Quantitative Methods - I (MB151): April 2006

1
Question Paper

Quantitative Methods - I (MB151): April 2006

• Answer all questions.
• Marks are indicated against each question.
1.
The value of a b c
1 1 1
log abc log abc log abc
+ +
is equal to
(a) a log abc (b) b log abc (c) abc log a (d) 0 (e) 1.
(1 mark)
< Answer >
2. If logex . log5625 = log1016. loge10 then the value of x is
(a) 1 (b) 2 (c) 5 (d) 10 (e) 16.
(2 marks)
< Answer >
3.
The value of 2
log 1
128 is equal to
(a) 0 (b) 1 (c) – 14 (d) – 1/2 (e) 7.
(1 mark)
< Answer >
4.
The value of
7 log 16 5 log 25 3 log 81
15 24 80
+ +
is equal to
(a) 0 (b) 1 (c) log 2 (d) log 3 (e) log 4.
(2 marks)
< Answer >
5. How many numbers greater than a million can be formed with the digits 4, 5, 5, 0, 4, 5, 3?
(a) 36 (b) 60 (c) 306 (d) 360 (e) 603.
(1 mark)
< Answer >
6. In a group of 9 people, there are 4 females and 5 males. The number of 4 member committees
consisting of at least 1 female that can be formed, would be
(a) 12 (b) 21 (c) 112 (d) 121 (e) 221.
(1 mark)
< Answer >
7. Six papers are set in an examination, of which 2 are Statistics. In how many different orders can the
papers be arranged so that the two statistics papers are not together?
(a) 240 (b) 480 (c) 720 (d) 840 (e) 1460.
(1 mark)
< Answer >
8.
Convert ( )0.657 8 to its decimal equivalent
(a) (0.03275)10 (b) (0.40625)10
(c) ( )0.115756219 10 (d) ( )0.95703125 10
(e) ( )0.841796875 10 .
(1 mark)
< Answer >
9. The sum of three numbers in A.P is 27. The product of their extremes is 77. Find the third number.
(a) 5 (b) 11 (c) 18 (d) 21 (e) 25.
(2 marks)
< Answer >
2
10. The 20th term of the following series is 13, 8, 3, –2 …….
(a) 32 (b) –57 (c) – 82 (d) – 97 (e) 112.
(1 mark)
< Answer >
11.
If f :R+ →R such that ( ) 8 f x = log x , then f −1 (x)
is equal to
(a) 8x (b) x8 (c) x
1
log 8 (d) x
1
log 8
−
(e) x −log 8.
(1 mark)
< Answer >
12. A candidate is required to answer 7 out of 12 questions, which are divided into two groups, each
containing 6 questions. He is not permitted to attempt more than 5 questions from either groups. In how
many different ways he can choose the 7 questions?
(a) 180 (b) 210 (c) 600 (d) 780 (e) 792.
(1 mark)
< Answer >
13. The sales of the company during the year 2003 was Rs.30 lakh and during the year 2005 was Rs.46
lakh. The sales of the company during the year 2004 were not known. What is the estimated value of
sales for the year 2004, using the interpolation method?
(a) 32 lakh (b) 35 lakh (c) 38 lakh (d) 40 lakh (e) 44 lakh.
(1 mark)
< Answer >
14. Which of the following are the 6 geometric means between 5 and 640?
(a) 10, 15, 20, 25, 30 and 35
(b) 10, 30, 40, 60, 160 and 180
(c) 10, 20, 60, 90, 180 and 220
(d) 10, 20, 80, 160, 320 and 440
(e) 10, 20, 40, 80, 160 and 320.
(1 mark)
< Answer >
15. 1 1 1 12
2 3 4
x − y + z =
2 1 1 8
10 9 3
x − y + z =
3 2 7 2
10 9 12
x − y − z = −
The value of z which satisfies the above three simultaneous equations is
(a) 3 (b) 4 (c) 6 (d) 9 (e) 12.
(2 marks)
< Answer >
16. Which one of the following is a positional average?
(a) Moving average (b) Arithmetic mean
(c) Progressive average (d) Quartiles
(e) Harmonic mean.
(1 mark)
< Answer >
17. Algebraic sum of the deviations of a set of values from their arithmetic mean is
(a) Between 0 and 1 (b) Between –1 and 0
(c) Greater than one (d) Equal to 1
(e) Equal to 0.
(1 mark)
< Answer >
3
18. Find the weighted arithmetic mean of first n natural numbers, weights being corresponding numbers
(a)
(n 1)
2
+
(b)
2n+1
3 (c)
n(n 1)
2
+
(d)
n(n 1)
2
−
(e)
(n 1)
2
−
.
(2 marks)
< Answer >
19. Expression of geometric mean in terms of arithmetic mean and harmonic mean is:
(a) G.M= A.M×H.M (b) G.M≤ A.M×H.M
(c) G.M≥ A.M×H.M (d) G.M= A.M×H.M
(e) G.M≥ A.M×H.M
(1 mark)
< Answer >
20. In which of the following the simple harmonic mean is appropriate?
(a) A set of ratios using the numerators of the ratio data as weights
(b) A set of ratios using the denominators of the ratio data as weights
(c) A set of ratios which have been calculated with the same numerators
(d) A set of ratios which have been calculated with the same denominators
(e) Both (b) and (d) above.
(1 mark)
< Answer >
21. The mean of squares of first 23 natural numbers is
(a) 4324 (b) 188 (c) 1128 (d) 104 (e) 3424.
(2 marks)
< Answer >
22. Which of the following is not true about coefficient of variation?
(a) Coefficient of variation is the ratio of standard deviation and mean
(b) Coefficient of variation is the ratio of mean and standard deviation
(c) Coefficient of variation is a relative measure and is most suitable to compare the two series
(d) A series or a set of values having lesser coefficient of variation as compared to the other is more
consistent
(e) Usually coefficient of variation is expressed in terms of percentages.
(1 mark)
< Answer >
23. Which measure of dispersion ensures lowest degree of reliability?
(a) Standard deviation (b) Mean deviation (c) Variance
(d) Range (e) Quartile deviation.
(1 mark)
< Answer >
24. An analysis of monthly wages paid to the workers of two firms X and Y belonging to the same industry
gives the following results
Firm X Firm Y
Number of workers 500 600
Average daily wage Rs.186.00 Rs.175.00
Standard deviation of distribution of wages 9 10
The average daily wages of the workers in the two firms X and Y taken together is
(a) Rs.180.00 (b) Rs.180.50 (c) Rs.185.00 (d) Rs.175.00 (e) Rs.175.50.
(1 mark)
< Answer >
4
25. Which of the following statement is not true about standard deviation?
(a) Combined standard deviation of two or more groups can be calculated
(b) The sum of the squares of the deviations of items of any series from a value other than the
arithmetic mean would always be smaller.
(c) Standard deviation is independent of any change of origin
(d) Standard deviation is dependent on the change of scale
(e) The standard deviation is also called root mean square deviation.
(1 mark)
< Answer >
26. If each value of a set is divided by 10, then the coefficient of variation will be
(a) Decreased by 10 times (b) Increased by 10 times
(c) Same as original value (d) Increases by 0.10 times
(e) Decreased by 0.10 times.
