**56**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 marks) |

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

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

(a) | Find the value of t for which the vectors 3i + j + tk and 2i − 2j + 4k are perpendicular to each other. | (0) | ||||||||||||||||||||||||||||||||||||

(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) | The MR function of output x is MR = 3 (x + 1)^{2} + 3. Find the total revenue function. | (0) | ||||||||||||||||||||||||||||||||||||

(h) |
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(i) |
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(j) | A function is defined as follows:
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2. | (a) | Find the area of the triangle OAB (O being the origin). A = (1,2,3) and B = (−3, −2, 1). | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) | Show that
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(c) | A radio manufacturing company produces two types of items:
During a week the company produces 128 items, the total cost of these items being Rs. 37,280. Using matrix algebra, find the number of A type and B type items which have been produced during the week. | 5 | (0) | |||||||||||||||||||||||||||||||||||

3. | (a) |
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(b) |
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(c) |
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4. | (a) | Determine the points of inflexion of the function f(x) = x^{4} − 3x^{3} + 3x^{2} + 5x + 11. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) |
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(c) | Solve graphically the following LPP: Minimize Z = 4x + 6y subject to the constraints 2x + 3y ≥ 6, x + y ≤ 8, y ≥ 1 and (x, y) ≥ 0. | 5 | (0) | |||||||||||||||||||||||||||||||||||

5. | (a) | Solve the game by using dominance property
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(b) | Determine the initial basic feasible solution to the following Transportation Problem by Vogel‘s Approximation Method.
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(c) | In a tool crib manned by a single assistant, operators arrive at the tool crib at the rate of 10 per hour. Each operator needs 3 minutes on an average to be served. Find out the loss of production due to waiting of an operator in a shift of 8 hours if the rate of production is 100 units per shift. | 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. | Attempt any five of the followingChoose the correct alternative stating proper reason: | 2x5=10 | ||||||||||||||||||||||||||||||||

(a) | If A and B be two independent events such that P(A) = 0.4, P(A ∪ B) = 0.7 then the value of P(B) is
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(b) | If A, B and C be mutually exclusive and exhaustive events such that 1/3 P(C) = ½ P(A) = P(B) then the value of P(B) is
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(c) |
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(d) | If 6 fair coins are tossed simultaneously, then the probability of getting at least 4 heads is
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(e) | For the following probability distribution
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(f) | If the distribution of a random variable X is normal with mean 0 and variace 1 with P(0 ≤ X ≤ 1) = 0.3413, then P(| X | > 1) is
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(g) | If the correlation of a coefficient between x and y is 0.4, then the correlation coefficient between −3x and +5y is
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(h) | If a random sample of size 5 is drawn without replacement from a finite population of 41 units with σ = 10, then the SE of sample mean is
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(i) | A random sample of size 100 has mean 15, the population variance being 25, then the 95% confidence interval for the population mean is
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(j) | To test the null hypothesis H_{0}: μ = 0 against H_{1}; μ ≠ 0, 10 independent sample observations are drawn from a normal population with mean μ and unknown variance σ^{2}. If the sample mean and variance are 6 and 4 respectively, the value of the appropriate test statistic is
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7. | (a) | A purse contains 4 nickel coins and 9 copper coins, while another purse contains 6 nickel and 7 copper coins. A purse is chosen at random and a coin is drawn at random from it. What is the probability that it is a nickel coin? | 5 | (0) | ||||||||||||||||||||||||||||||

(b) | A problem is given to 5 students and their chances of solving it are ½, 1/3, ¼, 1/5 and 1/6. What is the probability that the problem will be solved? | 5 | (0) | |||||||||||||||||||||||||||||||

8. | (a) | A random variable X is binomially distributed with mean 4 and s.d. = √2.4. Find the probability that more than half the trials are successes. | 5 | (0) | ||||||||||||||||||||||||||||||

(b) | A book contains 100 misprints distributed at random throughout its 100 pages. What is the probability that a page observed at random contains at least three misprints? [Given e = 2.718]. | 5 | (0) | |||||||||||||||||||||||||||||||

9. | (a) | Calculate the coefficient of correlation for the following data;
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(b) | The regression lines of two correlated variables x and y are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Identify the regression lines x on y and y on x. Find the means and the correlation coefficient of the variables. | 5 | (0) | |||||||||||||||||||||||||||||||

10. | (a) | The coefficient of rank correlation of the marks obtained 10 students in two particular subjects was found to be 0.8. It was then detected that the difference in ranks in the two subjects obtained by one of the students was wrongly taken to be 7 in place of 9. What should be the correct rank correlation coefficient? | 5 | (0) | ||||||||||||||||||||||||||||||

(b) | In 120 throws of a single die the following distribution of faces was observed:
Can you that the die is biased? [Given: the value of X | 5 | (0) | |||||||||||||||||||||||||||||||

11. | (a) | Suppose a decision maker faced with three decision alternatives and two states of nature. Apply (1) maximin and (ii) minimax regret approach to the following pay–off table to recommend the decision.
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(b) | In one sample of 8 observations, the sum of squares of the deviation of the sample values from the sample mean was 84.4 and in the other sample of 10 observations, it was 102.6. Test whether these differences are significant at 5% level, given that F_{0.05;(7,9)} = 3.29 | 5 | (0) |

SECTION III(Economic Techniques — 30 marks) |

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

(a) | The demand and supply function under perfect competition respectively are y = 16 − x^{2} and y = 2(x ^{2} + 2). Find the market price. | (0) | ||||||||||||||||||

(b) | If the demand law is p = 1 + x, find the elasticity of demand in terms of x. | (0) | ||||||||||||||||||

(c) | The total daily cost (in Rs.) for producing x plastic chairs is TC = 25x + 4500. If each chair sells for Rs. 40, find the break even point. | (0) | ||||||||||||||||||

(d) | The price and quantity of video recorder in the year 2004 are Rs. 438 and 37 respectively. The corresponding values for the year 2005 are Rs. 462 and 18. Find the price relative and quantity relative for 2005. | (0) | ||||||||||||||||||

(e) | Convert fixed base index (FBI) number of the following three years into corresponding chain base index numbers.
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(f) | For the Cobb – Douglus production function Q = 3K^{1/5} L^{4/5}, write down the relative share of K and that of L. | (0) | ||||||||||||||||||

(g) | If r_{12} r_{23} and r_{13} are all equal, then what is the value of r_{13–2}? | (0) | ||||||||||||||||||

(h) | Determine the normal equations for fitting the exponential curve y = ab^{x} (where a, b are constants) to a set of observed data by the method of least squares. | (0) | ||||||||||||||||||

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

(a) |
average and marginal revenue at any out put. Verify this for linear demand law p = a + bx. | (0) | ||||||||||||||||||

(b) | Below are given the figure of production (in ’000 tonnes) of a sugar factory.
(i) Fit a straight line by the method of "least squares" and find the trend values. (ii) What is the months increase in production. | (0) | ||||||||||||||||||

(c) | Calculate the index number of price for 2005 on the basis of 2000 from the given data;
If the weights of commodities A, B, C, D are increased in the ratio 1 : 2: 3: 4 what will be increase in index number? | (0) | ||||||||||||||||||

(d) | Given r_{12} = 0.8, r_{13} = 0.6; find the maximum possible value of r_{23}. | (0) | ||||||||||||||||||

(e) | Find the 5 year weighted moving average with weights 1, 2, 2, 2, 1 respectively for necessary trend of the following time series:
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(f) | For the transaction matrix given below, find the matrix of technological coefficients and hence find out the total output for each industry if the new final demands are 1000 and 2000 units respectively.
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