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 used have their usual meanings. |
| SECTION I(Mathematical Techniques — 40 marks) |
Answer Question No. 1 (compulsory — 10 marks) and
two other questions (15x2 = 30 marks) from this section. |
| 1. | Attempt any five questions: | 2x5=10 | |
| | (a) | If a = 2i + j − k, b = i + 2j + k, c = 3i − 2j + 2k, find the unit vector parallel to a − 2b + c | | (0) |
| | (b) | | Compute A2 and A3 If A = | ( | 0
−i | −i
0 | ) | where i = √− 1 |
| | (0) |
| | (c) | | Prove that | | bc
ca
ab | a
b
c | a2
b2
c2 | | = | | 1
1
1 | a2
b2
c2 | a3
b3
c3 | |
| | (0) |
| | (d) | | Find the value of x when A = | | ( | 1
x | 1
1 | ) | is orthogonal |
| | (0) |
| | (e) | | If y = f(x) = | | , prove that f(y) = x. |
| | (0) |
| | (f) | Draw the graph: y ≤ |x| | | (0) |
| | (g) | | | (0) |
| | (h) | | Find | | , when x = at2 and y = 2at. |
| | (0) |
| | (i) | | | (0) |
| | (j) | | If f(x, y) = x2 – 3xy + y2, find the value of | | — | |
| | (0) |
| 2. | (a) | | If A = | | 1
2
2 | 2
1
2 | 2
2
1 | find f(A) if f(x) = x2 − 4x − 2. |
| 5 | (0) |
| | (b) | Given that a = 2i + 3j + 6k, b = 3i − 6j + 2k, and c = 6i + 2j − 3k, show that a x b = 7c. Also find a unit vector perpendicular to each of the vectors a and b. | 5 | (0) |
| | (c) | | If | | a
1
1 | 1
b
1 | 1
1
c | = 0, show that | | + | | + | | = 1. |
| 5 | (0) |
| 3. | (a) | A function f(x) is defined as follows: | If A = | { | x + 1 when x ≤ 1
3 − ax2 when x > 1 |
For what value of a will f(x) be continuous? | 5 | (0) |
| | (b) | | If y = (x + √x2 + a2)n, show that (x2 + a2) | | + x | | − n2y = 0. |
| 5 | (0) |
| | (c) | Find the extreme value of the function f(x, y) = x2 − y2 + xy + 5x subject to x + y + 3 = 0. | 5 | (0) |
| 4. | (a) | | 5 | (0) |
| | (b) | Find the area cut off from the parabota y2 = 12x by its latus rectum. | 5 | (0) |
| | (c) | | Solve the following LPP by simplex method: | maximize z = 4x + 6y | | | subject to constraints: | x + 2y ≤ 40
2.5x + 1.5y ≤ 60;
x, y ≥ 0. |
| 5 | (0) |
| 5. | (a) | Given the pay–off matrix for player A, obtain the optimum strategies for both the players and determine the value of the game: | Player B | | Player A | | 3
10
12 | 8
11
14 | −4
13
8 |
| 5 | (0) |
| | (b) | Determine the initial basic feasible solution of the following transportation problem by the least cost method: | | D1 | D2 | D3 | D4 | Capacity | | O1 | 3 | 2 | 1 | 5 | 12 | | O2 | 4 | 6 | 2 | 8 | 8 | | O3 | 7 | 4 | 3 | 5 | 10 | | O4 | 2 | 6 | 7 | 5 | 4 | | Demand | 8 | 10 | 6 | 10 | |
| 5 | (0) |
| | (c) | Customers arrive at the booking office window being manned by a single individual at a rate of 25 per hour. The time required to serve a customer has an exponential distribution with a mean = 120 seconds. Find the average waiting time of a customer (i) in the system (ii) in the queue. | 5 | (0) |
| SECTION II(Statistical Techniques — 30 marks |
Answer Question No. 6 (compulsory — 10 marks) and
two other questions (10x2 = 20 marks) from this section |
| 6. | Attempt any five of the following
choose the correct alternative stating proper reason: | 2x5 | |
| | (a) | | If P(A) = | | , P(B) = | | and P(A ∪ B) = | | , then P(A ∩ B) is |
| | (0) |
| | (b) | | If for two events A and B, P(B) = | | and P(A | B) = | | , then P(A − B) is |
| | (0) |
| | (c) | Let A and B be two events such that P(A) = 0.4, P(A ∪ B) = 0.7 and P(B) = p, then the choice of p for which A and B are independent is | | (0) |
