Victorin CertiÞcte of Euction 007 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors SPECIALIST MATHEMATICS Written exmintion Mony 5 November 007 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 0 0 40 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 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 007
007 SPECMATH EXAM This pge is blnk
3 007 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 Express 3 + i 3i in polr form. 4 mrks TURN OVER
007 SPECMATH EXAM 4 Question 3. Show tht 5 i is solution of the eqution z ( 5 i) z + 4z 4 5 + 4i= 0. mrk 3 b. Fin ll other solutions of the eqution z ( 5 i) z + 4z 4 5 + 4i= 0.
5 007 SPECMATH EXAM Question 3 3 Fin the eqution of the tngent to the curve x x y+ y = t the point P(, 3). 3 mrks TURN OVER
007 SPECMATH EXAM 6 Question 4 Fin the volume generte when the region enclose by the curve y = line x = is rotte bout the x-xis to form soli of revolution. x, the x-xis, the y-xis n the 4 mrks
7 007 SPECMATH EXAM Question 5 A block of mss 6 kg is given n initil push. As result of this push, the block s initil velocity is 4 m/s n it trvels cross horizontl ßoor in stright line. It comes to rest 3 metres from where it ws pushe ue to the frictionl force, F, between the block n the ßoor. N F 6 g. Clculte the ccelertion of the block cross the ßoor. b. Clculte the vlue of µ, the coefþcient of friction between the block n the ßoor. Give your nswer in the form b where b n c re positive integers. cg TURN OVER
007 SPECMATH EXAM 8 Question 6 A prticle moves so tht its velocity t time t is given by v t 4sin t i + 6cos t j for 0!!! ()= ( ) ( ) t π.. Given tht r( 0)= i, Þn the position vector r( t ) of the prticle t ny time t.!!! b. Fin the crtesin eqution of the pth followe by the prticle. Question 6 continue
9 007 SPECMATH EXAM c. Sketch the pth followe by the prticle on the xes below. y 4 3 4 3 O 3 4 x 3 4 TURN OVER
007 SPECMATH EXAM 0 Question 7. Use Euler s metho to Þn y if y =, given tht y x 0 = y () = n h = 0.. Express your nswer s frction. b. Solve the ifferentil eqution given in prt. to Þn the vlue of y which is estimte by y. Express your nswer in the form log e () + b, where n b re positive rel constnts.
007 SPECMATH EXAM Question 8. Sketch the slope Þel of the ifferentil eqution y x =,, 0,, on the xes below. y = + for y =,, 0,, t ech of the vlues y 3 3 O 3 x 3 b. If y = when x = 0, solve the ifferentil eqution given in prt. to Þn y in terms of x. c. Sketch the grph of the solution curve foun in prt b. on the slope Þel in prt. 3 mrks mrk TURN OVER
007 SPECMATH EXAM Question 9 A prticle moves in the crtesin plne with position vector r = xi + y j where x n y re functions of t. If its velocity vector is v = yi + x j, Þn the ccelertion! vector! of the! prticle in terms of the position!! vector r.!! 3 mrks Question 0 Given tht tn x 4 7 ( ) = where x 0, π 4, Þn the exct vlue of sin(x). 3 mrks END OF QUESTION AND ANSWER BOOK
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 007
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: ( + b) h 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
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 θ z r z n = r n cis(nθ) (e Moivre s theorem) Clculus x n ( )= nx n ( θ ) 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 ( ( x) )= x cos ( ( 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: ( uv)= u v + v u v u u v u v = v y y u = u If y = 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
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