**58**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) |
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(d) | Draw the graph of f(x) =[x] for −1 < x < 1. | (0) | ||||||||||||||||||||||||||||||||||||

(e) |
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(f) | Differentiate x^{5} w.r.t x^{2}. | (0) | ||||||||||||||||||||||||||||||||||||

(g) |
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(h) | Find the slope of the curve log xy = x^{2} + y^{2} at the point (1,1). | (0) | ||||||||||||||||||||||||||||||||||||

(i) |
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(j) |
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2. | (a) | If (5, 2, 3), (−1, −1, 5) and (2, 4, −3) be the position vectors of three points A, B and C respectively with respect to origin O (0, 0, 0) then show that they are vertices of a right angled isosceles triangle. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) |
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(c) | Solve by Cramer’s rule: x + y + z = 24, 2x + 3Y + z = 2 and 8y − 3z = 2. | 5 | (0) | |||||||||||||||||||||||||||||||||||

3. | (a) | Verify that the function u = 6x^{2} − 2y^{2} − 12x − 16y has a saddle point. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) | A function f(x) is defined as follows: f(x) = 2x − 1 if x < 3, = K if x = 3, = 8 − x if x > 3. For what value of k f(x) is continuous at x = 3? With this value of k draw the graph. | 5 | (0) | |||||||||||||||||||||||||||||||||||

(c) |
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4. | (a) | For a certain establishment cost and revenue functions are c= x^{3} − 12x^{2} + 48x + 11 and R = 83x − 4x^{2} − 21 both in rupees respectively, obtain the maximum profit, Here x = output. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) |
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(c) |
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5. | (a) | Maximize z = 5x_{1} + 7x_{2} subject to 2x_{1} + 3x_{2}< 13, 3x_{1} + 2x_{2}< 12, x_{1}> 0, x_{2}> 0 by the simplex method. | 5 | (0) | ||||||||||||||||||||||||||||||||||

(b) | Obtain the initial basic feasible solution to the following transportation problem by least cost method and determine the cost associated with this solution :
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(c) | In a bank cheques are cashed at a single “teller“ counter. Customers arrive at the counter in a Poisson manner at an average rate of 30 customers per hour. The teller takes on the average a minute and a half to cash a cheque. The service time has been shown to be exponentially distributed, Calculate the percentage of time the letter is busy and average time a customer is expected to wait in the system. | 5 | (0) |

SECTION II(Statistical Techniques — 30 marks) |

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

6. | Attempt any five of the following (choose the correct alternative stating proper reason): | 2x5=10 | |||||||||||||||||||||||||||||||||

(a) | Probability of obtaining an even number in a single throw of an unbiased die is
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(b) |
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(c) | If for two independent events A and B, P(A) = 0.4, P(A U B) = 0.7, the value of P(B) is
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(d) | If a random variable X assumes only two values + 1 and − 1 such that P(X = 1) = P(X = −1), then V(X) is
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(e) |
then the probability of at least one success is
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(f) | For a Poisson distribution with mean 4, the coefficient of variation is
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(g) | For a standard normal variable Z, P(0 <Z <1) = 0.34. Then P(Z >− 1) is equal to
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(h) | If for two variables X and Y, correlation coefficient = −0.6, V(X) = 9, V(Y) =16, then the regression coefficient of X and Y is
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(i) |
Where p is the probability of a head in tossing a coin once. H
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(j) | If a random sample of size 4 with mean 50 is drawn from a normal population with mean u and
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7. | (a) | Box 1 contains 2 white and 2 black balls, Box 2 contains 2 white and 1 black balls. One of the boxes is selected at random and one ball is drawn from it. Find the probabilities that (i) it turns out to be White, (ii) it is selected from Box 1 if the ball drawn is white. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | The probability that A speaks the truth is 0.4 and that B speaks the truth is 0.7. What is the probability that they will contradict each other? | 5 | (0) | ||||||||||||||||||||||||||||||||

