**56**answerable questions with

**0**answered.

I—12(QNM)Revised Syllabus | |

Quantitative Methods | |

Time Allowed : 3 Hours | Full Marks : 100 |

SECTION I(Mathematical Techniques — 40 marks) |

Answer Question No. 1 (compulsory — 10 marks) andtwo other questions (15x2 = 30 marks) from this section. |

1. | Attempt any five of the following: | 2x5=10 | ||||||||||||||||||||||

(a) |
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(b) |
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(c) | Show that the vectors corresponding to the positions of the points (2, 1, –1) and (1, 1, 3) are at right angles. | (0) | ||||||||||||||||||||||

(d) | Determine f(x – 1) when f(x + 3) = 2x^{2} – 1. | (0) | ||||||||||||||||||||||

(e) |
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(f) |
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(g) | Differentiate x^{5} W.r.t x^{2}. | (0) | ||||||||||||||||||||||

(h) |
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(i) |
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(j) | A function f(x) is defined as follows
Is f(x) continuous at x = 3? | (0) | ||||||||||||||||||||||

2. | (a) |
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(b) |
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(c) | A shop–keeper stocks four brands of bathsoaps. The costs of the four brands are given by the row vector A = (1, 15, 20, 25). The beginning inventory of the soaps is given by the vector
Assuming the purchase of inventory, determine the cost of goods sold during the period. | 5 | (0) | |||||||||||||||||||||

3. | (a) | Discuss the continuity of f(x) at x = – 2, where
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(b) |
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(c) |
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4. | (a) |
what is the value of q, when average cost is minimum ? Verify that at this level average cost = marginal cost. | 4 | (0) | ||||||||||||||||||||

(b) | Show that there is a saddle point for function z = 18x^{2} – 6y^{2} – 36x – 48y. | 5 | (0) | |||||||||||||||||||||

(c) | Marketing department of a company calculated the pay–offs in terms of yearly net profits for each of the strategies of expected sale prices in following table:
Which strategy should the marketing executive choose on the basis of –
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5. | (a) | Find the area of the region bounded by the curves 6y = x^{2} and 6x = y^{2} | 4 | (0) | ||||||||||||||||||||

(b) | Obtain a polynomial of degree 3 passing through points (0, 1), (1, 2), (2, 5) and (3, 16). | 5 | (0) | |||||||||||||||||||||

(c) | A Company produces two types of containers K and L. Each Product has resource requirements and profit contribution as follows:
In addition because of demand, a maximum of 4 units of container K will be produced. Obtain by gra0phical method, the optimal production plan that maximizes the profit. | 6 | (0) |

SECTION II(Statistical Techniques — 30 marks) |

Answer Question No. 6 (compulsory — 10 marks) and twoother questions (10x2 = 20 marks) |

6. | Answer any five of the following | 2x5=10 | |||||||||||||||||||||||

(a) | If A_{1}, A_{2}, A_{3}, A_{4} are equally likely, mutually exclusive and exhaustive events, then show that P( A_{1}) is:
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(b) | For two equally likely, exhaustive and independent events A and B, P(AB) is:
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(c) | Let A and B are two events such that P(A) = 0.4, P(A U B) =0.7 and P(B) = p. For what choice p, are A and B independent?
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(d) | If the mean and variance of a binomial distribution with parameters (n, p) are 40 and 30 respectively, then the parameters are:
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(e) | If the expectation of a Poisson variable is 1, then p(x > 1), is
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(f) | The p.d.f. of a continuous variable is given by f(x) = kx(x – 1), 0 < x < 1; the value of k is
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(g) | Rank correlation coefficient between the marks in Mathematics and Statistics obtained by a group of students is 2/3 and sum of the squares of the differences in ranks is 55, then number of students in the group is:
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(h) | Two lines of regression are given by x + 2y = 5 and 2x + 3y = 8. Then r_{xy} is :
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(i) | A random sample of size 5 is drawn without replacement from a finite population consisting of 41 units. If the standard deviation of the population is 6.25, then standard error of the sample mean is
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(j) | A sample random sample of size 100 has mean 15 and population variance 25. Then the 99% confidence interval for the population mean is:
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7. | (a) |
(i) Exactly one of A and B occurs and (ii) none of them occurs. Also examine whether the events A and B are independent. | 5 | (0) | |||||||||||||||||||||

