**61**answerable questions with

**0**answered.

1—12(QNM)Revised Syllabus | |

Time Allowed : 3 Hours | Full Marks : 100 |

The figures in the margin on the right side indicate full marks. |

(Notations and symbols used have their usual meanings.) |

SECTION I (Mathematical Techniques — 40) |

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

Marks |

1. | Attempt any five questions:Choose the correct options showing the proper reasons/calculations. | 2x5 | |||||||||||||||||||||||||||||||||||

(a) |
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(b) |
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(c) | The value of x, for which vectors 2i — j + k, i — 3j — 5k and xi — 4j — 4k are coplanar, is
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(d) |
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(e) |
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(f) |
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(g) | The gradient of the curve y = x + √x^{2} + a^{2} is
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(h) | The function y = x^{3} – 3x^{2} + 7 has a minimum value at
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(i) |
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(j) |
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2. | (a) | Show that the vectors i + 2j + k, i + j – 3k and 7i – 4j + k are mutually orthogonal. Determine length of the vectors. | 5 | (0) | |||||||||||||||||||||||||||||||||

(b) |
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(c) |
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3. | (a) | Write short note on any one: | 5 | ||||||||||||||||||||||||||||||||||

(i) | North–West corner method for obtaining initial BFS in transportation problem. | (0) | |||||||||||||||||||||||||||||||||||

(ii) |
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(b) | If f(x) = √ x^{2} + a^{2} + a log_{e} (x + √ x^{2} + a^{2}), prove that
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(c) | Find the extreme value of the function f(x, y) = x^{2} – y^{2} + xy + 5y subject to x + y + 1 = 0. | 5 | (0) | ||||||||||||||||||||||||||||||||||

4. | (a) |
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(b) |
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(c) | The demand and supply function under perfect competition are y = 16 – x^{2} and y = 2 (x^{2} + 2) respectively. Find the market price, consumer’s surplus and producer’s surplus. | 5 | (0) | ||||||||||||||||||||||||||||||||||

5. | (a) | Solve the following LPP by using Simplex method. Maximize z = 4x + 3y subject to 2x + y ≤ 30, x + 2y ≤ 24, x ≥ 0, y ≥ 0. | 5 | (0) | |||||||||||||||||||||||||||||||||

(b) | A customer repairing shop has one mechanic only. Customers are appearing at a rate of 20 per hour. The time required to serve a customer has exponential distribution with mean of 144 seconds. Find the number of customers to be expected in the shop. Also find average waiting time of a customer (i) in the system, (ii) in the queue. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(c) | Solve the following game using the maximin–minimax principle;
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SECTION II (Statistical Techniques — 30 marks) |

Answer Question No. 6 (Compulsory — 10 marks) and two other questions(10x2 = 20 marks) from this section. |

6. | Answer any five of the following: | 2x5 | |||||||||||||||||||||||||||||||

(a) |
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(b) |
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(c) |
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(d) |
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(e) |
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(f) | For a random variable X following Poisson distribution with mean 2 then P(X ≥ 1) is
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(g) |
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(h) | Proportion of defective items in a large lot of items is p. Taking a random sample of 6 items from the lot and accepting the null hypothesis if the number of defectives in the sample being 5 or less, then the probability of type II error when p = 0.3 is
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(i) | If two simple random samples each of size 5 are drawn with and without replacement from a finite population of size 21 with variance 25, have standard errors of sample means s_{1} and s_{2} respectively, then s_{1}/s_{2} is
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(j) | A simple random sample of size 100 has mean 10, population variance being 25 and
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7. | (a) | The personnel department of a company has records which show the following analysis of its 200 engineers:
If an engineer is selected at random find the probability that he is (i) a graduate; (ii) a post–graduate given that his age is over 40; and (iii) under 30 and a graduate. | 5 | (0) | |||||||||||||||||||||||||||||

(b) | A shopkeeper of some highly perishable type of fruit observes that the daily demand of this fruit in his locality has the following probability distribution:
He sells for Rs. 10 a dozen while be buys each dozen at Rs. 4. Unsold fruits at the end of the day are sold next day at Rs. 2 per dozen. Assuming he stocks the fruits in dozens, how many dozens should he stock so that his expected profit is maximum? | 5 | (0) | ||||||||||||||||||||||||||||||

8. | (a) | A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers from the purchase department, 2 officers of the sales department and 1 Chartered Accountant. Find the probability of forming the committee such that there must be (i) one from each category; (ii) at least one from the purchase department; and (iii) the chartered Accountant. | 5 | (0) | |||||||||||||||||||||||||||||

