2018 YEAR 6 PRELIMINARY EXAMINATION MATHEMATICS 9758/01

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RAFFLES INSTITUTION 018 YEAR 6 PRELIMINARY EXAMINATION MATHEMATICS 9758/01 Paper 1 Total Marks: 100 hours Additioal materials: Aswer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS FIRST Write

1 RAFFLES INSTITUTION 018 YEAR 6 PRELIMINARY EXAMINATION MATHEMATICS 9758/01 Paper 1 Total Marks: 100 hours Additioal materials: Aswer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS FIRST Write your ame ad CT group o all the work you had i. Write i dark blue or black pe o both sides of the paper. You may use a HB pecil for ay diagrams or graphs. Do ot use staples, paper clips, glue or correctio fluid. Aswer all the questios. Give o-exact umerical aswers correct to sigificat figures, or 1 decimal place i the case of agles i degrees, uless a differet level of accuracy is specified i the questio. You are expected to use a approved graphig calculator. Usupported aswers from a graphig calculator are allowed uless a questio specifically states otherwise. Where usupported aswers from a graphig calculator are ot allowed i a questio, you are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. You are remided of the eed for clear presetatio i your aswers. 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 or part questio. RI018 This documet cosists of 9 prited pages. RAFFLES INSTITUTION Mathematics Departmet [Tur over

2 1 Adult tickets for a parade are sold at three differet prices, depedig o the type of seats. Childre uder the age of 1 ad Seior Citizes aged 65 ad above ca ejoy 0% ad 10% off the adult ticket prices respectively. Those who are betwee 1 ad 64 years old (iclusive) will have to pay the full adult ticket price. The umber of tickets sold i each category for each group of people, together with the total cost of the tickets for each group, are give i the followig table. Adults (1 to 64 years old (iclusive)) Childre (uder 1 years old) Seior Citizes (65 years old ad above) Category 1 tickets (premier) Category tickets (sheltered) Category tickets (usheltered) Total cost $ $ $ Write dow ad solve equatios to fid the price of a adult ticket for each of the ticket categories. [4] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1

The regio eclosed by the circle C 1 is deoted by A 1 with area a 1 ad the regio betwee the circles C 1 ad C is deoted by A with area a for =,,4,..., N, as show i Fig.

3 Ade, a member of Raffles Art Club, is helpig to desig a graphic for the school s homecomig evet. The mai body of the graphic cosists of N cocetric circles C 1, C, C, K, CN with radii r,,, r r K, Nr, where r is a costat ad N is a eve iteger, as show i Fig. 1. CN 1 C 4 AN A4 A A A1 Fig. 1 Fig. The regio eclosed by the circle C 1 is deoted by A 1 with area a 1 ad the regio betwee the circles C 1 ad C is deoted by A with area a for =,,4,..., N, as show i Fig.. (i) Show that the sequece a1, a, a, K, an is a arithmetic progressio. [4] With the help of a graphic software, Ade fills A1, A, A, K, AN with two of the school colours: gree ad black. A will be filled gree if is odd ad will be filled black if is eve. Ade wishes to create a better visual effect by havig differet itesities of gree. He fills A with the gree colour that is the same as the school colour. He reduces the 1 itesity of this gree colour by te percet to fill A ; ad reduces the itesity of the gree colour used to fill A by te percet to fill A ; ad this process cotiues for all 5 odd values of. Whe he fiishes fillig up all the areas, Ade fids that the itesity of the gree colour that fills AN falls below oe quarter of the itesity of the school 1 colour for the first time. (ii) Fid the value of N. [4] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1 [Tur over

4 4 1 9x 4x (a) Fid dx, givig your aswer i the form p+ ql + rl 5 where p, q 0 9 4x ad r are ratioal umbers to be determied. [5] (b) Prove that si x= si x 4si x. Hece, or otherwise, fid si xsi xsixdx. [4] 1 4 (a) The diagram below shows the curve of y = f( x) where f( x ) is a polyomial. The 1 curve has a miimum poit at 1, 4 ad cuts the y-axis at (0,). The lies x = 1 ad x = are the vertical asymptotes ad the lie y = 0 is the horizotal asymptote to the curve. y O x Sketch o separate diagrams, the graphs of (i) y = f( x), [] (ii) y= f( x), [] labellig all relevat poit(s). H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1

5 5 4 (b) The diagram below shows the curve of y= g( x). The curve has a maximum poit at (, 9) ad a miimum poit at (, 1). The curve crosses the x-axis at (1, 0) ad (,0). The lie x = 0 is the vertical asymptote ad the lie y= x 4 is the oblique asymptote to the curve. y O x y = x 4 x = 0 (i) Sketch the graph of y = g'( x), labellig all relevat poit(s) ad statig the equatios of ay asymptotes. [] (ii) Fid the area bouded by the graph of y = g'( x), the lies x= 1, x= ad the x-axis. [] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1 [Tur over

