Quantitative Methods-I (MB151): July 2006

1
Question Paper
Quantitative Methods-I (MB151): July 2006
· Answer all questions.
· Marks are indicated against each question.
1. Which of the following is/are the properties of inverse matrix?
I. The inverse of a given square matrix, if it exists, is unique.
II. The inverse of the inverse of a matrix is the matrix itself.
III. The transpose of the inverse is equal to the inverse of the transpose of the given matrix.
IV. If A and B are two non-singular matrices and A–1 and B–1 are their respective inverses, then
(AB)–1 = B–1. A–1.
(a) Only (I) above (b) Only (IV) above
(c) Both (I) and (IV) above (d) Both (II) and (III) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
2.
If 5 7
é + - ù
= ê ú ë - + û
x y y z
A
t x and
é - - ù
= ê ú ë - + + û
t x z t
B
z y x z t then find the value of x when A = B.
(a) 1 (b) 2 (c) 3 (d) 4 (e) 5.
(2 marks)
< Answer >
3.
If
2 3
1 2
é ù
=ê ú
ë û
A
and
1 2
2 1
é - ù
= ê ú ë - û
B
then find (AB)–1.
(a)
1 4 7
3 5 8
é- ù
ê - ú ë û (b)
1 4 7
3 5 8
é- ù
ê- ú ë û (c)
1 4 5
3 7 8
é- - ù
ê ú
ë û
(d)
1 5 8
3 4 7
é- ù
ê- ú ë û (e)
1 8 7
3 5 4
é ù
ê- ú ë û.
(2 marks)
< Answer >
4. If A = diag [2, –5, 9], B = diag [–3, 7, 14] and C = diag [4, –6, 3] then find 3A + B – 2C.
(a) diag [–2, 5, 18] (b) diag [2, –5, 9]
(c) diag [–5, 4, 35] (d) diag [6, –5, 12] (e) diag [3, 6, 9].
(1 mark)
< Answer >
5.
If P =
9 2 15
2 4 3
é ù
êë úû and Q =
2 1 3
1 -1 2
é ù
êë úû , find the matrix R such that P – R = 4 Q.
(a)
17 6 27
6 0 11
é ù
êë úû (b)
1 2 3
2 8 5
-
- -
é ù
êë úû (c)
9 8 13
5 1 9
é ù
êë úû
(d)
4 2 3
3 12 6
-
- -
é ù
êë úû (e)
2 4 6
4 16 10
-
-
é ù
êë úû .
(1 mark)
< Answer >
2
6.
If A =
1 1
2 1
-
-
é ù
êë úû , B =
a 1
b -1
é ù
êë úû and (A + B)2 = A2 + B2 then the values of a and b are
(a) a = 0 and b = 1 (b) a = 1 and b = 0
(c) a = 1 and b = –2 (d) a = 1 and b = 4
(e) a = –1 and b = –2.
(2 marks)
< Answer >
7.
The value of the determinant
1 2 3
6 7 8
13 14 15
é ù
ê ú
ê ú
êë úû is
(a) 10 (b) 15 (c) 18 (d) 20 (e) 0.
(1 mark)
< Answer >
8.
The value of
3 5
5 3 1 2 4 2
2 log 256 2 64 16
-
- ´ - ´ is
(a) 1 (b) 0 (c) –7/5 (d) 2/3 (e) 2.
(1 mark)
< Answer >
9.
If x =1 + loga bc , y =1 + logb ca , z = 1 + logc ab then the value of xyz =
(a) xy + xz + yz (b) xy – xz – yz
(c) xy + xz – yz (d) xy – xz + yz (e) –xy + xz + yz.
(2 marks)
< Answer >
10. A man has 6 friends. In how many ways he can invite one or more of them to party?
(a) 63 (b) 64 (c) 119 (d) 120 (e) 720.
(1 mark)
< Answer >
11. What is the value of 23-log2 7 ?
(a) 1 (b) 8 (c) 7 (d) 7/8 (e) 8/7.
(1 mark)
< Answer >
12. According to the inverse property of addition, which of the following is true?
(a) For every real number there exists another real number such that the sum of the two real numbers
is equal to 1
(b) For every real number there exists another number such that the sum of the two real numbers is
equal to 0
(c) The addition of zero to any real number is equal to that real number
(d) For every real number there exists another real number such that the product of the two numbers
is equal to 1
(e) The product of any real number with 1 is equal to that real number.
(1 mark)
< Answer >
13. If the fifth term of a G.P. is 32 and the common ratio is 2, then the sum of first 15 terms of the G.P. is
(a) 52131 (b) 65534 (c) 32766 (d) 45251 (e) 22843.
(2 marks)
< Answer >
14.
The third term of a H.P. is
17
and the 15th term is
1
31 . Then the common difference of the series is
(a) 1 (b) 2 (c) 3 (d) 5 (e) 8.
(2 marks)
< Answer >
3
15.
Find x if
logx log36
log2 log4
=
?
(a) 0 (b) 2 (c) 4 (d) 6 (e) 8.
(1 mark)
< Answer >
16.
If 2log(a + b) +log(a - b)-log(a2 -b2 ) =logx,
then the value of x is
(a) a (b) b (c) a + b (d) a – b (e) a/b.
(2 marks)
< Answer >
17. 2x y z 7
y x z 6
x y z 12
- + =
- + =
+ + =
The value of (y + z) in the above simultaneous equations would be
(a) 5 (b) 6 (c) 7 (d) 8 (e) 9.
(1 mark)
< Answer >
18. q p
p q
m
x=
n ,
p ( q 1 )
q 3
n
m
-
y =
and
z = q m( 2- p) P n2
The value of xyz would be
(a)
q m
n (b)
p
m
n (c)
p
q
n
m (d)
q
p
m
n (e)
n
m .
(2 marks)
< Answer >
19. What is the value of 0!?
(a) 0 (b) 1 (c) –1 (d) 8 (e)
Indeterminate.
(1 mark)
< Answer >
20.
If
and -
n n n n
Pr = Pr+1 Cr = Cr 1
. The value of r would be
(a) 6 (b) 5 (c) 4 (d) 3 (e) 2.
(2 marks)
< Answer >
21.
The sum to infinity of the series
1 1 1 1 ...
2 6 18 54
+ + + +
is
(a) 1 (b) 1/2 (c) 2/3 (d) 3/4 (e) 4/5.
(1 mark)
< Answer >
22.
If
2n1 2 n 1
Pn 1 : Pn 3 : 5 + -
- = . Then the value of n is equal to
(a) 1 (b) 2 (c) 3 (d) 4 (e) 5.
