MID-YEAR EXAMINATION 2018 H2 MATHEMATICS 9758/01. Paper 1 JUNE 2018

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1 MID-YEAR EXAMINATION 08 H MATHEMATICS 9758/0 Paper JUNE 08 Additioal Materials: Writig Paper, MF6 Duratio: hours DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO READ THESE INSTRUCTIONS FIRST Write i dark blue or black pe. DO NOT USE ERASABLE PEN. Write your idex umber / FIN umber o all the work you had i. Do ot use staples, paper clips, highlighter, glue or correctio fluid. Aswer all questios. Give o-exact umerical aswers correct to sigificat figures, or decimal place i the case of agles i degrees, uless a differet level of accuracy is specified i the questio. The use of a approved graphic calculator is allowed. Usupported aswers from a graphic calculator are allowed uless a questio specifically state otherwise. Where usupported aswers from a graphic calculator are ot allowed i a questio, you are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. Write all your aswers o separate papers provided. Start each questio o a FRESH NEW page. At the ed of the examiatio, faste all your work securely together. The umber of marks is give i brackets ( ) at the ed of each questio / part of a questio. The total marks for this paper is 00. This paper cosists of 7_ pages icludig the cover page. For Examier's Use Questio /5 Questio /7 Questio /7 Questio 4 /9 Questio 5 /8 Questio 6 /8 Questio 7 /9 Questio 8 /0 Questio 9 / Questio 0 / Questio / Total /00 NAME: SEAT NO: NRIC/FIN: BATCH: 9758/MYE/I/A/8 Page of 7

2 . A teddy bear plush toy is beig sold o differet olie stores Aazo, Busybuy ad Cold-. The delivery charges of Aazo ad Cold- are $5 ad $ respectively. The total price of the product sold o Aazo plus delivery charge is 0% more tha the price of the product sold o Busybuy. Durig a holiday period, the price of the plush toy after a 5% discout o Cold- plus delivery charge is the same as the price sold o Busybuy. Give that the product prices are each less tha $70 ad to the earest dollar, fid the prices of the product (to the earest itegers) o all olie stores. [5] ( ) ( ) + pi. The complex umber z is give by z =, where p < 0. + i (i) Give that z =, show that p =. [] (ii) π Hece show that arg ( z) =. [] (iii) Fid the smallest positive iteger such that z is purely imagiary ad z > 000. [] 4x. By usig a algebraic method, fid the solutio of the iequality x 4. x + [4] 4x + 4 Hece, solve the iequality x 5. x + [] 4. Fuctios f ad g are defied by 4 x f : x a, 4x + 0 x R, x g : x a π ta x +, 5 + x R. (i) Usig a algebraic method, fid the rage of f. [5] (ii) Show that g does ot have a iverse. [] (iii) Fid the rage of fg. [] 9758/MYE/I/A/8 Page of 7

3 5. (i) Verify that =. 4 ( ) ( + ) [] (ii) Hece fid. 4r [] r= (iii) Fid the value of. [] 4r r= (iv) By cosiderig 4r > 4r ad usig the result i part (ii), deduce that r for all positive itegers. [] r= < 6. The diagram shows the graph of y = f ( x) which has a vertical asymptote x = ad a horizotal asymptote y = 0. The curve crosses the x-axis at (, 0) ad has a maximum poit,. at ( ) Sketch, o separate diagrams, the graphs of (i) y = f ( x), [] (ii) (iii) y =, [] f ( x) f ( x) for x y = g( x), where g( x) =,showig clearly all relevat f ( x) for x > asymptotes, itercepts ad turig poit(s), where possible. [] 9758/MYE/I/A/8 Page of 7

4 7. Referred to the origi O, the poits A ad B have positio vectors a ad b respectively, such that a ad b are o-zero vectors that are ot parallel to each other. The poit C has positio vector λ a + µ b, where λ ad µ are o-zero costats. The poit D is such that ABCD is a parallelogram. a b (i) Give a geometrical iterpretatio of. [] a (ii) Fid the positio vector of D i terms of λ, µ, a ad b. [] (iii) The poit E lies o BD produced such that the ratio of BD : BE is : 5. Fid the positio vector of E i terms of λ, µ, a ad b. Hece, or otherwise, determie the area of the triagle BCE, givig your aswer i the form K a b, where K is a costat i terms of λ ad µ i its simplest form. [6] 8. The diagram shows a circle with equatio ( x ) + y = 4. (i) By usig the substitutio x = + siθ, fid the exact area bouded by the circle ad the y-axis for x 0. [6] (ii) The circle cuts the axes at the poits A ad B as show i the diagram. The regio bouded by the mior arc AB ad the lie segmet AB is rotated through 4 right agles about the x-axis. Fid the exact volume of the solid of revolutio. [4] 9758/MYE/I/A/8 Page 4 of 7

