0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred 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 80
Total marks 0 Attempt Questions 8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Etra writing booklets are available. Question (5 marks) Use a SEPARATE writing booklet. (a) Find ln d. (b) Evaluate + d. 0 (c) (i) Find real numbers a, b and c such that a b c = + +. ( ) (ii) Hence, find ( ) d. (d) Find cos θ dθ. (e) Evaluate 5 t + t dt.
Question (5 marks) Use a SEPARATE writing booklet. (a) Let w = i and z = + 4i. (i) Find w + z. (ii) Find w. w (iii) Epress in the form a + ib, where a and b are real numbers. z (b) On the Argand diagram, the comple numbers 0, + i, form a rhombus. + i and z z + i θ + i NOT TO SCALE O (i) Find z in the form a + ib, where a and b are real numbers. (ii) An interior angle, θ, of the rhombus is marked on the diagram. Find the value of θ. (c) Find, in modulus-argument form, all solutions of z = 8. (d) (i) Use the binomial theorem to epand (cosθ + i sinθ). (ii) (iii) Use de Moivre s theorem and your result from part (i) to prove that cos θ = cos θ + cos θ. 4 4 Hence, or otherwise, find the smallest positive solution of 4cos θ cosθ =.
Question (5 marks) Use a SEPARATE writing booklet. (a) (i) Draw a one-third page sketch of the graph y = sin π for 0 < < 4. (ii) Find lim. 0 π sin (iii) Draw a one-third page sketch of the graph y = for 0 < < 4. sin π (Do NOT calculate the coordinates of any turning points.) (b) The base of a solid is formed by the area bounded by y = cos and y = cos for 0 π. Vertical cross-sections of the solid taken parallel to the y-ais are in the shape of isosceles triangles with the equal sides of length unit as shown in the diagram. y O Find the volume of the solid. Question continues on page 5 4
Question (continued) n n! ( ) (c) Use mathematical induction to prove that n! for all positive integers n. ( ) y (d) The equation = represents a hyperbola. 6 9 (i) Find the eccentricity e. (ii) Find the coordinates of the foci. (iii) State the equations of the asymptotes. (iv) Sketch the hyperbola. (v) For the general hyperbola hyperbola as e. a y =, b describe the effect on the End of Question 5
Question 4 (5 marks) Use a SEPARATE writing booklet. (a) Let a and b be real numbers with a b. Let z = + iy be a comple number such that z a z b =. (i) a + b Prove that = +. b a ( ) (ii) Hence, describe the locus of all comple numbers z such that z a z b =. (b) In the diagram, ABCD is a cyclic quadrilateral. The point E lies on the circle through the points A, B, C and D such that AE BC. The line ED meets the line BA at the point F. The point G lies on the line CD such that FG BC. F A B C D G E Copy or trace the diagram into your writing booklet. (i) Prove that FADG is a cyclic quadrilateral. (ii) Eplain why GFD = AED. (iii) Prove that GA is a tangent to the circle through the points A, B, C and D. Question 4 continues on page 7 6
Question 4 (continued) (c) A mass is attached to a spring and moves in a resistive medium. The motion of the mass satisfies the differential equation d y dt dy + + y = 0, dt where y is the displacement of the mass at time t. (i) Show that, if y = ƒ () t and y g t are both solutions to the differential equation and A and B are constants, then = () y A ƒ t Bg t = ()+ () is also a solution. (ii) A solution of the differential equation is given by y = e kt for some values of k, where k is a constant. Show that the only possible values of k are k = and k =. (iii) A solution of the differential equation is y = Ae t + Be t. dy When t = 0, it is given that y = 0 and. dt = Find the values of A and B. End of Question 4 7
Question 5 (5 marks) Use a SEPARATE writing booklet. (a) A small bead of mass m is attached to one end of a light string of length R. The other end of the string is fied at height h above the centre of a sphere of radius R, as shown in the diagram. The bead moves in a circle of radius r on the surface of the sphere and has constant angular velocity ω > 0. The string makes an angle of θ with the vertical. θ R F N h r R θ mg O Three forces act on the bead: the tension force F of the string, the normal reaction force N to the surface of the sphere, and the gravitational force mg. (i) By resolving the forces horizontally and vertically on a diagram, show that (ii) and Show that N = mg F sinθ N sinθ = mω r F cosθ + N cosθ = mg. secθ m ω r cosecθ. (iii) Show that the bead remains in contact with the sphere if ω g h. Question 5 continues on page 9 8