(2 marks)
< Answer >
4
23. In a textile factory there are 50 skilled workers, 125 semiskilled workers and 75 unskilled workers. It
has been observed that on average a unit length of a particular fabric is woven by a skilled worker in 4
hours, by a semiskilled worker in 5 hours and by an unskilled worker in 6 hours. After three years of
experience the semiskilled workers are expected to become skilled and the unskilled workers to
become semiskilled. It is assumed that there will be no turnover of the workers within the next three
years.
What will be the change in time taken for weaving an unit length of the same fabric, after three years?
(a) No change (b) Reduction by 1.60 hours
(c) Reduction by 0.80 hours (d) Increase by 0.80 hours
(e) Increase by 1.60 hours.
(1 mark)
< Answer >
24. The following details are available with regard to a data set:
Sx = 33; Sx2 = 199; N = 6
If each observation in the data set is multiplied by 2 then the standard deviation of the resulting values
in the data set will be equal to
(a) 3.42 (b) 11.67 (c) 3 (d) 25 (e) 35.
(2 marks)
< Answer >
25. Which of the following is /are mathematical average(s)?
I. Arithmetic mean. II. Median.
III. Mode. IV. Geometric mean.
V. Harmonic mean.
(a) Only (I) above (b) Only (II) above
(c) (I), (II) and (IV) above (d) (I), (III) and (V) above
(e) (I), (IV) and (V) above.
(1 mark)
< Answer >
26. Which of the following is not true about mean absolute deviation?
(a) Mean deviation is obtained by calculating the absolute deviations of each observation from mean
(b) Mean deviation is a more comprehensive measure compared to range
(c) It is conducive to further algebraic treatment
(d) It cannot be computed for distributions with open end classes
(e) It can also be computed for grouped or continuous frequency distribution.
(1 mark)
< Answer >
27. The common averages used in statistical analysis are
I. Mean.
II. Median.
III. Mode.
If the data type is interval/ratio then which of the above average(s) is/are appropriate?
(a) Only (I) above (b) Only (II) above
(c) Both (I) and (II) above (d) Both (II) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
28. Which one of the following is false with regard to the geometric mean?
(a) Geometric mean is used to find the rate of population growth and the rate of interest
(b) Geometric mean is used in the construction of index numbers
(c) It is affected much by fluctuations of sampling
(d) If any one of the observations is zero, geometric mean becomes zero
(e) If any one of the observations is negative, geometric mean becomes imaginary regardless of the
magnitude of the other items.
(1 mark)
< Answer >
5
29. Sum of absolute deviations about median is
(a) 0 (b) 1 (c) –1
(d) Minimum (e) Maximum.
(1 mark)
< Answer >
30. Suitable measure of central tendency for qualitative data is
(a) Arithmetic mean (b) Geometric mean
(c) Median (d) Mode (e) Coefficient of variation.
(1 mark)
< Answer >
31. When a data series is arranged in ascending or descending order, what is the middle value of the series
called?
(a) Mean (b) Median (c) Mode (d) Quartiles (e) Octiles.
(1 mark)
< Answer >
32. Two samples are drawn from the population, the second sample contains 50 items with mean 16. If the
whole group has 125 items with mean 20, then the mean of the first sample is
(a) 15.00 (b) 18.50 (c) 22.66 (d) 25.42 (e) 28.01.
(2 marks)
< Answer >
33. The following distribution shows the ages of 100 persons in a group:
Age group (in years) Number of persons
25-30 9
30-35 14
35-40 21
40-45 14
45-50 13
50-55 12
55-60 9
60-65 5
65-70 3
What is the average age of the persons in the group?
(a) 25.5 years (b) 28.4 years (c) 32.6 years
(d) 39.3 years (e) 43.7 years.
(2 marks)
< Answer >
34. If any constant value ‘k’ is subtracted from each observation of a data set, then the variance is:
(a) Reduced by ‘k’ (b) Reduced by k 2 (c) Increased by ‘k’
(d) Increased by k 2 (e) Unaltered.
(1 mark)
< Answer >
35. The empirical relationship between range (R) and mean deviation (M.D) is
(a) 2R=15 M.D (b) 3R=17 M.D (c) R=17 M.D
(d) 3R= M.D (e) R= 15 M.D.
(1 mark)
< Answer >
36. Which of the following is true?
(a) Coefficient of variation expresses the standard deviation as a percentage of the mean
(b) Population coefficient of variation is always equal to sample coefficient of variation
(c) Population variance is always equal to sample variance for small samples
(d) Coefficient of variance is equal to coefficient of determination
(e) Population standard deviation is always equal to sample standard deviation for small samples.
(1 mark)
< Answer >
6
37.
If the quartile 1 Q =110 and 3 Q =150 then the coefficient of quartile deviation is equal to
(a) 0.121 (b) 0.133 (c) 0.312 (d) 0.153 (e) 0.512.
(1 mark)
< Answer >
38. If the standard deviation of a distribution is 21, the quartile deviation of the distribution is
(a) 7 (b) 14 (c) 28 (d) 32 (e) 42.
(1 mark)
< Answer >
39. The range of a set of values is 68 and the maximum value is 118. The coefficient of range is
(a) 0.861 (b) 0.159 (c) 0.550 (d) 0.405 (e) 0.642.
(1 mark)
< Answer >
40. A security analyst studied hundred companies and obtained the following return on investment (ROI)
data for the year 2005. The standard deviation in ROI of the companies is
Returns % 0-10 10-20 20-30 30-40
No. of companies 15 33 39 13
(a) 0 (b) 1 (c) 9 (d) 20 (e) 22.
(2 marks)
< Answer >
41. When the distribution is symmetrical and has one mode, the highest point on the curve is referred to as
the
I. Mean. II. Median. III. Mode. IV. Range.
(a) Only (I) above (b) Only (II) above
(c) Only (IV) above (d) (I), (II) and (III) above
(e) (II), (III) and (IV) above.
(1 mark)
< Answer >
42. Which of the following is/are the relative measure(s) of dispersion?
I. Variance.
II. Standard deviation.
III. Coefficient of variation.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (III) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
43. The coefficient of variation cannot be meaningfully used to compare the variability of two or more
sets of data, when
(a) The mean and standard deviation are equal for one or more sets of data
(b) The standard deviation is zero for one or more sets of data
(c) The standard deviation is one for one or more sets of data
(d) The mean is one for one or more sets of data
(e) The mean is zero for one or more sets of data.
(1 mark)
< Answer >
7
44. The following details are available with regard to four sets of data A, B, C and D.
Data set Number of observations Sum of observations Variance
A 15 90 16
B 22 110 4
C 26 208 36
D 30 210 25
Which data set is more consistent?
(a) A (b) B (c) C
(d) D (e) All are equally consistent.
(2 marks)
< Answer >
45. If the objective function of a profit maximizing linear programming problem is parallel to an edge of
the feasible region, which is in the direction of the optimal movement of the objective function, then
which of the following statements is true?
