2017 VCE Mathematical Methods 2 examination report

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1 7 VCE Mathematical Methods examination report General comments There were some excellent responses to the 7 Mathematical Methods examination and most students were able to attempt the four questions in Section B. Question was very well answered. Some students found Questions f., g., h., g, 4h. and 4i. challenging. Many students wrote their answers in the correct form. Exact answers needed to be given unless otherwise specified. Approximate answers were usually required in the probability questions. Most students answered everything that was required within a question. It is important for students to re-read questions as often they require more than one piece of information in the response. Advice to students Technology should be updated well before the examination. Technology is best put in mode radians. Occasionally the mode may need to be changed according to the question, such as in Question d. Chec that answers mae sense. By looing at the graph in Question bi., the gradient was negative and liewise in Question d., the a value was positive. Define functions on the technology at the start of each question in Section B. This saves time, especially when dealing with probability questions that involve hybrid functions, such as Question. If the functions have been defined at the start of the question, it is acceptable to use the function name, such as f(x), throughout the question rather than writing out the entire expression. This avoids missing out on mars if bracets are not inserted when finding the area between two curves, for example, in Question dii. and Question 4c. It also saves time and avoids transcription errors. Show a method for questions worth more than one mar. A small number of students did not show their method in the probability question, Question. Tae time when setching graphs, lie in Question a. If it is a linear graph use a ruler. Chec that the points have been positioned correctly if a grid has been given. Setch along an axis if the function is defined for those points. Use bracets with questions involving logarithms, such as Question 4b., ylog ( x ). Learn the correct wording to describe transformations. Not nowing this led to students maing errors in Question 4g. Specific information This report provides sample answers or an indication of what answers may have included. Unless otherwise stated, these are not intended to be exemplary or complete responses. The statistics in this report may be subject to rounding, resulting in a total more or less than per cent. VCAA

2 7 VCE Mathematical Methods examination report Section A The table below indicates the percentage of students who chose each option. The correct answer is indicated by the shading. Question % A % B % C % D % E % No answer Comments 95% confidence interval is (.9,.). The sample proportion is in the middle of the confidence interval ( ) 4 5 p x x p ( p ) x 4x 5 p The discriminant is negative for no real solutions. 6 4( p)( p5) 4p 4 p 4 Divide by 4 and change the inequality p 6p x x T y y x x x, x y y, y y ysinx 4 x y sin 4 ysinx y cos x 5 7 VCAA Page

3 7 VCE Mathematical Methods examination report Question % A % B % C % D % E % No answer Comments sin( x) x, x, x,,, sum: hx ( ) x h( x) hx x x x x X Bi 5, p Pr X p ( p), p 4 X X Pr Pr 4.469, correct to four decimal places VCAA Page

4 7 VCE Mathematical Methods examination report Question % A % B % C % D % E % No answer Comments b c d Area = f ( x) dx f ( x) dx f ( x) dx b a b c f ( x) dx f ( x) dx a c b f ( x) dx f ( x) dx a b bc, as b c, since f ( x) f ( x) np np( p) n p np p ( ) np np p, np np p, p n., n 99 n b cos( x) sin( x) tan( x), x 6 B, 6 A triangle shaded A sin( x) dx cos( x) dx 6 A : A shaded triangle : 8 VCAA Page 4

5 7 VCE Mathematical Methods examination report Section B Question a. Mars Average % f : R R, f ( x) x 5x, solve f( x) for x, x ,,, 9 9 5, turning points This question was answered well. Exact answers were required to obtain full mars. Some students gave their answers as (.9, 4.) and (.9, 4.). Others gave only the x values. Some mixed up the signs. Question bi. Mars Average % gradient f() f( ) 4, y 4 4( x ), y 4x This question was answered well. A common incorrect answer was y 4x. By inspection of the graph, the gradient was negative. Some students made arithmetic errors when calculating the gradient and/or y-intercept. Question bii. Mars Average % 78.8 ( ) ( ) ( ) ( ) f() f( ) d x x f x f x 68 7 The distance formula was used well. Some students applied this by hand and made arithmetic errors. Others had an incorrect distance formula using a multiplication operation between the two bracets instead of a plus. A common incorrect answer was 64. Question ci. Mars Average % ( ) ( ) ( ) d x x g x g x ( ( )) g() g( ) As in Question bii. some students did their solutions by hand and made arithmetic errors, especially sign errors. This would have been time consuming. sometimes given. Some incorrect answers contained. When defining technology, a multiplication sign must be inserted between and x. ( ) was g() x x x on the VCAA Page 5

