2007 Mathematical Methods & Mathematical Methods (CAS) GA 2: Exam 1

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1 7 7 Mathematical Methods & Mathematical Methods (CAS GA : Exam GENERAL COMMENTS The number of students who sat f the 7 examination was 5 77, which was 8 fewer than the 6 6 who sat in 6. Around 8% sced 9% me of the available marks, compared with 6% in 6, and % received full marks, compared with % in 6. The overall quality of responses was not as strong as that of recent years; however, there were many very good responses and it was rewarding to see the quite substantial number who wked through the questions to obtain full marks. It was, however, disappointing to find that a significant number of students were unable to obtain any marks on the paper. There appeared to be a wider gap between those students who understood the material of the course and those who did not than in the previous year. Too many students were unable to crectly evaluate simple arithmetic calculations, including fractions, simple algebraic expressions. obability was an area of weakness and many students did not sce well on these questions. While there was no significant difference evident between the responses of Mathematical Methods students and Mathematical Methods (CAS students, the mean perfmance on individual questions by Mathematical Methods (CAS students was slightly better than their Mathematical Methods counterparts. Students again need to be aware that the instruction to show appropriate wking when me than one mark is available is applied rigously when marking the papers. Failure to show appropriate wking results in marks not being awarded where only an answer is given in response to the question. Similarly, increct, careless and sloppy notation is penalised: in particular, the dropping of dx from anti-differentiation and integration expressions. Graphs should be drawn showing crect features, such as smoothness and endpoints. Students need also to be reminded that material from Units and of the study can be drawn on; Question 6, which related to basic probability, was an example of this. Question b. on conditional probability was another example and this area is often seen in either examination questions. As noted in previous assessment repts, there continues to be difficulty with algebraic skills, setting out, graphing skills and the proper use of mathematical notation. This was evident again this year in almost every question on the paper. SPECIFIC INFORMATION Question Marks Average % x sin( x x cos( x f ( x sin ( x Many students were able to obtain the crect answer; however, some students failed to gain both marks because they then increctly factised the crect expression. Students should be advised against proceeding beyond what is explicitly asked f in a question. Students who attempted to use the product rule rather than the quotient rule often could not find the derivative of ( sin ( x. Increct responses included simply differentiating the numerat and x denominat separately to obtain, cancelling sin(x adding numerat terms to give cos( x x sin( x + x cos( x f ( x. sin x ( Question a. Marks Average % Maths Methods and Maths Methods (CAS GA Exam VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 8

2 7 ( ( x ln + 5 e x 6 This question was generally well answered, with many students knowing at least one logarithm law. Common mistakes included simply cancelling the logarithms, adding to obtain x + 7, multiplying to get 6x + 5. Question b. Marks Average % 6. ( x g sec tan ( x ( x π sec π g π tan Given that this was a standard application of the chain rule f differentiation, the question was not well answered. Some students attempted to use the product rule. Common responses were, and. It was not tan x sec x cos x ( ( ( uncommon to see students substitute π into their expression but then fail to evaluate it, possibly due to not knowing the relevant exact values. As seen in recent years, students ability to deal effectively with both circular functions and logarithmic functions was an area of concern. Question a. Marks Average % 9 5. Many students had difficulty sketching the graph and including all the required infmation. Common errs were drawing the cusp point at x as a closed circle as well as the point at x, not indicating endpoints showing them as open circles at x at x, showing endpoints at extreme left and right, and excluding the section to the right of x altogether. Some students were happy to draw many-to-many relationships inverse relations. Other students drew only the section (, crectly. This section could also have been drawn with a suitable curve. Question b. Marks Average % R\{,} Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8

3 7 In this question, students often gave responses that were inconsistent with the graph they had drawn in part a. As always, R was a popular response regardless. Use of notation was slightly better than in previous years; however, union and intersection were sometimes confused as was the use of round, square curly brackets. Question Marks Average % 8.7 dv 6x dx dx dx dv dt dv dt 8 6x Some students had difficulty deciding on how to label variables; x was sometimes interchangeable with d h and little no consistency was evident through the question. Most students were able to crectly differentiate V with respect to x but then either increctly used the chain rule could not manage the arithmetic required. Common errs were to simply substitute into V to substitute into the derivative. Question 5 Marks Average % ( X > ( X + ( X ( C.5 (.5 + C (.5 ( Students generally either attempted to use the binomial distribution a tree diagram to answer the question. Some were unable to evaluate n n the binomial coefficients. Students should know that C n and Cn n. Notation 6 was po and often made it difficult to ascertain if the student really understood what was required. Common responses were , a combinations of such, and incomplete tree diagrams. obabilities greater than were also seen. Question 6 Students generally either did very well on this question not well at all. F a basic probability question it was very poly done. obabilities greater than were seen too often and arithmetic was again an issue f some. Question 6a. Marks Average % ( A B PR( B ( A B Students who used a Venn diagram Karnaugh map usually obtained the crect answer. Many students either just added multiplied and B. Some students attempted unsuccessfully to develop a fmula. ( A ( Question 6b. Marks Average % 77. Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8

