Use the multiple-choice answer sheet for Questions 0. Let z = 5 i and w = + i. What is the value of z + w? (A) + i (B) + i (C) i (D) 5i The equation + + = 0 defines implicitl as a function of. d What is the value of at the point (, )? d (A) (B) (C) (D)
The comple number z is shown on the Argand diagram below. z Which of the following best represents i z? (A) i z (B) i z (C) (D) i z i z
The graph = ƒ ( ) is shown below. Which of the following graphs best represents = ƒ? (A) (B) (C) (D)
5 The equation 5 = 0 has roots α, β and γ. What is the value of? α β γ ( A ) 8 ( B ) ( C ) 8 8 ( D ) 8 6 What is the eccentricit of the hperbola =? 6 (A) (B) (C) (D) 0 5 5
7 A particle P of mass m attached to a string is rotating in a circle of radius r on a smooth horizontal surface. The particle is moving with constant angular velocit ω. The string makes an angle α with the vertical. The forces acting on P are the tension T in the string, a reaction force N normal to the surface and the gravitational force mg. a T N r mg P Which of the following is the correct resolution of the forces on P in the vertical and horizontal directions? (A) T cosα + N = mg and T sinα = mrω (B) T cosα N = mg and T sinα = mrω (C) T sinα + N = mg and T cosα = mrω (D) T sinα N = mg and T cosα = mrω 8 The following diagram shows the graph = P (), the derivative of a polnomial P(). = P () 5 Which of the following epressions could be P()? (A) ( )( ) (B) ( + )( ) (C) ( )( + ) (D) ( + )( + ) 6
9 The diagram shows the graph = ( ) for 0. The region bounded b the graph and the -ais is rotated about the line = to form a solid. = Which integral represents the volume of the solid? (A) π ( ) d 0 (B) π 0 ( ) d (C) π ( )( + ) d 0 (D) π ( )( ) d 0 7
0 Without evaluating the integrals, which one of the following integrals is greater than zero? π (A) d π + cos π (B) sin d π (C) e d (D) tan d 8
Section II 90 marks Attempt Questions 6 Allow about hours and 5 minutes for this section Answer each question in a SEPARATE writing booklet. Etra writing booklets are available. In Questions 6, our responses should include relevant mathematical reasoning and/or calculations. Question (5 marks) Use a SEPARATE writing booklet. (a) Epress 5 + i 5 i in the form + i, where and are real. (b) Shade the region on the Argand diagram where the two inequalities z + and z i both hold. d (c) B completing the square, find. + + 5 (d) (i) Write z = i in modulus argument form. (ii) Hence epress z 9 in the form + i, where and are real. e (e) Evaluate d. e + 0 (f) Sketch the following graphs, showing the - and -intercepts. (i) = (ii) = 9
Question (5 marks) Use a SEPARATE writing booklet. (a) Using the substitution t = θ tan, or otherwise, find dθ. cos θ (b) The diagram shows the ellipse + = with a > b. The ellipse has a b focus S and eccentricit e. The tangent to the ellipse at P( 0, 0 ) meets the -ais at T. The normal at P meets the -ais at N. P( 0, 0 ) N S T (i) Show that the tangent to the ellipse at P is given b the equation = 0 b 0 ( 0 ). a 0 (ii) Show that the -coordinate of N is 0 e. (iii) Show that N T = S. Question continues on page 0
Question (continued) (c) For ever integer n 0 let I = e n e n (log ) d. Show that for n I = e n ni n n. (d) n the Argand diagram the points A and A correspond to the distinct comple numbers u and u respectivel. Let P be a point corresponding to a third comple number z. Points B and B are positioned so that APB and AB P, labelled in an anti-clockwise direction, are right-angled and isosceles with right angles at A and A, respectivel. The comple numbers w and w correspond to B and B, respectivel. B w ( ) P(z) B (w ) A (u ) A (u ) (i) Eplain wh w = u + i(z u ). (ii) Find the locus of the midpoint of B B as P varies. End of Question
Question (5 marks) Use a SEPARATE writing booklet. (a) An object on the surface of a liquid is released at time t = 0 and immediatel sinks. Let be its displacement in metres in a downward direction from the surface at time t seconds. The equation of motion is given b dv v = 0, dt 0 where v is the velocit of the object. (i) t t 0 e Show that v = e +. (ii) dv dv 00 Use = v to show that = 0log dt d e 00 v. (iii) How far does the object sink in the first seconds? (b) The diagram shows S SP. The point Q is on S S so that PQ bisects S PS. The point R is on S P produced so that PQ RS. R P a a S Q S (i) Show that PS = PR. (ii) Show that PS QS = PS QS. Question continues on page