6 7 VCE Mathematical Methods examination report Question cii. Mars Average Solve % for, or Students who answered Question ci. correctly were generally able to answer this question. Some students gave only one value for. Question di. Mars Average % Solve g( x) x for x, a, as a > A common incorrect answer was a. By inspection of the graph, the answer was positive. Some students found in terms of a, instead of a in terms of. Question dii. Mars Average % ( ) x g( x) dx 4 x x xdx was a common error, leaving out the bracets in ( ) these errors it would have been better to use the expression ( ) x x x dx. To avoid x g x dx. Some students overcomplicated the question by breaing up the areas into different sections. The easiest approach was to use upper function subtract lower function. There was evidence that students ( )( ) substituted instead of and this resulted in the answer of. 4 Question a. Mars Average % 89.9 t ht ( ) 65 55cos 5 height is m, range 55 65,55 65, This question was well answered., minimum height is m and maximum VCAA Page 6

7 7 VCE Mathematical Methods examination report Question b. Mars Average % Period =. He was in the capsule for minutes. 5 A common incorrect answer was 5 minutes. Question c. Mars Average % t h( t) sin 5, solve h() t for t, t = 7.5 minutes Most students were able to find the derivative. There were occasions when y x y x was attempted (average rate of change). Some students had their technology in degree instead of radian mode, t sin 5 giving h() t. Many could not find the maximum rate of change. A common incorrect 54 answer was 5 minutes. Many found the value of t for the maximum value of h. Others gave a general solution or two t values. Question d. Mars Average % tan( ), 7.4, correct to two decimal places 5 65 Many students new to get tan but they did not specify degree for their technology. A common incorrect answer was tan Some used tan 5 instead of tan 5. Others used sin 5. Students should be familiar with the relevant functionality for the context and select it appropriately. Question e. Mars Average dy dx % 9.9 x 5 x VCAA Page 7

8 7 VCE Mathematical Methods examination report This question was answered well. Some students wrote rationalise the denominator. dy dx x 5 x. There was no need to Question f. Mars Average % m PB u 5 u 65 or, solve 5 u 5 u u 5 u 65 = 5 u 5 u for u u , v , correct to two decimal places Many students were able to find the gradient of the line segment in terms of u, using their answer from Question e. Others used h(t) or y 5 x instead of y 5 x 65. Many students were unable to find the second gradient expression where they were required to use rise over run for the line segment P B. Question g. Mars Average % tan.67, correct to two decimal places Some students used radians instead of degrees. Question h. Mars Average % Angle difference = 9 ( ) , minutes, to the 6 nearest minute This question was not answered well. Some students wrote 7 minutes without showing any woring. As indicated in the instructions on the examination, for questions worth more than one mar, appropriate woring must be shown. VCAA Page 8

9 7 VCE Mathematical Methods examination report Question a. Mars Average % Many students did not draw their graphs along the t-axis, ignoring f(t) =. Some had an open circle at (45,.4). Others had an open circle over a closed circle at (45,.4). Many students did not use rulers to draw the line segments. Some graphs looed lie parabolas. Question b Mars Average % f () t dt 5 This question was answered well. Some students had the incorrect terminals. 44 instead of 45 was occasionally given, for example, ( ) ( ) of 5. Question c. f t dt f t dt. Others used as the lower limit instead 5 44 Mars Average % 49. Pr T 5 PrT 5 T 55 Pr T f () t dt f () t dt 4 Many students were able to use the conditional probability formula. A common incorrect answer was 4. VCAA Page 9

