Victorin Certificte of Euction 04 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 7 November 04 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 8 8 40 Stuents re permitte to bring into the exmintion room: pens, pencils, highlighters, ersers, shrpeners n 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 0 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 04
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3 04 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 (5 mrks) Consier the vector = 3 i j k, where i, j n k re unit vectors in the positive irections ~ ~ ~ ~ ~ ~ ~ of the x, y n z xes respectively.. Fin the unit vector in the irection of ~. mrk b. Fin the cute ngle tht ~ mkes with the positive irection of the x-xis. mrks c. The vector b = 3 i + m j 5 k ~ ~ ~ Given tht b ~ is perpeniculr to ~, fin the vlue of m. ~. mrks TURN OVER
04 SPECMATH EXAM 4 Question (5 mrks) The position vector of prticle t time t 0 is given by ( ) ()=( ) + + r t t i t 4t j ~ ~ ~. Show tht the crtesin eqution of the pth followe by the prticle is y = x 3. mrk b. Sketch the pth followe by the prticle on the xes below, lbelling ll importnt fetures. mrks y O x c. Fin the spee of the prticle when t =. mrks
5 04 SPECMATH EXAM Question 3 (5 mrks) 4 3 Let f be function of complex vrible, efine by the rule f ( z)= z 4z + 7z 4z + 6.. Given tht z = i is solution of f ( z)= 0, write own qurtic fctor of f ( z). mrks b. Given tht the other qurtic fctor of f ( z) hs the form z + bz + c, fin ll solutions of z 4 4z 3 + 7z 4z + 6 = 0 in crtesin form. 3 mrks TURN OVER
04 SPECMATH EXAM 6 Question 4 (3 mrks) Fin the grient of the norml to the curve efine by y = 3e 3x e y t the point (, 3).
7 04 SPECMATH EXAM Question 5 (5 mrks) ( )= ( ) ( ). For the function with rule f x 96cos 3x sin 3x, fin the vlue of such tht f x sin 6x. ( )= ( ) mrk b. Use n pproprite substitution in the form u = g( x) to fin n equivlent efinite integrl for π π 36 ( ) ( ) ( ) 96cos 3x sin 3x cos 6x in terms of u only. 3 mrks π π 36 ( ) ( ) ( ) c. Hence evlute 96cos 3x sin 3x cos 6x, giving your nswer in the form k, k Z. mrk TURN OVER
04 SPECMATH EXAM 8 Question 6 (5 mrks). Verify tht 4 = + 4 4. mrk Prt of the grph of y = x ( x 4) is shown below. y 5 4 3 O 3 4 x b. The region enclose by the grph of y = x = 4 is rotte bout the x-xis. x ( x 4) n the lines y = 0, x = 3 n Fin the volume of the resulting soli of revolution. 4 mrks
9 04 SPECMATH EXAM Question 7 (5 mrks) ( )= ( ) Consier f x 3xrctn x.. Write own the rnge of f. mrk 6x f x 3rctn x 4x. mrk b. Show tht ( )= ( )+ + ( )= ( ) c. Hence evlute the re enclose by the grph of g x rctn x, the x-xis n 3 the lines x= n x=. 3 mrks TURN OVER
04 SPECMATH EXAM 0 Question 8 (7 mrks) A boy of mss 5 kg is hel in equilibrium by two light inextensible strings. One string is ttche to ceiling t A n the other to wll t B. The string ttche to the ceiling is t n ngle θ to the verticl n hs tension T newtons, n the other string is horizontl n hs tension T newtons. Both strings re me of the sme mteril. A ceiling θ wll 5 kg boy B. i. Resolve the forces on the boy verticlly n horizontlly, n express T in terms of θ. mrks ii. Express T in terms of θ. mrk ( )< ( ) for 0 b. Show tht tn θ sec θ < θ < π. mrk c. The type of string use will brek if it is subjecte to tension of more thn 98 N. Fin the mximum llowble vlue of θ so tht neither string will brek. 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 04
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 3 π r h 3 Ah 4 3 π r3 bcsin A b c = = sina sin B sin C 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 Mechnics momentum: p= mv ~ ~ eqution of motion: R = m ~ ~ friction: F µn END OF FORMULA SHEET