(1 mark)
< Answer >
27. The selected items of a sample resulted into same values pertaining to a character. The variance of the
sample is
(a) Not determined (b) ∞ (c) 1
(d) – 1 (e) 0.
(1 mark)
< Answer >
28. The mean deviation from the A.M for the following distribution is
Class Interval 10-20 20-30 30-40 40-50 50-60
Frequency 8 15 25 9 3
(a) 32.33 (b) 8.289 (c) 25.68 (d) 5.264 (e) 15.67.
(2 marks)
< Answer >
29. Which of the following is not the measure of central value?
(a) Harmonic mean (b) Mode (c) Quartiles
(d) Percentiles (e) Interquartile range.
(1 mark)
< Answer >
30. Which of the following is/are true with respect to geometric mean?
I. Geometric mean cannot be calculated if any of the value in the set is zero.
II. Geometric mean is appropriate for averaging the ratios of change, for average of proportions, etc.
III. Geometric mean is considered most suitable average for index numbers.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
31. A salesman whose annual sales figures are most consistent will have
(a) Minimum coefficient of variation of annual sales figures
(b) Maximum coefficient of variation of annual sales figures
(c) Largest average sales per year
(d) Maximum variance of annual sales figures
(e) Minimum standard deviation of annual sales figures.
(1 mark)
< Answer >
32. The Bienayme-Chebyshev theorem is not applicable when the difference about the mean is
(a) 1.0 standard deviation (b) 1.5 standard deviations
(c) 2.0 standard deviations (d) 2.5 standard deviations
(e) 3.0 standard deviations.
(1 mark)
< Answer >
5
33. The average of a sample consisting of 65 items is 8 and the sum of the squares of the items is 4560. The
sample standard deviation is
(a) 2.50 (b) 6.25 (c) 64 (d) 400 (e) 570.
(1 mark)
< Answer >
34. The mean of a set of hundred values is 10. If every item is increased by 5, then the mean of the resulting
set of values will be equal to
(a) 5 (b) 10 (c) 15 (d) 20 (e) 50.
(1 mark)
< Answer >
35. For a distribution of data, mean =12, median =10 and mode = 8. It can be inferred that the distribution is
(a) Symmetrical
(b) Positively skewed
(c) Negatively skewed
(d) Skewed, though it can not be inferred whether it is positively or negatively skewed
(e) Not skewed.
(1 mark)
< Answer >
36. The arithmetic mean of a data set containing 60 observations is 3. The combined arithmetic mean of this
data set and another data set containing 40 observations is 3.80. The arithmetic mean of the second data
set, which contains 40 observations, is
(a) 3.00 (b) 3.80 (c) 4.00 (d) 5.00 (e) 5.50.
(2 marks)
< Answer >
37. The average monthly salary of the supervisors and workmen of Alpha Ltd. is Rs.6,000. The average
monthly salary of the supervisors is Rs.9,000 and the average monthly salary of the workmen is
Rs.5000. Moreover, the average monthly salary of the middle level managers of the company is
Rs.14,000. The percentage of supervisors among the total number of workers and supervisors in the
company is
(a) 10% (b) 20% (c) 25% (d) 30% (e) 40%.
(2 marks)
< Answer >
38.
If A=
4 2
1 1
⎡ ⎤
⎢− ⎥ ⎣ ⎦ then (A –2I)(A – 3I), is equal to (where I is the identity matrix)
(a)
0 0
0 0
⎡ ⎤
⎢ ⎥
⎣ ⎦ (b)
1 0
0 1
⎡ ⎤
⎢ ⎥
⎣ ⎦ (c)
0 1
1 0
⎡ ⎤
⎢ ⎥
⎣ ⎦ (d)
1 0
0 1
⎡− ⎤
⎢ − ⎥ ⎣ ⎦ (e)
0 1
1 0
⎡ − ⎤
⎢− ⎥ ⎣ ⎦.
(2 marks)
< Answer >
39.
If A – B =
1 0
0 1
⎡ ⎤
⎢ − ⎥ ⎣ ⎦, A + B =
3 4
2 5
⎡ ⎤
⎢ ⎥
⎣ ⎦ then AB is equal to
(a)
2 2
1 2
⎡ ⎤
⎢ ⎥
⎣ ⎦ (b)
1 2
1 3
⎡ ⎤
⎢ ⎥
⎣ ⎦ (c)
4 4
2 4
⎡ ⎤
⎢ ⎥
⎣ ⎦ (d)
3 2
1 1
⎡ ⎤
⎢ ⎥
⎣ ⎦ (e)
4 10
3 8
⎡ ⎤
⎢ ⎥
⎣ ⎦.
(2 marks)
< Answer >
6
40.
If n is an even natural number then
n 3 1
4 3
⎡ − ⎤
⎢ − ⎥ ⎣ ⎦ is equal to
I. Null matrix of order 2.
II. Unit matrix of order 2.
III. Scalar matrix of order 2.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (II) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
41.
Find the matrix X such that
1 2 3 8
2X
3 4 7 2
⎡ ⎤ ⎡ ⎤
+ ⎢ ⎥ = ⎢ ⎥
⎣ ⎦ ⎣ ⎦.
(a)
3 1
1 2
⎡ ⎤
⎢− ⎥ ⎣ ⎦ (b)
3 3
1 1
⎡ ⎤
⎢− − ⎥ ⎣ ⎦ (c)
3 1
1 2
⎡ − ⎤
⎢ ⎥
⎣ ⎦ (d)
1 3
2 1
⎡ ⎤
⎢ − ⎥ ⎣ ⎦ (e)
1 1
2 1
⎡− ⎤
⎢− − ⎥ ⎣ ⎦.
(1 mark)
< Answer >
42.
If
9 1 1 5
A ,B
7 8 7 12
⎡ ⎤ ⎡ ⎤
= ⎢ ⎥ = ⎢ ⎥
⎣ ⎦ ⎣ ⎦ , find the matrix C such that 5A+3B+ 2C is a null matrix.
(a)
0 0
0 0
⎡ ⎤
⎢ ⎥
⎣ ⎦ (b)
12 5
14 29
⎡− − ⎤
⎢− − ⎥ ⎣ ⎦ (c)
36 15
48 57
⎡− − ⎤
⎢− − ⎥ ⎣ ⎦
(d)
24 10
28 38
⎡− − ⎤
⎢− − ⎥ ⎣ ⎦ (e)
1 0
0 1
⎡ ⎤
⎢ ⎥
⎣ ⎦.
(2 marks)
< Answer >
43. The matrix A is said to be an identity matrix of order 3x3. Then the rank of matrix A
(a) Is more than three (b) Is less than three
(c) Is equal to the three (d) Is equal to one
(e) Does not exist.
