MATHEMATICAL METHDS (CAS) PILT STUDY Written emintion (Fcts, skills nd pplictions) Frid 5 November 004 Reding time: 9.00 m to 9.5 m (5 minutes) Writing time: 9.5 m to 0.45 m ( hour 0 minutes) PART I MULTIPLE-CHICE QUESTIN BK This emintion hs two prts: Prt I (multiple-choice questions) nd Prt II (short-nswer questions). Prt I consists of this question book nd must be nswered on the nswer sheet provided for multiple-choice questions. Prt II consists of seprte question nd nswer book. You must complete both prts in the time llotted. When ou hve completed one prt continue immeditel to the other prt. Number of questions Structure of book Number of questions to be nswered Number of mrks 7 7 7 Victorin Certificte of Eduction 004 Students re permitted to bring into the emintion room: pens, pencils, highlighters, ersers, shrpeners, rulers, protrctor, set-squres, ids for curve sketching, up to four pges (two A4 sheets) of pre-written notes (tped or hndwritten) nd one pproved CAS clcultor (memor m be retined) nd/or one scientific clcultor. For the TI-9, Voge 00 or pproved computer bsed CAS, their full functionlit m be used, but other progrms or files re not permitted. Students re NT permitted to bring into the emintion room: blnk sheets of pper nd/or white out liquid/tpe. Mterils supplied Question book of pges, with detchble sheet of miscellneous formuls in the centrefold. Answer sheet for multiple-choice questions. Instructions Detch the formul sheet from the centre of this book during reding time. Check tht our nme nd student number s printed on our nswer sheet for multiple-choice questions re correct, nd sign our nme in the spce provided to verif this. Unless otherwise indicted, the digrms in this book re not drwn to scle. At the end of the emintion Plce the nswer sheet for multiple-choice questions (Prt I) inside the front cover of the question nd nswer book (Prt II). You m retin this question book. Students re NT permitted to bring mobile phones nd/or n other electronic communiction devices into the emintion room. VICTRIAN CURRICULUM AND ASSESSMENT AUTHRITY 004
MATH METH (CAS) EXAM PT Working spce
MATH METH (CAS) EXAM PT Instructions for Prt I Answer ll questions in pencil on the nswer sheet provided for multiple-choice questions. Choose the response tht is correct for the question. A correct nswer scores, n incorrect nswer scores 0. Mrks will not be deducted for incorrect nswers. No mrks will be given if more thn one nswer is completed for n question. Question The grph of probbilit distribution for the rndom vrible W is shown below. p(w) 0.5 0.4 0. 0. 0. 0 The epected vlue of W is equl to A. 0.5 B. C.. D..5 E..5 0 w Question The continuous rndom vrible X hs probbilit densit function given b if 0 < < f ( ) = 0 elsewhere The vlue of such tht Pr(X < ) = 0.5 is A. 0.5 B. 0.04 C. 0.500 D. 0.750 E. 0.794 PART I continued TURN VER
MATH METH (CAS) EXAM PT 4 Question Christopher hs five pirs of identicl purple socks nd three pirs of identicl green socks. His socks re rndoml mied in his drwer. He tkes two individul socks t rndom from the drwer in the drk. The probbilit tht he obtins mtching pir is A. B. C. D. E. 8 5 4 8 8 Question 4 A binomil rndom vrible hs men 0 nd vrince 4. The vlues of n nd p respectivel re A. nd 0.9 B. 5 nd 0. C. 5 nd 0.8 D. 00 nd 0. E. 00 nd 0.8 Question 5 The rndom vrible X hs norml distribution with men. nd stndrd devition.4. If Z hs the stndrd norml distribution, then the probbilit tht X is greter thn 5 is equl to A. Pr (Z < ) B. Pr (Z > ) C. Pr (Z > ) D. Pr (Z > ) E. Pr (Z < ) PART I continued
