Additional Math (4047) Paper 1(80 marks)

Transcription

1 Aitional Math (07) Prepare b Mr Ang, Nov 07 A curve is such that 6 8 an the point P(, 8) lies on the curve. The graient of the curve at P is. Fin the equation of the curve. [6] C where C is an integration constant C 6 8 C 8 When =,, C 8 C 8 D where D is an integration constant D 8 D When =, 8, D 8 D Topic : Paper (80 marks) Time : hours Name :

2 Aitional Math (07) (i) Sketch the graph of for 6. [] Fin the coorinates of the points of intersection of the curve an the line 7. [] (i) Consier 0 6, when 6 6, 6 6 O Let u, u, 7u 6u 0 7u u 0 u or u 7 7 or 9 or 7 8 The points of intersections are 9, 7 an, 8. Note b author: the graph mabe ifferent from the mark scheme. Prepare b Mr Ang, Nov 07

3 Aitional Math (07) The variables an are such that when values of are plotte against a straight line is obtaine. It is given that = 0. when = 0.0 an that = 0. when =.00. Fin the value of when = 9. [] Let m c m c an m c an m c Since c m, then 6 m. So c. We get The roots of the quaratic equation 7 0 are an. Fin a quaratic equation with roots an. [6] 7 7 an 7 an 7 an Prepare b Mr Ang, Nov 07

4 Aitional Math (07) (i) Show that sec cosec tan. [] sec cosec tan Hence fin, for 0, the value of in raians for which sec cosec sec cosec. [] (i) L.H.S. sec cosec sec cosec sin sec cosec sin sec cosec tan = R.H.S. tan sec cosec sec cosec tan tan tan tan 7 tan 7 tan 7 tan 7 or tan.7 or. ( s.f.) Prepare b Mr Ang, Nov 07

5 Aitional Math (07) 6 A tennis club makes three equall size tennis courts, positione net to each other as shown in the iagram below. Each tennis court is rectangular an has sies of length m an m. The lines in the iagram represent wire netting. The total length of wire netting use is 88 m. m (i) m Show that the total area, A m, of the three tennis courts is given b A 9 [] 6. Given that can var, fin the imensions of each tennis that make A a maimium. (You are not require to show that A is a maimum.) [] (i) 88 6 A A 9 A 6 A 6 9 A Let 0, the imensions of each tennis 6 m m Prepare b Mr Ang, Nov 07

6 Aitional Math (07) 7 The triangle ABC is such that its area is 9 cm, the length of AB is angle BAC is 60. Without using a calculator, fin cm an (i) the length, in cm, of AC in the form a b, where a an b are integers, [] an epression, in cm, for BC in the form a b, where c an are integers. [] (i) the height from C to AB AC sin 60 AC sin 60 9 AC The area of triangle ABC 9 AC AC AC AC 9 AC A 60 B C AC B cosine rule, BC BC BC cos Prepare b Mr Ang, Nov 07 6

7 Aitional Math (07) 8 (i) B using long ivision, ivie 7 9 b. [] Epress in partial fractions. [] (i) Let 6 A B C A 9 B C When When When, 6 A A 9 9 A, 0 B C B C (), 0 0 B C () + (), 6 C C B C () Substitute C into (), we get B Prepare b Mr Ang, Nov 07 7

8 Aitional Math (07) 9 A bungee jumper falls verticall from rest from a point O. Her velocit, v m/s, t secons v after leaving O, is such that 0. After s, she reaches a point X. B integration, fin t (i) her velocit at X, [] her istance OX. [] On reaching X, she then slows so that her velocit, V m/s, T secons after reaching X, is V such that 0 kt, where k is a constant. T 0 (iii) Given that she first comes to rest at a point Y when T =, show that k. [] 9 (i) v v t C t v 0 t C v 0 t C When t = 0, v 0, C 0 where C is an integration constant v 0t When t =, v 0. Let s be the istance from O, s s t D where D is an integration constant t s 0 tt D s t D When t = 0, s 0, D 0 s t When t =, s 80. OX 80 m Prepare b Mr Ang, Nov 07 8

9 Aitional Math (07) (iii) V V T E where E is an integration constant T V 0 kt T E V 0 T kt E When T = 0, V 0, therefore, 0 E V 0T kt 0 When T =, V 0, therefore, k 0 9 k 70 0 k 9 Prepare b Mr Ang, Nov 07 9

10 Aitional Math (07) 0 The iagram shows a circle O. Points A, B, C an D lie on the circle. The line AD is a iameter an BC is parallel to AD. The lines PBT an QDT are tangents to the circle at B an D respectivel. (i) Show that angle PBA = angle DAC. [] Show that angle CBT = 90 (angle PBA). [] (i) Since AD // BC, DAC ACB (alternate angles) PBA ACB (alternate segment theorem) or (tangent-chor theorem) Therefore, PBA DAC Prepare b Mr Ang, Nov 07 0

11 Aitional Math (07) X Let X be a point on the circle O. CBT BXC (alternate segment theorem) or (tangent-chor theorem) BOC BXC (angles at center is twice of angles at circumference) Therefore, BOC CBT Similarl, AOB ACB (angles at center is twice of angles at circumference) COD DAC (angles at center is twice of angles at circumference) Since AOD is a straight line, we have AOB BOC COD 80 ACB CBT DAC 80 CBT 90 ACB DAC From part (i), we get DAC ACB PBA, therefore CBT 90 PBA Prepare b Mr Ang, Nov 07

12 Aitional Math (07) The equation of a curve is. (i) Eplain, with working, wh the curve has no turning. [] The iagram below shows part of the curve the curve.. Points P(, ) an Q(, ) lie on b B epressing in the form a, where a an b are constants, fin, showing full working the area of the shae region. [] (i) Given that, Since, 0 0. for all other values of. Therefore, 0. As such 0, the curve has no turning point. Prepare b Mr Ang, Nov 07

13 Aitional Math (07) B long ivision, area of the shae region ln 8 ln 8 ln 8 ln 8 ln Prepare b Mr Ang, Nov 07

14 Aitional Math (07) The equation of normal to the circle at the point R is k. (i) Fin the value of the constant k. [] The normal to the circle at R meets the -ais at the point S. Given that R lies between S an the centre of the circle, fin the length of RS. [] (i) Centre of the circle (, ) The normal passess through the centre of the circle. Substitute, into k 0 k Prepare b Mr Ang, Nov 07

15 Aitional Math (07) When 0, 0 S, 0 Substitute into or Since R must lie in between an, rejects When, 0 8 R, 8. the length of RS = 0 7 units 8 8 Prepare b Mr Ang, Nov 07