| | (d) | 8 fair coins tossed simultaneously. The probability of getting at least 6 heads is | | (0) |
| | (e) | In a summer a truck driver gets an average of one puncture in 500 km, then the probability that he will have no puncture in a journey of 2000 km is | (i) | e−1 | (ii) | e−2 | (iii) | e−3 | (iv) | e−4 |
| | (0) |
| | (f) | A sample of 100 measurements of breaking strength of cotton threads gave a mean of 7.4 and s.d. of 1.2. Then the 95% upper confidence limit for the population mean breaking strength is | (i) | 7.3526 | (ii) | 7.6352 | (iii) | 7.6325 | (iv) | 7.6123 |
| | (0) |
| | (g) | If a random variable X is uniformly distributed with a probability density function f(x) = 1, 0 ≤ x ≤ 1 then v(x) is | | (0) |
| | (h) | If the rank correlation coefficient between height and weight of a group of students is 0.5 and the sum of squares of the rank differences is 60 then number of students is | (i) | 8 | (ii) | 9 | (iii) | 10 | (iv) | 11 |
| | (0) |
| | (i) | A coin is tossed 4 times to rest its unbiasedness and H0 is accepted when 2 heads are obtained. Then Probability [type 1 Error] is | (i) | | (ii) | | (iii) | | (iv) | None of the above |
| | (0) |
| | (j) | A simple random sample of size 6 is drawn without replacement from a finite population consisting of 10 units. If the variance of the population is 9, the standard error of the sample mean is | | (0) |
| 7. | (a) | In a bolt factory, machines X1, X2 and X3 manufacture respectively 20%, 30% and 50% of the total of their output. Of them 5, 4 and 2 percent respectively are defective bolts. A bolt is drawn at random from the products and is found defective. What is the probability that it was manufactured by machine X2 or X3? | 5 | (0) |
| | (b) | A can hit a target two times out of 3, B three times out of 7 and C four times out of 9. What is the probability that the target will be hit by 2 persons only when all three try? | 5 | (0) |
| 8. | (a) | Fit a binomial distribution to the following data: | 5 | (0) |
| | (b) | Marks obtained by a number of students are assumed to be normally distributed with mean 50 and variance 36. If 4 students are taken at random, what is the probability that exactly two of them will have marks over 62? | [Given | 2
∫
0 | Φ (z) dz = 0.4772 where z is N (0, 1)]. |
| 5 | (0) |
| 9. | (a) | In random samples of 600 and 1000 men from two cities, 400 and 600 men are found to be literate. Do the data indicate at 5% level of significance that the populations are significantly different in the percentage of literacy? | 5 | (0) |
| | (b) | If regression equations are 8x − 10y = 64 and 40x − 18y = 320, what are (i) the mean values of x and y, and (ii) the coefficient of correlation between x and y? | 5 | (0) |
| 10. | (a) | In an experiment on immunization of cattle from tuberculosis the following results were obtained: | Affected | Unaffected | Inoculated
Not inoculated | 12
13 | 28
7 |
Examine the effect of vaccine in controlling the incidence of the desease. | 5 | (0) |
| | (b) | A newspaper agents’ experience shows that the daily demand x of newspapers in his area has the following probability distribution: Daily demand (x)
Probability | :
: | 300
0.1 | 400
0.3 | 500
0.4 | 600
0.1 | 700
0.1 |
He sales the newspapers for Rs. 2.00 each while he buys each at Re. 1.00. Unsold copies are traded as scrap and each such copy fetches 10 paise. Assuming that he stocks the newspapers in multiple of 100 only, how many should he stock so that his expected profit is maximum? | 5 | (0) |
