Prelim Examination 2015/2016 (Assessing all 3 Units) MATHEMATICS. CFE Advanced Higher Grade. Time allowed - 3 hours

Transcription

1 Prelim Eamination /6 (Assessing all Units) MATHEMATICS CFE Advanced Higher Grade Time allowed - hours Total marks Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Write your answers clearly in the answer booklet provided. In the answer booklet, you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the eamination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. Pegasys

2 FORMULAE LIST Standard derivatives Standard integrals f () f () sin cos tan tan sec ln, > f () a a sec a e a sin f ( ) d c a tan a tan a e a c a a c a c e e Summations a r r (Arithmetic series) S n na n n (Geometric series) S n d n r r n n n n nn n n n, r r 6, r r Binomial theorem n n n nr r n n n! a b a b where Cr r r r r!n r! Maclaurin epansion iv f () f () f () f ( ) f () f ()...!!! De Moivre s theorem n n rcos isin r cos n isin n Vector product i j k a a a a a ab = a b sinnˆ a a a i j k b b b b b b b b a b Pegasys

3 Answer all the questions.. Find the term in a in the epansion of a. a 6. Given the equation z + i = 8 + 7i, epress z in the form a + ib.. (a) Differentiate and simplify tan, where <. sin (b) Use the substitution u cos to evaluate d (cos ). (a) Prove that is irrational. (b) Consider the following two statements S and T: S: If p and q are two odd prime numbers then p + q is not prime. T: If p and q are two odd prime numbers then p q is not prime. For each of S and T, give a proof if it is true, or give a counter eample if it is false.. Given that i is a root of the equation z 6z 7z 8, find the other roots. 6. The radius of a sphere is increasing at a rate of cm/s. Find, in terms of, the rate at which the volume of the sphere is increasing when the radius is cm. [You may assume that the volume of a sphere is given by: V r.]

4 7. (a) Write down the matri R representing reflection in the line y. (b) Write down the matri S representing an anticlockwise rotation of 9 about the origin. 8. A curve is defined by the parametric equations 8t y t 7t for all t. d y Find the coordinates of the stationary points of this curve and, by considering, d determine their nature Let: A = and B = Show that AB = ki for some constant k, where I is the identity matri. Hence obtain: i) The inverse matri A - ii) The matri A B y. A function is defined implicitly by e lny. Find, in terms of and y, a formula for. d

5 . Solve the differential equation: + y = given that y = and =, when =.. The function f is defined by a b c 6 f where a, b and c are constants. It is known that the graph of f passes through the point (, 7) and has a stationary point at (-, 7). (a) Deduce that a, b and c must satisfy the system of equations a + b + c = a b + c = a b + c = -. (b) Use Gaussian elimination to find the values of a, b and c.. (a) Show that cos sin. coseccosec (b) Use integration by parts to show that cos sin d sin c. (c) Hence, or otherwise, find the particular solution of the differential equation d e cosec ycosecy, given that y when. 6. Evaluate d. 8 ( )

6 . Water is being heated in a kettle. At time t seconds, the temperature of the water is T C. The rate of increase of the temperature of the water at any time t is given by the differential equation dt dt k T, T where k is a positive constant. (a) Given that T when t, show that T e kt. (b) When the temperature of the water reaches C, the kettle switches off. Given that k.9, find the time, to the nearest second, when the kettle switches off. 6. Let z cos isin. (a) Use de Moivre s theorem to epress z in terms of. (b) (c) Use the binomial theorem to epress Hence show that z in terms of sin and cos. (i) cos 6cos cos cos (ii) sin 6sin sin sin. (d) Use your answers to (c)(i) and (c)(ii) to show that tan tan cot. tan tan tan [END OF QUESTION PAPER]

7 Marking Scheme CFE Advanced Higher Grade /6 Prelim (Assessing all Units) Give one mark for each ans: 8a marks finds correct general term simplifies to find correct epression for power of a solves for r correctly finds correct term Illustrations for awarding each mark 6 6r a r a r r 8a a r (a) ans: marks differentiates tan continues correctly simplifies correctly correctly (b) ans: 7 marks starts correctly substitutes correctly integrates correctly du sind du u u Pegasys

8 substitutes correctly correct answer 7 (a) (b) ans: i ; marks starts correctly correct root correct root i i Pegasys

9 Give one mark for each 6 ans: cm /s marks starts correctly continues correctly continues correctly correct answer Illustrations for awarding each mark dv dv dt dr r... r dr dt 7(a) 7(b) ans: R mark correct answer ans: S mark correct answer 8 Give one mark for each, MaimumT. P.& ans:, MinimumT. P. 7 marks Illustrations for awarding each mark find correct first derivative solves for t correctly correct coordinates correct coordinates finds second derivative correctly correct nature correct nature d 7 t 8 t d,, d y t (or equivalent) d d y, < MaimumT.P. d d y, > MinimumT.P. d Pegasys

10 9 Give one mark for each Illustrations for awarding each mark ans: d e y e y y marks starts correctly continues correctly differentiates correctly correct answer d d d... y y d y e y... e e y d y y d e y e y y y y d (or equivalent) Pegasys

11 Pegasys Give one mark for each Illustrations for awarding each mark (a) ans: Proof marks obtains first equation differentiates correctly obtains second equation obtains third equation a + b + c = c b a f ) ( a b + c = a b + c = - (b) ans:,, c b a marks correct augmented matri correct first modified system correct second modified system correct third modified system correct solution,, c b a

12 (a) ans: Proof mark proves correctly (b) ans: Proof marks starts correctly completes proof sin sin coseccosec sin sin.sin sin cos cos sin. cos sin... sin cos sin cos sin d sin sin cos d sin d sin cos d d sin c (c) ans: y sin e marks 8 rearranges & starts to integrate substitutes & integrates correctly substitutes to find correct constant correct particular solution cos ecy cos ecy cos y sin y e d sin e y e, : sin e 6 6 y sin e 8 d c c c 8 ans: ln 8 marks starts correctly continues correctly a correct constant all constants correct substitutes correctly integrates correctly substitutes correctly correct answer A B C D A A, B, C ord A, B, C & D B C D d ln ln ln ln Pegasys

13 Give one mark for each (a) ans: Proof marks separate the variables & know to integrate integrate correctly substitute correctly to find correct constant completes proof Illustrations for awarding each mark dt kdt t ln T kt c T kt Ae Ae A (or equivalent) kt T... T e kt e (b) ans: 9 seconds marks substitutes correctly correct answer.9t e 9 seconds Pegasys

14 Give one mark for each 6(a) ans: z cos isin mark Illustrations for awarding each mark correct answer cos isin 6(b) ans: z cos i cos sin cos sin i cos sin cos sin isin epands correctly simplifies correctly marks cos cos i sin cos i sin cos i sin cos i sin i sin cos i cos sin cos sin i cos sin cos sin isin 6(c) (i) ans: Proof equates real parts correctly marks cos cos cos sin cos sin correct epression cos 6cos cos cos 6(c) (ii) ans: Proof equates imaginary parts correctly marks sin cos sin cos sin sin correct epression sin 6sin sin sin 6(d) ans: Proof marks starts correctly simplifies correctly substitutes correctly completes proof cos sin 6 sec sec 6 tan tan sec tan sec 6 tan tan 6 tan tan tan tan tan tan tan... tan tan tan TOTAL MARKS = Pegasys

15 Additional Questions Solutions form Quest Pegasys

16 Pegasys