SPECIALIST MATHEMATICS

Transcription

1 Victorin Certificte of Euction 00 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors SPECIALIST MATHEMATICS Written exmintion Friy 9 October 00 Reing time: 9.00 m to 9.5 m (5 minutes) Writing time: 9.5 m to 0.5 m ( 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 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 00

2 00 SPECMATH EXAM This pge is blnk

3 3 00 SPECMATH EXAM Instructions Answer ll questions in the spces provie. Unless otherwise specifie 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 Consier f (z) = z 3 + 9z + 8z + 0, z C. Given tht f ( ) = 0, fctorise f (z) over C. 3 mrks TURN OVER

4 00 SPECMATH EXAM 4 Question A boy of mss kg is initilly t rest n is cte on by resultnt force of v 4 newtons where v is the velocity in m/s. The boy moves in stright line s result of the force.. Show tht the ccelertion of the boy is given by v t v 4. b. Solve the ifferentil eqution in prt. to fin v s function of t. mrk 4 mrks

5 5 00 SPECMATH EXAM Question 3 Reltive to n origin O, point A hs crtesin coorintes (,, ) n point B hs crtesin coorintes (, 3, 4).. Fin n expression for the vector AB in the form ibj ck. b. Show tht the cosine of the ngle between the vectors OA n AB is 4 9. mrk c. Hence fin the exct re of the tringle OAB. mrk 3 mrks TURN OVER

6 00 SPECMATH EXAM 6 Question 4 Given tht z = + i, plot n lbel points for ech of the following on the rgn igrm below. i. z ii. z iii. z 4 Im(z) O Re(z) mrks Question 5 Given tht f (x) = rctn(x), fin f. 3 mrks

7 7 00 SPECMATH EXAM Question 6 3π 4 Evlute cos (x)sin(x). π 3 mrks TURN OVER

8 00 SPECMATH EXAM 8 Question 7 Consier the ifferentil eqution y 4x, < x <, ( x ) for which y 3 when x = 0, n y = 4 when x = 0. Given tht 4x x ( x ), fin the solution of this ifferentil eqution. 3 mrks

9 9 00 SPECMATH EXAM Question 8 The pth of prticle is given by r( t) tsin( t) i tcos( t) j, t 0. The prticle leves the origin t t = 0 n then spirls outwrs. 3. Show tht the secon time the prticle crosses the x-xis fter leving the origin occurs when t =. 3 b. Fin the spee of the prticle when t =. mrk 3 mrks Let be the cute ngle t which the pth of the prticle crosses the x-xis. 3 c. Fin tn() when t =. mrk TURN OVER

10 00 SPECMATH EXAM 0 Question 9. On the xes below sketch the grph with eqution x xes n give the equtions of ny symptotes y ( y ). Stte ll intercepts with the coorinte 4 3 O 3 x 3 4 b. Fin the grient of the curve with eqution x ( y ) t the point where x = n y < mrks 3 mrks

11 00 SPECMATH EXAM Question 0 Prt of the grph with eqution y( x ) x is shown below. y O x Fin the re tht is boune by the curve n the x-xis. Give your nswer in the form b c where, b n c re integers. 4 mrks END OF QUESTION AND ANSWER BOOK

12 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 00

13 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

14 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

15 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