(b) | From 20 tickets marked with the first 20 numerals, one is drawn at random. What is the probability that it is a multiple of 3 or 7 ? | 5 | (0) | ||||||||||||||||||||||

8. | (a) | The table below shows the respective heights in centimeter of 10 fathers and their eldest sons:
Find the rank correlation coefficient. | 5 | (0) | |||||||||||||||||||||

(b) | Is it likely that a sample of size 300 whose mean is 12, is a random, sample from a large population with mean 12.8 and s.d. 5.2 ? | 5 | (0) | ||||||||||||||||||||||

9. | (a) | Between the hours 2 p.m. and 4 p.m. the average number of phone calls per minute coming into the switch board of a company is 2.35. Find the probability that during one particular minute, there will be at most 2 phone calls. [Assume Poisson distribution. Given e | 6 | (0) | |||||||||||||||||||||

(b) |
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10. | (a) | A company has head office at Kolkata and a branch office at Mumbai. The personal director wanted to know if the workers at the two places would like the introduction of a new plan of work and a survey was conducted for this purpose. Out of sample of 500 workers at Kolkata 62% favoured the new plan. At Mumbai out of a sample of 400 workers 41% were against the new plan. Is there any significant difference between the two groups in their attitude towards the new plan at 5% level? | 5 | (0) | |||||||||||||||||||||

(b) | A Mumbai film director claims that his films are liked equally by males and females. An opinion survey of a random samples of 1,000 film-goers revealed the following results:
Is the film director’s claim supported by the data?
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11. | (a) | An I.Q. test was administered on 10 candidates before and after they were trained. The results are given below:
Test whether there is any improvement in I.Q. after the training. | 5 | (0) | |||||||||||||||||||||

(b) | You are given the following payoffs of three acts A_{1}, A_{2} and A_{3}, and the states of nature S_{1}, S_{2}, S_{3}.
The probability of the three states of nature are respectively o.1, 0.7 and 0.2. Calculate and tabulate E.M.V. and conclude which of the acts can be chosen as the best. | 5 | (0) |

SECTION III(Economic Techniques — 30 marks) |

12. | Attempt any five of the following: | 2x5=10 | |||||||||||||||||

(a) | Express MR in terms of price elasticity and price. | (0) | |||||||||||||||||

(b) | Show the elasticity of demand with respect to price for the demand function x = 3p^{—2} is constant, p and x being the price and quantity in demand respectively. | (0) | |||||||||||||||||

(c) | The daily cost of production c for x units of an assembly is given b c(x) = Rs. 12.5x + 6400; and selling price of each unit Rs. 25. If now the selling price is reduced by Rs. 2.5 per unit, what will be the break even point? | (0) | |||||||||||||||||

(d) | If 10% fall in price causes a 15% rise in demand, then find price elasticity of demand and its nature. | (0) | |||||||||||||||||

(e) | The trend of annual sales of a company is as follows: Y Convert the equation to a monthly trend equation. | (0) | |||||||||||||||||

(f) | Justify the statement: "the purchasing power of money decreases as the wholesale price index increases". | (0) | |||||||||||||||||

(g) | For a trivariate distribution r_{12} = 0.7 and r_{23} = r_{13} = 0.6. Calculate r_{123}. | (0) | |||||||||||||||||

(h) | Determine the new indices for following 3 years after shifting base year from 1970 to 1988:
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13. | Answer any four of the following | 5x4=20 | |||||||||||||||||

(a) | A demand law is given by x = 100 — 2p where x is the quantity demanded and p is the price. If the price elasticity of demand at p = 10 is increased by 50% then obtain the percentage variation in demand. | (0) | |||||||||||||||||

(b) | Calculate the index number of prices for 1995 on the basis of 1990 from the data given below:
If the weights of commodities A, B, C, D are increased in the ratio 1 : 2 : 3 : 4, then what will be the increase/decrease in Index numb | (0) | |||||||||||||||||

(c) | Calculate the seasonal from the following data using the average method:
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(d) | calculate the 4–yearly moving average from the following data:
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(e) | Given the following transaction matrix, find the gross output to meet the new final demand of 200 units of agriculture and 800 units of Industry:
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(f) | Write short notes on any one of the following :
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