NORTH SYDNEY GIRLS HIGH SCHOOL 05 TRIAL HSC 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 In Questions 4, show relevant mathematical reasoning and/or calculations Total marks 70 Section I Pages - 6 0 marks Attempt Questions 0 Allow about 5 minutes for this section Section II Pages 7 4 60 Marks Attempt Questions 4 Allow about hour and 45 minutes for this section NAME: TEACHER: STUDENT NUMBER: QUESTION MARK 0 /0 /5 /5 /5 4 /5 TOTAL /70
Section I 0 marks Attempt Questions 0 Allow about 5 minutes for this section Use the multiple-choice answer sheet for Questions 0. What is the value of lim 57 4 (A) (B) (C) (D) Which of the following is equivalent to sin cos? (A) (B) (C) (D) sin 6 sin 6 sin sin -
A curve is defined by t and y log e t. Which of the following is the value of dy d at the point (, 0)? (A) (B) 4 (C) (D) 4 What is the value of sin? lim 0 tan (A) (B) (C) 4 9 (D) -
5 What is the value of sin, given that ACD in the diagram below? (A) (B) (C) (D) 5 5 5 5 4 5 d 6 What is the correct epression for 4? (A) (B) (C) (D) sin sin 4 c c sin 4 4 c sin c - 4
7 The graph below represents the depth of water in a channel (in metres) as it changes over time (in hours). Which of the following is NOT true? (A) The centre of motion is at 8 m (B) The period of oscillation is 8 hours (C) The amplitude is 8 m (D) The rate of change in the depth of water is the fastest when the depth is 8 m 8 Which of the following are the roots of the equation 4 6 0? (A),, (B),, (C),, 6 (D),, 6-5
9 What is the value of cos sin where? (A) (B) (C) (D) 0 In solving within the natural domain, three students obtain the following inequalities. Student I: Student II: Student III: Which students will obtain the correct solution to the original inequality? (A) (B) (C) (D) Student I only Student II only Student III only Student II and Student III - 6
Question (5 marks) Use a SEPARATE writing booklet. (a) Angela is preparing food for her baby and needs to use cooled boiled water. The equation y = Ae kt describes how the water cools, where t is the time in minutes, A and k are constants and y is the difference between the water temperature and the room temperature at time t, both measured in degrees Celsius. The temperature of the water when it boils is 00C and the room temperature is a constant C. (i) Find the value of A. (ii) The water cools to 88C after 5 minutes. Find the value of k correct to three significant figures. (iii) Angela can prepare the food when the water has cooled to 50C. How much longer must she wait? (b) A particle s displacement satisfies the equation t 5 4, where is measured in cm and t is in seconds. Initially, the particle is 4 cm to the right of the origin. (i) Show that the velocity is given by v 5. (ii) Find an epression for the acceleration, a in terms of. (iii) Find the position of the particle 0 seconds after the start of the motion. (iv) Briefly describe the motion of the particle. Question continues on page 9-8
Question (continued) (c) BAC and BAD are two circles such that the tangents at C and D meet at T on AB produced. A C B T D T Copy or trace the diagram into your writing booklet. If CBD is a straight line prove that: (i) TCAD is a cyclic quadrilateral (ii) TC = TD End of Question - 9
Question (continued) (c) Consider the parabola P p, p and, 4y. Q q q lie on the parabola. Q q, q P p, p (i) Find the equation of the chord PQ. (ii) Show that if PQ is a focal chord then pq. (iii) T ( t,t ), t 0 and R( r,r ) are two other points on the parabola distinct from P and Q. If TR is also a focal chord and P, T, Q and R are concyclic, show that p + q = t + r. End of Question -
Question 4 (5 marks) Use a SEPARATE writing booklet. (a) A particle is undergoing simple harmonic motion such that its displacement centimetres from the origin after t seconds is given by : 4sin t. (i) Between which two positions is the particle oscillating? (ii) At what time does the particle first move through the origin in the positive direction? n n (b) Use the principle of mathematical induction to prove 7 4 for all integers n. Question 4 continues on page -
Question 4 (continued) (c) Consider the region enclosed by the circle a y a and the line b shown in the diagram below, where 0b a. y b a (i) Show that the volume of the spherical cap formed by rotating this region around the -ais is given by b V a b cubic units (ii) A spherical goldfish bowl of radius 0 cm is being filled with water at a constant rate of 75 cm per minute. Using part (i) or otherwise, find the rate at which the water level in the bowl is rising when the bowl is half full of water. Question 4 continues on page 4 -
Question 4 (continued) (d) Consider the function f whose graph is shown below. (i) By restricting the domain of the original function to 0, find the equation of f. (ii) Hence, without solving directly, find the value(s) of for which 6. Leave your answer in eact form. No marks will be awarded for solving the equation directly for. End of paper - 4
Mathematics Etension Trial HSC 05 Suggested Solutions Section I. D Degree of numerator and denominator is the same. The limit is the ratio of the leading coefficients ie.. B Using auiliary angle method, this is o the form Rsin where R ( ). tan. 6. B Use parametric differentiation. dy dt d ; t dt ; dy d t 4. C. At dy dy d. d dt dt dy,0, t ;. d sin sin lim lim 0 tan 0 tan sin lim lim 0 0 tan 4 9 6. B using standard integrals table 7. C The amplitude is the distance from the centre of motion to the etreme of motion which is 8 = 4. 8. B Sum of roots = 4 and product of roots is 6. 9. C cos sin cos sin ; acute cos cos ; acute cos cos ; acute Alternately sub a second quadrant angle into your calculator and verify which option works. 0. D As 0 it is not necessary to multiply by the square of only by the square of as Student II has done. However by multiplying by the square of Student III does not generate etra solutions because 0 is not an admissible solution. 5. A sin sin BCD ˆ ACB ˆ 5.. 5 5 5 using compound angle result
Mathematics Etension Trial HSC 05 Suggested Solutions Section II Question (a) Using chain rule, d cos e.cos e e d e e cos e e (b) Using double angle results, cos 6 sin d d sin6 = c 6 (c) Using a ratio of 4: 4 4 P,, 4 4 (d) y; m and y ; y' ; m 4 at m m 4 tan mm 8 9 (nearest degree) (e) du d u du d u ; u d. du tan u u = 4 6 6 (f) (i) (ii) The upper bound of m to ensure eactly three solutions is found by finding the gradient of the tangent at 0 and is. As we need positive values of m, then the required range of values for m is 0 m.