(1 mark)
< Answer >
44. The augmented matrix of the following system of equations
3y – 5x + 2z = 1
2x – 8z = 5
5y – 3x = 3 is given as:
(a)
3 5 2 1
2 8 0 5
5 3 0 3
⎡ − ⎤
⎢ − ⎥ ⎢ ⎥
⎢⎣ − ⎥⎦ (b)
5 3 2 1
2 0 8 5
3 5 0 3
⎡− − ⎤
⎢ − − ⎥ ⎢ ⎥
⎢⎣− − ⎥⎦
(c)
5 3 2 1
2 0 8 5
3 5 0 3
⎡− ⎤
⎢ − ⎥ ⎢ ⎥
⎢⎣− ⎥⎦ (d)
5 3 2 1
2 0 8 5
3 5 0 3
⎡− − ⎤
⎢ − ⎥ ⎢ ⎥
⎢⎣− − ⎥⎦
(e)
3 5 2 1
2 8 0 5
5 3 0 3
⎡ − − ⎤
⎢ − − ⎥ ⎢ ⎥
⎢⎣ − − ⎥⎦ .
(1 mark)
< Answer >
7
45. The rank of every non singular matrix of order ‘n’ is
(a) Zero
(b) ‘n’ itself
(c) Less than the rank of the original matrix
(d) More than the rank of the original matrix
(e) Not defined.
(1 mark)
< Answer >
46. If A = [aij] m x n is a square matrix of order m, then the elements aij for which i = j, are called
(a) The row elements of A (b) The column elements of A
(c) The diagonal elements of A (d) The elements of A
(e) The identity elements of A.
(1 mark)
< Answer >
47. When two rows or columns of a determinant are identical then the value of the determinant is
(a) –2 (b) –1 (c) 0 (d) 1 (e) 2.
(1 mark)
< Answer >
48. Which of the following statement(s) is/are false?
I. Linear programming treats all relationships among decision variables as linear.
II. While solving the linear programming model, there is guarantee that we will get integer valued
solutions.
III. Linear programming model does not take into consideration the effect of time and uncertainty.
IV. Linear programming deals with single objective.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) Both (II) and (IV) above.
(1 mark)
< Answer >
49. In solving the linear programming problem, if the type of constraint is less than or equal to
(<) then
(a) The extra variable called slack variable is added
(b) The extra variable called surplus variable is subtracted, and an artificial variable is added
(c) Only an artificial variable is added
(d) The coefficient of extra variable in the objective function is M
(e) The coefficient of extra variable in the objective function is –M.
(1 mark)
< Answer >
50. Which of the following is/are the assumptions that are made in the construction of linear programming
models?
I. Proportionality.
II. Additivity.
III. Divisibility.
IV. Certainty.
(a) Only (I) above (b) Both (I) and (II) above
(c) Both (I) and (III) above (d) Both (II) and (IV) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
8
51. The solution for the following linear programming problem is
Max.z = 15x1 +17x2
Subject to the constraints
1
2
1 2
1 2
x 6
x 12
x x 24
x ,x 0
≥
≥
+ ≥
≥
(a) Max.z = 215 (b) Data is insufficient
(c) Infeasible solution (d) Unbounded solution
(e) It has multiple optimal solutions.
(2 marks)
< Answer >
52.
In the following linear programming problem, the value of 1 2 3 x ,x andx are equal to:
Max. z = 3x1 + 2x2 + 5x3
Subject to constraints
x1 + 2x2 + x3 < 430
3x1 + 2x3 < 460
x1 + 4x2 < 420
x1, x2, x3 > 0.
(a) 1 2 3 x = 30, x = 50and x = 300 (b) 1 2 3 x = 0, x =100and x = 230
(c) 1 2 3 x = 80, x = 0and x = 60 (d) 1 2 3 x = 0, x = 0and x = 140
(e) 1 2 3 x =100, x = 80and x = 0 .
(2 marks)
< Answer >
53. If Sk, the slack variable corresponding to a less-than inequality constraint for resource ‘k’, is found to
have a positive value in the optimal solution then it can be understood that
(a) The solution is not feasible
(b) All of the resource ‘k’ is used up
(c) Part of the resource ‘k’ is not used up
(d) A better solution could be achieved by decreasing resource ‘k’
(e) Both (c) and (d) above.
(1 mark)
< Answer >
54. Which of the following is/are the requirement of a linear programming problem?
I. The problem must have a well-defined single objective to achieve.
II. There must be alternative courses of action, one which will achieve the objective.
III. The decisions variables must be continuous in nature.
IV. The objective and the constraints must be linear functions.
V. Resources must be limited in supply and the achievement of the objective function is restricted by
these constraints.
(a) Both (I) and (III) above (b) (I), (II) and (IV) above
(c) (I), (III) and (IV) above (d) (I), (III) and (V) above
(e) All (I), (II), (III), (IV) and (V) above.
(1 mark)
< Answer >
9
55. In some special linear programming problem situations, the variables cannot take fractional values, then
the linear programming problem is formulated as
I. Dual LP Problem.
II. Goal Programming.
III. Integer Programming.
IV. Transportation problem.
(a) Only (I) above (b) Only (III) above
(c) Both (I) and (II) above (d) Both (II) and (III) above
(e) Both (III) and (IV) above.
(1 mark)
< Answer >
56. Allied Engines Inc. produces two models of car engines, a X engine and a Y engine.
The following information is available:
X Y
Selling Price Rs.800 Rs.1,000
Variable cost per unit Rs.560 Rs.625
Assembly hours per engine 2 hours 5 hours
Testing hours per engine 1 hour 0.5 hours
Total assembly hours available 600 hours
Total testing hours available 120 hours
Allied Engines Inc. has to decide how many of the X and Y engines it should produce if it has to
maximize the profit.
The number of engines Y to be produced is
(a) 60 (b) 75 (c) 90 (d) 105 (e) 120.
(2 marks)
< Answer >
57. Which of the following is/are true with respect to index numbers?
I. Index numbers measure changes in some quantities, which cannot be observed directly.
II. Index numbers are expressed in percentages which make it feasible to compare any two or more
index numbers.
III. Index numbers are of comparable nature at any two timings or places or any other situation.
IV. In index numbers the two periods, one is known as the base period and the other, the current
period are always comparable with each other.
(a) Only (I) above (b) Only (II) above
(c) Both (I) and (II) above (d) Both (II) and (IV) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
58. The following details are available with regard to a basket of goods:
Weighted average price of goods in the current year = Rs.46.875
Weighted average price of goods in the base year = Rs.37.5
Both the weighted averages have been calculated using the current year consumption quantities as
weights.
Which of the following is correct?
(a) Laspeyres price index = 125
(b) Laspeyres price index = 80
(c) Laspeyres quantity index = 125
(d) Paasches price index = 125
(e) Paasches price index = 80.
(1 mark)
< Answer >
59. If Σ P0Q0 = 268; Σ P1Q0 = 291; Σ P1Q1 = 301 and Σ P0Q1 = 251, then the value index is
(a) 103.44 (b) 106.77 (c) 108.58 (d) 112.31 (e) 115.94.
(1 mark)
< Answer >
10
60. The following data pertains to the consumption of materials by a bakery.
Inputs Units Prices (in Rs. per unit) Quantities Used (Units) 2002 2005 2002 2005
Flour Kilogram 15 30 500 700
Eggs Dozen 8 14 100 70
Milk Litres 6 17 200 120
Sugar Kilogram 10 16 50 70
Paasche’s quantity index for the given data is
(a) 122.04 (b) 125.80 (c) 197.81 (d) 201.44 (e) 205.31.
