2008 Mathematical Methods (CAS) GA 3: Examination 2

Transcription

1 Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed tht the pper ws ccessible nd tht it provided n opportunity for students to demonstrte wht they knew. There is evidence to suggest tht the cohort performed better thn the 007 cohort. The men for ws 44, compred with 40 lst yer. The medin score for the pper ws 45 mrks. Of the whole cohort, % of students scored 85% or more of the vilble mrks, nd 6% scored 75% or more of the vilble mrks. The men score for the multiple-choice section ws (out of ). Only five of the multiple-choice questions were distinctive from the Mthemticl Methods pper, Questions,, 6, 9 nd. Questions 6, 9 nd were nswered correctly by less thn 46% of students. As stted in the instructions, students must show pproprite working for questions worth more thn one mrk. This ws poorly done by some students, especilly in Question. Writing out the expression to be evluted or eqution to be solved is considered sufficient working. Students should be encourged to ttempt to write out n expression or eqution, even if they think their nswer to previous question is incorrect, becuse mrks cn be wrded; for exmple, in Questions ii. nd cii. Students must ensure they red questions crefully so tht they give the solutions over the required intervl. Errors relting to this were mde in Questions ii., bii., d., 4b. nd 4dii. Correct mthemticl nottion should lwys be used nd is expected. Clcultor syntx should not be used, s occurred in Questions biii., cii., ciii., 4b., 4di. nd 4dii. In Questions nd 4dii., for exmple, nswers could esily hve been checked with CAS. Students cn solve equtions by hnd but need to be creful not to mke lgebric errors. Students need to use the vribles given in question. Incorrect vribles were used in Question g. In Question 4 some students did not give nswers in exct form where this ws clerly specified in the question nd then did not obtin the corresponding mrks. Techers should drw students ttention to the April 009 VCAA Bulletin rticle on the use of exct vlues. From 00 it will be ssumed tht students will provide exct vlue nswers unless specified otherwise. Students should tke more cre when drwing grphs nd use pproprite scles, mke them clerly visible, use the correct domin nd use ruler for liner grphs. They should re-red the question to mke sure they hve crried out ll of the instructions. Mths Methods (CAS) GA Exm VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 009

2 SPECIFIC INFORMATION Section The tble below indictes the percentge of students who chose ech option. The correct nswer is indicted by shding. Question % A % B % C % D % E % No Answer Comments This ws the best nswered question on the pper. b Averge vlue = f ( x) dx b = ( loge (x + ) ) dx log (7) 6 = e 6 % of the students found the verge rte of chnge, not the verge vlue. ( f ( x) )dx = f ( x) dx dx = 5 [ x] = = % of students did not find the ntiderivtive of. Option C ws ( f ( x) ) dx = 7. For infinitely mny solutions the lines must hve equivlent equtions. x + y = 0 x + ( + ) y = 0 Since the y-intercept for both equtions is the sme, i.e. zero, then the vlues of cn be found by letting the determinnt of the corresponding mtrix equl to zero nd solving for, without checking the solutions for prllel lines. = 0 + Hence = or =. Alterntively, using multiples of coefficients nd equtions leds to ( + ) = 6. Hence = - or =. % students found the vlues for which there ws unique solution. Mths Methods (CAS) GA Exm Published: 4 October 06

3 Let ( x, y ) be the imge of (x, y) under the trnsformtion x = 4 x + x = x 4 y = y + y y = y x Hence, = 4 ( x ) y = + The period is. Hence the mximum occurs t the turning point where x =. 4 f = sin = 8 4 f = sin = 4 Hence the rnge is,. 4% of students ssumed the mximum occurred close to the endpoint,, nd gve the rnge s, ). u u f (u) = e + e u u u u ( f ( u) ) = ( e + e ) = e + + e Hence, f ( u) = ( f ( u)). For independent events Pr( A B) = Pr( A) Pr( B) Let A = {, } nd B = {, 4, 6} Pr( A ) = nd Pr( B ) = Pr( A B) = 6 Pr( A ) Pr( B) = = 6 Hence Pr( A B) = Pr( A) Pr( B) 6% of students chose option B, which contined mutully exclusive events. Mths Methods (CAS) GA Exm Published: 4 October 06

