MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2

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1 Class Inde Number Name MRIS STELL HIGH SCHOOL PRELIMINRY EXMINTION DDITIONL MTHEMTICS 406/0 8 September 008 Paper hours 0minutes dditional Materials: nswer Paper (6 Sheets RED THESE INSTRUCTIONS FIRST Write your Centre number, inde number and name on all the work you hand in Write in dark blue or black pen You may use a pencil for any diagrams or graphs Do not use staples, paper clips, highlighters, glue or correction fluid nswer all questions Write your answers on the separate nswer Paper provided Give non-eact numerical answers correct to significant figures, or decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question The use of a scientific calculator is epected, where appropriate You are reminded of the need for clear presentation in your answers t the end of the eamination, fasten all your work securely together The number of marks is given in brackets [ ] at the end of each question or part question The total number of marks for this paper is 00 For Eaminer s Use This document consists of 6 printed pages Turn over

2 Mathematical Formulae Quadratic Equation LGER For the equation a + b + c 0, b ± b 4ac a inomial epansion n n n r n n n n n r r n ( a + b a + a b + a b + L + a b + L + b, where n is a positive integer and n n! n( n ( n r + r ( n r! r! r! Identities TRIGONOMETRY sin + cos cos cos sec + tan cosec + cot sin( ± sin cos ± cos sin cos( ± cos cos m sin sin tan ± tan tan( ± m tan tan sin sin cos sin tan cos tan tan sin sin + sin sin ( + cos ( sin sin cos ( + sin ( cos + cos cos ( + cos ( cos cos sin ( + sin ( Formulae for C a sin b sin c sin C a b + c bc cos bcsin

3 nswer all the questions (a Given that f ( d 4 and f ( d 7, find the value of, where 6 6 [ f ( + ] d (b Differentiate with respect to, and hence evaluate ( d 0 [5] ( The diagram shows a rectangular garden with a perimeter of 7mThe angle, θ, subtended by the diagonal and the length of the garden is such that tan θ Using variables and y to represent the length and breadth of the garden respectively, θ y (i formulate two algebraic equations [] (ii Hence, use the inverse matri method to find the length and breadth of the rectangular garden [5] Epress sin 5cos in the form R sin( α where R is a positive constant and α is acute Hence, or otherwise find the maimum and minimum values of (sin 5cos [8] 4 (a The variables P and V are related by the equation PV 00 Given that V is increasing at a rate of 4 units per second, calculate the rate of change of P at the instant when V 5 units (b If cos( sin( C, epress cot in terms of and C 5 Nicholas wants to construct a closed right cylindrical can of base radius, r cm and height, h cm with a thin sheet of material of area, 600π cm Determine the value of r and h, in order that Nicholas may construct a cylindrical can with the greatest possible volume [7]

4 4 6 The diagram, not drawn to scale, shows C and CD overlapping to form another triangle, CM The coordinates of C are (9,, (8,8 and C(,9 C(,9 (8,8 Given that M is the perpendicular bisector of C and MDM (9, O D Find (i the coordinates of M, [] (ii the equation of M, [] (iii the eact length of MD [] (iv Given that C and that the area of CM is k cm, show that area of the pentagon CDM is 4k cm [] 7 In the figure, tangent M touches the circle at T RE, RCT and RDM are straight lines 4 RD DM and RD RE 5 ED is parallel to M (i State any pairs of similar triangles, [] (ii find the value of rea of ET rea of RT, [] (iii show that 5( ER TM ( + cot 8 (a Prove the identity sec cot (b Find all the angles,, between 0 and 80, for which cos + cos 0 [5]

5 5 9 John left a cup of chocolate drink on a table to cool so that, t minutes later, its t temperature, T C, is given by T (09 Find (i its initial temperature, [] (ii its temperature after 8 minutes, [] (iii the time elapsed, to the nearest minute, when its temperature is 5 C, [] (iv the average rate of change of the temperature after the cup has stood for 0 minutes [] 0 particle moves in a straight line so that, t seconds after leaving a fied point O, its velocity, v ms -, is given by v t 8t + 5 Calculate (i the distance from O at which the particle first comes to an instantaneous rest, [6] (ii the velocity of the particle when the acceleration is 4 m/s, [] (iii the value of t when it first return to O, [] (iv the distance traveled during the third second [] (a Sketch the curve y e Shade a region, bounded by the curve, the line y ln8 and the y-ais Find the area of the region [6] (b Find the largest integer, k for which the quadratic equation k + (8 4k + k + 0 has no real roots [5]

Name: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination 3 Secondary 4

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