(2 marks)
< Answer >
61. Which of the following is not true with regard to a value index?
(a) It measures changes in the total monetary worth
(b) It combines price changes and quantity changes
(c) It is not as useful as price indices or quantity indices
(d) It cannot distinguish between the price changes and quantity changes
(e) An increase in the value index indicates with certainty that the prices have increased.
(1 mark)
< Answer >
62. Which of the following statements is false regarding Fisher’s ideal price index?
(a) It is the geometric mean of Laspeyeres and Paasches price indices
(b) It takes into account both current year and base year prices and quantities
(c) It satisfies both time reversal and factor reversal tests
(d) It is free from bias
(e) For the same basket of goods Fisher’s ideal price index will be more than both Laspayres and
Paaches price indices.
(1 mark)
< Answer >
63. If the ratio between Laspeyres index number and Paasches index number is 28 : 27, find the missing
figure in the following table:
Commodity Base Year Current Year Price Quantity Price Quantity
X 1 10 2 5
Y 1 5 ? 2
(a) 1 (b) 2 (c) 3 (d) 4 (e) 5.
(2 marks)
< Answer >
64. The ratio of the weighted average price of goods in a basket in the current year to their weighted
average price in the base year expressed as a percentage, where both the weighted averages have been
calculated by using the base year consumption quantities as weights, yields which of the following?
(a) Paasches quantity index (b) Paasches price index
(c) Laspeyres quantity index (d) Laspeyres price index
(e) Weighted average of relatives price index.
(1 mark)
< Answer >
11
65. The following details are available with regard to a basket of goods:
Weighted average price of goods in current year
using base year consumption quantities as weights = Rs.40
Weighted average price of goods in the base year
using base year consumption quantities as weights = Rs.32
Which of the following is correct?
(a) Paasches price index = 125
(b) Paasches quantity index = 125
(c) Laspeyres price index = 125
(d) Laspeyres quantity index = 125
(e) Unweighted average of relatives price index = 125.
(2 marks)
< Answer >
66. A time series of annual data can contain which of the following components?
I. Secular trend.
II. Cyclical fluctuation.
III. Seasonal variation.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
67. A time series for the years 1993-2004 had the following relative cyclical residuals (RCR), in
chronological order as follows:
Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
RC −1% −2% 1% 2% −1% −2% 1% 2% −1% −2% 1% 2%
What will be the relative cyclical residual for 2005?
(a) +3% (b) −1%
(c) −2% (d) +2%
(e) Cannot be determined on the basis of the given information.
(1 mark)
< Answer >
68. The following linear trend equations for estimation of national income and population have been
constructed for a hypothetical economy for the period 2001 to 2005:
National income, ˆY = 4.86 + 1.5x
Population, ˆZ = 5.25 + 1.25x
Assuming that the year 2003 is coded as 0 and time interval of 1 year is considered for developing the
equations, in which year was the estimated per capita income highest?
Per capita income Nationalincome
Population
⎛ = ⎞ ⎜ ⎟
⎝ ⎠
(a) 2001 (b) 2002 (c) 2003 (d) 2004 (e) 2005.
(1 mark)
< Answer >
12
69. The following data pertains to the number of televisions sold by a company from 1996 to 2002.
Years (x) 1996 1997 1998 1999 2000 2001 2002
Number of televisions sold (y) 45 51 57 60 63 66 69
Calculate the estimated number of televisions that will be sold in 2003.
(a) 75 (b) 78 (c) 80 (d) 82 (e) 85.
(2 marks)
< Answer >
70. The time series analysis helps:
I. To compare the two or more series.
II. To know the behavior of business.
III. To make predictions.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
71. The estimated value of a variable according to trend equation is 67.2 in a particular year. The actual
value of the variable is 69.4. The relative cyclical residual is
(a) 0.86 (b) 1.49 (c) 2.83 (d) 3.27 (e) 4.18.
(2 marks)
< Answer >
72. Interpolation provides good estimates of missing values if and only if
I. The change of values is consistent.
II. The series does not refer to abnormal periods.
III. The arguments are equidistant.
(a) Only (I) above (b) Only (III) above
(c) Both (I) and (II) above (d) Both (I) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
73. The problems of interpolation are simpler than prediction because
I. There is no restriction in case of interpolation.
II. Interpolation is based on more stringent restrictions than prediction.
III. Interpolation has fewer restrictions than prediction.
(a) Only (I) above (b) Only (III) above
(c) Both (I) and (II) above (d) Both (II) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
74. In financial analysis extrapolation is widely used for
I. Forecasting future sales, cost and profit.
II. Long-term capital requirements.
III. Production of financial statements for financial institutions, banks, etc.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (II) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
13
75. In financial analysis, interpolation is used widely in
I. Determination of internal rate of return of a project.
II. Finding out the yield to maturity of a bond or debenture.
III. Other situations where the time value of money is considered and interpolations have to be made
while using the present and future values.
(a) Only (I) above (b) Only (III) above
(c) Both (I) and (II) above (d) Both (II) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
76. Which of the following statements is/are true?
I. Interpolation is a technique for obtaining most likely missing links.
II. Interpolation approach of estimations is probabilistic.
III. Interpolation means estimating a value which lies within the given range of arguments.
IV. Estimation under interpolation is more accurate than extrapolation.
(a) Both (I) and (II) above (b) Both (I) and (III) above
(c) Both (II) and (IV) above (d) (I), (III) and (IV) above
(e) (II), (III) and (IV) above.
(1 mark)
< Answer >
77. The purpose of using simulation technique is to
I. Reduce the cost of experiment on a model of real situation.
II. Imitate a real world situation.
III. Understand properties and operating characteristics of complex real-life problems.
(a) Only (I) above
(b) Only (II) above
(c) Only (III) above
(d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
78. In a transportation problem, a supplier can serve any destination and a destination can be served by any
suppler. If one suppler can serve only one destination and one destination can be served by only one
suppler, this type of problem is called
(a) Transportation problem
(b) Transportation problem with constraints
(c) General allocation problem
(d) Assignment problem
(e) Linear programming problem.
(1 mark)
< Answer >
79. Which of the following statement(s) is/are true with respect to the inventory?
I. Inventory helps in smooth and efficient running of business.
II. Inventory provides service to the customers immediately or at a short notice.
III. Inventory helps in maintaining the economy by absorbing some of the fluctuations when the
demand for an item fluctuates or seasonal.
IV. Maintaining of inventory may earn price discount because of bulk-purchasing.
(a) Only (I) above
(b) Only (III) above
(c) Both (I) and (II) above
(d) Both (II) and (IV) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
14
80. Large complicated simulation models are appreciated because
I. It is difficult to create the appropriate events.
II. They may be expensive to write and use as an experimental device.
III. Their average costs are not well defined.
(a) Only (I) above
(b) Only (II) above
(c) Only (III) above
(d) Both (I) and (II) above
(e) Both (II) and (III) above.
(1 mark)
< Answer >
15
Suggested Answers
Quantitative Methods - I (MB151): April 2006
1. Answer : (e)
Reason : a b c
1 1 1
log abc log abc log abc
+ +
abc abc abc
abc
log a log b log c
log abc 1.
