SPECIALIST MATHEMATICS

Transcription

1 Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Tuesy 5 June 08 Reing time:.00 pm to.5 pm (5 minutes) Writing time:.5 pm to 3.5 pm ( 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 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 (NHT) THIS PAGE IS BLANK o n o t w r i t e i n t h i s r e

3 3 08 SPECMATH EXAM (NHT) 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 (3 mrks) A light inextensible string hngs over frictionless pulley connecting msses of 3 kg n 7 kg, s shown below. o n o t w r i t e i n t h i s r e 7 kg 3 kg. Drw ll of the forces cting on the two msses on the igrm bove. mrk b. Clculte the tension in the string. mrks TURN OVER

4 08 SPECMATH EXAM (NHT) 4 Question (3 mrks) Let = 3i j+ m k n b= i j+ 3 k, where m R. Fin the vlue(s) of m such tht the mgnitue of the vector resolute of prllel to b is equl to 4. Question 3 (3 mrks) Fin sin(t) given tht t = + 3 rccos rctn. 3 4 o n o t w r i t e i n t h i s r e

5 5 08 SPECMATH EXAM (NHT) Question 4 (4 mrks) Throughout this question, use n integer multiple of stnr evitions in clcultions. The stnr evition of ll scores on prticulr test is.0. From the results of rnom smple of n stuents, 95% confience intervl for the men score for ll stuents ws clculte to be (44.7, 5.7). Clculte the men score n the size of this rnom smple. mrks o n o t w r i t e i n t h i s r e b. Determine the size of nother rnom smple for which the enpoints of the 95% confience intervl for the popultion men of the prticulr test woul be.0 either sie of the smple men. mrks TURN OVER

6 08 SPECMATH EXAM (NHT) 6 Question 5 (4 mrks) Evlute 3 x. x + x+ 5 Question 6 (4 mrks) Given tht y = (x )e x is solution to the ifferentil eqution y + b y = y, fin the vlues of n b, where n b re rel constnts. x x o n o t w r i t e i n t h i s r e

7 7 08 SPECMATH EXAM (NHT) Question 7 (4 mrks). Fin ( x ). x mrks o n o t w r i t e i n t h i s r e b. Hence, fin the length of the curve specifie by y = x from x = to x = 3. Give your nswer in the form kπ, k R. mrks TURN OVER

8 08 SPECMATH EXAM (NHT) 8 Question 8 (6 mrks) A circle in the complex plne is given by the reltion z i =, z C.. Sketch the circle on the Argn igrm below. mrk Im(z) O b. i. Write the eqution of the circle in the form (x ) + (y b) = c n show tht the grient of tngent to the circle cn be expresse s y x = x y. mrks Re(z) o n o t w r i t e i n t h i s r e ii. Fin the grient of the tngent to the circle where x = in the first qurnt of the complex plne. mrk Question 8 continue

9 o n o t w r i t e i n t h i s r e 9 08 SPECMATH EXAM (NHT) c. Fin the equtions of ll rys tht re perpeniculr to the circle in the form Arg(z) = α. mrks TURN OVER

10 08 SPECMATH EXAM (NHT) 0 Question 9 (9 mrks). i. Given tht cot(θ) =, show tht tn (θ) + tn(θ) = 0. mrks ii. Show tht tn( θ ) = ± +. mrk π iii. Hence, show tht tn = 3, given tht cot( ), θ = 3 where θ (0, π). mrk π b. Fin the grient of the tngent to the curve y = tn(θ) t θ =. mrks o n o t w r i t e i n t h i s r e Question 9 continue

11 o n o t w r i t e i n t h i s r e 08 SPECMATH EXAM (NHT) c. A soli of revolution is forme by rotting the region between the grph of y = tn(θ), π π the horizontl xis, n the lines θ = n θ = bout the horizontl xis. 3 Fin the volume of the soli of revolution. 3 mrks END OF QUESTION AND ANSWER BOOK

12 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

13 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)

14 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

15 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