5 MATH METH (CAS) EXAM PT Question 6 A sine function f hs n mplitude of nd period of 0. The rule for f could be π t A. f (t) = sin 5 B. f (t) = sin (0ϖt) C. f (t) = sin (0t) π t D. f (t) =.5 sin 5 E. f (t) =.5 sin (0ϖt) Question 7 The depth of wter ner the Lorne Pier chnges with the tides ccording to the rule ( ) h t 4π t π = 7 sin + 5 where t is the time in hours fter low tide nd h is the depth in metres. A low tide occurred t midnight. The time of the net high tide is A. 6.00 m B. 6.5 m C..0 pm D. 6.0 pm E..00 m on the net d Question 8 The sum of the solutions of cos = π A. 6 π B. π C., for 0 4ϖ, is D. 4ϖ E. 8ϖ PART I continued TURN VER
MATH METH (CAS) EXAM PT 6 Question 9 Which one of the following is complete set of liner fctors of the third degree polnomil b, where nd b re positive rel numbers? A., b B., b, + b C., b D., b, + b E., b, + b Question 0 Which one of the following functions does not hve n inverse function? A. f : [, 4) R, f () = B. g : R\{0} R, g () = C. h : R + R, h () = D. k : (, 0] R, k () = + E. m : R + R, m () = + Question If f () = e for ll R, then [ f ()] = f (), where is equl to A. B. 4 C. D. E. 4 PART I continued
7 MATH METH (CAS) EXAM PT Question Prts of the grphs of the functions with equtions = log e ( + ) nd = re shown below..0.0 The solution of the eqution log e ( + ) = is closest to A. 0.44 B. 0.45 C. 0.55 D. 0.56 E. 0.57 Question The following shows prt of the grph of the curve with eqution = P( Q) +. 5 4 4 5 The vlues of P nd Q respectivel could be P Q A. B. C. D. E. PART I continued TURN VER
MATH METH (CAS) EXAM PT 8 Question 4 Let f be polnomil function of degree. The grph of the curve with rule = f () either intersects the -is or touches the -is t ectl two points (, 0) nd (b, 0). A possible rule for f could be A. f () = ( )( b) B. f () = ( )( + b) C. f () = ( )( b) D. f () = ( + ) ( b) E. f () = ( + ) ( + b) Question 5 The miml domin of the function f with rule f () = log e ( ) + is A. R \ {0} B. (, ) C. R D. (0, ) E. (, 0) Question 6 The grph of the curve with rule = f (), where f is one-to-one function, hs ectl one smptote whose eqution is =. The grph of the curve with rule = f (), where f is the inverse function of f, will hve A. horizontl smptote t = B. horizontl smptote t = C. horizontl smptote t = D. verticl smptote t = E. verticl smptote t = Question 7 The grph of the function f with rule f () = cos() is trnsformed to the grph of the function g with rule g() = 5 cos() b A. diltion from the -is b scle fctor of 5 nd diltion from the -is b scle fctor of B. diltion from the -is b scle fctor of nd diltion from the -is b scle fctor of 5 C. diltion from the -is b scle fctor of nd diltion from the -is b scle fctor of 5 D. diltion from the -is b scle fctor of 5 nd diltion from the -is b scle fctor of E. diltion from the -is b scle fctor of 5 nd diltion from the -is b scle fctor of PART I continued
