Student Number ABBOTSLEIGH AUGUST 007 YEAR ASSESSMENT 4 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes. Working time 3 hours. Write using blue or black pen. Board-approved calculators ma be used. A table of standard integrals is provided. All necessar working should be shown in ever question. Total marks 0 Attempt Questions -8. All questions are of equal value. Answer each question in a new booklet. P/Yr Et Maths Trial 8.07
Outcomes assessed HSC course E E E3 E4 E5 E6 E7 E8 E9 appreciates the creativit, power and usefulness of mathematics to solve a broad range of problems chooses appropriate strategies to construct arguments and proofs in both concrete and abstract settings uses the relationship between algebraic and geometric representations of comple numbers and of conic sections uses efficient techniques for the algebraic manipulation required in dealing with questions such as those involving conic sections and polnomials uses ideas and techniques from calculus to solve problems in mechanics involving resolution of forces and resisted motion combines the ideas of algebra and calculus to determine the important features of the graphs of a wide variet of functions uses the techniques of slicing and clindrical shells to determine volumes applies further techniques of integration, including partial fractions, integration b parts and recurrence formulae, to problems communicates abstract ideas and relationships using appropriate notation and logical argument Harder applications of the Etension Mathematics course are included in this course. Thus the Outcomes from the Etension Mathematics course are included. From the Etension Mathematics Course Preliminar course PE appreciates the role of mathematics in the solution of practical problems PE uses multi-step deductive reasoning in a variet of contets PE3 solves problems involving inequalities, polnomials, circle geometr and parametric representations PE4 uses the parametric representation together with differentiation to identif geometric properties of parabolas PE5 determines derivatives that require the application of more than one rule of differentiation PE6 makes comprehensive use of mathematical language, diagrams and notation for communicating in a wide variet of situations HSC course HE appreciates interrelationships between ideas drawn from different areas of mathematics HE uses inductive reasoning in the construction of proofs HE3 uses a variet of strategies to investigate mathematical models of situations involving projectiles, simple harmonic motion or eponential growth and deca HE4 uses the relationship between functions, inverse functions and their derivatives HE5 applies the chain rule to problems including those involving velocit and acceleration as functions of displacement HE6 determines integrals b reduction to a standard form through a given substitution HE7 evaluates mathematical solutions to problems and communicates them in an appropriate form P/Yr Et Maths Trial 8.07
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. Find 3 sin d. (b) (i) Epress 3 in the form a b c. ( )( ) 3 Hence find. ( )( ) (c) Use the substitution sin, or otherwise, to evaluate 3 d. 3 4 (d) Find 3 d. 3 (e) Evaluate tan d. 3 0 P/Yr Et Maths Trial 8.07 3
QUESTION (5 marks) f ( ) 0 3 4 The diagram above is a sketch of the function f ( ). On separate diagrams sketch: (i) ( f ( )) f ( ) (iii) ln f ( ) (iv) f ( ) (b) (i) If f '( ) and f () 0, find f ''( ) and f ( ). 3 Eplain wh the graph of f ( ) has onl one turning point and find the value of the function at that point, stating whether it is a maimum or a minimum value. (iii) Show that f (4) and f (5) have opposite signs and draw a sketch of f ( ). P/Yr Et Maths Trial 8.07 4
QUESTION 3 (5 marks) Epress ( 3 ) 8 i in the form i. 3 (b) On an Argand diagram, sketch the region where the inequalities 3 z 3 and arg ( z ) both hold. 3 6 (c) Show that sin i cos sin i cos. 3 sin i cos (d) (i) Epress i z in modulus-argument form. 3 i Hence evaluate 7 cos in surd form. (e) The Argand diagram below shows the points A and B which represent the comple numbers z and z respectivel. B A O Given that BOA is a right-angled isosceles triangle, show that ( z z ) z z. P/Yr Et Maths Trial 8.07 5