(a) There are multiple optimal solutions to the problem
(b) The constraint equations have been formulated incorrectly
(c) There cannot be any optimal solution to the problem
(d) There is only one optimal solution to the problem
(e) The problem cannot be solved graphically.
(1 mark)
< Answer >
46. Which of the following is not a requirement of a linear programming problem?
(a) The problem must have a well defined single objective to achieve
(b) There must be alternative courses of action, one of which will achieve the objective
(c) The decision variables should be able to take any non-negative value
(d) The objective and constraints must be linear functions of the decision variables
(e) The resources need not be limited in supply.
(1 mark)
< Answer >
47. While plotting constraints on a graph paper, terminal points on both the axes are connected by a
straight line because
I. The constraints are linear equations or inequalities.
II. The resources are limited in supply.
III. The objective function is a linear function.
(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 >
48. A furniture manufacturer makes two products: chairs and tables. Processing of these products is done
on two machines A and B. A chair requires 2 hours on machine A, 6 hours on machine B. A table
requires 5 hours on machine A and no time on machine B. There are 16 hours per day available on
machine A and 30 hours on machine B. Profit gained by the manufacturer from a chair and table is
Rs.2 and Rs.10 respectively. What should be the daily production of each of the two products? (x1 -
chairs ; x2 - tables).
(a) x1 = 0 and x2 = 3.2 (b) x1 = 0 and x2 = 0
(c) x1 = 3.2 and x2 = 0 (d) x1 = 1 and x2 = 2
(e) x1 = 2 and x2 = 1.
(2 marks)
< Answer >
8
49. A company makes two kinds of leather belts. Belt A is of high quality and belt B is of lower quality.
The respective profits are Rs.4 and Rs.3 per belt. The production of each of type A requires twice as
much time as a belt of type B, and if all belts were of type B, the company could make 890 per day.
The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires
a fancy buckle and only 400 per day are available. There are only 650 buckles per day available for
belt B. In formulating this problem as an LP model which of the following is not correct?
(a) Maximize Z = 4x1 + 3x2
(b) Availability of time constraint : 1 2 2x + x £ 890
(c) Supply of leather constraint : 1 2 x + x £ 800
(d) Buckles availability constraints : 1 x 400 ³ , 2 x£ 650
(e) Both (a) and (b) above.
(2 marks)
< Answer >
50.
In the optimal simplex table, j j C - Z = 0value indicates
(a) Alternative solution (b) Infeasible solution
(c) Unbounded solution (d) Cycling (e) Cannot be determined.
(1 mark)
< Answer >
51. A variable which does not appear in the basic variable column of simplex table is
(a) Always equal to 1 (b) Always equal to –1
(c) Always equal to 0 (d) Never equal to 0
(e) Called a basic variable.
(1 mark)
< Answer >
52. The value of the dual variable
I. Can be obtained by examining j j C - Z row of primal optimal simplex table.
II. Can be obtained by examining the j Z row of primal optimal simplex table.
III. Represents marginal profit of each additional unit of resource.
(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 >
53. Which of the following statements is/are true with respect to duality?
I. The dual of the dual LP is again the primal problem.
II. If either the primal or the dual problem has an unbounded objective function value, the other
problem has no feasible solution.
III. If either the primal or dual problem has a finite optimal solution, the other one also possesses the same
and the optimal value of the objective function of the two problems are equal.
(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 >
54. 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 >
9
55. 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 >
56. Network models have advantage in terms of project
I. Planning.
II. Scheduling.
III. Controlling.
(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 >
57. Which of the following is false with regard to a linear programming problem?
(a) The objective function is a linear function of the decision variables
(b) The resource constraints represent the consumption pattern of the resources and the amount of
available resources
(c) The decision variables can assume only integer values
(d) There may be a specific combination of values of the decision variables for which the objective
is accomplished within the given constraints
(e) The resources are in limited supply.
(1 mark)
< Answer >
58. Which of the following is true with regard to the simplex method of solving a linear programming
problem on profit maximization?
(a) The values of slack variables indicate whether the solution is optimal or not
(b) At the optimal solution all the Zj – Cj values will be zero
(c) The values in the solution column indicate the variable to enter solution
(d) The values in the Zj – Cj row indicate the variable to leave solution
(e) The value at the bottom of the solution column indicates the profit in that solution.
(1 mark)
< Answer >
59. In simplex method of solving linear programming problem, the initial basic feasible solution is
obtained by
(a) Assigning zero to the slack variable
(b) Assigning one to all the decision variables
(c) Assigning zero to the surplus variable
(d) Assigning minus one to the surplus variable
(e) Assigning zero to all the decision variables.
(1 mark)
< Answer >
60. Linear trend of time series indicates towards
I. Change in geometric progression.
II. Constant rate of growth.
III. Constant rate of change.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) Both (I) and (III) above.
(1 mark)
< Answer >
10
61. For given five values 15, 24, 18, 33 and 42 the three year moving averages are
(a) 90, 117, 135 (b) 57, 75, 93 (c) 25, 38, 45
(d) 19, 25, 31 (e) 39, 42, 75.
(1 mark)
< Answer >
62. In ratio to moving average method for seasonal indices, the ratio of an observed value to the moving
average remove the influence of
I. Cyclic variation.
II. Irregular variation.
III. Trend.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) Both (I) and (III) above.
(1 mark)
< Answer >
63. Monthly fluctuation observed in a time series data are termed as
(a) Irregular variations (b) Seasonal variations
(c) Cyclic variations (d) Secular trend
(e) Both (c) and (d) above.
(1 mark)
< Answer >
64. The standard deviation of a data set
(a) Is expressed in the same unit as the observations in the data set
(b) Is expressed in the square of the unit of the observations in the data set
(c) Is expressed in the square root of the unit of the observations in the data set
(d) Is expressed in a different unit from the unit in which the observations in the data set are
expressed
(e) Is always expressed as a percentage of the mean of the data set.
(1 mark)
< Answer >
65. Networks Solutions Ltd. develops and sells network software products. Its sales have been increasing
in the recent years. The company requires an estimate for the sales in the year 2005. The following
data are collected:
Year Sales (Rs. in lakhs)
1999 35
2000 60
2001 85
2002 105
2003 135
2004 160
The sales for the year 2005, estimated on the basis of a linear trend equation, is
(a) Rs.150.16 lakh (b) Rs.168.84 lakh
(c) Rs.175.34 lakh (d) Rs.183.68 lakh
(e) Rs.195.25 lakh.
(2 marks)
< Answer >
66. Which of the following components of time series can be separated using the percent of trend
measure?
(a) Irregular variation (b) Secular trend
(c) Seasonal variation (d) Cyclical variation
(e) Both seasonal and irregular variations.