6 6 5 C 15 cm D B y cm E 0 cm A x cm G F Tom has a rectagular piece of paper ACDF with legth 0 cm ad breadth 15 cm. He folds the lower left-had corer, A, to reach the rightmost edge of the paper at E. After that he will cut out the triagle EFG ad the trapezium BCDE to obtai a kite shaped figure ABEG with AB of legth y cm ad AG of legth x cm. (i) Fid the legth of EF i terms of x. [] (ii) Show that y = x 15. [] x 15 (iii) Usig differetiatio, fid the exact values of x ad y which give the miimum area of the kite ABEG. [5] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1

7 7 y 6 (a) It is give that e = ta x + π. (i) Show that d y y y = e + e. [] dx (ii) π Usig differetiatio, fid the Maclauri series for l ta x + up to ad icludig the term i x. [] (b) Let f ( x) = x 4x+ 5 ( 1+ x)( 1 x). A B C 1+ x 1 x 1 x where A, B ad C are costats to be determied. [] (i) Express f ( x ) i the form + + ( ) ( ) ( ) (ii) Hece fid the expasio of f ( x ) up to ad icludig the term i x 4. [] (iii) Write dow the coefficiet of r x i the expasio of ( ) f x i terms of r. [1] 7 Do ot use a calculator i aswerig this questio. (a) Fid the roots of the equatio w (1 i) + 4 w+ ( i) = 0, givig your aswers i cartesia form a+ ib. [] (b) It is give that z = + i. (i) Fid a exact expressio for z 5 i. Give your aswer i the form re θ, where r > 0 ad π < θ π. [] (ii) z Fid the three smallest positive whole umber values of for which i* z is purely imagiary. [4] (iii) p Give that 1+ = z * 7, fid exactly the possible values of the real umber p. [] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1 [Tur over

8 8 8 The fuctio ( ) ( ) 1 costat. f x = a 1+ x (i) Solve the iequality ( x) (ii) Show that the graph of y f ( x) is defied for 0 x 7, where a is a positive real f a 1. [] = has a egative gradiet at all poits o the graph. Fid the rage of f. [] Use a = i the rest of the questio. 1 (iii) Fid f ( x). [] 1 (iv) Sketch o the same diagram the graphs of y = f ( x) ad y f ( x) =, showig clearly the geometrical relatioship betwee the two graphs ad the lie y = x. [] (v) Show that the x-coordiate of the poit of itersectio of the graphs of y = f ( x) 1 ad y = f ( x) satisfies the equatio x 9x + 8x 6= 0. 1 Hece fid the solutio of the equatio f ( x) = f ( x). [] 9 (a) The variables x ad y are related by the differetial equatio d y x xe dx the substitutio u= x, show that y= k u ed u where k is a costat to be determied. Hece fid the geeral solutio of the differetial equatio. [5] (b) Ms Frugal bought a secod-had ove which has a broke timer. Each time she bakes she uses the timer o her hadphoe. Oe day, she decides to make a loaf of bread. She takes the fermeted dough out of her refrigerator ad checks that the iteral temperature of the dough is 4 C. She puts it ito the ove which has bee pre-heated to a costat temperature of 180 C ad forgets to set the timer o her hadphoe. She also does ot ote the time. At am, she realizes that the timer has ot bee set. She checks her dough which ow has a iteral temperature of 80 C. Twety miutes later, she checks the dough agai ad the iteral temperature has rise to 10 C. Newto s Law of Coolig states that the rate of icrease of the temperature θ C of a object after t miutes is directly proportioal to the differece i the temperatures of the object ad its surroudig. The dough has to be baked at a costat ove temperature of 180 C for oe ad a half hour to cook through. By formig a differetial equatio, fid the time (to the earest miute) that Ms Frugal should remove the dough from the ove. [8] H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1

9 9 10 Two childre Hasel ad Gretel are participatig i the juior category of the Festival of Lights competitio. They decided to set up a structure cosistig of a pyramid with a rectagular base OABC ad vertex V. The mai power supply switch is positioed at O 1,, 1 ad with coordiates ( 0, 0, 0 ) ad relative to O, the coordiates of A ad C are ( ) ( 1, 1, 1) respectively. The vertex V is fixed at a height 8 metres above the rectagular base OABC such that it is equidistat to the poits O, A, B ad C. All the surfaces of the pyramid, excludig the base OABC, are completely covered with LED (light emittig diodes) light strips so that it illumiates i the dark. (i) (ii) Poit D is the foot of the perpedicular from V to the rectagular base OABC. Show that the coordiates of D is ( 0,1.5, 0 ). [1] Fid a vector perpedicular to the rectagular base OABC. Hece, fid the positio vector of V, give that its k-compoet is positive. [4] Hasel ad Gretel decide to istall two differet colour display schemes o the pyramid ad the cotrol switch is to be istalled iside the pyramid at a poit E with positio vector αi j αk, where α is a costat. (iii) Show that VE is perpedicular to the rectagular base OABC ad explai why < α < 0. [] (iv) Fid the distace betwee E ad the surface OVA i terms of α, simplifyig your aswer. [4] (v) Give that the ratio of the distace betwee E ad the rectagular base OABC to the distace betwee E ad the surface OVA is 10 : 105, fid the value of α. [] ******* Ed of Paper ******* H MA 9758/018 RI Year 6 Prelimiary Examiatio Paper 1

CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01

CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01 CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