(b) | Fit a binomial distribution to the following data and find expected frequencies:
Where x = number of heads, f = number of times x heads occur in 400 times of 4 tosses of a coin. | 5 | (0) | ||||||||||||||||||||||||||||||

9. | (a) | Find the correlation coefficient between x and y, the value of x when y = 10 and the value of y when x = 100 from the regression lines 2y = x + 50 and 3y = 2x + 10. | 5 | (0) | |||||||||||||||||||||||||||||

(b) | If X be a Poisson variable such that P(X = 0) + P(X = 1) = 4P(X = 2), find the probability that (i) X is positive, (ii) X is atmost one. | 5 | (0) | ||||||||||||||||||||||||||||||

10. | (a) | A random sample of size 10, drawn from a normal population with variance 4, has mean 48. Test at 5% level of significance the hypothesis that the population mean is 50.
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(b) | In an infantile paralysis epidemic, 500 persons contracted the disease. 200 received no serum treatment and of these 75 became paralysed. Of those who received serum treatment, 65 became paralysed. Was the serum treatment effective?
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11. | (a) | A population consists of 5 height 60, 61, 62, 64, 62 in feet. Drawing all possible samples without replacement of size 2 from the population find the sampling distribution of the sample mean. Also find mean of the sampling distribution. | 5 | (0) | |||||||||||||||||||||||||||||

(b) | Write short note on any one: | 5 | |||||||||||||||||||||||||||||||

(i) | Normal distribution and its uses; | (0) | |||||||||||||||||||||||||||||||

(ii) | Scatter diagram and its uses. | (0) |

SECTION III (Economic Techniques — 30 marks) |

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

(a) |
price and quantity of demand respectively. | (0) | ||||||||||||||||||||||

(b) | The marginal cost function when output is x units is given by MC = x^{2} – x + 4. Find the total cost function C and average cost function AC if fixed cost is 30. | (0) | ||||||||||||||||||||||

(c) |
are the prices and quantities demanded of the two commodities. | (0) | ||||||||||||||||||||||

(d) | If the seasonal index is 90 and the deseasonalised value of production is 30 find the production there. | (0) | ||||||||||||||||||||||

(e) | If the trend equation of production (y) is y = 81 + 2i with origin at 1992 and time unit = t unit = 1 year.Unit of y is 1,000 tonnes. What is monthly increase in production? | (0) | ||||||||||||||||||||||

(f) | If the Paasche’s price index number is written a weighted harmonic mean of price relatives what will be the weights? | (0) | ||||||||||||||||||||||

(g) | Net monthly income of an employee was Rs. 8,000/- per month in 2000. It rises to Rs. 10,000/- per month in 2008 by giving additional dearness allowance for compensating rightly. If the price index number in 2000 is 160, what was it in 2008? | (0) | ||||||||||||||||||||||

(h) | If r_{12}, r_{13}, r_{23} are equal to r (# 1) then find the value of multiple correlation coefficient R_{1.23}. | (0) | ||||||||||||||||||||||

(i) | The total daily cost for x articles is TC = 3x + 2,000 rupees. If each article is sold at Rs. 5 what is the break even point? | (0) | ||||||||||||||||||||||

13. | Answer any four of the following: | 5x4 | ||||||||||||||||||||||

(a) |
in terms of output x. Find the output for which AC is minimum. Find the total cost (C) and marginal cost (MC) for that output. | (0) | ||||||||||||||||||||||

(b) | A consumer demand curve is x = 100 – 2p where p and x denote price and quantity demanded respectively. If the price elasticity of demand when the price at p = 10 is increased by 40% then obtain the percentage decrease in demand. | (0) | ||||||||||||||||||||||

(c) | Calculate the trend values by the method of 4–yearly moving averages with weights 1 : 2 : 2 : 1 from the following data:
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(d) | Quarterly sales in thousand rupees of a company is given below:
Calculate seasonal indices if there is no appreciable trend. | (0) | ||||||||||||||||||||||

(e) | The price relatives and weights of a set of commodities are given below:
If the index for the set is 122 and the sum of weights is 40 find w | (0) | ||||||||||||||||||||||

(f) | On the basis of observations on some trivariate case r_{12} = 0.8, r_{13} = 0.7 and r_{23} = 0.6 prove that
where R | (0) | ||||||||||||||||||||||

(g) | Write short note on any one: | |||||||||||||||||||||||

(i) | Least square theory in fitting a straight line; | (0) | ||||||||||||||||||||||

(ii) | Leontief’s input output model. | (0) |