| 11. | (a) | Choose the best product applying Hurwitz method with coefficient of optimism α = 0.60 to the following data: | State of nature→ | Good | Fair | Poor | Acts
↓ | | Product A
Product B | 10000
12500 | 8000
9000 | 2000
−1000 |
| 5 | (0) |
| | (b) | A bakery keeps stock of a popular brand of cakes. Previous experience shows that daily demand pattern for the items with associated probabilities as given below: Daily demand (numbers)
Probability | 0
0.01 | 10
0.20 | 20
0.15 | 30
0.50 | 40
0.12 | 50
0.02 |
Use the following sequence of random numbers to simulate the demand for the next 10 days: 48, 78, 19, 51, 56, 77, 15, 14, 68, 09 Also (i) estimate the daily average demand for the cakes on the basis of the simulated data and (ii) find out the stock situation if the owner of the bakery decides to make 30 cakes every day. | 5 | (0) |
| SECTION III(Economic Techniques — 30 marks) |
| 12. | Attempt any five of the following: | 2x5 | |
| | (a) | | If the demand law is p = | | Complete the elasticity of demand in terms of x. |
| | (0) |
| | (b) | Write down the normal equations in fitting a parabolic equation y = a + bx + cx2 by the method of least squares. | | (0) |
| | (c) | If the production is 28 and seasonal index is 90, then find the deseasonalised value. | | (0) |
| | (d) | Calculate Laspeyre’s Index Number from the following table: | Item | p0 | p1 | q0 | q1 | A
B
C
D | 5
4
3
2 | 7
6
5
4 | 6
5
4
3 | 8
7
6
5 |
| | (0) |
| | (e) | If r12 = 0.4, r23 = 0.5, r31 = 0.6 then R1.23 = ? | | (0) |
| | (f) | Net monthly income of an employee was Rs. 800 per month in 1980. The price index number was 160 in 1980. It rises to 200 in 1982. Calculate the additional dearness allowance to be paid to the employee if he has to be rightly compensated. | | (0) |
| | (g) | The total daily cost (in Rs.) for producing x toys is TC = 2.5x + 3000. If each toy is sold for Rs. 4, what is the break even point? | | (0) |
| | (h) | Given total cost = 1400 + 30x −; 12x2 + x3; find at what level of x diminishing marginal return begins. | | (0) |
| 13. | Answer any four of the following: | 5x4 | |
| | (a) | The demand function is x = 20 + 2p + 3p2 where x is demand for the commodity at price p. Find the marginal quantity demand, average quantity demand and the elasticity of demand. Also compute elasticity of demand at a price p = 2. | | (0) |
| | (b) | The marginal cost function when output is x units, is given by MC = x2 − 2x + 5. Find the total cost function C and average cost function AC, if the fixed cost is 30. Also show that the slope of the average cost curve is | | (0) |
| | (c) | Fit a straight line trend by the method of least squares from the following figures of production of sponge iron in a factory and estimate the production (in tons) in 2001: Year
Production | :
: | 1995
15 | 1996
18 | 1997
16 | 1998
18 | 1999
20 | 2000
19 |
| | (0) |
| | (d) | The price relatives and weights of a set of commodities are given in the following table: Commodity
Price relative | :
: | A
120 | B
127 | C
125 | D
119 | | Weight | : | 2w1 | w2 | w1 | w2 + 3 |
If the index for the set is 122 and the sum of weights is 40; find w1 and w2. | | (0) |
| | (e) | The demand of function of a particular brand of watch is p = 75 − 0.3Q − 0.05Q2. Find the consumer's surplus at a quantity of 10 watches. | | (0) |
| | (f) | Write short notes on any one of the following: | | |
| | | (i) | Uses and limitations of index number; | | (0) |
| | | (ii) | Least squares theory. | | (0) |
