Quantitative Methods (MB151): January 2005

Quantitative Methods (MB151): January 2005

· Answer all questions.
· Marks are indicated against each question.
1. If all the terms of an arithmetic progression are multiplied by a constant quantity the resulting terms
will always form
(a) A geometric progression
(b) A harmonic progression
(c) An arithmetic progression
(d) Either a geometric progression or a harmonic progression
(e) Either a geometric progression or an arithmetic progression.
(1 mark)
< Answ er >
2. If the first term in a geometric progression is greater than 1 and the common ratio is less than 1, then
the consecutive terms will be
(a) The consecutive terms will be in increasing order
(b) The consecutive terms will be in decreasing order
(c) The consecutive terms will be same
(d) All the consecutive terms will be less than 1
(e) All the consecutive terms will be greater than 1.
(1 mark)
< Answer >
3. Which of the following is false with regard to the graphical method of solving linear programming
problems?
(a) It is applicable when there are two decision variables
(b) The decision variables are represented by the horizontal and vertical axes
(c) Straight lines are used to demarcate the feasible region
(d) The feasible region shows the solutions that satisfy all the constraints
(e) One of the corner points of the feasible region will always be at the origin.
(1 mark)
< Answer >
4. logab + loga c = 0
This implies that
(a) b = c (b) b = -c (c) b + c =
1
(d) b - c = 1 (e) b and c are reciprocals.
(1 mark)
< Answer >
5. Which of the following is true with regard to the classical approach to probability?
(a) It assumes that the outcomes are not equally likely
(b) The probability of an event is determined after performing the experiment large number of times
(c) The probability of an event is determined before performing the experiment
(d) It assumes that all possible outcomes of the experiment are not known
(e) The classical approach cannot be used to find out the probability of mutually exclusive events.
(1 mark)
< Answer >
6. If every item in a set of data is divided by a constant, then which of the following measures will
remain unchanged?
(a) Arithmetic mean (b) Geometric mean (c) Mode
(d) Variance (e) Coefficient of variation.
(1 mark)
< Answer >
2
7. The reciprocal of the harmonic mean of a data set is equal to
(a) Arithmetic mean
(b) Sum of all the observations
(c) Sum of the reciprocals of the observations
(d) Average of the reciprocals of the observations
(e) Product of the reciprocals of the observations.
(1 mark)
< Answer >
8. The coefficient of variation cannot be meaningfully used to compare the variability of two or more
sets of data, when
(a) The standard deviation is zero for one or more sets of data
(b) The standard deviation is 1 for one or more sets of data
(c) The mean is zero for one or more sets of data
(d) The mean is 1 for one or more sets of data
(e) The mean and standard deviation are equal for one or more sets of data.
(1 mark)
< Answer >
9. Events A and B are dependent. The joint probability of the events A and B is
(a) Equal to the product of the marginal probabilities of the events A and B
(b) Not equal to the product of the marginal probabilities of the events A and B
(c) Equal to the sum of the marginal probabilities of the events A and B
(d) Equal to the difference between the marginal probabilities of the events A and B
(e) Is always equal to 1.
(1 mark)
< Answer >
10. Which of the following is false with regard to linear programming problems?
(a) The contribution of each unit of the decision variables towards the objective to be achieved is
known
(b) The consumption of resources by each unit of the decision variables is known
(c) The values of the decision variables in the optimal solution will always be whole numbers
(d) Linear programming is used either to maximize or to minimize the value of the objective
function
(e) The optimal solution to any linear programming problem is one of the possible feasible solutions.
(1 mark)
< Answer >
11. Which of the following is false with regard to the simplex method of solving linear programming
problems?
(a) It involves an iterative procedure for arriving at the optimal solution
(b) Slack variables are used to represent the unused resources
(c) Slack variables make zero contribution towards the objective to be achieved
(d) The Zj – Cj values indicate the variable to leave solution
(e) The Zj – Cj values indicate whether the solution is optimal or not.
(1 mark)
< Answer >
12. If two events A and B are independent then, the conditional probability of event A given that event B
has occurred, is equal to
(a) Joint probability of events A and B
(b) Conditional probability of event B given event A
(c) Marginal probability of event B
(d) Marginal probability of event A
(e) Zero.