4 7 ( A B ( B A lot of increct attempts were seen f this question. A very popular increct response was, obtained by 5 confusing independent events with mutually exclusive events. Question 7 Marks Average % ( x ( cos( x f ( x xsin sin ( x xcos( x xsin ( x dx is an antiderivative. 9 This question should have been a standard and familiar calculus problem, but was not well answered. Some students differentiated, while others igned x and proceeded to anti-differentiate regardless. The instruction hence was often igned. Setting out and notation were very po in this question and dx often did not appear. Many students were unable to rearrange the expression. Question 8a. Marks Average % 5. π x sin πx π 5π, 5 x, This question was handled reasonably well. The biggest hurdle f many students was not knowing the crect basic angle f the exact value and therefe being unable to obtain the crect result f x. Other common errs were giving answers outside the given domain, including negative values, only finding x. Question 8b. Marks Average % π x The graph of y sin has first maximum turning point at π x. The graph is translated unit to the π 7 right, so x + is where the maximum first occurs. The maximum first occurs when ( π ( x sin π ( x π 7 x Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8

5 7 ( x π g ( x π cos π ( x π π, x 7 f minimum, f maximum Students found this question quite challenging, and a significant number did not know what was required. Many students encountered problems when attempting to set up an equation: f ( x + became f ( x and this led to x. Many variations on this theme were seen. Some students attempted to find the derivative of an almost crect ( π ( x expression, which they then equated to zero. Other students attempted to find when sin ; this was sometimes successful. Only a small number of students attempted to find the answer by transfmation. This question proved to be a good discriminat of overall student perfmance. Question 9 This should have been a straightfward question. Question 9a. Marks Average % 5. ( ( f x.5e f.5 x nmal: y x + Students often had difficulty crectly finding the y intercept; (, was a common answer, leading to the nmal being y x +. Quite a few students differentiated but then did not find the value of the derivative at x and simply substituted the expression f the derivative into the equation of the nmal, which led to increct attempts at algebraic manipulation and non-linear nmals. Some students found a tangent instead. Question 9b. Marks Average % x Area e + + e ( x x e x+ x dx Most students crectly set up an area expression but some were unable to proceed due to their nmal not being a linear expression. Many had the crect terminals, although instead of was a popular increct value. Arithmetic errs appeared frequently and subtraction mistakes in the integrand were also common. Some students also lost from the equation of the curve. The use of dx was surprisingly good. A few students used the area under the curve minus the triangle with some success. Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8 5

6 7 Question Marks Average % kx dx 7 kx 7 k 9 This question was generally well answered. Many students were able to set up the crect expression and integrate crectly. However the sometimes became instead of. Too many students were unable to evaluate the expression 9 crectly. Some students who had little idea simply set the given expression equal to 7. 9 kx 9dx 7 was also quite common. Question Some students who were unable to obtain any other marks on the paper crectly answered both parts of this question. However, overall this was not well answered. Question a. Marks Average % 7 9. ( T ( T F ( F + ( T F. ( F , using a tree: Common increct responses included increct values on the branches of the tree just taking one branch to obtain.8... Multiplication of decimals was not well done by some students. Question b. Marks Average % Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8 6

7 7 ( FT ( F T ( T ( TF ( F ( T This was a poly answered question, with many students obtaining no marks. Again, students struggled with conditional probability. Despite copying the fmula from the fmula sheet, students did not apply it to their values. ( F Common responses included ( TF ( F. Quite a few students didn t use their previous answer to part a. answers at all. Question Marks Average % The equation of line OP is y x (OP is perpendicular to y x f minimum length The point of intersection x x gives x. The codinates of P are (,, hence the minimum length is 5 ( ( ( l x + x ( 5 + l x x d l ( x dx so x ( ( l x + x 5x x+ so applying the quadratic fmula and symmetry, similar consideration, to 5x x+ value as occurring at x. identifies the minimum This proved to be a challenging question f most students. Many did not know what they were required to find, how to go about it if they did. Many different methods of solution were attempted, the most common being to use the nmal, the distance expression. A common, increct response was to find intercepts (, and (5, on the line then use the distance between two points to find the distance to be 5 5. Other common, increct responses involved taking P as the midpoint of the line, guessing and checking solutions without justification, increctly differentiating the distance equation, using an increct diagram. A small number of students used a vect approach to solve the problem. Some attempts at geometrical solutions were also seen with varying degrees of success. Maths Methods and Maths Methods (CAS GA Exam Published: 5 September 8 7