This Paper has

**55**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 have their usual meanings.) |

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) |
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(d) |
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(e) |
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(f) |
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(g) |
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(h) |
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(i) | f (x, y) = x^{3} + y^{3} + 3xy, show f_{xy} = f_{yx}. | (0) | |||||||||||||||||||||||||||||||||||||||||||||||||||||

(j) |
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2. | (a) | Using properties of determinant, show that
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(b) | If a = 3i + j + 2k and b = 2i − 2k + 4k, find: (i) a x b, (ii) | a x b |, (iii) unit vector perpendicular to both a and b. | 5 | (0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||

(c) |
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3. | (a) |
^{3} in the form pA + qI. | 5 | (0) | |||||||||||||||||||||||||||||||||||||||||||||||||||

(b) |
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(c) | Two firms are competing for business under the condition so that one firm’s gain is another firm’s loss. Firm A’s pay–off matrix is given below:
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4. | (a) | Verify that a saddle point exists for the function z = 18x ^{2} − 6y^{2} − 36x − 48y. | 5 | (0) | |||||||||||||||||||||||||||||||||||||||||||||||||||

(b) | Solve the following L.P.P. by simplex method.
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(c) | A T.V. repair–man finds that the time spent on his job has an exponential distribution with mean 30 minutes. If he repairs sets on "first–come first–served" basis and if the arrival of sets is approximately Poisson with an average 10 per 8 hours a day, then what is the repair–man’s expected idle time each day? Also obtain average number of units in the system. | 5 | (0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||

5. | (a) | The supply function of a product is y = 3x^{2} − 4, Find the producer’s surplus when 4 units are supplied. | 5 | (0) | |||||||||||||||||||||||||||||||||||||||||||||||||||

(b) | Obtain an initial B.F.S. to the transportation problem by north–west corner method.
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(c) | The total cost function of a company is c = 1/3 x^{3} − 5x^{2} + 28x + 10, where c is the total cost and x is output. A tax at the rate of Rs. 2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by p = 2530 − 5x, where Rs. p is the price per unit of output, find the maximum profit. | 5 | (0) |

SECTION II(Statistical Techniques — 30 marks) |

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

6. | Answer any five of the followingchoose the correct alternative stating proper reason: | 2x5=10 | ||||||||||||||||||||||||||

(a) | Probability of getting 2 heads in a single throw of 4 perfect coins is
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(b) | P {4 < x < 6} of a Poisson variate with mean parameter 1 (given e^{−1} = 0.3678) is
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(c) | Probability of getting 8 points by throwing 2 unbiased dice is
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(d) | Given P {0 ≤ z ≤ 0.45} = 0.1736 and P {0 ≤ z ≤ 1.92} = 0.4726 then P {−0.45 ≤ z ≤ 1.92} is
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(e) | For the two regression lines 9x + 3y = 46 and 3x + 12y = 19, the correlation coefficient is
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(f) | In order to test whether a coin is fair or not it is tossed 5 times. The null hypothesis of fairness is rejected if and only if the number of heads is 0 or 5. The probability of type −I error of the test is
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(g) | Probability of getting a zero value of a binomial distribution with mean 4 and variance 3 is
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(h) |
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(i) | A random sample of size 25 has been drawn from a normal population with variance 9. If the sample mean is 5 the 95% confidence interval for the population mean is
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7. | (a) | If P(A) = ¼ P(B) = 2/5 and P(A ∪ B) ½ find (i) P(A ∩ B) and (ii) P(A ∪ B), where A, B are mutually exclusive. | 5 | (0) | ||||||||||||||||||||||||

(b) | A class consists of 50 students out of which the number of girl students is 10, In the class 2 girls and 5 boys are rank holders in the previous examination. If a student is selected at random from the class and is found to be a rank holder, what is the probability that the student selected is a girl. | 5 | (0) | |||||||||||||||||||||||||