Question 5 (continued) (b) If p, q and r are positive real numbers and p + q r, prove that p q r + 0. + p + q + r (c) y The diagram shows the ellipse + =, where a > b. The line l is the a b tangent to the ellipse at the point P. The foci of the ellipse are S and S. The perpendicular to l through S meets l at the point Q. The lines SQ and S P meet at the point R. l y R b P Q S O S a Copy or trace the diagram into your writing booklet. (i) Use the reflection property of the ellipse at P to prove that SQ = RQ. (ii) Eplain why S R = a. (iii) Hence, or otherwise, prove that Q lies on the circle + y = a. End of Question 5 9
Question 6 (5 marks) Use a SEPARATE writing booklet. (a) Jac jumps out of an aeroplane and falls vertically. His velocity at time t after his parachute is opened is given by v(t), where v(0) = v 0 and v(t) is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is kv, where k is a positive constant. Let m be Jac s mass and g the acceleration due to gravity. Jac s terminal velocity with the parachute open is v T. Jac s equation of motion with the parachute open is dv m = mg kv. (Do NOT prove this.) dt (i) Eplain why Jac s terminal velocity v T is given by mg. k (ii) By integrating the equation of motion, show that t and v are related by the equation v v v v v t = ln. g T ( T + )( T 0 ) ( v v )( v + v ) T T 0 (iii) Jac s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes. Jac opens his parachute when his speed is v T. Gil opens her parachute when her speed is v T. Jac s speed increases and Gil s speed decreases, both towards v T. Show that in the time taken for Jac s speed to double, Gil s speed has halved. Question 6 continues on page 0
Question 6 (continued) (b) Let ƒ ( ) be a function with a continuous derivative. (i) Prove that y ƒ ( a )= 0. ( ) = ƒ ( ) has a stationary point at = a if ƒ ( a)= 0 or (ii) Without finding, eplain why point of infleion at = a if ƒ ( a)= 0 ƒ ( ) ( ) y ƒ has a horizontal and ƒ ( a ) 0. = ( ) (iii) The diagram shows the graph y = ƒ ( ). y O Copy or trace the diagram into your writing booklet. ( ( )) On the diagram in your writing booklet, sketch the graph y = ƒ, clearly distinguishing it from the graph y =. ƒ ( ) (c) On an Argand diagram, sketch the region described by the inequality +. z End of Question 6
Question 7 (5 marks) Use a SEPARATE writing booklet. (a) The diagram shows the graph of ( ) ƒ = for 0. + 4 y O The area bounded by y = line = to form a solid. ƒ ( ), the line = and the -ais is rotated about the Use the method of cylindrical shells to find the volume of the solid. (b) Let I = cos π 8 (4 ) d. (i) Use the substitution u = 4 to show that π sin u 8 I = du. u(4 u) (ii) Hence, find the value of I. Question 7 continues on page
Question 7 (continued) (c) The diagram shows the ellipse eccentricity of the ellipse. a + y b =, where a > b. Let e be the l Q y c b P Q S S a The line l is the tangent to the ellipse at the point P. The line l has equation y = m + c, where m is the slope and c is the y-intercept. The points S and S are the focal points of the ellipse, where S is on the positive -ais. The perpendiculars to l through S and S intersect l at Q and Q respectively. (i) (ii) By substituting the equation for l into the equation for the ellipse, show that a m + b = c. Show that the perpendicular distance from S to l is given by mae + c QS =. + m (iii) It is given that mae c QS =. + m Hence, prove that QS Q S =b. End of Question 7
Question 8 (5 marks) Use a SEPARATE writing booklet. (a) For every integer m 0 let Prove that for m I m = I m 0 m ( m = m + ) 5 d I m.. (b) A bag contains seven balls numbered from to 7. A ball is chosen at random and its number is noted. The ball is then returned to the bag. This is done a total of seven times. (i) (ii) (iii) What is the probability that each ball is selected eactly once? What is the probability that at least one ball is not selected? What is the probability that eactly one of the balls is not selected? Question 8 continues on page 5 4
Question 8 (continued) (c) Let β be a root of the comple monic polynomial n n ( ) = + + + + n 0 P z z a z a z a. Let M be the maimum value of a, a,, a. n n 0 (i) Show that n n n β M β + β + + β +. (ii) Hence, show that for any root β of P( z) β < + M. (d) Let S( ) = c +, where the real numbers c k satisfy k k = 0 for all k < n, and c n 0. n k c k c n Using part (c), or otherwise, show that S( )= 0 has no real solutions. End of paper 5
STANDARD INTEGRALS n n+ d =, n ; 0, if n < 0 n + d = ln, > 0 a a e d = a e, a 0 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 = tan, a 0 a + a a d = sin, a > 0, a < < a a a d = ln( ) + a, > a > 0 a d = ln( ) + + a + a NOTE : ln = log, > 0 e 6 Board of Studies NSW 0