(1 mark)
< Answer >
11
67. Weighted average of relatives price index calculated using base year values as weights is equal to
(a) Paasches price index (b) Laspeyres price index
(c) Fisher’s ideal price index (d) Unweighted average of relatives price index
(e) Unweighted aggregates price index.
(1 mark)
< Answer >
68. Which of the following statement is false?
(a) An unweighted aggregates price index does not attach any importance to the consumption or
usage levels of the items covered
(b) Laspeyres price index is an unweighted aggregates price index
(c) Paasches price index is a weighted aggregates price index
(d) A price index reflects the overall change in price for the basket of goods considered
(e) Fisher’s ideal price index is the geometric mean of Laspeyres and Paasches price indices.
(1 mark)
< Answer >
69. For a basket of commodities the price of each commodity in a year, X, is three times the price in the
base year. What is the Paasches price index for the year X?
(a) 120 (b) 150 (c) 175 (d) 300 (e) 600.
(1 mark)
< Answer >
70. For a basket of commodities the price of each commodity in a year, X, is twice the price in the base
year. What is the Marshall Edgerworth price index for year X?
(a) 170 (b) 100 (c) 200 (d) 250 (e) 150.
(1 mark)
< Answer >
71. Which of the following test(s) is/are not used for testing the consistency of index numbers?
I. Chi-square test.
II. Time reversal test.
III. Circular test.
IV. Factor reversal test.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (III) and (IV) above
(e) Both (II) and (IV) above.
(1 mark)
< Answer >
72. The link relatives are used in
(a) Price index (b) Circular test
(c) Factor reversal test (d) Chain index
(e) Laspeyres index.
(1 mark)
< Answer >
73. The following details are available with regard to a basket of commodities:
0
1
Q
P 1000
2
æ ö
S ç ÷ =
è ø
1
1
Q
P 650
2
æ ö
S ç ÷ =
è ø
0
0
Q
P 800
2
æ ö
S ç ÷ = è ø
1
0
Q
P 515
2
æ ö
S ç ÷ =
è ø
What is the Marshall-Edgeworth price index?
(a) 155.34 (b) 136.21 (c) 125.48 (d) 143.07 (e) 194.17.
(2 marks)
< Answer >
12
74. In the year Y the prices of the commodities A, B and C were in the ratio 2:5:7. In the year X prices of
the commodities A, B and C were in the ratio 2:4:6. Year X is the base year and year Y is the current
year. If the price relative for the commodity A is 120 then what is the unweighted aggregate price
index for the year Y?
(a) 120 (b) 130 (c) 150 (d) 140 (e) 160.
(2 marks)
< Answer >
75. Weighted average of relatives price index for various years can be readily compared if
(a) Base year values are used as weights
(b) Current year values are used as weights
(c) Base year prices are used as weights
(d) Current year prices are used as weights
(e) Sum of base year and current year quantities are used as weights.
(1 mark)
< Answer >
76. The average price of a group of commodities in the base year is Rs.40. The total of the prices for the
commodities in the group in the current year is Rs.400. The unweighted aggregates price index for the
current year is 125. How many commodities are there in the group?
(a) 4 (b) 5 (c) 6 (d) 7 (e) 8.
(1 mark)
< Answer >
77. Which of the following statements regarding interpolation is false?
(a) It allows us to forecast a value for some future date
(b) It allows a method of statistical estimation
(c) The figures obtained by interpolation are fairly correct as long as underlying assumption holds
good
(d) It allows us to make insertions
(e) It is a statistical technique.
(1 mark)
< Answer >
78. Which of the following can be done with the help of interpolation?
I. Finding the internal rate of return.
II. Filling in the gaps in time series data.
III. Completing the incomplete records.
IV. Computing the yield to maturity of a bond.
(a) Only (I) above (b) Only (II) above
(c) Both (I) and (II) above (d) Both (I) and (IV) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
79. The minor of an element of the matrix is defined as in one of the following ways:
(a) Deleting the row in which the element is present
(b) Deleting the rows and columns in which the element is not present
(c) Deleting the diagonal elements of the element present
(d) Deleting the row and column in which the element is present
(e) Deleting the column in which the element is present.
(1 mark)
< Answer >
80. Which of the following measures represents the scatter of the values in a data set?
(a) Arithmetic mean (b) Geometric mean (c) Mode
(d) Median (e) Standard deviation.
(1 mark)
< Answer >
13
Suggested Answers
Quantitative Methods-I (MB151): July 2006
1. Answer : (e)
Reason : The following are the properties of Inverse Matrix.
I. The inverse of a given square matrix, if it exists, is unique.
II. The inverse of the inverse of a matrix is the matrix itself i.e., for the square martix A,
we have, (A–1)–1 = A when A–1exists.
III. The transpose of the inverse is equal to the inverse of the transpose of the given matrix
i.e., for the martix A, (A–1)T = (AT)–1.
IV. If A and B are two non-singular matrices and A–1 and B–1 are their respective inverses,
then (AB)–1 = B–1. A–1.
Therefore (e) is the correct answer.
< TOP >
2. Answer : (a)
Reason : Given 5 7
x y y z
A
t x
é + - ù
= ê ú ë - + û and
t x z t
B
z y x z t
é - - ù
= ê ú ë - + + û
and A = B i.e., 5 7
x y y z
t x
é + - ù
ê - + ú ë û=
t x z t
z y x z t
é - - ù
ê - + + ú ë û
therefore, x + y = t – x or, 2x = t – y ………...(1)
y – z = z – t or, 2z – t = y ………...(2)
5 – t = z – y or, z + t =5 + y ……….(3)
And 7 + x = x + z + t or, z + t = 7 ……………(4)
From (3) and (4) we get, 5 + y = 7 hence, (2) reduces to 2z – t = 2 ……………(5)
Solving (4) and (5) we get, z = 3 , t = 4.
Finally, from (1) we get, 2x = 4 – 2 =2 or, x = 1
Hence, the value of x, y, z, and t are 1, 2, 3, and 4 respectively.
Therefore (a) is the correct answer.
< TOP >
3. Answer : (b)
Reason : AB =
( ) ( )
( ) ( )
2 3 1 2 2.1 3.2 2. 2 3. 1 8 7
1 2 2 1 1.1 2.2 1. 2 2. 1 5 4
é ù é - ù é + - + - ù é - ù
ê ú ê ú = ê ú = ê ú ë û ë - û ë + - + - û ë - û
8 7
32 35 3 0
5 4
AB
-
\ = =- + = ¹
-
Hence (AB)–1exists.
( ) ( )
( ) 1 ( )
4 7 4 7
Now, Adj. (AB) =
5 8 5 8
. 1 4 7
3 5 8
-
é+ - - - ù é- ù
ê ú = ê ú ë - + û ë- û
é- ù
= = ê ú ë- û
Adj AB
AB
AB
< TOP >
4. Answer : (c)
Reason : Given A = diag [2, –5, 9], B = diag [–3, 7, 14] and C = diag [4, –6, 3].
