SPECIALIST MATHEMATICS

Transcription

1 Victorin Certificte of Euction 06 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 4 November 06 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 8 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 06

2 06 SPECMATH EXAM THIS PAGE IS BLANK

3 3 06 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) A tut rope of length m suspens mss of 0 kg from fixe point O. A horizontl force of 3 P newtons isplces the mss by m horizontlly so tht the tut rope is then t n ngle of θ to the verticl.. Show ll the forces cting on the mss on the igrm below. mrk O θ rope 0 kg mss P b. Show tht sin( θ ) = 3. 5 mrk c. Fin the mgnitue of the tension force in the rope in newtons. mrks TURN OVER

4 06 SPECMATH EXAM 4 Question (3 mrks) A frmer grows peches, which re sol t locl mrket. The mss, in grms, of peches prouce on this frm is known to be normlly istribute with vrince of 6. A bg of 5 peches is foun to hve totl mss of 65 g. Bse on this smple of 5 peches, clculte n pproximte 95% confience intervl for the men mss of ll peches prouce on this frm. Use n integer multiple of the stnr evition in your clcultions. Question 3 (4 mrks) Fin the eqution of the line perpeniculr to the grph of cos (y) + y sin (x) = x t π 0,.

5 5 06 SPECMATH EXAM Question 4 (4 mrks) Chemicls re e to continer so tht prticulr crystl will grow in the shpe of cube. The sie length of the crystl, x millimetres, t ys fter the chemicls were e to the continer, is given by x = rctn (t). Fin the rte t which the surfce re, A squre millimetres, of the crystl is growing one y fter the chemicls were e. Give your nswer in squre millimetres per y. Question 5 (4 mrks) Consier the vectors = 3i + 5j k, b= i j+ 3kn c = i + k, where is rel constnt.. Fin the vector resolute of in the irection of b. mrks b. Fin the vlue of if the vectors re linerly epenent. mrks TURN OVER

6 06 SPECMATH EXAM 6 Question 6 (3 mrks) Write ( 3 i ) + 3i 4 in the form + bi, where n b re rel constnts. Question 7 (4 mrks) 3 Fin the rc length of the curve y = ( x + ) from x = 0 to x =. 3

7 7 06 SPECMATH EXAM Question 8 (6 mrks) The position of boy with mss 3 kg from fixe origin t time t secons, t 0, is given by ( ) + ( ( )) r = 3sin( t) i 3 cos t j, where components re in metres.. Fin n expression for the spee, in metres per secon, of the boy t time t. mrks b. Fin the spee of the boy, in metres per secon, when t = π. mrk c. Fin the mximum mgnitue of the net force cting on the boy in newtons. 3 mrks TURN OVER

8 06 SPECMATH EXAM 8 Question 9 (3 mrks) Given tht cos( x y) = 3 5 n tn (x) tn (y) =, fin cos (x + y). Question 0 (5 mrks) Solve the ifferentil eqution x y = x y, given tht y () = 0. Express y s function of x. END OF QUESTION AND ANSWER BOOK

9 Victorin Certificte of Euction 06 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 06

10 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)

11 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

12 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