= + +
= =
< TOP >
2. Answer : (b)
Reason : e 5 10 log x .log 625 = log 16.loge 10
4
e 5
e e
4 4
e e
log16 log10
log x .log 5 .
log10 log e
4 log x log 16.
log x log 2 .
x 2.
=
=
=
⇒ =
< TOP >
3. Answer : (c)
Reason :
( ) ( ) 1/ 2
1 7
2 2 2
log 128 log 2 7 .log 2 14.
1/ 2
− − −
= = = −
< TOP >
4. Answer : (c)
Reason :
7 log 16 5log 25 3log 81
15 24 80
+ +
( ) ( ) ( )
4 2 4
3 4
7 log 2 5log 5 3log 3
5 3 3 2 5 2
7 4log 2 log 5 log 3 5 2log 5 log 3 3log 2 3 4log 3 log 5 4log 2
log 2.
= + +
× × ×
− − + − − + − −
=
Therefore (c) is the correct answer.
< TOP >
5. Answer : (d)
Reason : Each number must consist of 7 or more digits. There are 7 digits in all, of which there
are 2 fours, 3 fives, and the rest different. Therefore the total numbers are 7!/(2!3!) =
420.
Of these numbers, some begin with zero and are less than one million and must be
rejected.
The numbers beginning with zero are 6!/(2!3!) = 60.
Therefore, the required numbers are 420 – 60 = 360.
< TOP >
6. Answer : (d)
Reason : The possibilities are: 1 female and 3 males
or 2 females and 2 males
or 3 females and 1 male
or 4 females and 0 males
= (4 C1 x 5 C3 ) + (4 C2 x 5 C2 ) + (4 C3 x 5 C1 ) + (4 C4 x 5 C0 )
= 40 + 60 + 20 + 1 = 121.
< TOP >
7. Answer : (b)
Reason : Number of ways in which 6 papers can be arranged = 6! ways.
If two statistics papers are to be kept together, then the six papers can be arranged in
< TOP >
16
5!2! ways.
Hence the number of arrangements in which six papers can be arranged so that two
statistics papers are not together = 6! – (5!2!) = 720 – 240 = 480.
8. Answer : (e)
Reason : 8
(0.657) = 6×8-1+5×8-2+7×8-3
=6×0.125+5×0.015625+7×0.001953125
( )= 0.841796875 10
< TOP >
9. Answer : (b)
Reason : Let the numbers be a-d, a a+d.
Sum of the numbers =27
i.e. (a-d) +a+(a+d) = 27
∴3a = 27
⇒ a = 9.
Product of the extremes = 27
i.e. (a-d) (a+d) = 77
= (a-d) (a+d) = 77
81-d2 = 77
d2 = 4 ⇒d = ±2
∴The required numbers are 7, 9 and 11.
< TOP >
10. Answer : (c)
Reason : Here a = 13
d = 8-13= –5
( ) t20 = a + 20 − 1 d
=a+19d
= 13+19(-5) = – 82.
< TOP >
11. Answer : (a)
Reason : f :R R + →
( )
( )
( )
8
y
1 y
1 x
let y f x log x
x 8
and x f y 8
f :R R
such that f x 8
−
+
−
= =
⇒ =
= =
∴ →
=
< TOP >
12. Answer : (d)
Reason : The number of ways the candidate can choose 7 questions is
(5A, 2B), (4A, 3B), (3A, 4B) and (2A, 5B), where 5A, 2B means 5 questions from part
A and 2 question from part B.
6 6 6 6 6 6 6 6
5 2 4 3 3 4 2 5 C × C + C × C + C × C + C × C
= 90 + 300 + 300 + 90 = 780
< TOP >
13. Answer : (c)
Reason : Sales during the year 2004 =
30 46 30 (2004 2003)
2005 2003
−
+ −
− =38 lakh.
< TOP >
14. Answer : (e)
Reason : Taking into account the first and the last terms 5 and 640, we have 8 terms in all. We
are given the first and the eighth terms. The first term ‘a’=5 and the eight term T8 =
< TOP >
17
a.r7
But T8= 640 = 5. r7
r7 = 640/5 = 128
r = (128)1/7 =2
now a =5 and r = 2, the six geometric means are 10, 20, 40, 80, 160 and 320.
15. Answer : (e)
Reason : By subtracting the second equation and the third equation from the first equation we
get
z
12
7
3
1
4
y 1
9
2
9
1
3
x 1
10
3
10
2
2
1 ⎟⎠
⎞
⎜⎝
+ ⎛ − + ⎟⎠
⎞
⎜⎝
− ⎛ − − ⎟⎠
⎞
⎜⎝
⎛ − −
= 12 – 8 – (–2)
or 0.x – 0.y +
z
12
3− 4 + 7
= 6 or z =
x12
6
6
= 12.
< TOP >
16. Answer : (d)
Reason : Moving average, progressive average and composite average are commercial averages.
Arithmetic mean and Harmonic mean are mathematical averages.
Median and quartiles are positional averages. Therefore (d) is the correct answer.
< TOP >
17.
Answer : (e)
Reason : Algebraic sum of deviations of a set of values from their arithmetic mean is zero,
Σ(X−X) = 0.
< TOP >
18. Answer : (b)
Reason : The first n natural numbers are 1,2,3…n
The sum of first n natural
numbers is = 1+2+3+…+n =
n(n + 1)
2
The sum of squares of first n natural numbers is
12+22+32 +...+ n2 =
n(n + 1)(2n + 1)
6
Weighted arithmetic mean =
wx ... n
w ... n
+ + +
=
+ + +
Σ
Σ
12 22 2
1 2 =
x w wx
1 1 12
2 2 22
3 3 32
. . .
. . .
. . .
< TOP >
19. Answer : (a)
Reason : Expression of geometric mean in terms of arithmetic mean and harmonic mean is
G.M.= A.M.×H.M.
< TOP >
20. Answer : (c)
Reason : The appropriate mean for a set of ratios using the numerators of the ratio data as
weights is the weighted harmonic mean.
The appropriate mean for a set of ratios using the denominators of the ratio data as
weights is the weighted arithmetic mean
The appropriate mean for a set of ratios, which have been calculated with the same
numerators, is the simple harmonic mean. Therefore (c) is the correct answer.
The appropriate mean for a set of ratios, which have been calculated with the same
denominators, is the simple arithmetic mean.
< TOP >
21. Answer : (b)
Reason : The sum of squares of first n natural numbers is
12+22+32 +...+ n2 =
n(n 1)(2n 1)
6
+ +
.
< TOP >
18
Here n=23
23(23 1)(46 1)
6
+ +
⇒
=4324
∴Mean of squares of first 23 natural numbers is =4324 / 23=188.
22. Answer : (b)
Reason : Coefficient of variation is the ration of standard deviation and mean. Options (a), (c),
(d) and (e) are correct with respect to coefficient of variation.
< TOP >
23. Answer : (e)
Reason : Quartile deviation ensures the lowest degree of reliability. Standard deviation ensures
the highest degree of reliability.
< TOP >
24. Answer : (a)
Reason : The average daily wages of all the workers in the two firms x and Y taken together is
given by
1 1 2 2
1 2
x n x n x
n n
+
=
+ =
500X186+600X175
500+600 =Rs.180
< TOP >
25. Answer : (b)
Reason : The sum of the squares of the deviations of items of any series from a value other than
the arithmetic mean would always be greater. The option (a), (c), (d) and (e) are
correct.
