Mathematics Extension 1 Time allowed: 2 hours (plus 5 minutes reading time)

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1 Name: Teacher: Class: FORT STREET HIGH SCHOOL 0 HIGHER SCHOOL CERTIFICATE COURSE ASSESSMENT TASK : TRIAL HSC Mathematics Extension Time allowed: hours (plus 5 minutes readin time) Syllabus Assessment Area Description and Markin Guidelines Outcomes Chooses and applies appropriate mathematical techniques in order to solve problems effectively HE, HE4 Manipulates alebraic expressions to solve problems from topic areas such as inverse functions, trionometry and polynomials HE, HE5 Uses a variety of methods from calculus to investiate HE6 mathematical models of real life situations, such as projectiles, kinematics and rowth and decay HE7 Synthesises mathematical solutions to harder problems and communicates them in appropriate form Questions 0, 4 Total Marks 70 Section I 0 marks Multiple Choice, attempt all questions, Allow about 5 minutes for this section Section II 60 Marks Attempt Questions 4, Allow about hour 45 minutes for this section Section I Total 0 Marks Q Q0 Section II Total 60 Marks Q /5 Q /5 Q /5 Q4 /5 Percent General Instructions: Questions 4 are to be started in a new booklet. The marks allocated for each question are indicated. In Questions 4, show relevant mathematical reasonin and/or calculations. Marks may be deducted for careless or badly arraned work. Board approved calculators may be used.

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3 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 solution to the equation x x? (A) x (B) x (C) x (D) x A parabola has the parametric equations x t and What are the coordinates of the focus? (A) ( 6,0) (B) (0, 6) (C) (6,0) (D) (0,6) y 6t. What is the acute anle to the nearest deree that the line x y5 0 makes with the y- axis? (A) 7 (B) 4 (C) 56 (D) 6 4 What are the coordinates of the point that divides the interval joinin the points A (,) and B (5,) externally in the ratio :? (A) ( 7, ) (B) ( 7,) (C) (,) (D) (, ) 0 Trial Examination Extension Mathematics Pae

4 5 Which of the followin is an expression for Use the substitution u e x. (A) (B) (C) (D) ( e ) x ( e ) x lo ( ) x e x e c c e x e x c e lo ( ) e x e x c e e e dx x? x 6 What is the domain and rane of (A) Domain: x y cos ( )? x. Rane: 0 y (B) Domain: x. Rane: 0 y (C) Domain: x. Rane: y (D) Domain: x. Rane: y x dx? 7 What is the indefinite interal for cos x sec (A) x sin x tan x c 4 (B) x sin x tan x c 4 (C) (D) x sin x tan x c 4 x sin x tan x c 4 0 Trial Examination Extension Mathematics Pae 4

5 8 A football is kicked at an anle of to the horizontal. The position of the ball at time t seconds is iven by x Vtcos and y Vtsin t where m/s is the acceleration due to ravity and v m/s is the initial velocity of projection. What is the maximum heiht reached by the ball? (A) (B) (C) (D) V sin sin V V sin sin V 9 At a dinner party, the host, hostess and their six uests sit at a round table. In how many ways can they be arraned if the host and hostess are separated? (A) 70 (B) 440 (C) 600 (D) The velocity of a particle movin in a straiht line is iven by v x where x metres is the distance from fixed point O and v is the velocity in metres per second. What is the acceleration of the particle when it is 4 metres from O? (A) a ms - (B) a 9.5 ms - (C) a ms - (D) a 7 ms - 0 Trial Examination Extension Mathematics Pae 5

6 Section II 60 marks Attempt Questions 4 Allow about hour and 45 minutes for this section Answer each question in a SEPARATE writin booklet. Extra writin booklets are available. In Questions 4, your responses should include relevant mathematical reasonin and/or calculations. Question (5 marks) Use a SEPARATE writin booklet. dx (a) Evaluate. [] x x (b) For what values of x is? [] x (c) Differentiate x lnx with respect to x. [] (d) In how many ways can a committee of men and women be chosen from a roup of 7 men and 9 women? [] (e) Let f x x 5x 7x 0. The equation 0 f x has only one real root. i. Show that the root lies between 0 and. [] ii. Use one application of Newtons Method with an initial estimate of x0 to find a better approximation of the root (to decimal places). [] (f) Evaluate x dx usin the substitution u x. [] x 0 End of Question 0 Trial Examination Extension Mathematics Pae 6

