MATHEMATICAL METHODS (CAS) Written Examination 1

Transcription

1 The Mthemticl Assocition of Victori Tril Exm 011 MATHEMATICAL METHODS (CAS) STUDENT NAME Written Exmintion 1 Reing time: 15 minutes Writing time: 1 hour QUESTION AND ANSWER BOOK Structure of book Number of questions Number of questions to be Number of mrks nswere Note Stuents re permitte to bring into the exmintion room: pens, pencils, highlighters, ersers, shrpeners, rulers. Stuents re NOT permitte to bring into the exmintion room: notes of ny kin, blnk sheets of pper, white out liqui/tpe or clcultor of ny type. Mterils supplie Question n nswer book of 8 pges, with etchble sheet of miscellneous formuls t the bck. Working spce is provie throughout the book. Instructions Detch the formul sheet from the bck of this book uring reing time. All written responses must be in English. Stuents re NOT permitte to bring mobile phones n/or ny other unuthorise electronic evices into the exmintion room.

2 PAGE 011 MAV MATHS METHODS CAS EXAM 1 THIS PAGE IS BLANK

3 011 MAV MATHS METHODS CAS EXAM 1 PAGE Instructions Answer ll questions in the spces provie. In ll questions where numericl nswer is require n exct vlue must be given unless otherwise specifie. In questions where more thn one mrk is vilble, pproprite working must be shown. Unless otherwise inicte, the igrms in this book re not rwn to scle. Question 1. ( x tn(x) ). x x b. i. ( e + x). x ii. Hence, fin n ntierivtive of x 4( e + 1) x. e + x = Question. Sketch the grph of f :[ 1, ],where f ( x) = ( x 1) + on the set of xes below. Clerly lbel the enpoints, intercept n shrp point with their coorintes. y x TURN OVER

4 PAGE MAV MATHS METHODS CAS EXAM 1 b. Fin the verge vlue of f, over the intervl [ 0, ]. mrks Question For wht vlue(s) of k, where k is rel constnt, o the simultneous equtions kx + y = 6 n x + ( k 1) y = 6 hve no solution? mrks Question 4 Solve log (x!1) + log (x +1) = 0 for x. 4 mrks

5 011 MAV MATHS METHODS CAS EXAM 1 PAGE 5 Question 5 x Let f : R R, where f ( x) = 1 e.. Fin 1 f. mrks b. Stte the coorintes of the point where 1 f = f. 1 mrk Question 6 If f ( x) = x +, ( x) = x 1 g n ( x) g( f ( x) ) h =, efine h (x) s hybri function. mrks TURN OVER

6 PAGE MAV MATHS METHODS CAS EXAM 1 Question 7 A trnsformtion is escribe by the eqution x = y 0 0 x 1 +. y 1. Fin the imge of the curve with eqution y = 1 uner this trnsformtion. Give your x + 1 nswer in the form y = + b where n b re rel constnts. x mrks b. Hence, escribe how the grph of y = 1 cn be trnsforme to the grph of the imge. x + 1 mrks Question 8 The tble below represents probbility istribution of rnom vrible X. x Pr X = x p p q ( ). If p q = 0, show tht the vlue of p is 0.16.

7 011 MAV MATHS METHODS CAS EXAM 1 PAGE 7 b. Fin Pr ( < X < ) X. 1 mrk Question 9 The length of time, t (hours) tht certin se nemones survive is rnom vrible whose ( k! cos!! t $ $ * " # % & +1 * " # % & 0 ' t ' probbility ensity function cn be moelle by f ( t) = ) * * + 0 elsewhere n k is rel constnt.. Show tht k =. b. Evlute the probbility tht prticulr se nemone will survive for more thn two yers. TURN OVER

8 PAGE MAV MATHS METHODS CAS EXAM 1 Question cos (x) x Solve ( ) = 0 for 0 x π. 4 mrks END OF QUESTION AND ANSWER BOOK

9 Mthemticl Methos (CAS) Formuls Mensurtion re of trpezium: 1 b h volume of pyrmi: 1 curve surfce re of cyliner: π rh volume of sphere: volume of cyliner: π r h re of tringle: volume of cone: 1 π r h Ah 4 π r 1 bcsin A Clculus x x n nx n1 n 1 n1 xx x c, n1 n 1 x e x e x x 1 e x e x c log e( x) 1 1 x x x x loge x c 1 sin( x) cos( x) sin( x) x cos( x) c x 1 cos( x) = sin( x) cos( x) x sin( x) c x tn( x) = sec ( x) x cos ( x) prouct rule: x uv u v v u v u u v x x quotient rule: u x x x v v chin rule: y x y u u x pproximtion: f x h f x hf x Probbility Pr(A) = 1 Pr(A) Pr( A B) = Pr(A) + Pr(B) Pr(A B) Pr(A B) = Pr A B Pr B trnsition mtrices: S n = T n S 0 men: μ = E(X) vrince: vr(x) = = E((X μ) ) = E(X ) μ probbility istribution men vrince iscrete Pr(X = x) = p(x) μ = x p(x) = (x μ) p(x) continuous Pr( < X < b) = f( x) x b μ xf( xx ) σ ( xμ) f( x) x END OF FORMULA SHEET