< TOP >
26. Answer : (c)
Reason : If each value of a set is divided by a constant then the coefficient of variation is same
as original value
< TOP >
27. Answer : (e)
Reason : Since the items of a sample resulted into same values pertaining to a character, the
variance of the sample is zero. Therefore (e) is the correct answer.
< TOP >
28. Answer : (b)
Reason :
C.I Midvalue(X) Frequency (f) f.x X − X f. X − X
10-20 15 8 120 –17.33 138.64
20-30 25 15 375 –7.33 109.95
30-40 35 25 –875 2.67 66.75
40–50 45 9 405 12.67 114.03
50–60 55 3 165 22.67 68.01
60 1940 497.38
fx
Mean .
f
1940
32 33
60
= = = Σ
Σ
Mean Deviation =
1 f X X
N
= −
=
.
= .
497 38
8 289
60
Therefore (b) is the correct answer.
< TOP >
29. Answer : (e)
Reason : Harmonic mean , mode , quartiles and percentiles are the measures of central value.
Interquartile range is a measure of dispersion.
< TOP >
30. Answer : (e)
Reason : I. Geometric mean cannot be calculated if any of the value in the set is zero.
II. Geometric mean is appropriate for averaging the ratios of change, for average of
proportions, etc.
III. Geometric mean is considered most suitable average for index numbers
Therefore (e) is the correct answer
< TOP >
19
31. Answer : (a)
Reason : A salesman whose annual sales figures are most consistent will have minimum
coefficient of variation of annual sales figures. i.e., if the coefficient of variation is
minimum then it is said to be consistent.
< TOP >
32. Answer : (a)
Reason : According to the Bienayme-Chebyshev rule, the percentage of data points lying within
± k standard deviation of the mean is at least 2
1 1
k
⎛ − ⎞ ⎜ ⎟
⎝ ⎠× 100.
If k = 1, then it will be 0 percent which is impossible. So this rule is not applicable for
k = 1
< TOP >
33. Answer : (a)
Reason : Sample standard deviation = n 1
nx
n 1
x 2 2
−
−
−
Σ
= 64
65 x 8
64
4560 2
−
= 71.25− 65
= 6.25
= 2.50.
< TOP >
34. Answer : (c)
Reason : Mean, x = n
Σx
If every item is increased by 5 then,
Mean of the modified set of data = n
Σ(x + 5)
= n
5
n
x Σ
+
Σ
= n
n .5
x +
= x + 5
∴ In this case the mean of the modified data = 10 + 5 = 15.
< TOP >
35. Answer : (b)
Reason : For a positively skewed distribution mean > median > mode.
< TOP >
36. Answer : (d)
Reason : The combined arithmetic mean, x12 = 1 2
1 1 2 2
n n
n x n x
+
+
Given : n1 = 60 x1 = 3
n2
= 40 x 2 = ?
x12 = 3.80
∴3.80 = 60 40
(60 3) 40x 2
+
× +
or 380 = 180 + 40x 2 or x2 = 40
380 −180
= 5.00.
< TOP >
37. Answer : (c)
Reason : Let N1 denote the number of supervisors and N denote the total number of supervisors
and workmen
< TOP >
20
N1X1 (N N1)X2 X12 N
+ −
=
or
9000N1 5000(N N1) 6000
N
+ −
=
or
9000N1 5000N 5000N1 6000
N
+ −
=
or
4000N1 5000N 6000
N
+
=
or
4000N1 5000 6000
N
+ =
or
4000N1 1000
N
=
or
N1 1000 1 0.25
N 4000 4
= = =
∴Percentage of supervisors = 0.25 × 100 =25%.
38. Answer : (a)
Reason : A=
4 2
1 1
⎡ ⎤
⎢− ⎥ ⎣ ⎦ then (A –2I) (A – 3I)=
4 2 1 0 4 2 1 0
2 3
1 1 0 1 1 1 0 1
4 2 2 0 4 2 3 0
1 1 0 2 1 1 0 3
2 2 1 2
1 1 1 2
0 0
.
0 0
⎡⎡ ⎤ ⎡ ⎤⎤ ⎡⎡ ⎤ ⎡ ⎤⎤
⎢⎢ ⎥ − ⎢ ⎥⎥ ⎢⎢ ⎥ − ⎢ ⎥⎥ ⎣⎣− ⎦ ⎣ ⎦⎦ ⎣⎣− ⎦ ⎣ ⎦⎦
⎡⎡ ⎤ ⎡ ⎤⎤ ⎡⎡ ⎤ ⎡ ⎤⎤
= ⎢⎢ ⎥ − ⎢ ⎥⎥ ⎢⎢ ⎥ − ⎢ ⎥⎥ ⎣⎣− ⎦ ⎣ ⎦⎦ ⎣⎣− ⎦ ⎣ ⎦⎦
⎡ ⎤⎡ ⎤
= ⎢ ⎥ ⎢ ⎥ ⎣− − ⎦ ⎣− − ⎦
⎡ ⎤
=⎢ ⎥
⎣ ⎦
< TOP >
39. Answer : (e)
Reason : A – B =
1 0
0 1
⎡ ⎤
⎢ − ⎥ ⎣ ⎦ ………..(1)
A + B =
3 4
2 5
⎡ ⎤
⎢ ⎥
⎣ ⎦…………(2)
Add (1) and (2)
< TOP >
21
1 0 3 4
A B A B
0 1 2 5
4 4
2A
2 4
2 2
A
1 2
substituteAin (1)
1 0
A B
0 1
2 2 1 0
B
1 2 0 1
1 2
B
1 3
2 2 1 2 4 10
now AB .
1 2 1 3 3 8
⎡ ⎤ ⎡ ⎤
⇒ − + + = ⎢ ⎥ + ⎢ ⎥ ⎣ − ⎦ ⎣ ⎦
⎡ ⎤
⇒ =⎢ ⎥
⎣ ⎦
⎡ ⎤
⇒ =⎢ ⎥
⎣ ⎦
⎡ ⎤
− = ⎢ ⎥ ⎣ − ⎦
⎡ ⎤ ⎡ ⎤
⎢ ⎥ − = ⎢ ⎥ ⎣ ⎦ ⎣ − ⎦
⎡ ⎤
⇒ =⎢ ⎥
⎣ ⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
40. Answer : (c)
Reason :
n 3 1
4 3
⎡ − ⎤
⎢ − ⎥ ⎣ ⎦
take n=2
then
2 3 1 3 1 3 1 5 0
4 3 4 3 4 3 0 5
⎡ − ⎤ ⎡ − ⎤ ⎡ − ⎤ ⎡ ⎤
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ − ⎦ ⎣ − ⎦ ⎣ − ⎦ ⎣ ⎦
which is a scalar matrix
now take n = 4
4 3 1 25 0
4 3 0 25
⎡ − ⎤ ⎡ ⎤
⎢ ⎥ = ⎢ ⎥ ⎣ − ⎦ ⎣ ⎦
which is also a scalar matrix
Therefore, if n is an even natural number then
n 3 1
4 3
⎡ − ⎤
⎢ − ⎥ ⎣ ⎦ is equal to scalar matrix of
order 2.
