Level 2 Certificate Further MATHEMATICS 83601 Paper 1 non-calculator Report on the Examination Specification 8360 June 2013 Version: 1.0
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General Most students were well prepared for this examination and were able to make good attempts at all questions in the time allowed. There were many examples of students showing good algebraic manipulation skills and most set out their work in a clear, logical way. Areas of study tested on this paper included calculus, the equation of a circle, solving an equation given in matrix form, and the factor theorem, and there were some more difficult questions on geometric proof and laws of indices. Although some of these questions proved to be challenging, some very elegant and novel solutions were seen. Topics that were well done included: calculating the gradient of a curve using calculus solving an equation containing square root symbols differentiation expanding brackets and simplifying changing the subject of a formula. Topics which students found difficult included: working out the equation of a tangent using and applying the factor theorem surds manipulation completing the square when the coefficient of x 2 is greater than 1 geometric proof solving equations using the laws of indices. Question 1 Part (a) was generally well answered. A number of students did not recognise that dy dx represents the gradient function. Some differentiated the given expression. Part (b) was less well answered. A significant number of students seemed to think that x equalled 1 rather than dy. There were a dx significant number of non-attempts at part (b). Question 2 Part (a) was well answered. In part (b) some students used 6 2 + 8 2 to find the diameter and occasionally forgot to divide by 2 for the radius. Mistakes in part (c) included forgetting to write + between (x + 1) 2 and (y 7) 2 and sometimes writing x in both brackets. There were also a significant number of non-attempts. Part (d) was answered correctly by many students. 3 of 6
Question 3 Part (a) was usually well answered with clear working. Part (b) was not so well done. Some students correctly found the length of 9 but failed to subtract 4. Some recognised the ratio but made arithmetical errors. A common misconception was to try to use Pythagoras' theorem. There were a significant number of non-attempts at part (b). Question 4 This question was well answered, with only a few students failing to square 6 as a first step. Question 5 Part (a) was a good discriminator. 'Show that' questions require every step of the working to be shown. Many students did not do this, frequently failing to show the change of signs clearly. Many students did not appreciate that part (a) would help them answer part (b), and instead worked out 107 2 and 97 2, with varying degrees of success. Question 6 Common errors were evaluating 3 4 as 12 and forgetting to apply the power of 4 to the term in x. Question 7 Some students made a sign error in one of the terms or an error in one of the powers, but overall this question was well answered. Some students tried to take out a common factor from their final expression. Question 8 Part (a) was well answered. Part (b) was a good discriminator. Many students began correctly by substituting x = 2 in their expression for dy to work out the gradient, although a large number dx produced the wrong answer for 4 8, with 24 and 36 being equally common. Some students then used 12 as the y-intercept, producing a number from somewhere for m. A few decided m was 1 / 12. To work out the value of c was a two-step process and many students did not use a correct method. Some substituted 0 into the equation of the curve. More able students knew to substitute x = 2 into the equation of the curve but some then thought that the answer (y = 5) would be the value of c, giving an answer of y = 12x + 5 4 of 6
Question 9 This question was a good discriminator. Most students chose to use the quadratic formula rather than completing the square. Most of these went on to accurately substitute in the correct values and calculate as far as 8. However, many then divided the number inside the square root by 2. Some only divided the 6 by 2 to get 3 ± 8. Of those who completed the square, the most common error was to get as far as (x + 3) 2 but did not work out the constant correctly. Quite a few students tried to factorise the given expression and some tried to rearrange the equation. Question 10 Most students knew to multiply by (a x) and then expand and rearrange the terms. Sign errors were quite common. Many students factorised the two terms in x and were able to obtain a fully correct solution Question 11 In part (a), there were many students who gave the x-axis points of intersection as the three factors. Some then proceeded to use the correct three factors in their answer to part (b). The most common method in part (b) was to substitute the values 5 and 3 in the expression for f(x) and try to solve a pair of simultaneous equations. There were some correct solutions but many more incorrect ones. There were also a significant number of non-attempts at part (b). Question 12 This question was a good discriminator. Most students obtained the two equations from the matrices and made a good attempt at solving them. Of those who solved the quadratic in x, the majority were able to obtain the correct x values. However, sign errors were often made in working out the corresponding y values. Those who chose to do the quadratic in y rarely managed to factorise and solve it correctly. Some students used a trial and improvement method and were occasionally successful, but this is not a method to be encouraged. Some presentation was rather untidy with work not set out in a logical manner. Question 13 There were some very elegant solutions to this question but for the majority it proved to be a 8 challenging question. Successful students usually started by rationalising. Many students 3 1 wrote this fraction but did not progress further. Some multiplied out the original expression and then tried to square it but this approach was rarely successful. 5 of 6
Question 14 Some very good and successful attempts were seen, using the many different approaches that were possible. For the majority, however, it proved too challenging. Proof questions demand a rigorous approach with facts supported by reasons. Many students gave reasons that were neither supported by working nor relevant to a logical solution. A few students correctly used the alternate segment theorem within their solution, although many used it incorrectly, often stating that angle BOC = 2x. The most common error was to assume a symmetry of the diagram that did not exist, and thus deciding that all the angles at the base of triangles OCD and OBC must be x. Question 15 This question was a good discriminator. The fact that the coefficient of x was not 1 made this a challenging question for many students. Those who removed the 2 as a factor were often successful. Errors in the use of brackets were common. Some tried to expand the expression a(x + b) 2 + c and attempted to equate coefficients. There were very few correct solutions from this approach. Question 16 This question was a good discriminator. Only the most able students were able to process the negative power, the need to square root and the need to cube. Poor presentation was a feature of many answers. It was possible to do the correct steps of the process in any order, even the converting of the mixed number to an improper fraction, although it was advisable to leave the cubing of the final fraction as the last step since 64 3 is not easy to evaluate without a calculator. One fairly common error was to assume that the power of 2 / 3, became a power of 3 / 2. Some students managed the square root and cube steps successfully but forgot that the power was initially negative and so gave an answer that was the reciprocal of the correct one. There were a significant number of non-attempts at this question. Mark Ranges and Award of Grades Grade boundaries and cumulative percentage grades are available on the Results Statistics page of the AQA Website. Converting Marks into UMS marks Convert raw marks into Uniform Mark Scale (UMS) marks by using the link below. UMS conversion calculator www.aqa.org.uk/umsconversion 6 of 6