7 Question (5 marks) Use a SEPARATE writin booklet. (a) Two points P ap, ap and, Q aq aq lie on the parabola x 4ay. The eneral tanent at any point on the parabola with parameter t is iven by y tx at (do NOT prove this). i. Find the co-ordinates of the point of intersection T of the tanents to the parabola at P and Q. [] ii. You are iven that the tanents at P and Q intersect at an anle of 45. Show that [] p q pq iii. By evaluatin the expression x 4ay, or otherwise, find the locus of the point T when the tanents at P and Q meet as described in part ii above. [] (b) For the function iven by f x x 4 i. State the domain for the function f x. [] ii. Find the inverse function f x for the iven function f x. [] iii. Find the restrictions on the domain and rane for f x to be the inverse function of f x. [] (c) Prove by Mathematical Induction that n n is divisible by, for all positive inteer n. [] (d) For 0, find all the solutions of sin cos. [] End of Question 0 Trial Examination Extension Mathematics Pae 7

8 Question (5 marks) Use a SEPARATE writin booklet. (a) The rate at which a body warms in air is proportional to the difference in temperature T of the body and the constant temperature A of the surroundin air. This rate is iven by the differential equation dt kt A dt where t is the time in minutes and k is a constant. kt i. Show that T A Ae 0, where A 0 is a constant, is a solution of this equation. [] ii. A cold body, initially at 5 C, warms to 0 C in 0 minutes. The air temperature around the body is 5 C. Find the temperature of the body after a further 40 minutes have elapsed. Give your answer to the nearest deree. [] (b) The acceleration of a particle movin in a straiht line is iven by d x x dt where x is the displacement, in metres, from the oriin O and t is the time in seconds. Initially the particle is at rest at x 4. i. If the velocity of the particle is v m/s, show that [] v x x 4 ii. Show that the particle does not pass throuh the oriin. [] iii. Determine the position of the particle when v 0. Justify your answer. [] (c) For the raph of x y x i. Find the horizontal asymptote of the raph. [] ii. x Without the use of calculus, sketch the raph of y, showin the x asymptote found in part (i). [] Question continues on pae 9 0 Trial Examination Extension Mathematics Pae 8

9 (d) The velocity v m/s of a particle movin in simple harmonic motion alon the x-axis is iven by v 8x x i. Between which two points is the particle oscillatin? [] ii. What is the amplitude of the motion? [] iii. Find the acceleration of the particle is terms of x. [] iv. Find the period of oscillation. [] End of Question 0 Trial Examination Extension Mathematics Pae 9

10 Question 4 (5 marks) Use a SEPARATE writin booklet. (a) Let ABPQC be a circle such that AB=AC, AP meets BC at X and AQ meets BC at Y, as shown below. Let BAP and ABC. i. Copy the diaram into your writin booklet, markin the information iven above, and state why AXC. [] ii. Prove BQP. [] iii. Prove BQA. [] iv. Prove the quadrilateral PQYX is cyclic. [] Question 4 continues on pae 0 Trial Examination Extension Mathematics Pae 0

11 (b) Two yachts, A and B, subtend an anle of 60 at the base C of a cliff C. From yacht A, the anle of elevation to point P, 00 m vertically above C, is 0. Yacht B is 600m from C. i. Calculate lenth AC. [] ii. Calculate the distance between the two yachts. [] Question 4 continues on pae 0 Trial Examination Extension Mathematics Pae