< TOP >
41. Answer : (d)
Reason :
1 2 3 8
2X
3 4 7 2
⎡ ⎤ ⎡ ⎤
+ ⎢ ⎥ = ⎢ ⎥
⎣ ⎦ ⎣ ⎦
3 8 1 2
2X
7 2 3 4
2 6
2X
4 2
1 3
X .
2 1
⎡ ⎤ ⎡ ⎤
⇒ = ⎢ ⎥ − ⎢ ⎥
⎣ ⎦ ⎣ ⎦
⎡ ⎤
⇒ = ⎢ ⎥ ⎣ − ⎦
⎡ ⎤
⇒ = ⎢ ⎥ ⎣ − ⎦
< TOP >
42. Answer : (d)
Reason :
9 1 1 5
A ,B
7 8 7 12
⎡ ⎤ ⎡ ⎤
= ⎢ ⎥ = ⎢ ⎥
⎣ ⎦ ⎣ ⎦,
< TOP >
22
given 5A + 3B+ 2C =
0 0
0 0
⎡ ⎤
⎢ ⎥
⎣ ⎦
9 1 1 5 0 0
5 3 2C
7 8 7 12 0 0
45 5 3 15 0 0
2C
35 40 21 36 0 0
48 20 0 0
2C
56 76 0 0
24 10
C .
28 38
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⇒ ⎢ ⎥ + ⎢ ⎥ + = ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⇒ ⎢ ⎥ + ⎢ ⎥ + = ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤
⇒ ⎢ ⎥ + = ⎢ ⎥
⎣ ⎦ ⎣ ⎦
⎡− − ⎤
⇒ = ⎢ ⎥ ⎣− − ⎦ =
0 0
0 0
⎡ ⎤
⎢ ⎥
⎣ ⎦
43. Answer : (c)
Reason : Given that the matrix A is an identity matrix of order 3x3. Then the rank of such
matrix is equal to number of rows or number of columns.
i.e. rank = 3.
< TOP >
44. Answer : (c)
Reason : The given systems of equations can be written as
5 3 2 1
2 0 8 5
3 5 0 3
−
−
−
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢⎣ ⎥⎦ . This is the
augmented matrix of the systems of equations.
< TOP >
45. Answer : (b)
Reason : The rank of every non-singular matrix of order ‘n’ is ‘n’ itself.
< TOP >
46. Answer : (c)
Reason : If A = [aij] m x n is a square matrix of order m, then the elements aij for which i = j, are
called the diagonal elements of matrix A.
< TOP >
47. Answer : (c)
Reason : When two rows or columns of a determinant are identical then the value of the
determinant is zero.
< TOP >
48. Answer : (b)
Reason : The option (II) is false because, while solving the linear programming model, there is
no guarantee that we will get integer valued solutions
The options (I), (III) and (IV) are correct (these are the limitations of linear
programming).
< TOP >
49. Answer : (a)
Reason : In solving the linear programming problem, if the type of constraint is less than or
equal to (<) then the extra variable called slack variable is added. Therefore (a) is the
correct answer. The coefficient of extra variable in the objective function
(Max.z/Min.z) is 0.
< TOP >
50. Answer : (e)
Reason : These are some of the main assumptions that are made in the construction of linear
programming models:
I. Proportionality.
II. Additivity.
III. Divisibility.
IV. Certainty.
Therefore(e) is the correct answer.
< TOP >
51. Answer : (d)
Reason : 1 2 Max.z = 15x +17x
< TOP >
23
1
2
1 2
1 2
subject to the constra int s
x 6
x 12
x x 24
x ,x 0
≥
≥
+ ≥
≥
the solution for the above linear programming problem is unbounded solution because
the feasible region extends infinitely to the right.
x
1
1
3
X =12
2
x
2
2 4 6 8 10 12 14 16 18 20 22 24
22
24
20
18
16
14
12
10
8
6
4
2
x + x = 24
1 2
x = 6
1
52. Answer : (b)
Reason :
3 2 5 0 0 0
Basic
variabl
es
CB XB X1 X2 X3 S1 S2 S3 Min Ratio
(XB/Xt)
S1 0 430 1 2 1 1 0 0 430/1 = 430
← S2 0 460 3 0 2 0 1 0 460/2 = 230
←
S3 0 420 1 4 0 0 0 1 –
X1=X2=X3=0 Z = 0 –3 –2 –5
↑
0 0 0 ←Δj
← S1 0 200 1
2
−
2 0 1 1
2
−
0 200/2 = 100
←
→ X3 5 230 3/2 0 1 0 1
2
0 –
S3 0 420 1 4 0 0 0 1 420/4 = 105
X1=X2=S2=0 Z = 1150 9/2 –2
↑
2 0
↓
5/2 0 ← Δj
X2 2 100 1
4
−
1 0 1
2
1
4
−
0
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24
X3 5 230 3/2 0 1 0 1
2
0
S3 0 20 2 0 0 –2 1 1
X1=S2=S3=0 Z = 1350 4 0 0 1 2 0 ←Δj > 0
53. Answer : (c)
Reason : The positive value of a slack variable in the optimal solution of a linear programming
problem indicates that part of that resource has not been used up.
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54. Answer : (e)
Reason : I. The problem must have a sell-defined single objective to achieve.
II. There must be alternative courses of action, one which will achieve the objective.
III. The decisions variables must be continuous in nature.
IV. The objective and the constraints must be linear functions.
V. Resources must be limited in supply and the achievement of the objective
function is restricted by these constraints.
The concept of linear programming can be applied to those problems which satisfy the
above requirements.
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55. Answer : (b)
Reason : In cases where fractional values of the variables do not make a sense, the linear
programming problem can be formulated and solved as an integer programing.
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56. Answer : (c)
Reason : 2X+5Y≤ 600 The equation of line is 2X + 5Y = 600 …….(i)
Testing time:
1
X + Y 120
2
≤
Or, 2X + Y ≤ 240 The equation of line is
2X + Y = 240….(ii)
So, the vertices and the corresponding profits are
(120, 0) : 240×120 = Rs.28800
(75, 90) : 75× 240 + 90×375= Rs.51750
(0, 120) : 120× 375=Rs.45000
So, the optimal point is (75, 90) and the company should produce 90 numbers of
Engines.
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57. Answer : (e)
Reason : I. Index numbers measure changes in some quantities, which cannot be observed
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25
directly.
II. Index numbers are expressed in percentages, which make it feasible to compare
any two or more index numbers.
III. Index numbers are of comparable nature at any two timings or places or any other
situation.
IV. In index numbers the two periods, one is known as the base period and the other,
the current period are always comparable with each other.
The options (I), (II), (III) and (IV) are the characteristics of an index numbers.
Therefore (e) is the correct answer.
58. Answer : (d)
Reason : Paasches price index =
1 1
0 1
PQ
100
P Q
Σ
×
Σ =
1 1
1
0 1
1
PQ
Q
100
P Q
Q
⎡Σ ⎤
⎢ Σ ⎥ ⎣ ⎦×
⎡Σ ⎤
⎢ Σ ⎥ ⎣ ⎦ =
46.875 100 125.