QUESTION 4 (5 marks) If z i is a root of the equation z 3 pz qz 6 0 where p and q are real, 3 find p and q. (b) Show that if the polnomial 3 f ( ) p q has a multiple root, then 3 4 p 7q 0. 3 (c) The base of a solid is the region in the first quadrant bounded b the curve sin, 3 the -ais and the line. Find the volume of the solid if ever cross-section perpendicular to the base and the ais is a square. (d) (i) Find the five roots of the equation z 5. Give the roots in modulus-argument form. Show that 5 z can be factorised in the form : 4 ( )( cos )( cos ) 5 5 5 z z z z z z (iii) Hence show that 4 cos cos. 5 5 P/Yr Et Maths Trial 8.07 6
QUESTION 5 (5 marks) The ellipse ( ) is rotated about the ais. 4 Use the method of slicing to find the volume of the solid formed b the rotation. 4 (b) In the triangle ABC, AD is the perpendicular from A to BC. E is an point on AD and the circle drawn with AE as diameter cuts AC at F and AB at G. 4 B G E D A F C Prove B, G, F and C are concclic. (c) The diagram below shows the part of the circle a in the first quadrant. L M O (i) If the horizontal line LM through L(0, b ), where 0 b a, divides the area between the curve and the coordinates aes into two equal parts, show that sin b b a b. 3 a a 4 If the radius of the circle is unit, show that b can be found b solving the equation sin, where sin b. 3 (iii) Without attempting to solve the equation, how could (and hence b ) be approimated? P/Yr Et Maths Trial 8.07 7
QUESTION 6 (5 marks) An ellipse has equation the ellipse. with vertices A(,0) and A'(,0). P is a point (, ) 4 3 on (i) Find its eccentricit, coordinates of its foci, S and S ', and the equations of its directrices. 3 Prove that the sum of the distances SP and S ' P is independent of the position of P. (iii) Show that the equation of the tangent to the ellipse at P is. 4 3 (iv) The tangent at P(, ) meets the directri at T. Prove that angle PST is a right angle. 3 (b) If a b c, (i) Prove a b ab. Prove 9. a b c (iii) Prove ( a )( b )( c ) 8abc. P/Yr Et Maths Trial 8.07 8
QUESTION 7 (5 marks) c The point T( ct, ) lies on the hperbola t c. The tangent at T meets the ais at P and the ais at Q. The normal at T meets the line at R. Q R T not to scale P c You ma assume that the tangent at T has equation t ct. (i) Find the coordinates of P and Q. Find the equation of the normal at T. (iii) Show that the coordinate of R is c t ( t ). (iv) Prove that PQR is isosceles. 3 (b) (i) If I n d prove that I n ( n 3) I n n ( n ) ( ). 4 n Hence evaluate d. 0 P/Yr Et Maths Trial 8.07 9
QUESTION 8 (5 marks) A plane of mass M kg on landing, eperiences a variable resistive force due to air resistance of magnitude Bv newtons, where v is the speed of the plane. That is, M Bv. (i) Show that the distance ( D ) travelled in slowing the plane from speed V to speed U under the effect of air resistance onl, is given b: 4 M V D ln( ) B U After the brakes are applied, the plane eperiences a constant resistive force of A Newtons (due to brakes) as well as a variable resistive force, Bv. That is, M ( A Bv ). After the brakes are applied when the plane is travelling at speed U, show that the distance D required to come to rest is given b: 4 M B D ln U B A. (iii) Use the above information to estimate the total stopping distance after landing, for a 00 tonne plane if it slows from 90m/s to 60 m/s under a resistive force of 5v Newtons and is finall brought to rest with the assistance of a constant braking force of magnitude 75 000 Newtons. (b) n A B 0 3 n n n The diagram above represents the curve n sin, 0 n, where n is an integer n. n The points O(0,0), A( n, n) and B( n,0) lie on this curve. (i) B considering the areas of the lower rectangles of width from 0 to n, prove that 3 ( n ) n sin sin sin... sin. 3 n n n n Hence or otherwise, eplain wh n r n n sin. n r END OF PAPER P/Yr Et Maths Trial 8.07 0