4 f ( x) = sin(4x) + Reflection in the x-xis h ( x) = sin(4x) Diltion by fctor of 4 from the y-xis g ( x) = sin( x) Note the diltion by fctor of 4 from the 0 4, 4 = y-xis ffects the domin, [ 0, ] Section Question i. Mrks 0 Averge % Pr( X = 8) = 0.8 = 0.678, correct to four deciml plces This question ws well done. Some students gve the nswer correct to only three deciml plces nd others rounded incorrectly, giving s the nswer. ii. Mrks 0 Averge % X ~ Bi(8, 0.8), Pr( X = 6) = C = 0.96, correct to four deciml plces This question ws generlly nswered well. Some students gve the nswer without showing ny working. For questions worth more thn one mrk, pproprite working must be shown. bi. Mrks 0 Averge % = 0.95, correct to four deciml plces Mny students tried to use the trnsition mtrix. bii. Mrks 0 Averge % Pr(GGN) + Pr(GNG) + Pr(NGG) = = 0.849, correct to four deciml plces Mny students did not know or relise tht there were three cses, while others did not show dequte working. Some students used 0.6 s one of the probbilities. Mths Methods (CAS) GA Exm Published: 4 October 06 4

5 biii. Mrks 0 Averge % , , correct to four deciml plces Mny students evluted Some students gve 0.8 s the nswer. Most students hd the correct trnsition mtrix. Correct mthemticl nottion must be used nd is expected. Students should not use clcultor syntx in their responses; for exmple, [0.84, 0.64; 0.6, 0.6]^8*[; 0]. biv. Mrks 0 Averge % x x = 0.6 x x, x = 0.8, 80% Some students gve 0.8 s the nswer when percentge ws required. As this ws one mrk question, working ws not required. ci. Mrks 0 Averge % Mximum is (4., 0.8) This question ws done resonbly well. The locl mximum hd to be lbelled with its coordintes. Some students did not drw the horizontl lines long the x-xis corresponding to the elsewhere prt of the domin of f, while others rounded incorrectly or gve their nswers correct to only one deciml plce when the question sked for two. The x-xis must be scled. Mths Methods (CAS) GA Exm Published: 4 October 06 5

6 cii. Mrks 0 Averge % f ( x) dx = 0., correct to four deciml plces This question ws nswered well by mny students. Some students worked out Pr(X = ) insted of Pr(X < ). Others simply gve the nswer. Clcultor syntx should not be used; for exmple, ( f ( x), x,, ) is to be written s f ( x) dx. ciii. Mrks 0 Averge % xf ( x) dx = 4., correct to four deciml plces Some students worked out the medin or the verge vlue of the function. It ws plesing to see tht mny students hd the correct nottion, including the dx. Some students simply gve the nswer without showing ny working. Question i. Mrks 0 Averge % f ( ) f () 7 = = f ( ) f () f ( ) f () Some students used nd others left the grdient in the form. when simplifying their nswer; this could esily hve been checked with CAS. Some students mde errors ii. Mrks 0 Averge % dy 7 7 = =, x = dx x, s x > Mny students did not ttempt this question. Some students lso wrote x = s solution. Some students mde lgebric errors when solving the eqution by hnd, while others found the ntiderivtive. bi. Mrks 0 Averge % e f(x)dx = 7 This question ws nswered extremely well. Some students left their nswer s 7 loge ( e) or 7ln( e). Mths Methods (CAS) GA Exm Published: 4 October 06 6

7 bii. Mrks 0 Averge % dx = 7, b = e b x Some students simply wrote down the nswer. Others lso gve positive. b = e s solution. The question stted tht b ws ci. Mrks 0 Averge % ( ) Are = Atrpezium = (7 + )( ) = = or 7 7 Are = A tringle + A rectngle = ( )(7 ) + ( ) or 7 7 7( ) Are = x dx = This question ws done poorly. Very few students used the first two methods. Mny students tried unsuccessfully to find the eqution of the line segment CA nd others did not evlute the integrl, s the question sked for the nswer in terms of. Some students found the re under the curve of f, evluting f ( x) dx, while others found the re between the line segment CA nd the curve of f. cii. Mrks 0 Averge % ( ) = 7, = + Mny students did not ttempt this question. Students should be encourged to write out the eqution, even if they know their nswer to the previous question is incorrect. ciii. Mrks 0 Averge % The re under the curve is less thn the re of the trpezium. Hence f ( x) dx < 7. From b.i. f ( x) dx = 7 but f ( x) dx < 7, so < e. This ws difficult question nd two sttements were required to get one mrk. Mny students did not ttempt to nswer this question. e Mths Methods (CAS) GA Exm Published: 4 October 06 7