37.5
× =
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59. Answer : (d)
Reason : Value Index is
100
268
100 301
P Q
P Q
0 0
1 1 × = ×
Σ
Σ
= 112.31
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60. Answer : (a)
Reason : Paasche’s Quantity Index for the given data is
1 1
0 1
Q .P
Q .P
Σ
Σ x 100 i.e.,
700 30 70 14 120 17 70 16
500 30 100 14 200 17 50 16
× + × + × + ×
× + × + × + × x 100 =
25140 100
20600
×
= 122.04.
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61. Answer : (e)
Reason : A value index can not distinguish between the price changes and quantity changes.
Hence an increase in the value index cannot indicate with certainty that the prices have
increased.
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62. Answer : (e)
Reason : Fisher’s Ideal Index is the geometric mean of Laspeyres Index and Paasche’s Index. So
its value falls between the values of Laspeyres and Paasches indices. Hence, from
above discussion, we can infer that option (e) is false regarding Fisher’s Ideal Index.
Options (a), (b), (c) and (d) are all true regarding Fisher’s Ideal Index.
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63. Answer : (d)
Reason : Let the misting value be ‘x’
Given that
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26
1 0
0 0
1 1
0 1
L 28
P 27
PQ 100
P Q or, 28
PQ 100 27
P Q
20 5x
or, 10 5 28
10 2x 27
5 2
or, 20 5x 7 28
15 10 2x 27
or, 20 5x 20
10 2x 9
or, x 4.
=
Σ
×
Σ
=
Σ
×
Σ
+
+ =
+
+
+
× =
+
+
=
+
=
64. Answer : (d)
Reason : Laspeyres price index =
1 0
0 0
PQ
100
P Q
Σ
×
Σ
=
1 0
0
0 0
0
PQ
Q
100
P Q
Q
⎛ Σ ⎞
⎜ Σ ⎟ ⎝ ⎠×
⎛ Σ ⎞
⎜ Σ ⎟ ⎝ ⎠
=
Weighted average price in the current year
100
Weighted average price in the base year
×
Note: Base year quantities are used as weights in the weighted averages.
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65. Answer : (c)
Reason : Laspeyres price index =
1 0
0 0
PQ
100
P Q
Σ
×
Σ =
1 0
0
0 0
0
PQ
Q
100
P Q
Q
⎡Σ ⎤
⎢ Σ ⎥ ⎣ ⎦×
⎡Σ ⎤
⎢ Σ ⎥ ⎣ ⎦ =
40 100
32
×
= 125.
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66. Answer : (d)
Reason : A time series of annual data can contain only secular trend and cyclical fluctuation
among the alternatives mentioned in the question. It does not contain seasonal
variations, as these variations take place within a year.
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67. Answer : (e)
Reason : Based on the given information, it is not possible to determine the value, as it is
difficult to forecast the cyclical fluctuations. It is because of the fact that time period of
the cyclical variations varies from time to time.
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68. Answer : (e)
Reason : By careful observation of equations, it can be interpreted that as the time (i.e., the
value of x increases by one unit) passes on , national income is increasing by 1.5 units,
where as population is increasing only by 1.25 units. In other words, growth rate in
national income is more than that in population. Hence, as the time passes on, per
capita income goes on increasing. Hence, it will be highest in the year 2005. This can
be proved through calculations.
Year 2001 2002 2003 2004 2005
National income (A) 1.86 3.36 4.86 6.36 7.86
Population (B) 2.75 4 5.25 6.5 7.75
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27
Per capita income (A/B) 0.676 0.84 0.926 0.978 1.014
69. Answer : (d)
Reason : Computing the secular trend(odd number of observations)
The given series is having odd number of observation, n=7
X 13993/ 7 1999
X x X
= =
= −
Year(x) Number of Telivisions(y) X XY
1996 58 -3 -174 9
1997 63 -2 -126 4
1998 65 -1 -65 1
1999 69 0 0 0
2000 71 1 71 1
2001 76 2 152 4
2002 79 3 237 9
TOTAL 481 0 95 28
X2
Since ΣX = 0 we can use the formula
2
XY
b=
X
Σ
Σ , and
a Y
n
= Σ
Therefore, substituting the values we have,
b 95 3.39
28
= =
and
a 481 68.7
7
= =
Therefore the regression equation showing the trend is Y = a + bX
e Y = 68.7 + 3.39X
from the above equation, we can estimate the number of the televisions that will be
sold bin 2003.
( )
e
X 2003 1999 4
Y 68.7 3.39X
68.7 3.39 4
82.26or 82
= − =
∴ = +
= +
=
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70. Answer : (e)
Reason : The time series analysis helps:
I. To compare the two or more series.
II. To know the behavior of business.
III. To make predictions.
Therefore (e) is the correct answer.
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71. Answer : (d)
Reason : Relative cyclical residual measure =
(Y Yˆ )
100 ˆY
−
×
(69.4 67.2)
100
67.2
3.27
−
= ×
=
Therefore (d) is the correct answer.
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72. Answer : (e)
Reason : Interpolation provides good estimates of missing values if and only if
I. The change of values is consistent.
II. The series does not refer to abnormal periods.
III. The arguments are equidistant.
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28
Therefore (e) is the correct answer
73. Answer : (b)
Reason : The problems of interpolation are simpler than prediction because Interpolation has
fewer restrictions than prediction. Therefore (b) is the correct answer.
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74. Answer : (e)
Reason : In financial analysis extrapolation is widely used for
I. Forecasting future sales, cost and profit.
II. Long-term capital requirements.
III. Production of financial statements for financial institutions, banks, etc. Therefore
(e) is the correct answer.
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75. Answer : (e)
Reason : In financial analysis, interpolation is used widely in:
I. Determination of internal rate of return of a project.
II. Finding out the yield to maturity of a bond or debenture.
III. Other situations where the time value of money is considered and interpolations
have to be made while using the present and future values.
It must be noted that in all the above, interpolation is used to insert figures in a series
other than a time series.
Therefore (e) is the correct answer.
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76. Answer : (d)
Reason : Interpolation approach of estimation is non-probabilistic. Options (I), (III) & (IV) are
true. Therefore (d) is the correct answer.
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77. Answer : (e)
Reason : The purpose of using simulation technique is to
I. Reduce the cost of experiment on a model of real situation.
II. Imitate a real world situation.
II. Understand properties and operating characteristics of complex real-life
problems.
Therefore (e) is the correct answer.
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78. Answer : (d)
Reason : If each job requires exactly one resource and each resource can be used on only job,
the resulting problem, is an assignment problem. Therefore (d) is the correct answer.
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79. Answer : (e)
Reason : Inventory maintenance is necessary because of the following reasons:
I. Inventory helps in smooth and efficient running of business.
II. Inventory provides service to the customers immediately or at a short notice.
III. Inventory helps in maintaining the economy by absorbing some of the
fluctuations when the demand for an item fluctuates or seasonal.
IV. Maintaining of inventory may earn price discount because of bulk-purchasing.
Therefore (e) is the correct answer.
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80. Answer : (b)
Reason : Large complicated simulation models are appreciated because they may be expensive
to write and use as an experimental device.
Therefore (b) is the correct answer.
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