(1 mark)
< Answer >
3
13. 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 >
14. The geometric mean between two given quantities is equal to
(a) The sum of the arithmetic mean and harmonic mean between the two quantities
(b) The difference between the arithmetic mean and harmonic mean between the two quantities
(c) The geometric mean of the arithmetic mean and the harmonic mean between the two quantities
(d) The product of the arithm etic mean and the geometric mean between the two quantities
(e) The ratio of the arithmetic mean to the geometric mean between the two quantities.
(1 mark)
< Answer >
15. The appropriate average for a set of ratios using the denominators of the ratio data as weights is
(a) Simple arithmetic mean (b) Weighted arithmetic mean
(c) Simple harmonic mean (d) Weighted harmonic mean
(e) Geometric mean.
(1 mark)
< Answer >
16. The sum of an arithmetic progression (A.P) consisting of n terms is zero. Which of the following is
true? (t1 represents the first term and tn represents the nth term of the A.P.)
(a) t1 = –tn (b) t1 = tn (c) t 1 / tn = 0 (d) t1 / tn > 1 (e) t n / t1 >
1.
(1 mark)
< Answer >
17. The smallest number that can be divided by each of a group of numbers without leaving a remainder is
called
(a) A factor of the quantities
(b) The highest common factor of the quantities
(c) The least common multiple of the quantities
(d) The average of the quantities
(e) The sum of all the quantities.
(1 mark)
< Answer >
18. 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 Z j – 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 >
19. In which of the following conditions two events, A and B, are said to be mutually exclusive?
(a) 0 < P(A or B) < 1 (b) P(A or B) = 1
(c) P(A) = P(B) = 1 (d) P(A/B) = 0 and P(B/A) = 0
(e) 0 < P(A and B) < 1.
(1 mark)
< Answer >
20. Which of the following is true?
(a) In a logarithmic function the logarithmic operation is applied on the dependent variable
(b) In a logarithmic function the base of the logarithm is always 10
(c) In a logarithmic function the independent variable cannot be negative
(d) In a logarithmic function the base of the logarithm can be any real number
(e) In a logarithmic function the dependent variable cannot be negative.
(1 mark)
< Answer >
4
21. Which of the following is not an assumption underlying linear programming?
(a) The objective to be accomplished can be expressed as a linear function of the decision variables
(b) The constraints on the use of resources can be expressed as linear equations or inequations
(c) The amount of resources consumed by each unit of the decision variables is uncertain
(d) The decision variables can take non-negative values only
(e) The total consumption of a resource is the sum of the resources consumed by each decision
variable.
(1 mark)
< Answer >
22. The reciprocals of the terms in a harmonic progression are
(a) In geometric progression (b) In harmonic progression
(c) In arithmetic progression (d) Always in decreasing order
(e) Always in increasing order.
(1 mark)
< Answer >
23. Which of the following are the conditions for applying the Bayes’ theorem for computing posterior
probabilities of certain events?
(a) The events must be non-mutually exclusive
(b) The events must be mutually exclusive
(c) The events must not be collectively exhaustive
(d) The events must be collectively exhaustive
(e) Both (b) and (d) above.
(1 mark)
< Answer >
24. When the chi-square test is used as a test of independence, the number of degrees of freedom is
determ ined by
(a) The sample size
(b) The ratio of sample size and the population
(c) Number of rows in the contingency table, only
(d) Number of columns in the contingency table, only
(e) Both the number of rows and the number of columns in the contingency table.
(1 mark)
< Answer >
25. The weighted average of price relatives using base values as weights is same as the
(a) Unweighted aggregates price index (b) Unweighted aggregates quantity index
(c) Laspeyres price index (d) Laspeyres quantity index
(e) Paasche’s price index.
(1 mark)
< Answer >
26. The standard error of estimate of a simple regression line is
(a) Equal to the slope of the regression line
(b) Equal to the square root of the coefficient of determination
(c) Equal to the Y intercept of the regression line
(d) The measure of the variability of the observed values around the regression line
(e) Equal to the coefficient of correlation between two variables.
