This Paper has

**57**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) and two(15x2 = 30 marks) from this section . other questions |

1. | Attempt any five questions: | 2x5=10 | ||||||||||||||||||||

(a) |
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(b) | If a = 2i + 5j + 3k, b = i – 2j – k, find the value of| 3a+ 2 b| | (0) | ||||||||||||||||||||

(c) |
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(d) | For what value(s) of x, the function √(x – 1) (x – 2) is not defined? | (0) | ||||||||||||||||||||

(e) | If ƒ(x) = e^{px + q}, (p, q are constants), then show that ƒ(a).ƒ(b).ƒ(c) = ƒ(a + b + c) e^{2q} | (0) | ||||||||||||||||||||

(f) |
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(g) |
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(h) |
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(i) | Draw the graph: ƒ(x) = 1, x ≥ 0 = — 1, x < 0 and show if ƒ(x) exists at x = 0. | (0) | ||||||||||||||||||||

(j) |
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2. | (a) | Show that the vectors i – 2j + 3k, 2i + 4k and 3j form a right–angled triangle where i, j, k are the unit vectors perpendicular to each other. | 5 | (0) | ||||||||||||||||||

(b) | Solver by Cramer’s Rule: x + y + z = 6, 2x – y + 3z = 9 and x + 3y – 2z = 1. | 5 | (0) | |||||||||||||||||||

(c) | A shopkeeper stocks 4 brands of bath–soap. The cost of the four brands are given by row vector A = (10 12 15 20). The beginning inventory of the shop is given by the vector.
Assuming no purchase of inventory, determine the cost of goods sold during the period. | 5 | (0) | |||||||||||||||||||

3. | (a) |
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(b) |
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(c) | The slope of a curve at each point is proportional to the square of the abscissa of the point. Find the equation of the curve if it passes through the point (2, 2) with a slope of 1. | 5 | (0) | |||||||||||||||||||

4. | (a) | Solve by matrix method: 4x + 3y + z = 8, 2x + y + 4z = –4 and 3x + z = 1. | 5 | (0) | ||||||||||||||||||

(b) | For a certain establishment the total cost function C and the total revenue function R are given by C = x ^{3} – 12x^{2} + 48x + 11 and R = 83x – 4x^{2} – 21 where x = output. Obtain (i) the output for which profit is maximum and (ii) the maximum profit. | 5 | (0) | |||||||||||||||||||

(c) | Solve the following LPP graphically: Maximise z = 7x + 5y subject to the constraints: x + 2y ≤ 6, 4x + 3y ≤ 12, x, y ≥ 0. | 5 | (0) | |||||||||||||||||||

5. | (a) | Two firms are competing for business under the conditions so that one firm’s gain is another firm’s loss. Firm A’s pay–off matrix is given below:
Suggest optimal strategies for the two firms and the net outcome thereof. | 5 | (0) | ||||||||||||||||||

(b) | Obtain initial BFS to the following TP by NWCM:
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(c) | In a bank, cheques are cashed at a single ‘teller’ counter. Customers arrive at the counter in Poisson manner at an average rate of 30 customers per hour. The ‘teller’ takes on an average, a minute and a half to cash cheque. The service time has been shown to be exponentially distributed. Calculate (i) the percentage of time the ‘teller’ is busy and (ii) the average time a customer is expected to wait. | 5 | (0) |

SECTION II(Statistical Techniques — 30 marks) | |||

Answer Question No. 6 (compulsory — 10 marks) and twoother questions (10 x 2 = 20 marks) from this section. |

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

(a) | In a throw of two unbiased dice, the probability of throwing total seven points is
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(b) |
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(c) | A card is drawn from each of two well–shuffled pack of cards. The probability that at least one of them will be an ace is
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(d) | An oil exploration firm finds that 5% of the test wells it drills yields a deposit of natural gas. If the firm drills 6 wells, then the probability of at least one well will yield gas is
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(e) | Assuming that on an average 2% of the output in a factory manufacturing certain bolts is defective and that 200 units are in a package. The probability of at most 3 defective bolts may be found in that package is [Given e^{–4} = 0.0183]
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(f) | A box contains 4 white and 6 red balls. If 2 balls are drawn at random from it, thus the mathematical expectation of the number of red balls is
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(g) | A random sample of size 5 is drawn without replacement from a finite population of size 54. If s.d. of population is 5.5, then the standard error (S.E.) of sample mean is
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(h) | Karl Pearson’s coefficient of correlation between x and y is 0.52, their covariance is +7.8. If the variance of x is 16, then s.d. of y will be
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(i) | Two lines of regression are given byx + 2y = 5 and 2x + 3y = 8. Then r_{xy} is
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(j) | In order to test whether a coin is fair or not it is tossed 4 times. The null hypothesis of fairness is rejected if and only if the number of heads is 0 or 4. The probability of type–I error of the test is
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7. | (a) | A speaks the truth in 60% and B in 90% cases. In what percentage of cases are they likely to contradict each other in stating the same fact? | 5 | (0) | |||||||||||||||||||||||