8. | (a) | Fit a Poisson distribution to the following data:
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(b) | Find the mean and s.d. of a normal distribution where 8% of the items are over 64 and 31% are under 45. Given : P{0 < z < 0.495 = 0.19} and P{0 < z < 1.405} = 0.42, where z is a N(0, 1) variate. | 5 | (0) | |||||||||||||||||||||||||

9. | (a) | A group of 5 patients treated with medicine A, weight 42, 39, 48, 60, 41 kg. while a second group of 7 patients from the same hospital treated with medicine B, weight 38, 42, 56, 64, 68, 69, 62 kg. Do you agree with the claim that medicines increase the weight equally? [Value of t at 5% level of significance of 10 d.f. is 1.812]. | 5 | (0) | ||||||||||||||||||||||||

(b) | A die was thrown 90 times with the following results:
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10. | (a) | A confectioner sells confectionery items. Past data of demand per week (in ‘00 kg) with frequency is given below:
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(b) | The following are the marks obtained by 7 students in two subjects. Compute the rank correlation coefficient.
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11. | (a) | A bag contains defective articles the exact number of which is unknown. A sample of 100 from the bag gives 8 defective articles. Find the possible limits for the proportion of defective articles in that bag. | 5 | (0) | ||||||||||||||||||||||||

(b) | An educational entrepreneur in order to run one out of two software courses CUR and FUT, has obtained following estimates from the software experts:
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SECTION III(Economic Techniques — 30 marks) |

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

(a) |
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(b) | If Laspeyre’s index number is 114.4 and Paasche’s index number is 114.6, find out the Fisher’s ideal index number. | (0) | ||||||||||||||||||||||

(c) | The total daily cost (in Rs.) for producing x dot–pens is TC = 2.5x + 3000. If each dot–pen sells for Rs. 5, what is the break–event point? | (0) | ||||||||||||||||||||||

(d) | Write down the normal equations in fitting a parabolic equation y = a + bx + cx^{2} by the method of least square. | (0) | ||||||||||||||||||||||

(e) | If 12½% fall in price causes a 25% rise in demand, then find price elasticity of demand and its nature. | (0) | ||||||||||||||||||||||

(f) | If r_{12} = 0.7, r_{23} = 0.4 and r_{31} = 0.6, calculate R_{1.23}. | (0) | ||||||||||||||||||||||

(g) | With the base year 1960 as the base, the C.L.I in 1972 stood at 250. X was getting a monthly salary of Rs. 500 in 1960 and Rs. 750 in 1972. How much should X have received as extra allowance in 1972 to maintain his standard of living in 1960? | (0) | ||||||||||||||||||||||

(h) | Given total cost = 250 − 15x − 12x^{2} + x^{3}, find at what level of x diminishing marginal return begins. | (0) | ||||||||||||||||||||||

13. | Answer any four of the following: | 5x4=20 | ||||||||||||||||||||||

(a) | The total revenue from the sale of q units is given by R = 40q − 5q^{2}. Calculate the price elasticity of demand, when the marginal revenue is 10. | (0) | ||||||||||||||||||||||

(b) | From the given data, show that Fisher’s ideal index number satisfies the Time Reversal Test.
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(c) | The table below gives the number of production of a commodity:
Using the method of least squares fit a straight line. The company has physical facilities to produce only 70.6 units (in ‘000 tons) a year and it believes that at least for the next decade the trend will continue as before. | (0) | ||||||||||||||||||||||

(d) | If r_{12} = 0.8, r_{13} = 0.6, r_{23} = −0.2 then examine whether these are inconsistent. | (0) | ||||||||||||||||||||||

(e) | Suppose the demand curve is given by q = 10 − p. What is the total consumer’s surplus from consuming 6 units of the item? If the price changes from Rs. 4 to Rs. 6, what is the change in consumer’s surplus? | (0) | ||||||||||||||||||||||

(f) | Quarterly sales in (Rs. ’000) of a company are given below:
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