SPECIALIST MATHEMATICS

Transcription

1 Victorin Certificte of Euction 07 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written emintion Thursy 8 June 07 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 40 Stuents re permitte to bring into the emintion room: pens, pencils, highlighters, ersers, shrpeners n rulers. Stuents re NOT permitte to bring into the emintion 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 0 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 emintion You my keep the formul sheet. Stuents re NOT permitte to bring mobile phones n/or ny other unuthorise electronic evices into the emintion room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 07

2 07 SPECMATH EXAM (NHT) THIS PAGE IS BLANK

3 3 07 SPECMATH EXAM (NHT) Instructions Answer ll questions in the spces provie. Unless otherwise specifie, n ect 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 5 kg mss on smooth plne incline t 30 is hel in equilibrium by horizontl force of mgnitue P newtons, s shown in the igrm below. P 30. On the igrm bove, show ll other forces cting on the mss n lbel them. mrk b. Fin P. mrks TURN OVER

4 07 SPECMATH EXAM (NHT) 4 Question (3 mrks) Fin given tht 8 = 6 6 log e( ), (, 4). Question 3 (3 mrks) y Fin the grient of the curve with eqution = sin when =. Give your nswer in the form b, where, b Z

5 5 07 SPECMATH EXAM (NHT) Question 4 (4 mrks) Fin the vlues of n b given tht z i is fctor of z 3 + ( + b)z + (b )z 4 = 0, where n b re rel constnts. Question 5 (4 mrks) Prt of the grph of y = 4 + is shown below. y Fin the volume generte if the region boune by the grph of y = n the -is is rotte bout the -is. 4 +, the lines = n =, TURN OVER

6 07 SPECMATH EXAM (NHT) 6 Question 6 (3 mrks) Fin ll rel solutions of tn() = tn().

7 7 07 SPECMATH EXAM (NHT) Question 7 (5 mrks) Let y = ( 4 y).. Epress y in terms of, where y(0) = 3. 3 mrks b. Epress y in terms of y. mrks TURN OVER

8 07 SPECMATH EXAM (NHT) 8 Question 8 (3 mrks) A 3 kg mss hs velocity v ms, where v= rctn when it hs isplcement metres from the origin, > 0. Fin the net force, F newtons, cting on the mss in terms of.

9 9 07 SPECMATH EXAM (NHT) Question 9 (4 mrks) The rnom vribles X n Y re inepenent with μ X = 4, vr(x) = 36 n μ Y = 3, vr(y) = 5.. The rnom vrible Z is such tht Z = X + 3Y. i. Fin E(Z). mrk ii. Fin the stnr evition of Z. mrk b. Reserchers hve reson to believe tht the men of X hs ecrese. They collect rnom smple of 64 observtions of X n fin tht the smple men is X = 38. i. Stte the null hypothesis n the lterntive hypothesis tht shoul be use to test tht the men hs ecrese. mrk ii. Clculte the men n stnr evition for istribution of smple mens, X, for smples of 64 observtions. mrk TURN OVER

10 07 SPECMATH EXAM (NHT) 0 Question 0 (4 mrks) Consier the vectors = i j+ 3k n b = i + cj+ k. Fin the vlue of c, c R, if the ngle between n bis π 3. Question (4 mrks) Fin the length of the curve specifie prmetriclly by = θ sin(θ), y = cos(θ) from π θ = to θ = π, where R +. Give your nswer in terms of. 3 END OF QUESTION AND ANSWER BOOK

11 Victorin Certificte of Euction 07 SPECIALIST MATHEMATICS Written emintion 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 emintion room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 07

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 () + sin () = + tn () = sec () cot () + = cosec () sin ( + y) = sin () cos (y) + cos () sin (y) sin ( y) = sin () cos (y) cos () sin (y) cos ( + y) = cos () cos (y) sin () sin (y) tn( ) + tn ( y) tn( + y) = tn( )tn ( y) cos ( y) = cos () cos (y) + sin () sin (y) tn( ) tn ( y) tn( y) = + tn( )tn ( y) cos () = cos () sin () = cos () = sin () tn( ) sin () = sin () cos () tn( ) = tn ( )

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 (comple numbers) z = + iy = r( cos( θ) + isin ( θ) )= r cis( θ ) z = + 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 ) pproimte confience intervl for μ z s 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 n ( )= n n n n+ = + c, n n + e e ( )= e e = + c ( log e() )= = loge + c ( sin( ) )= cos( ) sin( ) = cos( ) + c ( cos( ) )= sin ( ) cos( ) = sin ( ) + c ( tn( ) )= sec ( ) sin ( ( ) )= cos ( ( ) )= ( tn ( ) )= + prouct rule quotient rule chin rule Euler s metho ccelertion sec ( ) = tn ( ) + c = sin c 0 +, > = cos + c, > 0 = tn c + + ( b n ) n ( ) ( b ) n+ + = + + c, n + ( + b) = loge + b + c ( uv)= u v + v u v u u v u v = v y y u = u If y = f( ), 0 = n y 0 = b, then n + = n + h n y n + = y n + h f ( n ) v v v = = = = v t t t rc length + f ( ) or () t y () t t ( ) ( ) + ( ) t Vectors in two n three imensions Mechnics r= i+ yj+ zk r = + y + z = r i r y z r = = i+ j+ k t t t t r. r = rr cos( θ ) = + yy + zz momentum END OF FORMULA SHEET eqution of motion p= mv R = m