(1 mark)
< Answer >
27. In many situations managers resort to sampling to draw some conclusions about a population. Which
of the following is not an advantage of sampling over a census?
(a) The population may be too large to be studied in full, hence sampling is the only feasible means
(b) A study of sample is usually cheaper than a census
(c) Sampling usually provides information quicker than a census
(d) In destructive testing sampling is the only available course
(e) The conclusions obtained from sampling are more accurate than census.
(1 mark)
< Answer >
5
28. If the covariance of two random variables X and Y is equal to zero then it can be said that
(a) The random variables are dependent on each other
(b) The random variables have a positive correlation
(c) The random variables have a negative correlation
(d) The random variables are independent of each other
(e) The variances of the two random variables are always zero.
(1 mark)
< Answer >
29. Which of the following is true with regard to Fisher’s ideal price index?
(a) It does not consider the base year prices
(b) It does not consider the base year quantities
(c) It does not consider the current year prices
(d) It does not consider the current quantities
(e) It is the geometric mean of the Laspeyres’ and Paasche’s price indices.
(1 mark)
< Answer >
30. Which of the following is the graphical plot of the values of the dependent and independent variables,
in the context of regression analysis?
(a) Scatter diagram (b) Frequency polygon
(c) Histogram (d) p chart (e) Ogive.
(1 mark)
< Answer >
31. Which of the following is false for the multiple correlation coefficient between dependent variable Y
and two independent variables X1 and X2?
(a) It depends on the correlation coefficient between Y and X1
(b) It depends on the correlation coefficient between X1 and X2
(c) It depends on the correlation coefficient between Y and X2
(d) It will take values between 0 and 1
(e) It will take values between –1 and +1.
(1 mark)
< Answer >
32. The Y intercept in the simple linear regression equation (X is the independent variable and Y is the
dependent variable) represents the
(a) True value of Y when X = 0
(b) Change in average value of Y per unit change in X
(c) Expected value of Y when X = 0
(d) Standard deviation of the values of X
(e) Mean of the values of X.
(1 mark)
< Answer >
33. Which of the following is false with regard to hypothesis testing?
(a) The purpose of hypothesis testing is to find out whether there is a significant difference between
the sample statistic and the hypothesized population parameter
(b) Rejection of null hypothesis implies the acceptance of alternative hypothesis
(c) The acceptance of null hypothesis proves that it is true
(d) The rejection of null hypothesis does not prove that it is false
(e) The null hypothesis is rejected if the test statistic falls in the critical region.
(1 mark)
< Answer >
34. In a binomial distribution the probability of getting zero or more number of successes is equal to
(a) 0
(b) The probability of getting zero success
(c) The probability of getting successes in all the trials
(d) 1 minus the probability of getting successes in all the trials
(e) 1.
(1 mark)
< Answer >
6
35. In a right tailed test of hypothesis the
(a) Acceptance region lies under the right tail
(b) Rejection region lies under the right tail
(c) Acceptance region lies under the right and left tails
(d) Rejection region lies at the center of the distribution
(e) Entire region lying on the right of the mean of the distribution is the rejection region.
(1 mark)
< Answer >
36. The standard error of estimate is developed from the variations of the
(a) Observed values of the dependent variable around a fixed quantity
(b) Observed values of the dependent variable around the mean of the observed values of the
independent variable
(c) Observed values of the dependent variable around the fitted regression line
(d) Observed values of the independent variable around their mean
(e) Observed values of the independent variable around the mean of the observed values of the
dependent variable.
(1 mark)
< Answer >
37. If the dependent variable decreases as the independent variable increases in an estimating equation
then, the coefficient of correlation will be in the range of
(a) 0 to –2.00 (b) 0 to –1.00 (c) 0 to 0.50 (d) 0 to 0.25 (e) 0 to
1.00.
(1 mark)
< Answer >
38. Which of the following represents the proportion of variation in the dependent variable that is
explained by the regression line?
(a) Coefficient of determination (b) Coefficient of correlation
(c) Coefficient of variation (d) Standard error of estimate
(e) Standard error of mean.
