QUANTITATIVE METHODS

General:

The overall performance of the candidates was satisfactory. However, one thing was quite evident that the students were good as far as formulas were concerned but were weak in concepts and interpretation. The teaching faculty should put emphasis in their teaching on the understanding of the mathematical/statistical phenomena as well so that the candidates are atleast able to retain the concepts even if they forget the fomulas.

Question-wise comments are as under:

Q.1

This question consisted of 3 parts. All three were quite simple and the candidates scored well on this question.

(a)

In this part some candidates failed to establish the correct equation of cost i.e. (Cost + Profit = Sale price), and therefore arrived at all sorts of incorrect figures most commonly Rs. 255 (300-15% of 300).

(b)

The best way to solve this part of the question was to take log of both sides of the equation. Majority of the candidates did the same. However, in the second step, some of the students did not place the value (2x-1) in bracket and lost easy marks.

(c)

In this part some students did not use the quadratic formula instead they solved the equation through factorization. Though their method was correct, they could not secure any marks, as they were required to solve the equation by using quadratic formula. Few of those who used quadratic formula, placed the denominator under "" only instead of putting it under the entire numerator. Some of the students put b = -81 and c = 0 whereas the correct values of b and c were 0 and -81 respectively. This clearly shows that the candidates memorize the formulas but are poor in their application.

Q.2

Considerable number of candidates failed to establish the correct cost function by adding the levy of Rs. 0.20 per unit i.e. 0.2x in the cost function.

In part (b) the students were required to find the minimum price per unit at break-even level. They could have solved it by taking the number of units as 6000 in the cost function as well as the revenue function and then equating the two functions to find the price. Very few students could do it correctly which showed their inability to apply their knowledge to practical situations.

Q.3

(a)

It was a very simple question. Ordinarily it was required to be solved in two steps. However, some of the good students managed to solve it in one step also. While solving in two steps the compound amount at the end of year 1 should have been taken as the Principal amount for calculating the compounded value at the end of year 3. Most students couldn’t solve it correctly once again due to their inability to adopt to the situation although they knew the formula for compound interest.

(b)

Surprisingly, some of the candidates were unable to distinguish between compound interest and simple interest and used the formula for compound interest instead of simple interest.

·

Formula for present value was used. 

Q.5

(a)

Most students didn’t attempt this question. Very few knew the method of finding derivatives of expressions involving the value “e”.

(b)

This was one of the questions which majority of the candidates found difficult to complete. It could have been solved easily by first differentiating the expression on the left hand side and then simplifying it to arrive at the expression on the right hand side. Such questions need a lot of practice which was definitely lacking.

Q.6

(a)

In this part, the candidates were first required to find the marginal cost by taking the first derivative of the cost function. Almost all candidates carried out this step. Thereafter the first and second derivative of the marginal cost function should have been calculated. The value of x at which first derivative is zero was the output where marginal cost was minimum subject to the second derivative test. Quite a large number of candidates failed to take these steps. Instead, they equalized the marginal cost function to zero to find the value of x and incorrectly considered it to be the output at which marginal cost is minimum.

(b)

Surprisingly, many candidates were unable to determine the average cost function from the total cost function and lost some easy marks. Many of those who did manage to arrive at the average cost function were unable to find the output where average cost was minimum the procedure for which was the same as discussed in part (a) above.

(c)

In this part average cost was to be equated with marginal cost to get the required output. Surprisingly even after equating the two functions correctly, many students were unable to simplify successfully.

Q.7

All three parts of this question relates to basic rules and operations of matrices such as subtraction and multiplication.

(a)

In this part, larger bracket was to be solved first and the resultant matrix was to be multiplied with the first matrix. Majority of the candidates were able to solve this part successfully.

(b)

This was one of the poorest attempted questions. Most of the candidates were unable to build the required relationship from the information given in the question. A careful reading of the question yields the following set of equations:

x + 5 = y      (since the product AB exists)

11 – y = x     (since the product BA exists)

And the rest is nothing but solving the above two equations simultaneously.

(c)

This was one of the easiest and highest scoring question of the paper. Majority of candidates got it right.

Q.8

The performance in this question was mixed. Since this area had not been tested in the past many attempts, significant number of students did not attempt it at all. Majority of those who applied the correct formula to compute the co-efficient of skewness were unable to explain as to why it was positively skewed. Whereas, a number of candidates whose results were incorrect came up with correct justification that since mean > median > mode, the distribution is positively skewed.

Q.9

It was an easy question from a commonly tested area. But surprisingly, it proved to be a very low scoring question. Cost of living Index (CLI) could be computed by using either price–quantity relationship or expense–price relationship i.e.

CLI =                 or          CLI =   

Where,

W = expenses for the base year;

P₁ = Price for the year 2005;

Q₀ = Quantity for the base year; and

P₀ = Price for the base year

Although majority of the candidates knew the correct formula, they were unable to substitute correct values in the formula.

Since Q₀ was not given, the most common mistake committed by those who solved the question by applying the price quantity relationship was that they took the expenses of the base year as Q₀ instead of finding the base year quantities by dividing the expenses on each item by its base year price. Apparently they lacked conceptual understanding of the method and were unable to tackle small deviation from the direct application of formula.

Q.10

(a)

Both regression lines were required to be calculated. Majority of the candidates did it correctly. However, many students calculated just one equation and used the same to calculate the height as well as weight. Correct interpretation of the regression co-efficients was rarely given.

(b)

As is common with such questions very few students had clear cut idea about the type of relationship depicted by various values of ‘r’ as given in the question.

Q.11

Both parts of this question were quite easy and majority of the candidates scored high on this question. However, in part (ii) of (a) and in part (b) some candidates faced difficulties, as they did not seem to understand the meaning of the term “at least”.

Q.12

This question required simple application of normal distribution and most of the candidates managed to find the area under normal curve correctly. However, only a few of them calculated the expected number of days, which shows that they did not read the question carefully. Some of them did not round off the number of days to the nearest whole number.

Q.13

(a)

It was a fairly well attempted question. However, some candidates reversed the values of defect free items with defected items which lead them to wrong conclusion.

(b)

This part was successfully solved by a large number of candidates. However,  some of them could not find the correct value of Z at 98% level of confidence.

Q.14

The main mistake made by the candidates in this question was the use of z-distribution instead of t-distribution. The candidates must know that when the sample size is small and standard distribution of the population is not given,   t-distribution should be used. Some of the candidates who correctly applied the t-distribution were unable to find the correct table value of t. As always, a very small number of candidates could interpret the result correctly.

Q.15

Majority of the candidates successfully solved this question except the conclusion. Surprisingly, some candidates rejected the null hypothesis but said that the machine is working properly, a self-contradiction. A large number of students used the formula
Z =  which in this case was incorrect.