**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) | Find the direction cosines of the vector 2i − 3j + 5k. | (0) | |||||||||||||||||||||||||||||||||||||||

(d) | If f(x) = x + |x| show that f(3) ≠ f(− 3) | (0) | |||||||||||||||||||||||||||||||||||||||

(e) | Draw the graph of the function y =x |x| where x is real | (0) | |||||||||||||||||||||||||||||||||||||||

(f) | Show that f(x) = |x| is continuous at x = 0 | (0) | |||||||||||||||||||||||||||||||||||||||

(g) |
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(h) |
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(i) |
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(j) |
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2. | (a) | Express – 6i + 7j – 8k, as a linear combination of the vectors 3i + 2j − K and 4i – j + 2k. | 5 | (0) | |||||||||||||||||||||||||||||||||||||

(b) | A factory finds that it makes a profit of Rs. 20 on each article if the number of articles produced, is less or equal to 800 daily. The profit per article decreases by 2 paise per article produced in excess of 800. How many articles should be produced per day to yield maximum profit? | 5 | (0) | ||||||||||||||||||||||||||||||||||||||

(c) | A company is to employ 40 labourers from either A or B comprising of persons in different age groups as under:
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3. | (a) |
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(b) |
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(c) |
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4. | (a) | Minimize z = 4x + 6y subject to the constraints: 2x + 3y > 6, x + y < 4, y > and x, y > 0. | 5 | (0) | |||||||||||||||||||||||||||||||||||||

(b) | The Parker Flower shop promises its customers to deliver within 4 hours on all flower orders. All flowers are purchased on the previous day and delivered to Parker by 8 a.m. in the next morning. Parker's daily demand for roses is as follows:
Parker purchases roses for Rs. 10 per dozen and sells them at Rs. 30 All unsold roses are donated to a local hospital. How many dozens of roses should parker order each evening to maximise its profits? What is the optimum expected profit? | 7 | (0) | ||||||||||||||||||||||||||||||||||||||

(c) | The demand function for a commodity is p = 20 − 3q and the supply function on the market is p = 2q. Find the consumer’s surplus under pure competition. | 3 | (0) | ||||||||||||||||||||||||||||||||||||||

5. | (a) | Obtain an initial basic feasible solution to the given transportation problem by North–West Corner Rule:
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(b) | Customers arrive at a sales counter manned by a single per according to a Poisson process with a mean rate of 10 per hour. The time required to serve a customer has an exponential distribution with a mean of 200 seconds. | 5 | (0) | ||||||||||||||||||||||||||||||||||||||

(c) | Solve the following game using the dominance property:
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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 (choose the correct alternative stating proper reason): | 2x5=10 | |||||||||||||||||||||||||||||||||

(a) |
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(b) | If the events A and B are independent and P(A) = ½, P(B) = 1/3, then the probability of occurrence of almost one of A and B is
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(c) | An unbiased coin is tossed thrice. If first two tosses show tail the probability that other toss shows a head is
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(d) | When two unbiased coins are tossed once, the expected number of heads is
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(e) | A random variable X is binomially distributed with mean 4 and standard deviation √2. Then the set of parameters (n, p) is
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(f) | If a random variable X follows a Poisson distribution with parameter 1 then P(0 < X < 3) is
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(g) | If a random variable X follows normal distribution with mean 0 and variance 1 and P(x < 1.96) = 0.95, then P(X > 1.96) is
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(h) | The results of a bivariate distribution are Σx = 30, Σy = 60, Σ y ^{2} = 760, Σxy = 360, n = 10, Σ x^{2} = 190. Then regression coefficient of x on y is
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(i) | For a simple random sample of size 25, drawn without replacement from a population of size 50 with variance 16, the standard error of the sample mean 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 1 error of the test is
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7. | (a) | There are two identical boxes containing 4 white and 3 red balls, and 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is red. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | In pen factory machines M_{1’}, M_{2’}, M_{3’} manufacture respectively 30, 40 and 45 percent of the total output. Of their output 4, 3 and 3 percent respectively are defective pens. One pen is drawn at random from the product and is found to be defective. What is the probability that it is manufactured in the machine M_{3}? | 5 | (0) | ||||||||||||||||||||||||||||||||