10 7 VCE Mathematical Methods examination report Question d. Mars Average % a 45 f ( t) dt.7 or f ( t) dt. or f ( t) dt., a 9.649, correct to four decimal places a A number of correct approaches were used. 75 answer. VCAA Page a a f ( t) dt.7, a = 5.65 was a common incorrect 7 t dt.7 was occasionally given. Some students attempted to use the inverse 65 a normal as a method. Question ei. X Mars Average % Bi7, 5 Pr X.54, correct to four decimal places, Many students recognised that the distribution was binomial and gave the correct n and p values. Pr X. Some used Question eii. Mars Average % 59. Pr X Pr X X Pr X , correct to four decimal places Many students were able to set up the conditional probability. Some wrote Pr X Pr X X Pr X Question f. Mars Average % q( p) p ( p) p ( p) 5 4. Others rounded incorrectly, giving.765 as the answer. 4 7 p ( p ) (p ) Of those who attempted this question, some students did not realise that the binomial distribution was required. Question gi. Mars Average % 64.6 Solve q( p) for p, p.59, q.5665, correct to four decimal places

11 7 VCE Mathematical Methods examination report Some students new to solve q( p) if they had an equation in Question f. Others found only p. Some gave exact values for their answers. Question gii. 7 Mars Average % 9 7. f ( t) dt d minutes, to the nearest minute d 7 Some students used q instead of p in their equation, solving solved d Question 4a. f ( t) dt , obtaining d = 4 minutes. Mars Average % x, the graph of x f( x) d f ( t) dt for d. Others y has been translated one unit to the left and two units down to x x c get the graph of f, c, d or T y y d, x x c, x x c, y y d, y y d, xc xc y d, y d, c, d This question was answered reasonably well. Some students made sign errors. Question 4b. Mars Average Let % x y, inverse swap x and y, f :, R, f ( x) log ( x ) y x, x y, ylog ( x ) There are other acceptable expressions for the inverse. Some students found the rule but did not give the domain. Some students did not use bracets, leaving their answer as y log x. VCAA Page

12 7 VCE Mathematical Methods examination report Question 4c. Mars Average % 49, Solve f ( x) f ( x), x or x f ( x) f x ( x) dx and ( ) x f ( x) dx f ( x) f ( x) dx log e() f x dx were common incorrect expressions. x x f ( x) dx should be written as ( ) ( ) is better to use the expression ( ) ( ) often given. Question 4d. Mars Average % f () log e(), f () log () e This question was answered reasonably well. Question 4e. Mars Average % x g ( x) e, solve g ( x) f ( x) for, log () This question was answered reasonably well. Question 4f. g Mars Average % x ( x) loge f x dx. To avoid this type of error it f x f x dx. An exact answer was required..96 was e Some students used their answer to Question 4e. incorrect response. x g ( x) loge log () was a common e Question 4gi. Mars Average % 69. x x g ( x ) e, g ( x) e, dilation of a factor of from the y-axis VCAA Page

13 Many students were unable to describe the transformation correctly, for example, dilation of a factor in the y-axis. Others put their answer in terms of log e () instead of. Some gave two transformations. Question 4gii. Mars Average % 7. g x ( x) loge, x g ( x) loge, dilation of a factor of from the x-axis Students who answered Question 4gi. correctly tended to answer this question well. Question 4h. Mars Average % 77. x g ( x) e, g x ( x) loge, x L: y x, L: y, tan 6 or tan or This question was not answered well. Some students found one answer only. Others gave approximate answers. Question 4ii. Mars Average % 95. Solve for,,, p Many students tried to solve g ( x) g ( x) and then attempted to find the discriminant. Question 4iii. As Mars Average % 98 the graphs of g and g will intersect in the third quadrant, lim x f ( x) dx 4, b 4, as g has a vertical asymptote with equation x and equation y, the area will approach 4 as increases. This question was not answered well. g has a horizontal asymptote with VCAA Page

2016 VCE Specialist Mathematics 2 examination report

2016 VCE Specialist Mathematics 2 examination report 016 VCE Specialist Mathematics examination report General comments The 016 Specialist Mathematics examination comprised 0 multiple-choice questions (worth a total of 0 marks) and six extended-answer questions