Scholarship 2013 Calculus

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1 930Q 930 S Scholrship 013 Clculus.00 pm Mondy 18 Novemer 013 Time llowed: Three hours Totl mrks: 40 QUESTION BOOKLET There re six questions in this ooklet. Answer ANY FIVE questions. Write your nswers in Answer Booklet 930A. Pull out Formule nd Tles Booklet S CALCF from the centre of this ooklet. Show ALL working. Answers developed using CAS clcultor require ALL commnds to e shown. Correct nswers only will not e sufficient. Strt your nswer to ech question on new pge. Crefully numer ech question. Check tht this ooklet hs pges 8 in the correct order nd tht none of these pges is lnk. YOU MAY KEEP THIS BOOKLET AT THE END OF THE EXAMINATION. New Zelnd Qulifictions Authority, 013. All rights reserved. No prt of this puliction my e reproduced y ny mens without the prior permission of the New Zelnd Qulifictions Authority.

2 You hve three hours to complete this exmintion. This exmintion consists of SIX questions. Answer ny FIVE questions. QUESTION ONE (8 Mrks) Prince Rupert s drops re mde y dripping molten glss into cold wter. A typicl drop is shown in Figure 1. Figure 1: A seventeenth century drwing of typicl Prince Rupert s drop. Imge from The Art of Glss p 354, trnslted nd expnded from L Arte Vetrri (161) y Antonio Neri. A mthemticl model for drop s volume of revolution uses y φ( e x e x ) for x 0, nd is is shown in Figure, where φ is the golden rtio φ y (cm) x (cm) Figure : A mthemticl model for drop s volume of revolution. Where is the modelled drop widest, nd how wide is it there? () The drop chnges shpe ner B in Figure 1, where the concvity of the revolved function is zero. y Use d dx x x (e 6e + 4) φ to find theexct x coordinte of B. 4x y e (c) The volume formed y rotting curve y f (x) etween x nd x is given y y dx. Show tht the volume of the drop etween x 0 nd x ln( p) is V p 1 p Hence or otherwise, explin why the volume of the drop is never more thn some upper limit V L, no mtter how long its til..

3 3 QUESTION TWO (8 Mrks) This question defines four new terms relted to functions nd their definite integrls over specified intervl. 1 The innerproduct of twocontinuous functions f nd g over the intervl x is f, g f( x) gx ( )dx The norm of thecontinuous function f over the intervl x is f f, f 3 The ngleθ etween twofunctions f nd g over the intervl x is given y cosθ f f, g g where f 0, g 0 4 Two functions f nd g re orthogonl over the intervl x if thengle etween them is π Find the exct vlues of k for which f (x) kx + 1 nd g(x) x + k re orthogonl over 0 x 1. () Consider the functions p(x) 3x 4 nd q(x) 9x 5 over 0 x 1. Find the exct ngle etween the two functions. (c) For wht positive integers n nd m re sin (nx) nd sin (mx) orthogonl over 0 x π?

4 4 QUESTION THREE (8 Mrks) A function f is even if f ( x) f (x) for ll x in its domin. A function f is odd if f ( x) f (x) for ll x in its domin. (i) Recll tht polynomil is function in the form p(x) 0 x x n x n. Descrie which polynomils re even, nd which re odd, nd which re neither. (ii) Suppose tht g is ny even differentile function defined for ll rel numers (not necessrily polynomil). Use the limit definition of the derivtive to prove tht dg dx is n odd function. () Suppose y e x sin(kx), where k is non-zero constnt. 3 d y Find the vlues of k for which Cy, nd hence find the vlue of C. 3 dx

5 5 QUESTION FOUR (8 Mrks) Find ll the points which stisfy z n z, where z is complex numer, nd n is whole numer where n 9. How mny different solutions re there ltogether? () (i) The reltivistic rocket eqution is elow. m 0 m 1 1+ Δv c 1 Δv c c u Show tht this eqution rerrnges to Δv c tnh u c ln m 0 where the hyperolic tngent function is tnh( x) m 1 x e 1 x e + 1. (ii) The reltivistic rocket eqution is derived from the following differentil eqution, where u nd c re constnts. M dm v dv u 1 c Show tht ln M c 1+ v u ln c 1 v c is solution of this differentil eqution.

6 6 QUESTION FIVE (8 Mrks) A got is tethered with rope of length R, to point P, due west of the end of fence, which runs due north. The region of grss the got cn rech is shown in Figure 3. P θ R Figure 3: The re reched y got tethered ner the end of fence. The following eqution gives the re the got cn rech, A, s function of the ngle θ shown, π where 0 θ. π θ 1 A( ) R R R 4 π (1 cos ) 1 θ + θ + sinθcosθ Find the vlue of θ where A(θ ) is minimised. All working must e shown. () Answer ONE of the following options. EITHER Consider the following liner inequlities, where is positive constnt. x y 3 x + y 4 Drw non-liner ojective function P(x,y) for which there re exctly two points in the fesile region which mximise the ojective function. Justify your nswer crefully.

7 OR Figure 4 shows the tsks tht need to e completed in pinting two smll rooms. Estimted durtions for the tsks re shown in rckets (in minutes). 7 strt (0) uy mteril (30) move (40) prepre rooms (30) pint rooms (80) tidy rooms (30) pint dries (40) return (30) end (0) Figure 4: Flow chrt nd expected durtions (in minutes) of tsks for pinting two rooms. The pinters relise tht they could complete the jo fster y working on ech room seprtely, nd strt new pln, s shown in Figure 5. strt (0) uy mteril move move prepre prepre pint pint tidy pint dries tidy pint dries return return end (0) Figure 5: Incomplete flow chrt of tsks for pinting two rooms. Discuss how much fster the pinters might complete the tsk, nd ny rel-world limittions tht my restrict them. Figure 5 is reproduced on pge 7 of the nswer ooklet so tht you cn show ny working. Question Six strts on the following pge.

8 8 QUESTION SIX (8 Mrks) By considering the expnsion of (cis θ) 5, or otherwise, prove oth of the following identities. cos 5θ cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ sin 5θ 5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ () Answer ONE of the following options. EITHER A techer sets 99 homework questions for her Clculus clss ech week, of three different types: esy, difficult, nd impossile. The numer of questions of ech type, given in week n, re represented y x n, y n, nd z n respectively. The techer uses the following system of liner equtions to vry the numer of questions of ech type given ech week. x n x n + 0.7y n + 0.6z n y n x n + 0.y n + 0.4z n z n x n + 0.1y n Her clss notice tht the numer of questions of ech type stilises fter severl weeks. Tht is, in the long run they notice tht x n + 1 x n, y n + 1 y n, nd z n + 1 z n. 930Q How mny questions of ech type will the techer give ech week once the numers stilise? OR Use your knowledge of ellipses to sketch ll points in the complex plne stisfying the following inequlities, where k is positive constnt. k z + i + z i k

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