8. | (a) | An unbiased coin is tossed 4 times. Let X be the number of heads. Find the probability distribution, mean and variance of X. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | Fit a Poisson distribution to the following data and calculate the expected frequencies [given e^{−0.5 = 0.607]: No. of deathsFrequency 0122 160 215 32 41 } | 5 | (0) | ||||||||||||||||||||||||||||||||

9. | (a) | Ten competitors in a voice contest are ranked by two judges in the following order:
Find the rank correlation coefficient. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | The heights of 10 students of a college are 68, 70, 62, 67, 61, 68, 70, 64, 66, 64 inches. Is it reasonable to9 believe that the average height is greater than 64 inches? Test at 5% level of significance considering population is normal. [Given: upper 5% points of t distribution with 9 and 10 degrees of freedoms are 1.83 and 1.81 respectively.] | 5 | (0) | ||||||||||||||||||||||||||||||||

10. | (a) | Test whether inoculation is effective in preventing tuberculosis from the following data:
[Given: upper 5% points of chisquare distribution with 1 and 4 degrees of freedoms are 3.84 and 9.49 respectively.] | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | A sample size of 600 persons selected at random from a large city shows that the percentage of male in the sample is 53%. It is believed that male to total population ratio in the city is ½. Test whether this belief is confirmed by the observation. | 5 | (0) | ||||||||||||||||||||||||||||||||

11. | (a) | Suppose that a decision maker faced with 3 decision alternatives and 2 states of nature. Apply (i) maximin and (ii) minimax regret approach to the following pay−off table to recommend the decisions:
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(b) | From a population of 5 numbers 11, 13, 15, 17 and 21 draw all possible samples of size 3 with SRSWOR. Obtain the sampling distribution of sample mean and its standard error. | 5 | (0) |

SECTION III(Economic Techniques — 30 marks) |

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

(a) | If the demand function is p = 48 − x^{2}, for what value of x be elasticity of demand will be unity where p and x represent price and quantity respectively? | (0) | |||||||||||||||||

(b) |
are average and marginal revenue respectively at any output. | (0) | |||||||||||||||||

(c) | If trend equation of annual sales of a company is y = 80 + 5.2t, origin 1988, t unit = 1 year, find the quarterly trend equation. | (0) | |||||||||||||||||

(d) | Show that Laspeyre’s price index number can be written as weighted arithmetic mean of price relatives. | (0) | |||||||||||||||||

(e) | If the cost of living index number in 1998 with 1995 as base year is 118% and an worker draws Rs. 2360 in 1998, find the amount, the worker draws in 1995 if he maintains same standard of living in 1995 and 1998. | (0) | |||||||||||||||||

(f) | The total cost function of a firm is c = Rs. 14.3x + 6000 where C is the total cost and x is the output. A tax at the rate of Rs. 2.80 per unit of output is imposed and the producer adds if to his cost. If the selling price of each unit is Rs. 24.60 what will be the break even point? | (0) | |||||||||||||||||

(g) | For a certain trivariate data the following results are obtained:
Estimate the value of x | (0) | |||||||||||||||||

(h) | Given Σ p_{0}q_{0} = 1360, Σ p_{n}q_{0} = 1900, Σ p_{0}q_{n} = 1344, Σp_{n}q_{n} = 1880. Show that Fisher’s ideal index number satisfies Factor Reversal Test. | (0) | |||||||||||||||||

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

(a) | If the demand curve is of the form p = ad^{−mx}, where p is the price and x is
curve p = 5e | (0) | |||||||||||||||||

(b) | Below are given the figures of production (in thousand tons) of a sugar factory:
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(c) | An enquiry into the budgets of the middle class families in a certain city are as follows:
Find the C.L.I. of 1976 considering 1975 as base period. | (0) | |||||||||||||||||

(d) | 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 800 and 1600 units respectively.
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(e) | From a trivariate distribution the following correlation coefficients are obtained r_{12} = 0.7, r_{13} = 0.6 and r_{23} = 0.4.
multiple correlation coefficient and partial correlation coefficient respectively. | (0) | |||||||||||||||||

(f) | On the basis of quarterly sales (in Rs. lakhs) of a certain commodity to the year 1971–75, the following calculations were made: Trend : y = 250 + 0.6 t with origin at 1^{st} quarter of 1971, t = time unit (one quarter), y = quarterly sales (Rs. in lakhs)
Estimate the quarterly sales for the year 1972 (Use multiplicative model). | (0) |