SPECIALIST MATHEMATICS

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1 Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 9 November 08 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 n rulers. Stuents re NOT permitte to bring into the exmintion room: ny technology (clcultors or softwre), notes of ny kin, blnk sheets of pper n/or correction flui/tpe. Mterils supplie Question n nswer book of 9 pges Formul sheet Working spce is provie throughout the book. Instructions Write your stuent number in the spce provie bove on this pge. Unless otherwise inicte, the igrms in this book re not rwn to scle. All written responses must be in English. At the en of the exmintion You my keep the formul sheet. Stuents re NOT permitte to bring mobile phones n/or ny other unuthorise electronic evices into the exmintion room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 08

2 08 SPECMATH EXAM THIS PAGE IS BLANK

3 3 08 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 ms, where g = 9.8 Question (4 mrks) Two objects of msses 5 kg n 8 kg re ttche by light inextensible string tht psses over smooth pulley. The 8 kg mss is on smooth plne incline t 30 to the horizontl. The 5 kg mss is hnging verticlly, s shown in the igrm below. 8 kg 30 5 kg. On the igrm bove, show ll forces cting on both msses. mrk b. Fin the mgnitue, in ms, n stte the irection of the ccelertion of the 8 kg mss. 3 mrks TURN OVER

4 08 SPECMATH EXAM 4 Question (4 mrks). Show tht + i = cis π. mrk 4 b. Evlute ( 3 i) ( + i) 0, giving your nswer in the form + bi, where, b R. 3 mrks Question 3 (4 mrks) Fin the grient of the curve with eqution x sin( y) + xy = π t the point 8 nswer in the form, where, b n c re integers. π b + c π π,. 6 6 Give your

5 5 08 SPECMATH EXAM Question 4 (4 mrks) X n Y re inepenent rnom vribles. The men n the vrince of X re both, while the men n the vrince of Y re n 4 respectively. Given tht n b re integers, fin the vlues of n b if the men n the vrince of X + by re 0 n 44 respectively. Question 5 (4 mrks) x + Sketch the grph of f( x)= on the xes provie below, lbelling ny symptotes with their equtions x 4 n ny intercepts with their coorintes. y x 4 TURN OVER

6 08 SPECMATH EXAM 6 Question 6 (3 mrks) A prticle of mss kg moves uner force F ~ so tht its position vector ~ r t ny time t is given by r = sin( t) i + cos( t) j+ t k. Distnces re mesure in metres n time is mesure in secons. ~ ~ ~ ~ Fin the chnge in momentum, in kg ms π, from t = t = π to. Question 7 (3 mrks) Given tht cot( x) + tn( x) = cot( x), use suitble ouble ngle formul to fin the vlue of, R.

7 7 08 SPECMATH EXAM Question 8 (4 mrks) A tnk initilly hols 6 L of wter in which 0.5 kg of slt hs been issolve. Pure wter then flows into the tnk t rte of 5 L per minute. The mixture is stirre continuously n flows out of the tnk t rte of 3 L per minute.. Show tht the ifferentil eqution for Q, the number of kilogrms of slt in the tnk fter t minutes, is given by mrk Q t 3Q = 6 + t b. Solve the ifferentil eqution given in prt. to fin Q s function of t. Express your nswer in the form Q =, where, b n c re positive integers. 3 mrks ( 6 + t ) b c TURN OVER

8 08 SPECMATH EXAM 8 Question 9 (5 mrks) A curve is specifie prmetriclly by r() t = sec( t) i + tn( t) j, t R. ~ ~ ~. Show tht the crtesin eqution of the curve is x y =. mrks b. Fin the x-coorintes of the points of intersection of the curve x y = n the line y = x. mrk c. Fin the volume of the soli of revolution forme when the region boune by the curve n the line is rotte bout the x-xis. mrks

9 9 08 SPECMATH EXAM Question 0 (5 mrks) The position vector of prticle moving long curve t time t secons is given by 3 ( ) t r() t = i+ rcsin () t + t t j, 0 t, where istnces re mesure in metres. ~ 3 ~ ~ The istnce metres tht the prticle trvels long the curve in three-qurters of secon is given by ( ) = t + bt+ c t Fin, b n c, where, b, c Z END OF QUESTION AND ANSWER BOOK

10

11 Victorin Certificte of Euction 08 SPECIALIST MATHEMATICS Written exmintion FORMULA SHEET Instructions This formul sheet is provie for your reference. A question n nswer book is provie with this formul sheet. Stuents re NOT permitte to bring mobile phones n/or ny other unuthorise electronic evices into the exmintion room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 08

12 SPECMATH EXAM Specilist Mthemtics formuls Mensurtion re of trpezium curve surfce re of cyliner ( + b) h π rh volume of cyliner volume of cone π r h 3 π r h volume of pyrmi 3 Ah volume of sphere re of tringle sine rule 4 3 π r3 bcsin( A) b c = = sin( A) sin ( B) sin( C) cosine rule c = + b b cos (C ) Circulr functions cos (x) + sin (x) = + tn (x) = sec (x) cot (x) + = cosec (x) sin (x + y) = sin (x) cos (y) + cos (x) sin (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 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)

13 3 SPECMATH EXAM Circulr functions continue Function sin or rcsin cos or rccos tn or rctn Domin [, ] [, ] R Rnge π π, [0, ] π π, Algebr (complex numbers) z = x+ iy = r( cos( θ) + isin ( θ) )= r cis( θ ) z = x + y = r π < Arg(z) π z z = r r cis (θ + θ ) z z r = cis θ r θ ( ) z n = r n cis (nθ) (e Moivre s theorem) Probbility n sttistics for rnom vribles X n Y E(X + b) = E(X) + b E(X + by ) = E(X ) + be(y ) vr(x + b) = vr(x ) for inepenent rnom vribles X n Y vr(x + by ) = vr(x ) + b vr(y ) pproximte confience intervl for μ x z s x z s, + n n istribution of smple men X men vrince E( X )= µ vr ( X )= σ n TURN OVER

14 SPECMATH EXAM 4 Clculus x x n ( )= nx n n n+ xx= x + c, n n + x e x e x x ( )= e x e x = + c ( log e() x )= x x x x = loge x + c ( sin( x) )= cos( x) sin( x) x = cos( x) + c x ( cos( x) )= sin ( x) cos( x) x = sin ( x) + c x ( tn( x) )= sec ( x) x sin ( ( x) )= x x cos ( ( x) )= x x ( tn ( x) )= x + x prouct rule quotient rule chin rule Euler s metho ccelertion sec ( x) x = tn ( x) + c x x = sin c 0 x +, > x x x = cos + c, > 0 x x x = tn c + + ( x b n ) x n ( ) ( x b ) n+ + = + + c, n + ( x + b) x = loge x + b + c ( x uv)= u v x + v u x v u u v u x x x v = v y y u = x u x If y = f( x), 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 x x t rc length + f ( x) x or x () t y () t t x x ( ) ( ) + ( ) t Vectors in two n three imensions Mechnics r= xi+ yj+ zk r = x + y + z = r i r x y z r = = i+ j+ k t t t t r. r = rr cos( θ ) = xx + yy + zz momentum END OF FORMULA SHEET eqution of motion p= mv R = m

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