9 MATH METH (CAS) EXAM PT Question 8 At =, the grph of the function f, with rule f () = ( + ) ( ) + 4 hs A. n -is intercept. B. -is intercept. C. locl minimum. D. locl mimum. E. point of inflection with zero grdient. Question 9 If = sin(), the rte of chnge of with respect to t = k, ϖ < k < ϖ, is A. cos(k) B. cos(k) C. sin(k) D. sin(k) E. k cos() Question 0 The number of nts, N, in colon vries with time ccording to the rule N(t) = 000e 0.t, where t is the time mesured in ds, nd t 0. The verge rte of chnge in the number of nts over the first 0 ds is closest to A. 7 B. 8 C. 7 D. 78 E. 78 Question Using the pproimtion formul f ( + h) f () + h f (), with f () = nd =, n pproimte vlue of (.8) is given b A. f () + 0. f () B. f (8) + 0. f (8) C. f () f (.8) D. f () 0. f () E. f (8) 0. f (8) PART I continued TURN VER
MATH METH (CAS) EXAM PT 0 Question The grph of the function f, with rule = f (), is shown below. Which one of the following could be the grph of the curve with eqution = f ()? A. B. C. D. E. PART I continued
MATH METH (CAS) EXAM PT Question The grph of the function f, with rule = f (), is shown below. Which one of the following is most likel to be the grph of n ntiderivtive function of f? A. B. C. D. E. PART I continued TURN VER
MATH METH (CAS) EXAM PT Question 4 The intervl [, ] is divided into n equl subintervls b the points 0,, n, n where = 0 < < < n < n =. Let δ = i i for i =,,, n. n Then lim δ i δ ( ) is equl to 0 A. 0 i= B. C. d D. d E. d Question 5 The grph of the function f, with rule = f (), is shown below. b c The totl re bounded b the curve with eqution = f () nd the -is on the intervl [, c] is given b c A. f ( ) d b B. f ( ) d + f ( ) d b C. f ( ) d + f ( ) d 0 D. f ( ) d + f ( ) d b c b c b c 0 E. f ( ) d f ( ) d f ( ) d 0 b c 0 PART I continued
MATH METH (CAS) EXAM PT Question 6 Prts of the grphs with equtions = e nd = re shown below. = = e The totl re of the region bounded b the -is, the line = nd the curve with eqution = e is given b e ( ) A. e d 0 e ( ) B. e d 0 loge ( ) C. ( e ) d 0 log e ( ) D. ( e ) d 0 ( ) E. e d 0 Question 7 If b f ( ) d =, then 4 f d A. 4(b ) 6 B. 4(b ) + 6 C. 4( b) + 6 D. E. 0 b ( ( )) is equl to END F PART I MULTIPLE-CHICE QUESTIN BK
Victorin Certificte of Eduction 004 SUPERVISR T ATTACH PRCESSING LABEL HERE MATHEMATICAL METHDS (CAS) PILT STUDY Written emintion (Fcts, skills nd pplictions) Frid 5 November 004 Reding time: 9.00 m to 9.5 m (5 minutes) Writing time: 9.5 m to 0.45 m ( hour 0 minutes) PART II QUESTIN AND ANSWER BK This emintion hs two prts: Prt I (multiple-choice questions) nd Prt II (short-nswer questions). Prt I consists of seprte question book nd must be nswered on the nswer sheet provided for multiple-choice questions. Prt II consists of this question nd nswer book. You must complete both prts in the time llotted. When ou hve completed one prt continue immeditel to the other prt. Structure of book Number of questions Number of questions to be nswered Number of mrks 6 6 Students re permitted to bring into the emintion room: pens, pencils, highlighters, ersers, shrpeners, rulers, protrctor, set-squres, ids for curve sketching, up to four pges (two A4 sheets) of pre-written notes (tped or hndwritten) nd one pproved CAS clcultor (memor m be retined) nd/or one scientific clcultor. For the TI-9, Voge 00 or pproved computer bsed CAS, their full functionlit m be used, but other progrms or files re not permitted. Students re NT permitted to bring into the emintion room: blnk sheets of pper nd/or white out liquid/tpe. Mterils supplied Question nd nswer book of 7 