This can be written as
= -
é ù
ê ú
ê ú
êë úû
2 0 0
A 0 5 0
0 0 9 ,
-
= = -
é ù é ù
ê ú ê ú
ê ú ê ú
êë úû êë úû
3 0 0 4 0 0
B 0 7 0 and c 0 6 0
0 0 14 0 0 3
Now,
= - = -
é ù é ù
ê ú ê ú
ê ú ê ú
êë úû êë úû
2 0 0 6 0 0
3A 3 0 5 0 0 15 0
0 0 9 0 0 27
< TOP >
14
= - = -
é ù é ù
ê ú ê ú
ê ú ê ú
êë úû êë úû
4 0 0 8 0 0
2C 2 0 6 0 0 12 0
0 0 3 0 0 6
Now,
-
- + - -
-
= = -
é ù é ù é ù
ê ú ê ú ê ú
ê ú ê ú ê ú
êë úû êë úû êë úû
é ù
ê ú
ê ú
êë úû
6 0 0 3 0 0 8 0 0
3A + B - 2C = 0 15 0 0 7 0 0 12 0
0 0 27 0 0 14 0 0 6
5 0 0
0 4 0 diag[ 5,4,35]
0 0 35
5. Answer : (b)
Reason : P =
9 2 15
2 4 3
é ù
êë úû Q =
2 1 3
1 -1 2
é ù
êë úû
Given that P – R = 4Q => P – 4Q = R i.e.,
9 2 15
2 4 3
é ù
êë úû –
2 1 3
4
1 -1 2
é ù
êë úû = R
9 2 15
2 4 3
é ù
êë úû –
8 4 12
4 -4 8
é ù
êë úû =
1 2 3
2 8 5
-
- -
é ù
êë úû
< TOP >
6. Answer : (d)
Reason : Given (A + B)2 = A2 + B2
(A + B) =
1 1
2 1
-
-
é ù
êë úû +
a 1
b -1
é ù
êë úû =
a 1 0
b 2 2
+
+ -
é ù
êë úû
(A + B)2 =
a 1 0
b 2 2
+
+ -
é ù
êë úû
a 1 0
b 2 2
+
+ -
é ù
êë úû =
( )( )
( )( ) ( )
a 1 a1 0
b 2 a 1 2 b 2 4
é + + ù
ê + + - + ú ë û-----(1)
A2 =
1 1
2 1
é - ù
ê - ú ë û
1 1
2 1
é - ù
ê - ú ë û=
1 2 1 1
2 2 2 1
é - - + ù
ê - - + ú ë û=
1 0
0 1
é- ù
ê - ú ë û
B2 =
a 1
b 1
é ù
ê - ú ë û
a 1
b 1
é ù
ê - ú ë û=
a2 b a 1
ab b b 1
é + - ù
ê - + ú ë û
Now A2 + B2 =
1 0
0 1
é- ù
ê - ú ë û+
a2 b a 1
ab b b 1
é + - ù
ê - + ú ë û=
2a
b 1 a 1
ab b b
é + - - ù
ê ú
ë - û ---(2)
From (1) and (2)
( )( )
( )( ) ( )
a 1 a1 0
b 2 a 1 2 b 2 4
é + + ù
ê + + - + ú ë û=
2a
b 1 a 1
ab b b
é + - - ù
ê ú
ë - û
0 = a – 1 => a = 1 and
4 = b
Therefore (d) is the correct answer.
< TOP >
7. Answer : (e) < TOP >
15
Reason :
1 2 3
6 7 8
13 14 15
( ) ( ) ( )
( ) ( )
1105 112 2 90 104 3 84 91
7 2 14 3 7
7 28 21
0
- - - + -
=- - - + -
=- + -
=
Therefore (e) is the correct.
8. Answer : (c)
Reason :
3 5
5 3 4 1 2 2
2 log 256 2 64 16
-
- ´ - ´
5 6 3 3
2 log 2- 2-
= 5 6 1
2 log 2 - 2-
=
5 7
2 log 2-
=
7 / 5
2 log 2-
=
2
2
7 7
log
5 5
- -
=
.
< TOP >
9. Answer : (a)
Reason : Given 1+ logabc = x
logaa + logabc = x
logaabc = x ---------- (1)
simillarly
logbabc = y ---------- (2)
logcabc = z --------- (3)
Consider
abc abc abc
1 1 1
log a log b log c
x y z
+ + = + +
=>
yz xz xy
xyz
+ +
= logabcabc = 1
=> xy + xz + yz = xyz
< TOP >
10. Answer : (a)
Reason : A man can invite some or all of his 6 friends to dinner. That is he can invite any number of
his friends to dinner
Therefore the total number of ways =26–1=63
< TOP >
11. Answer : (e)
Reason : 23-log2 7
2
3
log 7
3
2
2
2
7
8
7
=
=
=
< TOP >
12. Answer : (b)
Reason : a. This is not a property of the real numbers
b. According to the inverse property of addition for every real number there exists
another number such that the sum of the two real numbers is equal to 0.
< TOP >
16
c. This is the identity property of addition.
d. This is the inverse property of multiplication.
e. This is the identity property of multiplication.
13. Answer : (b)
Reason : Given the 5th term = ar 4 = 32 and r = 2.
\ a(2)4 = 32
16a = 32
\ a = 2
The sum of n terms of a G.P is given by
( n )
n
a r 1
s
r 1
-
=
-
( 15 )
15
2 2 1
s 65534
2 1
-
Þ = =
-
< TOP >
14. Answer : (b)
Reason : Let the H.P be
1 1 1
, , .......
a a d a 2d
+
+ +
The third term =
1 1
a 2d 7
=
+ ………(1)
\a+2d=7………….(1)
The 15th term =
1 1
a 14d 31
=
+ ………(2)
\by solving (1) and (2)
We get a = 3, d= 2
\The H.P is
1 1 1 1
, , , .....
3 5 7 9
< TOP >
15. Answer : (d)
Reason :
( )( )
( )
( )( )
( )
( )( )
( )
2
2
logx log36
log2 log4
log36 log2
logx
log4
log6 log2
logx
log2
2 log6 log2
logx
2 log2
logx log6
x 6.
=
=
=
=
=
Þ =
Therefore (d) is the correct answer.
< TOP >
16. Answer : (c)
Reason : Given ( ) ( ) ( ) 2log a + b + log a - b - log a2 - b2 = l o g x
< TOP >
17
( ) ( ) ( )
( ) ( ) (( )( ))
( ) ( ) ( ( ) ( ))
( ) ( ) ( ) ( )
( )
( )
( )
2log a b log a b log a2 b2
2log a b log a b log a b a b
2log a b log a b log a b log a b
2log a b log a b log a b log a b
log a b
log a b logx
a b x.