Question (continued) (c) Let P be a point on the hperbola given parametricall b = a secθ and = b tanθ, where a and b are positive. The foci of the hperbola are S(ae, 0) and S ( ae, 0) where e is the eccentricit. The point Q is on the -ais so that PQ bisects SPS. P(a sec q, b tan q) S ( ae, 0) Q S(ae, 0) (i) Show that SP = a(e secθ ). (ii) It is given that S P = a(e secθ + ). Using part (b), or otherwise, show a that the -coordinate of Q is. secθ (iii) The slope of the tangent to the hperbola at P is prove this.) b sec θ. (Do NT a tan θ Show that the tangent at P is the line PQ. End of Question
Question (5 marks) Use a SEPARATE writing booklet. (a) Find + 8 + d. (b) The diagram shows the graph =. The line is an asmptote. l (i) Use the above graph to draw a one-third page sketch of the graph = indicating all asmptotes and all - and -intercepts. (ii) B writing of the line. in the form a m + b +, find the equation Question continues on page 5
Question (continued) (c) The solid ABCD is cut from a quarter clinder of radius r as shown. Its base is an isosceles triangle ABC with AB = AC. The length of BC is a and the midpoint of BC is X. The cross-sections perpendicular to AX are rectangles. A tpical cross-section is shown shaded in the diagram. D r r r A B X a C r Find the volume of the solid ABCD. Question continues on page 6 5
Question (continued) (d) The diagram shows points A and B on a circle. The tangents to the circle at A and B meet at the point C. The point P is on the circle inside ABC. The point E lies on AB so that AB EP. The points F and G lie on BC and AC respectivel so that FP BC and GP AC. A G E P C B F Cop or trace the diagram into our writing booklet. (i) Show that APG and BPE are similar. (ii) Show that EP = FP GP. End of Question 6
Question 5 (5 marks) Use a SEPARATE writing booklet. (a) (i) a+ b Prove that ab, where a 0 and b 0. (ii) If, show that ( + ). (iii) Let n and j be positive integers with j n. Prove that n j( n j + ) n +. (iv) For integers n, prove that ( n n n n + ) n!. (b) Let P( z )= z kz + k z kz +, where k is real. Let α = + i, where and are real. Suppose that α and iα are zeros of P z, where α iα. (i) Eplain wh α and iα are zeros of P z. = ( ) + ( ) (ii) Show that P z z z k kz. ( )( + ) (iii) Hence show that if P z has a real zero then P( z )= z + z (iv) Show that all zeros of P z have modulus. ( ) or P( z )= z + z. (v) Show that k =. (vi) Hence show that k. 7
Question 6 (5 marks) Use a SEPARATE writing booklet. (a) (i) In how man was can m identical ellow discs and n identical black discs be arranged in a row? (ii) In how man was can 0 identical coins be allocated to different boes? (b) (i) Show that tan + tan = tan + for < and <. (ii) Use mathematical induction to prove j = tan for all positive integers n. n tan = n + j n n (iii) Find lim tan. n j j = Question 6 continues on page 9 8
Question 6 (continued) (c) Let n be an integer where n >. Integers from to n inclusive are selected randoml one b one with repetition being possible. Let P( k) be the probabilit that eactl k different integers are selected before one of them is selected for the second time, where k n. (i) Eplain wh ( n! ) k P k =. k n ( n k )! (ii) Suppose P( k ) P( k ). Show that k k n 0. (iii) Show that if n > k. n + >k then the integers n and k satisf (iv) Hence show that if n + is not a perfect square, then P( k) is greatest when k is the closest integer to n. You ma use part (iii) and also that k k n > 0 if P( k)< P( k ). End of paper 9
STANDARD INTEGRALS n d n+ =, n ; n + 0, if n < 0 d = ln, > 0 a e d a = e, a 0 a cosa d = sina, a 0 a sin a d = cosa, a 0 a sec a d = tana, a 0 a seca tana d = seca, a 0 a d a + = tan, a 0 a a a a + a d d d = sin, a > 0, a a = ln( + a ), = ln( + + a ) < < a > a > 0 NTE : ln = log, > 0 e 0 Board of Studies NSW 0