This preserves the relative weightings of the two sections of the Mathematics in Practice document. The remainder of this report consists of detailed comments on each of the free-answer questions for each article in the 1997 study.
Each calculation was quite involved and frequent errors occurred due to misinterpretation of the question. 1 point was awarded to students who calculated that $387 was the cost of the "Deluxe Suite" and did not go further to complete the question.
Students who answered (ii) in m2 mostly succeeded in answering this question correctly. 1 point was awarded if candidates correctly calculated that the bars cost $54 and then failed to add item (ii).
The majority of candidates ignored the fact that there was a double addition of 0.04 m2 and had an answer of 0.68 m2 which was awarded 1 mark. A large number of candidates did not use compasses and a freehand sketch received no marks.
In this question, students were asked to calculate the expected number of times a particular outcome might occur in 100 trials. This question asked students to complete a graph by adding two more columns using data provided in a table.
A number of students used the angle sum of the quadrilateral T U V S to show the desired result. This part of the question was not answered correctly, with less than 50% of the candidacy receiving the mark.
Students who substituted the given form of the sine rule failed to take the reciprocal of both sides. Many students did not understand the meaning of the term 'angle of depression' and simply repeated their part (iv) result or gave the answer 63◦.
The marking scheme took into account the two possible ways of interpreting this question, either using the original set of counters or those in the event space for 1. Many students could write the time in hours but could not convert it to years.
Students who understood the question still had problems with the concept of 'adjacent space'. d) Overall, a handful of students scored well in this part. This made it very difficult to find out whether the candidate wrote the question number or answered the question.
Students had difficulty with this question as they did not know the formula for the radius of a small circle and it was not given in the question. Another common mistake was to use the arc length of a small circle with 6400 as the radius.
The amount saved by lowering the interest rate to 7.8% for the third year in exchange for paying a $500 charge was to be calculated. The tax bracket that applies to taxpayers who pay 20% of their total income in taxes had to be found.
The second part of the question asked students to find the height of the Seng business block. This part required candidates to calculate the height of the ridge above the ceiling height. The main error in the 'formula-comparison' group was the identification of the focal length.
The most notable comment is that many students could not recognize the connection between parts (iii) and (iv). In general, the marking scheme awarded students one mark for recognizing the need to solve = 2x−9 and their answer to part (iii) simultaneously and another mark for correctly finding either the x ory coordinate of P from their two equations. Unfortunately, there were still some centers where the majority of graphs were much smaller and poorly drawn.
Unfortunately, many students only considered|x|equaltax and were happy to give solutions of x= 3, x=−2 even though x=−2 contradicted their graph. Many students attempted (iii) and (iv) using only algebraic techniques and did not see the connection or attempt to use their graphs from (i) and (ii). Although the diagram was provided, many students were unable to interpret it successfully, believing that they needed to evaluate.
Many students arrived at zero or in a negative area and tried to fix this by taking the absolute value. Definition and correct use of notation were often poor and it was sometimes difficult for examiners to know which parts of the student's solution were involved in arriving at the final answer.
Some students couldn't find exact values when substituting their trigonometric functions, thinking the values were in degrees instead of radians. Some students spread their work out too much with their solution written over several pages. Incorrect substitutions in their definite integrals (eg cos 0 = 0), ignoring 12 etc. in part of their evaluation, carelessness with negatives involved in subtractions.
Those who attempted an SSS or RHS test often did not provide enough information to support their case and gain full marks. Many students made careless transcription errors in naming angles, eg 6 BP Q was often written as 6 P P Q. Students should have seen the connection to part (i) and been able to derive a pair of corresponding angles of equal in congruent triangles.
Some students demonstrated correct logic but did not achieve full marks because they did not provide any justification for their statements. Candidates should also check that they have carefully copied the figures in the question into their notebooks.
They also received points for replacing their (often incorrect) solution of dAdx = 0 in 12 ·x√. However, it was disappointing to see that a significant number of students failed to see that negative answers for x or values of x > 2.5 are physically impossible. For this part of the question, the students had to think about a logo with stripes, the areas of which formed an arithmetic sequence. The most common approach has been to consider the difference in area of the then and n+1e square.
An alternative, but uncommon, geometric approach was to find the area of two rectangles or trapezoids. This method had the advantage of being algebraically simpler, although it was the student's burden. It is possible that some students were discouraged from doing parts (ii) and (iii) because they could not understand the concepts in part (i).
Most students achieved at least partial marks here, usually for the front line area.
The question was generally well answered, with many students receiving near full or full marks. Students did not score the second point if they failed to show the necessary connection on the test. Not all students integrated the constant and some changed the question from πR sec22x−1dx to πR sec2(2x−1)dx.
A common mistake was to extract 1/2 as a common factor from an exact expression without adjusting the second term to π3. One method that inevitably failed was to start with dvdt = -kv and use integration. Many students figured out that C = 100 without showing any function and then correctly plugged the value into part (iii) to find it.
In this section, many candidates revealed their lack of knowledge about the relationship between exponentials and logarithms. It was good to see that many students who had a forv value greater than 85 declared that their answer was wrong and also gave a reason for it.
The meaning of the interval 0 < t < 3π was missed by most candidates in all 3 parts of the question. Many who attempted this part did not realize that a particle at rest has zero velocity and that they should have used a derivative to find when this happens. In general, the graph of functiont+sint was completely unknown to the vast majority of 2 Unit candidates.
The concept of curvature and its relation to inflection points or the sign of the first derivative was obviously not understood by most candidates. Although there were very few completely correct solutions, the better candidates presented very good sketches with evidence of a clear understanding of the information from (i) and (ii) and of the requirements of the question. Generally, many candidates plotted marks for this part of the question and those who got decimal values for the x-values finished with good plots.
A sketch like this requires wording (horizontal fold, stationary point, fold) to clearly indicate the nature of the point, and it also requires that these types of points be drawn correctly. Many otherwise good sketches had no labels or had the correct labeling but no attempt to make the curve horizontal at (π, π).
Students had to recognize that the point where t = 1 on the velocity curve corresponds to a local maximum on the displacement curve. The better students indicated that it was a stationary point, a horizontal turning point or a turning point. The students had to determine the distance traveled in the first two seconds and thus the amount of ink used.
Students who could integrate with t = 0, 1, 2 did well, as did students who could draw a linear representation of x=f(t) or part of the graph of x=f(t).
In this part, students had to find the equation of the line connecting P to Q, given that tanβ = m. For this part, students had to show that the x-coordinates of P and Q were solutions of the given equation. However, less than half of the students who attempted part (b) went on to part (i) or (ii).
They were then asked what the probability was that exactly two dice showed a 2. The students were asked what the probability was that exactly two dice showed the same number. There were many students who scored 12 and many scored zero even though they attempted to answer several parts of the question.
Those who considered the speed of the plane relative to the sailor were more likely to complete this part correctly. There were many variations in (z + 2i)(z2 − 2iz − 4), and subsequent errors in the handling of the quadratic, such as the statement (−2i) 2 = 4 in the discriminant calculation. Most candidates who attempted this part took indefinite integrals of the functions in (ii), and then interchanged variables (sometimes adding constants of integration in different places).
The integration involves a simple standard integral and this part of the question was done well by most candidates.
Unit (Additional) and 3/4 Unit (Common)
Unit (Additional)