8 d. Mrks 0 Averge % m = e4 nd n = e4 OR m = e4 nd n = e4 Mny students tried unsuccessfully to solve the simultneous equtions by hnd. Other students gve only one solution, m stting tht m nd n hd to be positive but mn nd n hd to be positive. Some students gve the second solution only, possibly becuse this ws the first solution given on the clcultor nd they did not scroll cross to get the second solution. Question. Mrks 0 Averge % = 50loge ( + t), t.945 h = 9 minutes, to the nerest minute This question ws done quite well. Some students used 99 insted of 00, while others gve.9 minutes. b. Mrks 0 Averge % km t 5 km/h =.6 h,.6 h >.945 h, therefore he will not get the ntidote in time. Other methods were used to nswer this question, including working out the concentrtion. Some students did not give conclusion. c. Mrks 0 Averge % x (9 + ) Time(AM) = = Time(NY); Time(MN) = 5 (9 x ) 8 x T + = x 9 x 5 = x Some students ppered to simply unpck the formul nd did not relte their discussion to the digrm nd/or the context. d. Mrks 0 Averge % dt = 0, x =.5 km dx dy Some students simply gve the nswer without showing ny working, or x =.5 h, or used = 0. dt Mths Methods (CAS) GA Exm Published: 4 October 06 8

9 e. Mrks 0 Averge % When x =.5, T.49,.49 h <.9 h. Therefore, he gets the ntidote in time. Mny students did not give conclusion. Some went further nd clculted the concentrtion. f. Mrks 0 Averge % 4 4. A = (0, 6) C = (, 4) Mny students gve the correct coordintes for A but then gve the coordintes for B, not C. Some gve their nswers s 6 nd 4. Some hd A = (6, 0) or [0, 6]. g. Mrks 0 Averge % z = + 8 d 6 6 Some students used y nd x, insted of z nd d or wrote CD = + 8 or just + 8. Students should use the vribles d d s given in the question. Some found the eqution of the stright line CD. Mny students did not ttempt this question. h. Mrks 0 Averge % dys Mny students did not ttempt this question. Five nd seven dys were very populr nswers. Question 4 4i. Mrks 0 Averge % f = This question ws nswered quite well. 4ii. Mrks 0 Averge % m =,,, y = x+ + This question ws generlly well done. Some students did not give exct nswers nd some worked out the eqution of the tngent rther thn the norml. Mths Methods (CAS) GA Exm Published: 4 October 06 9

10 4iii. Mrks 0 Averge % x-intercept is +, y-intercept is+ Once gin, some students did not give the exct nswers for the intercepts. Some students were not creful bout where the norml should be drwn. 4b. Mrks 0 Averge % ,,, d x This question ws not done well. Mny students gve generl solution to tn = using clcultor dx (8en6 + ) x = syntx; for exmple,. Students should be fmilir with representing generl solutions using mthemticl nottion, involving suitble prmeter nd the substitution of relevnt vlues for the prmeter to determine solutions within the required intervl. 4c. Mrks 0 Averge % tn =, = Mny students tried to solve tn =. 4di. Mrks 0 Averge % x h ( x) = cos + sec x Mths Methods (CAS) GA Exm Published: 4 October 06 0

11 This question ws done quite well. Some students wrote 60 x x h ( x) = cos + sec. insted of nd others wrote 4dii. Mrks 0 Averge % h ( x) = 0, x = Some students gve the generl solution using clcultor syntx. 4e. Mrks 0 Averge % Sttionry point t (, ), correct shpe, equtions of the symptotes nd correct y-intercept It ppered tht some students did not tke enough cre when drwing this grph or possibly rn out of time. Mny did not show the sttionry point. Some students did not give the equtions of the symptotes or they wrote y = insted of x =. Some sketched the grph of the derivtive. The symptotic behviour ws shown better this yer thn in 007. Mths Methods (CAS) GA Exm Published: 4 October 06