8. | (a) | What is the probability of guessing correctly at least 6 of 10 answers in a TRUE–FALSE objective test? | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | If the random variable X follows a normal distribution with mean 18 and variance 625, find the value of (i) P(X > —31) and (ii) P(X < 67 / X > —31)
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9. | (a) | For the variables x and y the regression equations are 4x − y + 8 = 0 and 7x − 3y + 39 = 0. Identify the regression equation of x and y and that of y on x. Find the means of x and y correlation coefficient between x and y. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | From a population 3,5,5,7,9,10 of 6 units, find the sampling distribution of sample mean of simple random samples without replacement of size two. Then show that mean of sample means is exactly equal to the population mean. | 5 | (0) | ||||||||||||||||||||||||||||||||

10. | (a) | Pay−offs of acts X,Y and Z and the states of nature L, M and N are as follows :
The probabilities of the states of nature are respectively 0.1, 0.7 and o.2. Calculate EMV and conclude which of the acts can be chosen as the best. | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | The following results were obtained from the record of age (x) and blood pressure (y) of a group of 10 women:
Find the regression equation of y on x and use it to estimate the blood pressure of a woman of age 45. | 5 | (0) | ||||||||||||||||||||||||||||||||

11. | (a) | I.Q. test was administered to 5 persons before and after they are trained. The results are given below :
Test whether there is any improvement in I.Q. after the training. Given that t | 5 | (0) | |||||||||||||||||||||||||||||||

(b) | A die is thrown 120 times of which 1 comes 20 times, 2 or 3 comes 45 times, 4 or 5 comes 40 times and 6 comes 15 times. Test whether the die is perfect or not. Given upper 5% point of the Chisquare distribution at 3 and 5 d. f are 7.81 and 11.07 respectively. | 5 | (0) |

SECTION III(Economic Techniques — 30 marks) |

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

(a) | If (x + 3)^{3}p = 4, compute the elasticity of demand in terms of x. | (0) | ||||||||||||||||||||||||||||||||||||

(b) | Find the average cost function when the marginal cost function is 3x | (0) | ||||||||||||||||||||||||||||||||||||

(c) | If the quarterly trend equation of production of sugar in a company is y = 5.27 + 0.04 t with origin = 3rd quarter of 1983 and unit of t = 1 quarter, find the trend equation based on annual sales with origin 1983 and unit of t = 1 year. | (0) | ||||||||||||||||||||||||||||||||||||

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

(e) |
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(f) | Deseasonalise the data using multiplicative model of tea production:
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(g) | Find r_{23} if r_{23.1} = 0.6 and r _{13} = 0.5. | (0) | ||||||||||||||||||||||||||||||||||||

(h) | If the multiple regression equation of X_{1} on X_{2} and X_{3} is X_{1} = 0.6 X_{2} + 0.4_{3} − 14, r_{13.2} = 0.8 and V(X_{3}) = 16, find V(X_{1}). | (0) | ||||||||||||||||||||||||||||||||||||

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

(a) | A demand law is given by x = 80 − 4p where x is the quantity demanded and p is the price of a commodity. If the price elasticity of demand at p = 8 is increased by 50%, obtain the percentage decrease in demand. | (0) | ||||||||||||||||||||||||||||||||||||

(b) | Find the trend values from the following time series data by method of moving average with suitable period to be obtained by you from the data :–
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(c) | Calculate the price index number by Fisher’s method:
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(d) | In a trivariate distribution,
_{1} on x_{2} x_{3}. Estimate x_{1} for x_{2} = 80, x_{3} = 120. | (0) | ||||||||||||||||||||||||||||||||||||

(e) | Given the following input–output table:
Find the gross production of steel and coal and also the total labour required. | (0) | ||||||||||||||||||||||||||||||||||||

(f) | Writer short notes on any one of the following: | |||||||||||||||||||||||||||||||||||||

(i) | Elasticity of demand. | (0) | ||||||||||||||||||||||||||||||||||||

(ii) | Partial correlation coefficient. | (0) | ||||||||||||||||||||||||||||||||||||

(iii) | Least square theory | (0) |