+ + - - -
= + + - - + -
= + + - - + + -
= + + - - + - -
= +
Þ + =
+ =
Therefore (c) is the correct answer.
17. Answer : (e)
Reason :
2 7..................( )
6...................( )
12.................( )
( ) ( )
2 ( ) 6 12 18
( ) 9
- + =
- + =
+ + =
+
´ + = + =
+ =
x y z i
y x z ii
x y z iii
Bydoing ii iii weget
y z
y z
< TOP >
18. Answer : (c)
Reason :
p q 1
q p p (q 1) q p 2 p 2
q (2 p) p 2 q p
p q q 3 q 3
p q
p 3 2-p
+
q q q
q q 1 2
p p p
(p 3 p+2)
q
(q q +1 2)
p
1 1
q p p
1 1 q
p q
xyz= m × n × m n m ×n ×m n
n m n m
=m
n
m
=
n
m n n
= =
m
n m
-
- -
-
æ ö
ç - ÷
è ø
æ - ö
ç - - ÷
è ø
- -
- -
-
-
=
=
< TOP >
19. Answer : (b)
Reason : The value of 0! is unity.
< TOP >
20. Answer : (e)
Reason :
< TOP >
18
n n
r r+1
n n
r r-1
As,
n! n!
Or, =
(n - r)! (n - r - 1)!
(n - r - 1)!
Or, 1
(n - r)!
Or, n - r = 1 .............(i)
Second condition is
n! n!
Or, =
r!(n - r)! (r - 1)!(n - r 1)!
Or, (n-r + 1) = r
Or, n - 2r =
P = P
C = C
=
+
-1.......................(ii)
By (i) - (ii) we get
r = 1-(-1) = 2
21. Answer : (d)
Reason : The series is a G.P whose first term a = ½ and the common ration r = 1/3
a
s
1 r
1 /2 1 3 3
1 1 /3 2 2 4
¥ =
-
= = ´ =
-
< TOP >
22. Answer : (d)
Reason : We have 5.
(2n 1)! (2n 1)!
3.
(n 2)! (n 1)!
+ -
=
+ -
Þ 5.
(2n 1)2n(2n 1)! (2n 1)!
3.
(n 2)(n 1)n(n 1)! (n 1)!
+ - -
=
+ + - -
2
2
10(2n 1) 3(n 2)(n 1)
20n 10 3n 9n 6
3n 11n 4 0
Þ + = + +
Þ + = + +
Þ - - =
or, (n -4)( 3n + 1) = 0
Therefore , n = 4 .
< TOP >
23. Answer : (c)
Reason : Average time taken now =
( ) ( ) ( )
( )
50 4 125 5 75 6
5.1
50 125 75
´ + ´ + ´
=
+ + hours
Average time taken after three years =
( ) ( )
( )
4 50 125 75 5
4.3
50 125 75
+ + ´
=
+ + hours
Change in time taken = 5.1 – 4.3 = 0.80 hours reduction.
< TOP >
24. Answer : (a)
Reason : Original data set:
Sx = 33 Sx2 = 199 N = 6
After modification:
Sx = 2 ´ 33 = 66
Sx2 = 4 ´ 199 = 796
< TOP >
19
\Standard deviation of the resulting data set =
x2 x 2
N N
S æ S ö -ç ÷
è ø
=
2 796 66
6 6
-æ ö ç ÷
è ø
=
796 6 662
36
´ -
=
420
36 =
35
3 = 3.42
25. Answer : (e)
Reason : Arithmetic mean, geometric mean and Harmonic mean are mathematical averages.
Therefore (e) is the correct answer.
Median and mode are positional averages.
< TOP >
26. Answer : (c)
Reason : Mean deviation is not conducive to further algebraic treatment. All other statements are
true.
Therefore the correct answer is (c)
< TOP >
27. Answer : (e)
Reason : If the data type is interval/ratio then mean, median, modes are the suitable averages.
Therefore (e) is the correct answer.
< TOP >
28. Answer : (c)
Reason : Geometric mean is not affected much by fluctuations of sampling.
The options (a), (b), (d) and (e) are true
< TOP >
29. Answer : (d)
Reason : Sum of absolute deviations about median is minimum. Sum of deviations about mean is
zero. Therefore (d) is the correct answer.
< TOP >
30. Answer : (c)
Reason : Suitable measure of central tendency for qualitative data is median. Therefore (c) is the
correct answer.
< TOP >
31. Answer : (b)
Reason : The median, as the name suggests, is the middle value of a series arranged in either
ascending or descending order of magnitude.
< TOP >
32. Answer : (c)
Reason : Given the whole group N = 125 items with mean X = 20
i.e.,
X
20
N
X
20
125
X 2500
=
Þ =
Þ =
å
å
å
and
Second sample n2 = 50 items with mean 2 x = 16
i.e.,
2
2
2
2
x
16
n
x
16
50
x 800
=
Þ =
Þ =
å
å
å
the first sample contains n1 = N – n2 = 125 – 50 = 75 items.
< TOP >
20
1 2
1 2
1
X x x
x X x
x 2500 800 1700
= +
Þ = -
Þ = - =
å å å
å å å
å
1
1700
x 22.66
75
\ = =
33. Answer : (e)
Reason :
Class Mid-value
(m)
Frequency
(f)
f ´ m
25-30 27.5 9 247.5
30-35 32.5 14 455
35-40 37.5 21 787.5
40-45 42.5 14 595
45-50 47.5 13 617.5
50-55 52.5 12 630
55-60 57.5 9 517.5
60-65 62.5 5 312.5
65-70 67.5 3 202.5
Sf = 100 Sf ´ m = 4365
\ Mean =
f m 4365
f 100
S ´ =
S = 43.65 years
< TOP >
34. Answer : (e)
Reason : If any constant value ‘k’ is subtracted from each observation of a data set, then the variance
is unaltered.
< TOP >
35. Answer : (a)
Reason : The empirical relationship between range and mean deviation is 2R=15 M.D
< TOP >
36. Answer : (a)
Reason : Coefficient of variation expresses the standard deviation as a percentage of the mean.
Therefore (a) is the correct answer.
< TOP >
37. Answer : (d)
Reason : The coefficient of quartile deviation is
3 1
3 1
Q -Q
Q.D=
Q +Q
=
150 110
150 110
-
+ =
40
260 = 0.153.
< TOP >
38. Answer : (b)
Reason : The relation ship between the standard deviation and the quartile deviation is
3Q.D B 2S.D
Given S.D = 21
2
Q.D S.D
3
2
21)
3
14.