12 (c) A projectile, with initial speed V0 m/s, is fired at an anle of elevation from the oriin at O towards a taret T, which is movin away from O alon the x-axis. You may assume that the projectiles trajectory is defined by the equations x Vt cos y Vt sin t where x and y are the horizontal and vertical displacements of the projectile in metres at time t seconds after firin, and where is the acceleration due to ravity. i. Show that the projectile is above the x-axis for a total of V sin seconds. [] ii. Show that the horizontal rane of the projectile is V sin cos metres. [] iii. At the instant the projectile is fired, the taret T is d metres from O and it is movin away at a constant speed of u m/s. Suppose that the projectile hits the taret when fired at an anle of elevation. Show that [] d u V cos V sin iv. Deduce that the projectile will not hit the taret if [] V d u. V End of Question 4 0 Trial Examination Extension Mathematics Pae

13 Section I: Multiple Choice Worked Solutions 4 5 x x x x x Test solutions x x x (incorrect) Solution is x x t and y 6t or x (x) x x x a 6 and the parabola is concave downwards Focus is (0, 6) For xy5 0 then m Anle the line makes with the x-axis tan Anle the line makes with the y-axis mx nx x m n 5 7 The coordinates of point are ( 7, ) u e du e dx du x x x e dx Also x x u e or e u x x x e dx e e dx x x e e ( u) du u du u loe uuc e e c x x lo e( ) ( ) x x lo e( e ) e c my ny y m n x x x (correct) Mark: C Mark: B Mark: C Mark: A Mark: C

14 6 7 8 x Domain: or x. Rane: 0 y cos x sec xdx ( cos x) sec xdx x sin x tan xc 4 y Vtsin t y Vsin t Maximum heiht when y 0 0Vsin t V sin t Mark: A Mark: C Mark: C Maximum heiht V sin V sin hv sin V sin 9 0 With no restrictions there are 8 people Arranements = ( n )! 7! 5040 When the host and hostess sit next to each other. Arranements =!( n )!!6! 440 Number of arranements when host and hostess are separated vx v 4x x9 9 v x 6x d 9 a x 6x dx 4x 6 When x 4 then a Mark: C Mark: C

15 Section II: Free Response Worked Solutions Question dx (a) Evaluate x. [] dx x x sin uses standard interal correctly sin sin 4 4 correct inverse values correct answer x (b) For what values of x is? x [] Notin x : x. x x x x x x 0x xx x xx xx 4 finds boundaries y x justifies required values Hence x or x 4 correct answer x ln x with respect to x. [] (c) Differentiate Let y x ln x, then dy x dx x x. lnx. x correct use of product & chain rules xxln x ( x ln x ) correct answer (d) In how many ways can a committee of men and women be chosen from a roup of 7 men and 9 women? []

16 C C correct answer (e) Let f x x 5x 7x The equation 0 f x has only one real root. i. Show that the root lies between 0 and. [] f f justification correct Hence the root lies between 0 and. ii. Use one application of Newtons Method with an initial estimate of x0 to find a better approximation of the root (to decimal places). [] f ' x x 0x7 f ' f Hence f x0 x x0 f ' x0 0 values and formula correct correct answer x (f) Evaluate dx usin the substitution u x. x [] 0 u x du dx dx du u x x0, u x, u set-up values correct Then 0 x dx x u. du chane of variable correct u

17 du u uln u ln ( ln) ln ln correct answer

18 Question (a) Two points P ap, ap and Q aq, aq lie on the parabola x 4ay. The eneral tanent at any point on the parabola with parameter t is iven by y tx at (do NOT prove this). i. Find the co-ordinates of the point of intersection T of the tanents to the parabola at P and Q. [] Tanents are y px ap y qx aq and solvin simultaneously: 0 pxqxap aq q p xa q p a q p q p xa pq since p q correct value for x y p a pq ap ap apq ap apq correct value for y ii. You are iven that the tanents at P and Q intersect at an anle of 45. Show that [] p q pq m m tan, with 45, m p, m q: mm p q tan 45 pq p q pq pq pq correct use of formula and correct alebra to result iii. By evaluatin the expression x 4ay, or otherwise, find the locus of the point T when the tanents at P and Q meet as described in part ii above. [] x 4ay From x 4 4 a p q a apq a pq a pq 4 a p pq q a pq 4 a p pq q pq a p pqq a pq 4ay : a pq since p q pq