(b) | If a random variable x follows a Poisson distribution such that Prob. (x = 1) = Prob. (x = 2); find P(x = 3). Given e^{–2} = 0.1353] | 5 | (0) | ||||||||||||||||||||||||

8. | (a) | Suppose half of the population of town are consumers of rice. 100 investigators are appointed to find out its truth. Each investigator interviewed 10 individuals. How many investigators do you expect to report that three or less or the people interviewed are consumers of rice? | 5 | (0) | |||||||||||||||||||||||

(b) | Pay–offs of three acts A, B and C and states of nature X, Y, Z are given below:
The probabilities of the states of nature x, Y and Z are 0.3, 0.4 and 0.3 respectively. Calculate the EMV for trhe data gives and select the best act. Also find the expected value of perfect information (EVPI). | 5 | (0) | ||||||||||||||||||||||||

9. | (a) | For a certain bivariate data, following results are obtained: n = 25, Σx = 125, Σy = 100, Σx ^{2} = 650, Σy^{2} = 436, Σxy = 520. Find the regression coefficients and the correlation coefficient. | 5 | (0) | |||||||||||||||||||||||

(b) | A sample of 625 members has a mean 3.2 cm and s.d. 2.12 cm. Can the sample be regarded as drawn from a population with mean 3.08 cm? Find the 95% confidence limits for the population mean. [Given P _{r} {z ≥ 1.96} = 0.025 where z is N(0, 1) variable]. | 5 | (0) | ||||||||||||||||||||||||

10. | (a) | The marks scored by five students in a test of statistics carrying 100 marks are 50, 60, 50, 60 and 40. Draw all possible random samples of size 4 without replacement; find the sampling distribution of sample mean. Hence find the standard error of the sample mean. | 5 | (0) | |||||||||||||||||||||||

(b) | A Kolkata film director claims that his films are liked equally by males and females. An opinion survey of a random sample of 1000 film–goers revealed the following results:
Is the film director’s claim supported by the data?
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11. | (a) | Find the rank correlation coefficient (r) for following data of marks obtained by 8 students of mathematics and statistics:
Interpret r. | 5 | (0) | |||||||||||||||||||||||

(b) | A sample of 100 arrivals of customers in a departmental store is according to the following distribution:
Use the following random numbers to simulate for the next 10 arrivals: 49, 19, 89, 73, 05, 12, 76, 65, 39, 25. | 5 | (0) |

SECTION III(Economic Techniques — 30 marks) |

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

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

(b) | At what output level marginal cost is minimum when total cost C of output q is given by
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(c) | "I_{2007/2000} = 90". What does it mean? Is the reverse true? | (0) | ||||||||||||||||

(d) | Given (with usual notations): Σp_{0}q_{0} = 280, Σp_{0}q_{1} = 384, Σp_{1}q_{0} = 344, Σp_{1}q_{1} = 472 Determine (i) Laspeyre’s Index and (ii) Paasche’s Index. | (0) | ||||||||||||||||

(e) | Taking 2000 as base year, cost of living index of 2007 was 350. A person drawing a monthly salary of Rs. 800 in 2000, drew Rs. 3,000 and approached management for increased salary to maintain his standard of living as was in the year 2000. Should his demand be met? | (0) | ||||||||||||||||

(f) | If r_{12} = 0.6, r_{23} = 0.5 and r_{31} = 0.4, show that r_{12.3} is consistent. | (0) | ||||||||||||||||

(g) | Total daily cost (in Rs.) for producing x number of chairs is TC = 4.5x + 9100. If each chair sells for Rs. 50, find the break–even point. | (0) | ||||||||||||||||

(h) | Consumer’s demand for a commodity declines by 15% with price increasing from Rs. 4 to Rs. 5 per unit. Is demand elastic? | (0) | ||||||||||||||||

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

(a) | If the supply function is p = √9 + q and quantity sold in the market is 16, show that producer’s surplus is 14.7. | (0) | ||||||||||||||||

(b) | An equity into the budgets of the middle class families in a certain city are as follows:
Find CLI of 1976 as compared with that of 1975. | (0) | ||||||||||||||||

(c) | Given the production function Q = √LK where prices per unit of the variable inputs K and L are Rs. 3 and Rs. 4 respectively with total cost at Rs. 80. Determine the maximum output subject to the cost constraint. | (0) | ||||||||||||||||

(d) | Find the gross output, if the final demand for industry A and B are respectively 800 and 1600 units and input, output coefficient matrix is
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(e) | Find the trend values for the following series using a 3—year moving average with weights 3, 4 and 5.
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(f) | Write short notes on any one of the following: | |||||||||||||||||

(i) | Multiple correlation coefficient. | (0) | ||||||||||||||||

(ii) | Stratified sampling. | (0) |