(
\ =
=
=
Therefore (b) is the correct answer.
< TOP >
39. Answer : (d)
Reason : Range = maximum value – Minimum value
68 = 118 – X
=> X = 118 – 68 = 50.
< TOP >
21
Coefficient of range is =
Max.Value Min.Value
Max.Value Min.Value
-
+
=
68
168 =0.405
Therefore (d) is the correct answer.
40. Answer : (c)
Reason :
Returns % 0-10 10-20 20-30 30-40
No. of companies 15 33 39 13
Calculation of standard deviation
Return on
investment
Midpoint
No.of
companies deviations
% X f fX (X- X) f
(X -X)2
0-10 5 15 75 –15 3375
10-20 15 33 495 –5 825
20-30 25 39 975 5 975
30-40 35 13 455 15 2925
total 100 2000 8100
Mean =
fX 2000
20
f 100
= = å
å
Standard deviation =
( )2 f X X 8100
81 9
f 100
-
= = = å
å
Standard deviation for the return on investment is 9
< TOP >
41. Answer : (d)
Reason : Since the distribution is symmetrical the mean = median= mode. Therefore the highest
point on the curve is refereed to as the mean, median , mode. Therefore (d) is the correct
answer.
< TOP >
42. Answer : (c)
Reason : Coefficient of variation is a relative measure of dispersion. Therefore (c) is the correct
answer.
< TOP >
43. Answer : (e)
Reason : When the mean and standard deviation are equal for one or more sets of data, c.v. can be
meaningfully used for comparison of variability
A standard deviation is equal to zero implies that there is no deviation in the data set. The
same will be reflected by the c.v. provided mean is not equal to zero
When the standard deviation is one the c.v. can be meaningfully used for comparison of
variability provided mean in not equal to zero.
When the mean is equal to one the c.v. can be meaningfully used for comparison of
variability..
It cannot be meaningfully used for comparison of variability when mean of one or more
data sets is zero
< TOP >
44. Answer : (b)
Reason : Data set A
x 90
6
N 15
16 4
m= = =
s= =
å
< TOP >
22
Coefficient of variation =
100
s
´
m =
4 100 66.6%
6
´ =
Data set B
x 110
5
N 22
4 2
m= = =
s= =
å
Coefficient of variation =
100
s
´
m =
2 100 40%
5
´ =
Data set C
x 208
8
N 26
36 6
m= = =
s= =
å
Coefficient of variation =
100
s
´
m =
6 100 75%
8
´ =
Data set D
x 210
7
N 30
25 5
m= = =
s= =
å
Coefficient of variation =
s´100
m =
5 100 71.4%
7
´ =
The data set which has least coefficient of variance is most consistent.
Here data set B is least coefficient of variance . therefore dataset B is more consistant.
45. Answer : (a)
Reason : If the objective function of a profit maximizing linear programming problem is parallel to
an edge of the feasible region, which is in the direction of the optimal movement of the
objective function then there are multiple optimal solutions to the problem.
Therefore the correct answer is (a).
< TOP >
46. Answer : (e)
Reason : The resources should be limited in supply in a linear programming problem. All other
statements are correct for Linear programming problem.
Therefore the correct answer is (e).
< TOP >
47. Answer : (a)
Reason : While plotting constraints on a graph paper, terminal points on both the axes are connected
by a straight line because the constraints are linear equations or inequalities. Therefore (a) is
the correct answer.
< TOP >
48. Answer : (a)
Reason : x1 and x2 = number of chairs and tables produced, respectively
1 2 Max.Z = 2x +10x
Subject to constraints
1 2
1
1 2
2x 5x 16
6x 30
x ,x 0
+ £
£
³
cj 2 10 0 0
cB B xB X1 X2 S1 S2 Ratio
0 S1 16 2 5 1 0 16/5
0 S2 30 6 0 0 1 30/0
< TOP >
23
Zj–Cj=0 -2 -10 0 0
10 X2 16/5 2/5 1 1/5 0
0 S2 30 6 0 0 1
Zj–Cj=32 2 0 2 0
Since all the values of Zj–Cj are positive the optimal solution is obtained.
The daily production of each of the two products i.e., chairs X1 = 0 and tables X2 = 3.2 and
Max. Z = 32.
49. Answer : (d)
Reason : Let us define the following decision variables;
1 x and 2 x = number of belts of type A and B, respectively manufactured each day.
The LP model would be as follows:
Maximize (total profit ) Z = 4x1 + 3x2
Subject to the constraints
availability of time constraint : 1 2 2x + x £ 890
supply of leather constraint : 1 2 x + x £ 800
buckles availability constraints : 1 x400 £ , 2 x£ 650
and 1 2 x , x ³ 0
< TOP >
50. Answer : (a)
Reason : In the optimal simplex table, j j C - Z = 0 value indicates alternative solution. Therefore (a)
is the correct answer.
< TOP >
51. Answer : (c)
Reason : A variable which does not appear in the basic variable column of simplex table is always
equal to zero. Therefore (c) is the correct answer.
< TOP >
52. Answer : (c)
Reason : The value of the dual variable Represents marginal profit of each additional unit of
resource. Therefore (c) is the correct answer.
< TOP >
53. Answer : (e)
Reason : The statements I, II and III are the standard results on duality. Therefore (e) is the correct
answer.
< TOP >
54. 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.
< TOP >
55. 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.
III. Understand properties and operating characteristics of complex real-life problems.
Therefore (e) is the correct answer.
< TOP >
56. Answer : (e)
Reason : Network models have advantage in terms of project
I. Planning.
II. Scheduling.
III. Controlling.
Therefore (e) is the correct answer.
< TOP >
24
57. Answer : (c)
Reason : a. The objective function of a LPP is a linear function of the decision variables.
b. The resource constraints represent the consumption pattern of the resources and the
amount of available resources.
c. The decision variables can assume non-negative continuous values.
d. There may be a specific combination of values of the decis ion variables for which the
objective is accomplished with in the given constraints.
e. The resources are in limited supply.
< TOP >
58. Answer : (e)
Reason : (e) is true because the value at the bottom of the solution column indicates profit. (a) is false
because the Zj – Cj values indicate whether the solution is optimal or not. (b) is false because
at the optimal solution all the Zj – Cj need not be zero. (c) is false because the values in Zj – Cj
row indicate the variable to enter solution. (d) is false because the value to leave solution is
indicated by the ratio of the values in solution column to the corresponding values in the
column for the variable to enter solution.
< TOP >
59. Answer : (e)
Reason : To initiate the solution procedure from the origin, the initial feasible solution is set by
assigning zeros to all the decision variables.
< TOP >
60. Answer : (c)
Reason : Linear trend of time series indicates towards constant rate of change. Therefore (c) is the
correct answer.
