General:

The overall response of the candidates in this paper was mixed. Majority of the candidates attempted all questions. Some of the candidates performed well, others were ignorant of even basics. Majority of the candidates were deficient in the conceptual understanding of the subject and interpretation of their results. The teaching institutions must emphasize this aspect of the subject.

Question-wise comments are given below:

Q.1

This was an easy question and majority of the candidates scored high on this question. The solution of all three parts of the question was based on the knowledge of the following facts:

(i)

Two lines are perpendicular to each other if the product of their slopes is equal to 1.

(ii)

Two lines are parallel to each other if their slopes are equal.

Some candidates gave graphical display and if this was correct, it was accepted; however, it was a lengthy method and must have consumed lot of time.

Q.2

Both parts of this question were quite simple and majority of the candidates successfully solved both the parts.

(a)

This part could be solved through a number of alternative methods i.e., the candidates could solve this question either by factoring the left hand side of the equation or by transforming the 4^th degree equation into a quadratic equation by taking m² = x. Some candidates used the quadratic formula for the original equation and ignored the degree of the equation altogether, and calculated only two roots instead of four. Such candidates could also determine all roots of the equation if they would have taken the square root of the values obtained with the help of quadratic formula or if they had transformed the equation into second degree before applying the quadratic formula. A few candidates successfully arrived at the correct solution by converting the equation into perfect square.

(b)

This part of the question was also very simple for those candidates who knew the formulas. Majority of the candidates scored high on this question.

Q.3

(a)

This part of the question was simple and straightforward. Present value of the amount was to be determined using formula for simple interest. Majority of the candidates successfully solved it. Surprisingly some of the candidates found the present value larger than the future value, clearly displaying lack of conceptual understanding.

(b)

This part of the question has three sub parts. Part(i) was correctly answered by a high proportion of the candidates. Majority of the students did know the formula of present value of annuity and applied it correctly. However, some candidates used the formula for future annuity instead of the formula for the present value. It appears that candidates do remember various formulas but which one is to be used in a particular situation is the major stumbling block for them. Some candidates who found the correct answer to part (i) were unable to answer parts (ii) and part (iii), particularly part (ii), which clearly showed that conceptual understanding was lacking.

Q.4

(a)

The important step in this part of the question was to convert the demand function into total revenue function by multiplying the demand function with x. Some of the students who were not successful in solving this part were those who were unable to take this first step. Some of the candidates used lengthy procedure using quotient rule method of obtaining first derivative, because they did not know that 1/25,000 is a constant and not a variable.

(b)

In this part three main errors were made by the candidates, as described below:

(i)

After obtaining the first derivative, some of the candidates were unable to find the critical points, because x² was in the denominator.

(ii)

Many candidates did not know that minus value of x (the number of employees) was irrelevant in this situation.

(iii)

Some candidates did not check the critical point whether it is a maximum or minimum point. It could have been done by taking the second derivative and then putting the value of x in the expression so that they can show explicitly that the value of second derivative is positive and it is the point of minima.

Q.5

(a)

A very simple question and a very large number of the candidates obtained full marks. However, some candidates were unable to solve this question, because, they did not know that in the dimension of a matrix, the order is first number of rows and then number of columns.

(b)

This was a simple question on the multiplication of Matrix which was attempted correctly by majority of the students. 

Q.6

This question had three requirements i.e.,

(i)

To sketch the four lines and identify the feasible region;

(ii)

To find the corner points either through graph or by simultaneously solving the pairs of equations; and

(iii)

Determine the maximum profit by putting these values of the corner points in the profit function.

Surprisingly only a few candidates were able to draw all the lines, particularly the line 3y – 2x = 0, which passes through the origin. Since they were unable to draw all four lines, feasible region could not be determined.

Although corner points can be obtained from the graph if the lines are plotted carefully, but the best way was to solve the pair of equations. This was not followed by a number of candidates.

Q.7

(a)

This part required comparison of two scores by compiling their Z-values. Instead, majority of the candidates used co-efficient of variation (CV) which was not the correct approach.

(b)

Only a few candidates attempted this part correctly. This part could also have been solved by using basic probability concepts and logical reasoning.

Q.8

This was one of the easiest questions. Majority of the candidates attempted this question correctly as there was no complication whatsoever.

Q.9

A straightforward question of establishing a least square regression line of number of passengers and bus-ticket price.

Part (a) and part (c) were successfully solved by majority of the candidates, but only a few candidates, interpreted the result in part (b).

It has been observed that candidates generally do not know how to interpret the results. More emphasis should be placed on it by the educational institutions as this will strengthen the concepts of students.

Q.10

It was again a simple question of computing coefficient of correlation and coefficient of determination. The weakest area was the interpretation of the two coefficients as is mostly the case as discussed in Q.9 above.

Q.11

(a)

Majority of candidates were unable to find the area in each half of the curve that could be calculated by dividing the given area with two and finding the value of Z for the resultant. Also, a number of candidates did not use the symmetry property for finding the value of –Z. 

(b)

Candidates should note that the area to the right of x = 70 was 20% i.e. 0.20. Therefore, area under the curve between mean and x = 70 should have been 0.50 – 0.20 = 0.30. Many candidates did not perform this step and this was the main error made by them. However, a vast majority solved the question correctly.

(c)

This part was solved correctly by majority of the candidates.

Q.12

(a)

Majority of the candidates correctly reached the value of mean of sampling distribution; however, many candidates were unable to calculate the value of standard deviation of the sampling distribution. Such candidates simply took the standard deviation of the total population as the standard deviation of sampling distribution. Candidates should note that the standard deviation of the sampling distribution can be calculated by dividing the standard deviation of the entire population with the square root of the sample size. Majority of the candidates were able to solve second portion of this part correctly.

(b)

Although most candidates were able to find the value of Z at 95% confidence level correctly, only a small percentage did apply the correct formula for determining the sample size, which again shows lack of understanding of the concept.

Q.13

In this question the candidates were required to determine the 90% confidence interval for population proportion. However, a large number of candidates used the formula of confidence interval for population mean. Some of them could not find the Z values at the 90% confidence level; others did not use the proper denominator for finding the standard error of proportion. Only a few candidates gave the correct interpretation of the result.

Q.14

Majority of the candidates used Z-distribution as test statistic instead of t-distribution. The candidates should recall that t-distribution should be used when the sample size is small and the given variable has normal distribution in the population. It was also surprising that a large number of the candidates could not compute the sample standard deviation. Some of them even did not correctly state the null hypothesis and alternative hypothesis. Only a few candidates obtained full marks in this question.