19 a pq p q a a pqa p q a a apq apq a ay y since y apq Thus x 4ay a ay y, leadin to the equation of the locus of T bein x a 6ay y x 4 (b) For the function iven by f x x 4 i. State the domain for the function f x. [] ii. Find the inverse function f x for the iven function Consider y x 4 : swappin x and y ives x x y4 x y4 y4 swappin and squarin correct x y 4 y x x correct answer iii. Find the restrictions on the domain and rane for f x inverse function of f x. [] f x, with x 4this leads to y. For f x, these f x. [] to be the For reverse, so the restrictions on f x are x and f x 4 correct answer (c) Prove by Mathematical Induction that n n is divisible by, for all positive inteer n. [] To prove n n N, where N is an inteer: Let n : n n. which is divisible by, hence the statement is true for n. Assume true for n k i.e. assume k k M, M an inteer or k M k initial value and assumption correct Then show true for nk k k Q, Qan inteer i.e. show LHS k k k k kk k k 5k M kk 5k by assumption

20 M k k M k k Q Q an inteer. Hence, as true for n, by the principle of mathematical induction, the statement is true for all inteer n. correct resolution (d) For 0, find all the solutions of sin cos. [] sin cos cos sincoscos 0 cossin 0 correct factors Hence cos 0 sin 0, sin 7, 6 6

21 Question : (a) The rate at which a body warms in air is proportional to the difference in temperature T of the body and the constant temperature A of the surroundin air. This rate is iven by the differential equation dt kt A dt where t is the time in minutes and k is a constant. kt i. Show that T A Ae 0, where A 0 is a constant, is a solution of this equation. [] kt T A Ae 0 dt kt ka0 e dt kt But from, TA0e T A dt kt kt A, hence T A Ae 0 is a solution to the equation. dt correct resolution ii. A cold body, initially at 5 C, warms to 0 C in 0 minutes. The air temperature around the body is 5 C. Find the temperature of the body after a further 40 minutes have elapsed. Give your answer to the nearest deree. [] kt A 5, hence T 5 A0e When t 0, T 5, hence k0 55 A0e 0 A0 T 5 0e kt correct resolution of initial constants Now when t 0, T 0 ives 0 5 0e 0k 5 0e e 0k 4 0k 0k ln 4 k ln correct resolution of k 0 4 Then, when t 60 : T 5 0e 5 0e 60 ln 0 4 ln correct answer (b) The acceleration of a particle movin in a straiht line is iven by d x x dt where x is the displacement, in metres, from the oriin O and t is the time in seconds. Initially the particle is at rest at x 4.

22 d x i. If the velocity of the particle is v m/s, show that [] v x x 4 x dt d v x dx At x 4, v 0 : v x dx c x x c c 4 4 v x x or v x 6x 8 correct answer ii. Show that the particle does not pass throuh the oriin. [] At x 0: v 8, but this is impossible, hence the particle does not pass throuh the oriin. correct answer with justification iii. Determine the position of the particle when v 0. Justify your answer. [] When v 0 : 0 x 6x8 00 x 6x8 0 x x54 x 9x 6 correct solutions Hence, x 9 or x 6, but the particle starts at x 4 and never passes throuh the oriin ( x 0 ), so x 6 is not an acceptable answer. The position of the particle when v 0 is x 9. justification correct x (c) For the raph of y x i. Find the horizontal asymptote of the raph. [] x y b 4ac x x x x x x y x x y Now, as x y 0, then x, y ives a horizontal asymptote at y. asymptote correct

23 x ii. Without the use of calculus, sketch the raph of y x, showin the asymptote found in part (i). [] Notin asymptotes at y and x, and intercepts of x 0; 0y y y 0; x 0 x y x asymptotes, branches/intercepts correct (d) The velocity v m/s of a particle movin in simple harmonic motion alon the x-axis is iven by v 8x x i. Between which two points is the particle oscillatin? [] v 8xx 4x x Thus at x and x 4, v 0. The particle oscillates between x and x 4. values correct ii. What is the amplitude of the motion? [] 4 Amplitude is correct value iii. Find the acceleration of the particle is terms of x. [] v 8xx v 4 x x, then d v a dx d 4 x x dx x x correct acceleration