< TOP >
61. Answer : (d)
Reason : The three year moving average is calculated as fallows:
values Three years moving
totals
Three years moving
average
15
24 57 = (15+24+18) 19 = (57/3)
18 75 = (24+18+33) 25 = (75/3)
33 93 = (18+33+42) 31 = (93/3)
42
< TOP >
62. Answer : (e)
Reason : In ratio to moving average method for seasonal indices, the ratio of an observed value to the
moving average remove the influence of trend and cyclic variations. Therefore (e) is the
correct answer.
< TOP >
63. Answer : (b)
Reason : Monthly fluctuation observed in a time series data are termed as Seasonal variations. (shortterm
fluctuations observed in a time series data, particularly in a specified period are called
seasonal variations.). Therefore (b) is the correct answer.
< TOP >
64. Answer : (a)
Reason : (a) The standard deviation of a data set is expressed in the same unit as the observations in
the data set.
(b), (c), (d) and (e) are incorrect with regard to the standard deviation.
< TOP >
65. Answer : (d)
Reason : X : Year
x : Coded value for year
Y : Sales
Year x = (X– X )´ 2 Y xY x2
1999 –2.5 ´ 2 = –5 35 –175 25
< TOP >
25
2000 –1.5 ´ 2 = –3 60 –180 9
2001 –0.5 ´ 2 = –1 85 –85 1
2002 0.5 ´ 2 = 1 105 105 1
2003 1.5 ´ 2 = 3 135 405 9
2004 2.5 ´ 2 = 5 160 800 25
580 870 Sx2 = 70
X =
X
n
å
= 2001.5
ˆY
= a + bx
b = 2
xY
x
å
å =
870
70 = 12.43
a = Y =
Y
n
å
=
580
6 = 96.67
\ The linear trend equation is
ˆY
= 96.67 + 12.43x
The estimated sales for the year 2005:
X = 2005 \ x = (2005–2001.5) ´ 2 = 7
Estimated sales for the year 2005 = 96.67 + 12.43 ´ 7 = Rs.183.68 lakh.
66. Answer : (d)
Reason : The cyclical variations can be separated using the percent of trend measure.
< TOP >
67. Answer : (b)
Reason : The weighted average of relatives price index calculated by using the base year values as
weights is given by:
P1 100 P0Q0 P0
P0Q0
ìïæ ö ïü åíç ´ ÷ ´ ý ïçè ÷ø ï î þ
å =
P Q 1 0 100
P0Q0
å
´
å
which is same as Laspeyres price index.
< TOP >
68. Answer : (b)
Reason : Laspeyres price index is a weighted aggregates price index. Alternatives (a), (c) and (d) are
true.
< TOP >
69. Answer : (d)
Reason : Given P1 = 3P0, For each commodity
\ Paasches price index =
1 1
0 1
PQ
100
P Q
S
´
S =
0 1
0 1
3PQ
100
P Q
S
´
S
=
0 1
0 1
3 P Q
100
P Q
S
´
S = 300.
< TOP >
70. Answer : (c)
Reason : Given: P1 = 2P0
Marshall Edgerworth price index =
( )
( )
1 0 1
0 0 1
P Q Q
100
P Q Q
S +
´
S +
=
0 0 1
0 0 1
2P (Q Q )
100
P (Q Q )
S +
´
S + =
0 0 1
0 0 1
2 P (Q Q )
100
P (Q Q )
S +
´
S +
< TOP >
26
= 200.
71. Answer : (a)
Reason : Chi-square test is not a test of consistency for index numbers.
< TOP >
72. Answer : (d)
Reason : In chain index, the base year is not fixed but changes from year to year. Here, new relatives
are constructed for each year with the previous year as base. The new relatives enable
comparison between one period and another period which succeeds it and hence referred to
as ‘Link Relatives’.
< TOP >
73. Answer : (c)
Reason : Marshall-Edgeworth price index =
1 0 1
0 0 1
P (Q Q )
100
P (Q Q )
S +
´
S +
=
0 1
1
0 1
0
(Q Q )
P
2 100
(Q Q )
P
2
+
S
´
+
S
(Dividing both numerator and denominator by 2)
=
0 1
1 1
0 1
0 0
Q Q
P P
2 2 100
Q Q
P P
2 2
S æ ö +S æ ö çè ÷ø çè ÷ø ´
S æ ö +S æ ö ç ÷ ç ÷
è ø è ø =
1000 650 100 125.48
800 515
+ ´ =
+
< TOP >
74. Answer : (d)
Reason :
Year Y Commodity A B C
Price 2K 5K 7K
Year X Commodity A B C
Price 2L 4L 6L
Unweighted aggregates price index =
1
0
P
100
P
S
´
S
Let the price of commodity A in the year Y be P1A and in the year X be P0A.
\ SP1 = lA lA lA lA lA lA lA
P 5k p 7k P P 2.5P 3.5P 7P
2k 2k
+æ ö +æ ö = + + = çè ÷ø çè ÷ø
SP0 = P0A + oA 0A 0A 0A 0A 0A
4L P 6L P P 2P 3P 6P
2L 2L
æ ö + æ ö = + + = çè ÷ø çè ÷ø
\
1
o
P
100
P
S
´
S =
lA lA
0A 0A
7P 7 P 7 100 100 120 140
6P 6 P 6
´ = æ ö æ ´ ö = ´ = çè ÷ø ç ÷ è ø
< TOP >
75. Answer : (a)
Reason : Weighted average of relatives price index for various years can be readily compared if base
year values are used as weights.
< TOP >
76. Answer : (e)
Reason : Unweighted aggregates price index =
1
0
P
100
P
S
´
S = 125
\ 0
400
100
P
´
S = 125
< TOP >
27
or SP0 =
400 100
125
´
= 320
0 P=
40
or n =
0
0
P 320
8
P 40
S
= =
.
77. Answer : (a)
Reason : Interpolation doesn’t allows us to forecast a value for some future date. Options b,c,d and e
are correct with respect to interpolation.
< TOP >
78. Answer : (e)
Reason : In financial analysis, the technique of interpolation is widely used to find the internal rate of
return of a project, yield to maturity of a bond and in all investment decisions which require
the use of the present value and future value interest factor tables, whereas the technique of
extrapolation is widely used for long-term capital requirements, production of financial
statements for financial institution, banks, etc., and is helpful in those circumstances where
forecasting and prediction of future sales, cost and profit are required.
< TOP >
79. Answer : (d)
Reason : The minor of an element of the matrix is defined as deleting the row and column in which
the element is present then the matrix is called the minor of an element.
< TOP >
80. Answer : (e)
Reason : The standard deviation represents the scatter of the values in a data set.
Arithmetic mean, geometric mean, harmonic mean and median are measures of central
tendency.
< TOP >
< TOP OF THE DOCUMENT >