Scholarship 2014 Calculus

Transcription

1 93202Q S Scholarship 2014 Calculus 9.30 am Wednesda 19 November 2014 Time allowed: Three hours Total marks: 40 QUESTION BOOKLET There are five questions in this booklet. Answer ALL FIVE questions, choosing ONE option from part of each question. Write our answers in Answer Booklet 93202A. Pull out Formulae and Tables Booklet S CALCF from the centre of this booklet. Show ALL working. Answers developed using a CAS calculator require ALL commands to be shown. Correct answers onl will not be sufficient. Start our answer to each question on a new page. Carefull number each question. Check that this booklet has pages 2 7 in the correct order and that none of these pages is blank. YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION. New Zealand Qualifications Authorit, All rights reserved. No part of this publication ma be reproduced b an means without the prior permission of the New Zealand Qualifications Authorit.

2 2 This eamination consists of FIVE questions. Answer all FIVE questions, choosing ONE option from part of each question. QUESTION ONE (8 Marks) ( ) t (a) Find all solutions of t+5 + ( ) t2 5t+5 = 14. Give the solutions in eact form. ( ) Find all solutions of ln sin 1 e 5 = ln sin 1 e ( ), where is real. Give the solutions in eact form. OR A hospital has 150 rooms that it intends to upgrade into double or triple rooms. The hospital can spend no more than $ in upgrading the rooms. The cost to upgrade each double room is $6400, and the cost to upgrade each triple room is $8000. There are at least 4 requests for double rooms for ever 3 requests for triple rooms, and the hospital decided to consider this in its planning. The dail income on a double room is $1200 per room, and the dail income on a triple room is $1800 per room. Find the combination of double and triple rooms that maimises the potential dail income, assuming total occupanc of the beds. Comment on an limitation(s) with our answer, and suggest a wa to resolve the matter(s).

3 3 QUESTION TWO (8 Marks) (a) Consider the equation 4 2k 2 + q 2 = 0, where k and q are real numbers. Describe how the nature of the roots of this equation depends on k and q. Your answer should cover comple and repeated roots. Find eact epressions for the areas of the three labelled regions bounded b the two curves = 9 cosec 2 and = 16 sin 2 π between = = 6 and 5π.shown in Figure A B C Figure 1: The curves = 9cosec 2 and = 16 sin 2 π between = = 6 and 5π 6. OR The following sstem of linear equations with variables, and z contains a constant c. (c + 2) + (c 2)z = 0 (c + 3) + (c 3) = 0 (c + 5) + (c 5)z = 0 Investigate how the solutions of the linear sstem depend on the value of c.

4 4 QUESTION THREE (8 Marks) (a) A narrow rod 12 cm long is made of a material of varing densit. The densit of the rod at a point cm from one end A of the rod is given b ρ() = b r (12 ) in the interval 0 12, where b and r are positive constants. The centre of gravit of the rod is at the point c cm from A such that ρ()( c)d = 0. Find the centre of gravit of the rod, in terms of r A famil of functions is built from two functions f () and g (), with a new function h p () defined for each value of p, 0 p 1. These functions are shown in Figure 2. f () = 2 + sin g () = 26 + sin h p () = [ f ()] 1 p [ g ()] p The function S (p) represents the difference between the maimum and minimum values of h p (). Find the eact value of p that maimises S (p). Note that if a is a constant, d d (a ) = lna a. p = 0.8 p = Figure 2: Graphs of = f (), = g (), and = h p () as dashed lines with various values of p from 0.1 to 0.9.

5 OR 5 A to developer is designing a new product to be manufactured in its factor. The design team has identified 11 activities and their precedence relationships. The project needs to be completed in 38 weeks. Find the critical path, crash the project to fit the 38-week deadline, state the changes that ou make and their effect, and find the most economical crashing cost to the clients. Note: a network diagram with project duration times is in our answer booklet on pages 26 and 27. Table 1: Activit precedence and duration details for new product manufacture Activit Description Precedence Duration (weeks) A Design the product 4 B Design the manufacturing process A 7 C Purchase materials A 3 D Purchase manufacturing equipment B 5 E Install manufacturing equipment D 15 F Receive materials C 6 G Pilot production run E, F 2 H Evaluate product design G 2 I Evaluate manufacturing process performance G 3 J Obtain client approval H, I 4 K Make alterations and commence manufacturing J 3 Activit Duration (weeks) Table 2: Activit crash costs for new product manufacture Normal cost ($) Crash time (weeks) Crash cost ($) Ma. weeks of reduction Increased cost per week saved ($) A B C D E F G H I J K

6 6 QUESTION FOUR (8 Marks) (a) Find the eact values of for which cis( 2 ) = cis, between π and π. Figure 3 ma be useful cos( 2 ) cos() sin( 2 ) sin() Figure 3: Graphs of the real and imaginar parts of cis( 2 ) and cis with π π. The rule for the derivative of a triple product is given below. ( f gh)' = f 'gh + f g'h + f gh' Using this rule, or otherwise, find the values of the coefficients A to J in the following rule for the third derivative of a triple product. OR (uvw)''' = Au''' vw + Buv''' w + Cuvw''' + Du''v'w + Eu''vw' + Fu'v''w + Guv''w' + Hu'vw'' + Iuv'w'' + Ju'v'w' There are man integer solutions to the equation n r = n + 1 r 1, including n = r = 1. Find an epression for n in terms of r, and hence find another of the integer solutions.

7 7 QUESTION FIVE (8 Marks) 2 d (a) Show that = e ec 2 is a solution of the differential equation = c ln (1+ ln ). 2 d Consider the function F(, ) = ( A)( 1 + A) 1,where0< A< 1. The regions where F(, ) 0 for various values of A are shaded in Figure 4. Describe how the boundaries of the shaded regions arise from the solution of F(, ) = 0, and hence, or otherwise, find the eact values of A for which the shaded region is half the area of the circle A = A = A = 0.75 Figure 4: Regions showing the positive values of F(, ) shaded gre, for various values of A. OR 2 2 An ellipse with equation + = 1 is enclosed b the hperbolas given b = 1 and = a b Determine the largest area of an ellipse enclosed b the hperbolas. 2 2 Note that the area of the ellipse + = 1is A= π ab. 2 2 a b

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