pges. Instructions Detch the formul sheet from the centre of the Prt I book during reding time. Write our student number in the spce provided bove on this pge. All written responses must be in English. At the end of the emintion Plce the nswer sheet for multiple-choice questions (Prt I) inside the front cover of this question nd nswer book. Students re NT permitted to bring mobile phones nd/or n other electronic communiction devices into the emintion room. VICTRIAN CURRICULUM AND ASSESSMENT AUTHRITY 004
MATH METH (CAS) EXAM PT This pge is blnk
MATH METH (CAS) EXAM PT Instructions for Prt II Answer ll questions in the spces provided. A deciml pproimtion will not be ccepted if n ect nswer is required to question. In questions where more thn mrk is vilble, pproprite working must be shown. Unless otherwise indicted, the digrms in this book re not drwn to scle. Question The dimeters of circulr mts produced b mchine re normll distributed, with men cm nd stndrd devition.5 cm.. Sketch the probbilit densit curve for the dimeters of the circulr mts produced b the mchine. b. It is known tht 6.00 % of mts produced b the mchine hve dimeter less thn K cm. Find the vlue of K, correct to one deciml plce. + = mrks PART II continued TURN VER
MATH METH (CAS) EXAM PT 4 Question Find the reltionship between m nd k such tht the line with eqution = m intersects the curve with eqution = ( k) t the point (0, 0) nd one other point onl. mrks Question n the set of es below, sketch the grph of the function f with rule f () = Lbel ech smptote with its eqution. + mrks PART II continued
5 MATH METH (CAS) EXAM PT Question 4 π. n the set of es below, sketch the grph of the function with rule = sin t, π t π. 4 5 4 π π π 4 π 4 π 4 π π 4 π t 4 5 π b. Write down the generl solution of the eqution sin t 4 = where t R. + = 5 mrks PART II continued TURN VER
MATH METH (CAS) EXAM PT 6 Question 5 Let f : R R be continuous function with the following properties. f (0) = 0 f (0) = 0 f (4) = 0 f () = 0 f () < 0 for (, 0) (0, ) f () > 0 for (, ). n the set of es provided below, sketch possible grph of f. 5 4 4 5 b. If f () = ( b) nd the point (, 4) lso lies on the grph of f, find the vlues of nd b. c. Find the eqution of the tngent to the grph of f t the point where = 4. + + = 7 mrks PART II continued
7 MATH METH (CAS) EXAM PT Question 6 A brrel contins 00 blls, some of which re rinbow-coloured. Four blls re rndoml selected from the brrel, with replcement. This mens tht bll is selected, its colour noted, nd the bll replced before the net bll is selected. Let p be the proportion of rinbow-coloured blls in the brrel.. Write down n epression for the probbilit tht ectl one of the four blls selected is rinbowcoloured. b. Find the ect vlue of p for which this probbilit will be mimum. + = mrks END F PART II QUESTIN AND ANSWER BK
MATHEMATICAL METHDS (CAS) PILT STUDY Written emintions nd FRMULA SHEET Detch this formul sheet during reding time. Directions to students This formul sheet is provided for our reference. VICTRIAN CURRICULUM AND ASSESSMENT AUTHRITY 004