(1 mark)
< Answer >
39. If the regression equation is a perfect estimator of the dependent variable then which of the following
is false?
(a) The standard error of estimate is zero
(b) The coefficient of correlation is zero
(c) The coefficient of determination is 1.00
(d) All the data points fall on the regression line
(e) The coefficient of correlation is 1.00.
(1 mark)
< Answer >
40. Which of the following is true wit h regard to a given coefficient of correlation and its corresponding
coefficient of determination?
(a) The coefficient of determination is always greater than or equal to zero, and less than or equal to
1
(b) The coefficient of determination is always less than zero
(c) The coefficient of determination is always equal to 1
(d) The coefficient of determination always has the same sign as the coefficient of correlation
(e) The magnitude of coefficient of determination is always higher than the magnitude of coefficient
of correlation.
(1 mark)
< Answer >
7
41.
If
a2 , b2, c2 are in arithmetic progression, then which of the following is correct?
(a)
(a + b)(b + c)
a + b + c =
(c + a) (b)
2(a + b)(b + c)
a + b + c =
(c + a)
(c)
2(a + b)(b + c)
a + b + 2c =
(c + a) (d)
2(a + b)(b + c)
2a + b + c =
(c + a)
(e)
2(a + b)(b + c)
a + 2b + c =
(c + a) .
(2 marks)
< Answer >
42. Two drawings, each of three balls, are made from a bag containing 5 red and 8 black balls, the balls
not being replaced before the second drawing. What is the likelihood that the first drawing will give 3
red balls and the second drawing will give 3 black balls?
(a) 7/15 (b) 5/143 (c) 15/143 (d) 7/429 (e) 1/3.
(2 marks)
< Answer >
43. A box cont ains six balls of unknown colours. Three balls are drawn and found to be green. What is the
probability that no green ball is remaining in the box?
(a) 1 (b) 1/2 (c) 1/20 (d) 1/35 (e) 7/16.
(2 marks)
< Answer >
44. A fair die is thrown three times, and the sum of the three numbers obtained is 15. What is the
probability that the number obtained in the first throw was 4?
(a) 1/5 (b) 1/9 (c) 1/12 (d) 1/18 (e) 1/108.
(2 marks)
< Answer >
45. In a trial the judge is 65 percent confident that Sunil has committed a crime. Ramesh is a witness who
knows whether Sunil is innocent or guilty. However, Ramesh is Sunil’s friend and will lie in the court
with probability 0.25 if Sunil is guilty. He will tell truth if Sunil is innocent. What is the likelihood that
Ramesh will not lie in the court?
(a) 0.1625 (b) 0.8375 (c) 0.35 (d) 0.75 (e) 1.00.
(2 marks)
< Answer >
46. A number is randomly selected from the set of values {1, 2,…, 10000}and is observed to be odd.
What is the probability that it is divisible by 3?
(a) 0.056 (b) 0.6667 (c) 0.1667 (d) 0.3334 (e) 0.50.
(2 marks)
< Answer >
47. The sum of the first term through the tenth term of an arithmetic progression (A.P.) is 50 and the sum
of the eleventh term through the twentieth term of the A.P. is 250.
What is the sum of the twenty-sixth term through the thirtieth term of the A.P.?
(a) 50 (b) 300 (c) 550 (d) 250 (e) 500.
(2 marks)
< Answer >
48. What is the value of the following sum?
S =
log x + log 2 + log x2 + log 4 + log x3 + log 8 + .... + log x10 + log 1024
(a) log2x (b) 55 (c) 55 log x (d) 55 log (2x) (e) 5 log (2x) .
(2 marks)
< Answer >
8
49. For a class consisting of 60 students, a test on English was conducted. The marks obtained by the
students are given below:
Marks obtained out of 125 Numbers of students
5 - 25 7
25 - 45 15
45 - 65 18
65 - 85 12
85 -105 6
105 -125 2
What is the median marks obtained by the students in the class?
(a) 30.5 (b) 45 (c) 48.6 (d) 53.3 (e) 65.
