Mathematics Extension 1

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1 BAULKHAM HILLS HIGH SCHOOL TRIAL 04 YEAR TASK 4 Mathematics Etension General Instructions Reading time 5 minutes Working time 0 minutes Write using black or blue pen Board-approved calculators may be used Show all necessary working in Questions -4 Marks may be deducted for careless or badly arranged work Total marks 70 Eam consists of pages. This paper consists of TWO sections. Section Page -4 (0 marks) Questions -0 Attempt Question -0 Section II Pages 5-0 (60 marks) Attempt questions -4 Table of Standard Integrals is on page

2 Section I - 0 marks Use the multiple choice answer sheet for question -0. Given the equation 0, what is the value of given that 3.6 and 5. (A) 0.77 (B) 0.04 (C) 0.04 (D) A Not to scale P Q T 4 8 In the diagram above, is a tangent and is a chord produced to. The value of is (A) (B) 3 (C) 4 (D) How many distinct permutations of the letter of the word "" are possible in a straight line when the word begins and ends with the letter (A) (B) 80 (C) 360 (D) 70

3 4. The coordinates of the point that divides the interval joining 7,5 and, 7 eternally in the ratio : 3 are (A) 0,8 (B) 0, (C),8 (D), 5. What is the domain and range of cos? (A) :, : 0 (B) :, : 0 (C) :, :0 (D) :, :0 6. Which of the following is the general solution of 3tan 0, where is an integer? (A) (B) (C) (D) 7. The displacement of a particle moving in simple harmonic motion is given by 3cos where is the time in seconds. The period of oscillation is: (A) (B) (C) (D) 3 3

4 8. and are parallel chords in a circle, which are 0cm apart., 4cm and cm. A R B O Find the diameter of the circle to decimal place (A) 4.4cm (B) 8.cm (C) 4.8cm (D) 6.5cm C T D 9. The domain of log 45 is (A) 45 (B) 4, 5 (C) 45 (D) 4,5 0. Which of the following represents the derivate of sin? (A) (B) (C) (D) End of Section 4

5 Section II Etended Response All necessary working should be shown in every question. Question (5 marks) - Start on the appropriate page in your answer booklet Marks a) Evaluate cos 4 3 b) Find log, using the substitution log c) Prove the identity sincos cos sin cos d) Solve for e) (i) Show that a root of the continuous function ln lies between 0.8 and 0.9. (ii) Hence use the halving the interval method to find the value of the root correct to decimal place. f) i Find tan tan ii Hence sketch tan tan for End of Question 5

6 Question (5 marks) - Start on the appropriate page in your answer booklet Marks a) When a polynomial is divided by 4 the remainder is 3. What is the remainder when is divided by b) In the given diagram, and are tangents and,, are collinear. 3 P Not to scale Q T R S Copy or trace the diagram in to your writing booklet. Prove that the points,,, are concyclic. 6

7 Question (continued) Question continues on the following page c) y P = 4ay (ap,ap ) M Not to Scale Q(aq,aq ) O Points P, and, lies on the parabola 4. The chord subtends a right angle at the origin. (i) (ii) Prove 4 Find the equation of the locus of, the midpoint of. 3 d) Find the coefficient of in the epression of e) Prove by mathematical induction for positive integers 3 3 End of Question 7

8 Question 3 (5 marks) - Start on the appropriate page in your answer booklet Marks a) (i) Epress 3 sin cos in the form sin where 0 and 0. (ii) Hence state the least value of 3 sin cos and the smallest positive value of for this least value to occur. b) In the cubic equation the sum of the roots is equal to twice their product. Find the values of. 3 c) Find the number of arrangements of the letters of the word if there are 3 letters between and. d) Below is the graph of a function y - 3 Copy the diagram in your booklet, and on the same set of aes sketch a possible graph for. e) It is estimated that the rate of increase in the population of a particular species of bird is given by the equation where and are positive constants. (i) (ii) Verify that for any positive constant, the epression satisfies the above differential equation. What can be deduced about as increases? 3 End of Question 3 8

9 Question 4 (5 marks) - Start on the appropriate page in your answer booklet a) 30cm Not to Scale 30cm hcm Water is poured into a conical vessel at a constant rate of 4cm /s. The depth of water is cm at any time seconds. (i) Show that the volume of water is given by. (ii) (iii) Find the rate at which the depth of water is increasing when 6cm. Hence find that rate of increase of the area of surface of the liquid when 6. b) The acceleration of a particle is given by the equation 8, where is the displacement in centimetres from a fied point, after seconds. Initially the particle is moving from with speed cm/s in a negative direction. (i) (ii) (iii) Prove the general result. Hence show that the speed is given by cm/s. Find an epression for in terms of. Question 4 continues on the following page 9

10 Question 4 (continued) c) A projectile is fired from the origin with velocity with an angle of elevation, where. YOU MAY ASSUME cos, sin Where and y are the horizontal and vertical displacements from, seconds after firing (i) Show the equation of flight can be epressed as tan 4 tan where (ii) Show that a point, can be hit by firing at different angles and provided 4. (iv) Show that no point above the -ais can be hit by firing at different angles and satisfying both and. End of Paper. 0

11 STANDARD INTEGRALS n n d, n n ; 0, if n 0 d ln, 0 a e a d e, a 0 a cos ad sin a, a 0 a sin ad cos a, a 0 a sec ad tan a, a 0 a sec a tan ad sec a, a 0 a a d tan, a 0 a a a d sin a, a 0, a a a d ln( a ), a 0 a d ln( a ) NOTE: ln log, 0 e

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