SPECIALIST MATHEMATICS
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1 Victorin Certificte of Euction 009 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors SPECIALIST MATHEMATICS Written exmintion Friy 30 October 009 Reing time: 3.00 pm to 3.5 pm (5 minutes) Writing time: 3.5 pm to 4.5 pm ( hour) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be nswere Number of mrks 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, clcultor of ny type, blnk sheets of pper n/or white out liqui/tpe. Mterils supplie Question n nswer book of 9 pges with etchble sheet of miscellneous formuls in the centrefol. Working spce is provie throughout the book. Instructions Detch the formul sheet from the centre of this book uring reing time. Write your stuent number in the spce provie bove on this pge. 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. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 009
2 009 SPECMATH EXAM This pge is blnk
3 3 009 SPECMATH EXAM Instructions Answer ll questions in the spces provie. A eciml pproximtion will not be ccepte if n exct nswer is require to question. 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. Tke the ccelertion ue to grvity to hve mgnitue g m/s, where g = 9.8. Question Fin ll solutions to the eqution z 4 z 6 = 0, z C. 3 mrks Question A 50 kg stuent stns in lift which ccelertes ownwrs t rte of ms.. Fin the rection of the lift floor on the stuent correct to the nerest newton. mrks A few minutes lter the lift ccelertes upwrs t rte of ms. b. Fin the rection of the lift floor on the stuent, correct to the nerest newton, uring this secon stge of the motion. mrks TURN OVER
4 009 SPECMATH EXAM 4 Question 3 Resolve the vector 5i j 3kinto two vector components, one which is prllel to the vector i j k n one which is perpeniculr to it. 3 mrks Question 4 Given tht cos (θ ) = where,, fin cis (θ) in crtesin form. 4 4 mrks
5 5 009 SPECMATH EXAM Question 5 Consier the fmily of curves efine by the reltion 3x 3 y + kx + 5y xy = 4 where k R.. Verify tht every curve in the fmily psses through the point (0, 4), n fin the other point of intersection with the y-xis. b. Fin n expression for y in terms of x, y n k. mrks c. Hence evlute the grient of the curve t the point (, ). mrks mrks TURN OVER
6 009 SPECMATH EXAM 6 Question 6 Fin ll rel vlues of m such tht y = e mx is solution of y y 3 0y 0. mrks Question 7 A mss hs ccelertion ms given by = v 3, where v ms is the velocity of the mss when it hs isplcement of x metres from the origin. Fin v in terms of x given tht v = where x =. 4 mrks
7 7 009 SPECMATH EXAM Question 8 x. Show tht f( x) 4 x cn be written in the form f x 6 ( ) 4 x. mrk x b. Fin the exct re enclose by the grph of f( x) 4 x, the x-xis, n the lines x = n x =. 3 mrks TURN OVER
8 009 SPECMATH EXAM 8 Question 9 Let y ( y ) 4 n y 0 = y (0) = 0.. Solve the ifferentil eqution bove giving y s function of x. b. Apply Euler s metho to fin y, using step size of mrks mrks
9 9 009 SPECMATH EXAM Question 0 Let f ( x) x 3 rcsin.. Stte the implie omin n the rnge of f. b. Fin f ( x) giving your nswer in the form f ( x) bx( x c) where, b n c re integers. mrks 3 mrks END OF QUESTION AND ANSWER BOOK
10
11 SPECIALIST MATHEMATICS Written exmintions n FORMULA SHEET Directions to stuents Detch this formul sheet uring reing time. This formul sheet is provie for your reference. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 009
12 SPECMATH Specilist Mthemtics Formuls Mensurtion re of trpezium: curve surfce re of cyliner: volume of cyliner: volume of cone: volume of pyrmi: volume of sphere: re of tringle: sine rule: cosine rule: bh π rh π r h π r h 3 3 Ah 4 3 π r 3 bcsin A b c sin A sin B sinc c = + b b cos C Coorinte geometry ellipse: x h y k b hyperbol: x h y k b Circulr (trigonometric) functions cos (x) + sin (x) = + tn (x) = sec (x) cot (x) + = cosec (x) sin(x + y) = sin(x) cos(y) + cos(x) sin(y) cos(x + y) = cos(x) cos(y) sin(x) sin(y) tn( x) tn( y) tn( x y) tn( x) tn( y) sin(x y) = sin(x) cos(y) cos(x) sin(y) cos(x y) = cos(x) cos(y) + sin(x) sin(y) tn( x) tn( y) tn( x y) tn( x) tn( y) cos(x) = cos (x) sin (x) = cos (x) = sin (x) tn( x) sin(x) = sin(x) cos(x) tn( x) tn ( x) function sin cos tn omin [, ] [, ] R rnge π π, [0, ] π, π
13 3 SPECMATH Algebr (complex numbers) z = x + yi = r(cos θ + i sin θ = r cis θ z x y r π < Arg z π z r z z = r r cis(θ + θ ) cis z r θθ z n = r n cis(nθ) (e Moivre s theorem) Clculus x n nx n n n x x c, n n e x e x x e e x c log e( x) x x log x c e sin( x) cos( x) sin( x) cos( x) c cos( x) sin( x) cos( x) sin( x) c tn( x) sec ( x) sin cos ( x) x ( x) x sec ( x) tn( x) c x x x sin c, 0 x cos c, 0 tn ( x) x x x tn c prouct rule: quotient rule: chin rule: Euler s metho: ccelertion: uvu v v u u v v y y u u If y v u u v f x, x 0 = n y 0 = b, then x n + = x n + h n y n + = y n + h f(x n ) x v v v v t t constnt (uniform) ccelertion: v = u + t s = ut + t v = u + s s = (u + v)t TURN OVER
14 SPECMATH 4 Vectors in two n three imensions r xi yj zk ~ ~ ~ ~ r ~ = x y z r ~ r. r ~ = r r cos θ = x x + y y + z z r ~ y z r i j k ~ t t ~ t ~ t ~ Mechnics momentum: p mv ~ ~ eqution of motion: R m ~ ~ friction: F μn END OF FORMULA SHEET
SPECIALIST MATHEMATICS
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