MATH METH (CAS) PILT STUDY Mensurtion re of trpezium: Mthemticl Methods CAS Formuls + b h ( ) volume of prmid: curved surfce re of clinder: π rh volume of sphere: volume of clinder: π r h re of tringle: volume of cone: π r h Ah 4 π r bcsin A Clculus d d n n n n n+ ( ) = d = + c, n n + d d e e ( ) = e d e = + c d ( log e( ) ) = d d = loge + c d ( sin( ) ) = cos( ) sin( ) d = cos( ) + c d d ( cos( ) ) = sin( ) cos( ) d = sin( ) + c d d d ( tn( ) ) = = sec ( ) product rule: ( d cos ( ) d uv) = u dv v du d + d d d du pproimtion: f ( + h) f ( ) + h f ( ) chin rule: = d du d v du u dv b verge vlue: f d b d u ( ) quotient rule: d d d v = v Probbilit Pr(A) = Pr(A ) Pr(A B) = Pr(A) + Pr(B) Pr(A B) ( ) Pr(A B) = Pr A B trnsition mtrices: S Pr( B) n = T n S 0 men: µ = E(X) vrince: vr(x) = σ = E((X µ) ) = E(X ) µ Discrete distributions Pr(X = ) men vrince generl p() µ = p() binomil hpergeometric σ = ( µ) p() = p() µ n C p ( p) n np np( p) D N D C C n N C n n D N n D D N n N N N Continuous distributions Pr( < X < b) men vrince generl b f ( ) d norml If X is distributed N(µ, σ ) nd Z = X µ σ µ = f ( ) d σ = ( µ ) f ( ) d = f ( ) d µ, then Z is distributed N(0, ), f ( z) = e π z
MATH METH (CAS) PILT STUDY Tble Norml distribution cdf 0 4 5 6 7 8 9 4 5 6 7 8 9 0.0.5000.5040.5080.50.560.599.59.579.59.559 4 8 6 0 4 8 6 0..598.548.5478.557.5557.5596.566.5675.574.575 4 8 6 0 4 8 5 0..579.58.587.590.5948.5987.606.6064.60.64 4 8 5 9 7 5 0..679.67.655.69.6.668.6406.644.6480.657 4 8 5 9 6 0 4 0.4.6554.659.668.6664.6700.676.677.6808.6844.6879 4 7 4 8 5 9 0.5.695.6950.6985.709.7054.7088.7.757.790.74 7 0 4 7 4 7 0.6.757.79.74.757.789.74.7454.7486.757.7549 6 0 6 9 6 9 0.7.7580.76.764.767.770.774.7764.779.78.785 6 9 5 8 4 7 0.8.788.790.799.7967.7995.80.805.8078.806.8 6 8 4 7 9 5 0.9.859.886.8.88.864.889.85.840.865.889 5 8 0 5 8 0.0.84.848.846.8485.8508.85.8554.8577.8599.86 5 7 9 4 6 8..864.8665.8686.8708.879.8749.8770.8790.880.880 4 6 8 0 4 6 9..8849.8869.8888.8907.895.8944.896.8980.8997.905 4 6 7 9 5 6..90.9049.9066.908.9099.95.9.947.96.977 5 6 8 0 4.4.99.907.9.96.95.965.979.99.906.99 4 6 7 8 0.5.9.945.957.970.98.994.9406.948.949.944 4 5 6 7 8 0.6.945.946.9474.9484.9495.9505.955.955.955.9545 4 5 6 7 8 9.7.9554.9564.957.958.959.9599.9608.966.965.96 4 5 6 7 8.8.964.9649.9656.9664.967.9678.9686.969.9699.9706 4 4 5 6 6.9.97.979.976.97.978.9744.9750.9756.976.9767 4 4 5 5.0.977.9778.978.9788.979.9798.980.9808.98.987 0 4 4..98.986.980.984.988.984.9846.9850.9854.9857 0 4..986.9864.9868.987.9875.9878.988.9884.9887.9890 0..989.9896.9898.990.9904.9906.9909.99.99.996 0.4.998.990.99.995.997.999.99.99.994.996 0 0.5.998.9940.994.994.9945.9946.9948.9949.995.995 0 0 0.6.995.9955.9956.9957.9959.9960.996.996.996.9964 0 0 0 0.7.9965.9966.9967.9968.9969.9970.997.997.997.9974 0 0 0 0 0.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.998 0 0 0 0 0 0 0.9.998.998.998.998.9984.9984.9985.9985.9986.9986 0 0 0 0 0 0 0 0 0.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 0 0 0 0 0 0 0 0 0..9990.999.999.999.999.999.999.999.999.999 0 0 0 0 0 0 0 0 0..999.999.9994.9994.9994.9994.9994.9995.9995.9995 0 0 0 0 0 0 0 0 0..9995.9995.9995.9996.9996.9996.9996.9996.9996.9997 0 0 0 0 0 0 0 0 0.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998 0 0 0 0 0 0 0 0 0.5.9998.9998.9998.9998.9998.9998.9998.9998.9998.9998 0 0 0 0 0 0 0 0 0.6.9998.9998.9999.9999.9999.9999.9999.9999.9999.9999 0 0 0 0 0 0 0 0 0.7.9999.9999.9999.9999.9999.9999.9999.9999.9999.9999 0 0 0 0 0 0 0 0 0.8.9999.9999.9999.9999.9999.9999.9999.9999.9999.9999 0 0 0 0 0 0 0 0 0.9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0 0 0 0 0 0 0 0 0 END F FRMULA SHEET