24 iv. Find the period of oscillation. [] x x So n, and period T n secs. correct value

25 Question 4: (a) Let ABPQC be a circle such that AB=AC, AP meets BC at X and AQ meets BC at Y, as shown below. Let BAP and ABC. i. Copy the diaram into your writin booklet, markin the information iven above, and state why AXC. [] i) Well done, althouh most students did it the lon way AXC is the external anle to ABX, which is equal to the opposite interior anles. correct reason ii. Prove BQP. [] Construction: join BQ, PQ. BQP BAP (anles in same sement on arc BP) correct reason iii. Prove BQA. [] BQA BCA (anles in same sement on arc AB) BCA ABC ( ABC isosceles, iven AB=BC) Hence BQA correct reason iv. Prove the quadrilateral PQYX is cyclic. [] AXC (from (i)) PQA BQP BQA (from (ii) and (iii)) use of previous parts correctly AXC PQA PQXY is a cyclic quadrilateral (external eq. opp. int. ) correct reason ii) Well done, althouh students who used anles subtended by the same arc should also write at the circumference. iii) This was poorly done. Only a handful of students knew the rule equal chords subtend equal anles ath the circumference. iv) Well done, but aain most students did it the lon way.

26 (b) Two yachts, A and B, subtend an anle of 60 at the base C of a cliff C. Very well done. Only a handful of students lost any marks. The most common errors were: 00 tan 0 AC AC 00 tan 0 or usin the incorrect tri ratio. From yacht A, the anle of elevation to point P, 00 m vertically above C, is 0. Yacht B is 600m from C. i. Calculate lenth AC. [] In ACP : 00 tan 0 AC 00 AC tan m (to nearest m) correct answer ii. Calculate the distance between the two yachts. [] In ABC : AB AC BC. AC. BC.cos 60 AC 600 AC 600, then usin calculator memory for AC, correct substitutions AB m (to nearest m) correct answer (c) A projectile, with initial speed V0 m/s, is fired at an anle of elevation from the oriin at O towards a taret T, which is movin away from O alon the x-axis. You may assume that the projectiles trajectory is defined by the equations x Vt cos y Vt sin t

27 where x and y are the horizontal and vertical displacements of the projectile in metres at time t seconds after firin, and where is the acceleration due to ravity. i. Show that the projectile is above the x-axis for a total of V sin seconds. [] The particle returns to the x-axis when y=0. Hence 0Vt sin t tvsin t, and so t=0 or V sin t 0, which leads to ( correct solvin of quadratic) t V sin V sin t, thus the particle is above the x-axis for V sin seconds as reqd. ii. Show that the horizontal rane of the projectile is V sin cos metres. [] The horizontal rane is is the value of x for t found in (i), i.e. V sin x V. cos V sincos as reqd. correct substitution iii. At the instant the projectile is fired, the taret T is d metres from O and it is movin away at a constant speed of u m/s. Suppose that the projectile hits the taret when fired at an anle of elevation. Show that [] d u V cos V sin For the taret, dx u dt, hence x udt ut c, and at t 0, x d, so c d, x ut d derives taret equation correctly The projectile therefore hits the taret after time V sin (from part i) when i) Well done. ii) Well done. iii) Mixed results. Many students didn t derive x ut d or it s equivalent. V sin cos x (from part ii). Thus, substitutin these values in ives: V sin cos Vsin u d substitutes correct values

28 V sincos Vusin d Vu sin V sincos d V sincos d u Vsin Vsin d V cos correct alebra to required V sin result iv. Suppose the projectile is fired at an anle of. Deduce that 4 the projectile will not hit the taret if [] V d u. V V sin If, then the maximum rane of the projectile is xmax reached 4 V V sin in time t The taret then must move beyond xmax in this same time, i.e. V ut d derives condition for a miss correctly Vsin V u d, and with sin sin 4 V V u d V V u d V u d. V V d as reqd. correct alebra to required result. V iv) Most students could substitute to arrive at 4 RHS of inequality.