5 l + dy 9. (i) Give that y =, show that ( + x) + 6 y( + x) =. + dx By further differetiatio, fid the Maclauri series for y up to ad icludig the term i x. [5] (ii) Use your aswer i (i) to give a approximatio for 0. l + dx. [] l + (iii) Without the use of a calculator, fid dx i the form ( l 0 + a l ) where a ad b are itegers to be determied. [4] Hece, calculate the percetage error of the approximatio i (ii). [] 0. (a)a filter fuel is made from a thi flat sheet of metal i the shape of a sector of a circle with radius cm ad agle θ radias (see Diagram ). The two straight sides of the metal sector are the joied together, without overlap, to form a coe with height H cm ad radius R cm (see Diagram ). (i) Express R i terms of θ ad π. [] (ii) Hece, show that the volume, V, of the filter fuel is give by 4 584θ V = 4 ( 4π θ ). [] π (iii) Fid, i exact form, the value of θ such that the volume of the filter fuel is a maximum. You eed ot show that the volume is a maximum. [] [Volume of coe = π r h, where r is the base radius ad h is the height of the coe.] (b) A café prepares cosommé soup by pourig broth through the filter fuel ito a hemispherical bowl with radius 5 cm, as show i the figure above. 9758/MYE/I/A/8 Page 5 of 7

6 800 (i) Give that at time t miutes, soup fills the iitially empty bowl at a rate of 0t + cm per miute, fid the volume, S, of soup i the bowl after 4 miutes, leavig your aswer correct to decimal places. [] π (ii) The volume of soup i the bowl is give by S = ( 5 h h ), where h is the depth of the soup, as show i the diagram. Fid the rate of chage of the depth of the soup after 4 miutes. [4]. I a particular plaet ad moo orbit system, a moo A orbits aroud the plaet Regusti- Oe followig a ellipse path alog the x-y plae as show i diagram below. + z / 0,000 km Diagram Moo A Regusti-Oe O + y / 0,000 km + x / 0,000 km (a) Takig the cetre O of Regusti-Oe as the origi (0, 0), ad also makig use of the followig iformatio: Nearest distace from moo to cetre of Regusti-Oe = 0,000 km Furthest distace from moo to cetre of Regusti-Oe = 0,000 km (i) Determie the equatio of the path that the moo takes to orbit aroud Regusti-Oe. Leave your aswer i the form of y = P Qx, where P ad Q are real umbers betwee ad 0 to be determied. [] (ii) Hece, calculate the area of the plae surface map out by the moo after it completes oe cycle of the orbit, leave your aswer to s.f i km. [] (iii) Due to the o-uiform gravitatioal pull ad earby iter-plaetary attractive forces, after every 0 Earth years, the moo will shift slightly towards the positive y-axis by 00 km ad 00 km towards the egative x-axis. Determie the ew equatio of the path that the moo will move after 0 years, takig the path obtaied i part (i) as year zero. [] (iv) Usig part (iii) equatio, determie the distaces from the moo to Regusti Oe after 0 years whe the x-displacemet of the moo is +5,000 km. [] 9758/MYE/I/A/8 Page 6 of 7

7 (b) Regusti-Oe is actually oe of the two plaets i a biary-plaetary system. Its twi, Regusti-Two, is a plaet of idetical mass ad size, which is positio i the positive z axis from Regusti-Oe. It also has a moo B which orbits about a path parallel ad same size to the path maps out by the moo A of Regusti-Oe. See diagram. + z / 0,000 km Diagram Regusti-Two Moo B Lie l Distace betwee the cetres of Regusti-Oe ad Regusti-Two is 50,000 km. Regusti-Oe Moo A θ k j O + y / 0,000 km i + x / 0,000 km Takig i, j ad k as the uit vectors alog the x, y ad z axes o Regusti-Oe, ad at year zero, (i) Write dow the equatios of the plaes map out by the moos of Regusti-Oe ad Regusti-Two respectively i the scalar-product form. [] (ii) I diagram above, the two moos are at extreme opposite eds of their orbital paths. A dotted lie l is draw to coect the two moos. Determie the vector equatio of the lie. [] (iii) Usig part (b)(ii) aswer, determie the acute agle θ, betwee the lie ad the plae maps out by the moo of Regusti-Oe. [] Ed of Paper 9758/MYE/I/A/8 Page 7 of 7

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