Question (a) (i) At t 0; y00 77 0 77 Ae A 77 (ii) t 5; T 88 65 5k 65 77e 5k 65 e 77 65 5k ln 77 65 k ln 0.09 (4dp) 5 77 (iii) 50 77e kt kt 7 e 77 7 kt ln 77 7 t ln k 77 t 0.98 0m 55s Therefore, she must wait another 5 min 55 sec. (b) (i) t 5 4 dt 5 d d v dt 5 d d d a d 5 5 (ii) a v 5
(iii) When t 0 0 54 560 60, 6 Initially, 4 so v 0 (4) 5 So the particle is moving to the right. v can never be zero so the particle never turns around. So it can never be at. 6. (iv) Initially the particle is 4 units to the right and moving to the right. the acceleration at this time is negative, so the particle is slowing down. (c) (i) Let TCB and TDB TCB CAB (Angle between tangent and chord equal to angle in alternate segment) TDB DAB (Angle between tangent and chord equal to angle in alternate segment) CAD CAB DAB () CTD 80 (Angle sum of CTD ) () Now, CAD CTD 80 (adding () and ()) TCAD is a cyclic quadrilateral. (opposite angles are supplementary) (ii) CAT CDT (angles in the same segment in circle TCAD) This means that TCB TDB TC TD (equal sides opposite equal angles in TCD ) Question (a) (i) P 0 is a factor 6 P by inspection. [Alternately use long division]. P P (ii) QP k Q k k k If Q 0 for all, then Or k 6
(b) (i) BP hcot 45 h and DP h cot 0 h BP DP h h 4h h BD BDP is right angled at P (converse of Pythagoras Theorem) h (ii) DBP tan 60 h In BPF, BF h h 4h Using the cosine rule, PF BP BD. BP. BD.cos PBF h h 4..4. h h 7h 4h h PF h as required ˆ (c) (i) m PQ q p q p Equation of PQ is: p q y p p y p q p q y pq p q q p pq p pq p p q (ii) Focus is 0,. Sub into eqn of chord PQ 0 pq or pq (iii) PQ and TR are chords of circle PTQR and intersect at the focus S. PS SQ TS SR (product of intercepts of intersecting chords) But PS p using the locus definition; distance from focus = distance from directri Similarly, QS q etc p q t r pq p q tr t r But pq p q and tr t r p q t r
Question 4 (a) (i) Centre of motion is. Amplitude is 4. Hence, oscillates between 6 and +. (ii) Solving for 0 4sint sin t 5 t,,,... 6 6 6 t,,,... as t 0 6 6 t,,... 4 Graph of displacement is as below: Hence, crosses the origin in a positive direction the second time ie at crosses in a positive direction. t. Alternately, use v 0 to find when it n n (b) To prove 74 for n Test if true for n = : LHS = 777 4 and RHS = LHS < RHS, hence true for n. 4 64 Assume the result is true for some n = k where k ; k k ie assume that 7 4 k k k Prove true for n = k + ie prove that 7 4 k k 7 4 by assumption. Multiply both sides by. k k.4 k k 7 4.4 k k 7.4 k k 7 4.4 k k 74 Hence, the proposition is true for all n by Mathematical Induction.
y a a (c) (i) V b y d 0 b 0 a a d b 0 a aa a b ab b ab b V ab b 0 0 d dv (ii) a 0 and 75 dt b V 0 b0 b b dv 0 b b db dv dv db (Chain Rule) dt db dt db dv dv dt dt db db 75 0bb dt When the bowl is half full, b 0 db 75 75 00 00 dt 00 4 (d) (i) y For the inverse: y y Multiply by y y y y y0 4 y As y 0 for the inverse, then y 4
(ii) f 6 6 6 6 6 4 f 6 8 65 8 65 9 6 65 End of solutions