(2 marks)
< Answer >
50. The following details are available with regard to two populations, A and B:
Group Number of
observations
Mean Standard
deviation
A 30 10 4
B 20 15 6
What is the combined standard deviation for both the populations?
(a) 3.63 (b) 2.45 (c) 4.90 (d) 5.48 (e) 4.10.
(2 marks)
< Answer >
51. A group consists of 150 children. The group is divided into three subgroups viz., A, B and C, in the
ratio of 2:5:3 respectively. The average age of the children in the subgroup A is 8 years. The average
age of the children in the subgroup B is 10 years. The average age of the children in the subgroup C is
12 years.
What is the average age of all the 150 children in the group?
(a) 8.5 years (b) 9.3 years (c) 10.2 years (d) 10.9 years (e) 11.5 years.
(2 marks)
< Answer >
52. A firm manufactures steel pipes in three plants viz, A, B and C. The daily production volumes from
the three firms A, B and C respectively are 1000 units, 2000 units and 4000 units respectively. It is
known from p ast experience that 2% of the output from plant A, 3% of the output from plant B and
5% of the output from plant C are defective. A pipe is selected from a days total production and found
to be defective. What is the probability that the pipe is manufactured by plant B?
(a) 14.29% (b) 21.43% (c) 28.57% (d) 4% (e)
57.14%.
(2 marks)
< Answer >
53. Three numbers a, b, and c are in arithmetic progression (AP). p is the geometric mean between a and
b and, q is the geometric mean between b and c. What is the arithmetic mean between p2 and q2?
(a) a2 (b) b2 (c) c2 (d) p (e) q.
(2 marks)
< Answer >
54. The latest nationwide political poll indicates that for persons, who are randomly selected, the
probability that they support Party A is 0.55, the probability that they support Party B is 0.30, and the
probability that they support Party C is 0.15. It is assumed that these probabilities are accurate and
that the randomly selected persons have independent opinions towards the different parties.
What is the probability that out of a randomly chosen group of ten persons, less than eight persons do
not support Party A?
(a) 0.02739 (b) 0.02289 (c) 0.0045 (d) 0.99618 (e)
0.97261.
(2 marks)
< Answer >
9
55. The Traffic Police Department of Kolkata has gathered data on the number of traffic accidents and
number of cricket matches that occur in the city over a weekend. According to them, there is a close
relationship between the number of accidents and the number of cricket matches played during
weekends.
Number of cricket matches 18 28 8 10 13 23 32
Number of accidents 5 8 3 4 6 7 8
Assuming that there is a linear relation between these two variables, what would be your prediction
about the number of accidents that will occur on a weekend during which 25 cricket matches take
place in Kolkata?
(a) 6.13 (b) 7.08 (c) 8.07 (d) 9.45 (e) 10.52.
(2 marks)
< Answer >
56. The following details are available with regard to a regression analysis between two variables, X and
Y, with X as the independent variable:
Percentage of variations in Y that is explained by the variations in X = 84%.
S(Y- Y) 2 = 1369
Number of observations = 6
What is the standard error of estimate for the given regression relationship?
(a) 43.81 (b) 6.62 (c) 14.80 (d) 54.76 (e) 7.40.
(2 marks)
< Answer >
57. What is the sum of the following series?
p2 , (3p2 + q), (5p2 + 2q), ... ... ..., éë(2n + 1)p2 + nqùû
(a) (n + 1) éë2p2 + n (2p2 + q)ùû (b)
2 ( 2 ) n
2p + n 2p + q
2
éë ùû
(c)
( ) 2 ( 2 ) n + 1
2p + n 2p + q
2
éë ùû (d)
( ) 2 ( 2 ) n + 1
p + p 2p + q
2
éë ùû
(e)
2 ( ) n
p + n 2p + q
2
éë ùû .
(2 marks)
< Answer >
58. Popular Garments Company is in the business of trading men’s garments. It purchases readymade
garments from different manufacturers and sells them in the retail market. The company sells a
particular brand of trousers named “Boss” every year during the summer season. It has to decide on
the number of these trousers to stock for this season. Each trouser costs Rs.600 and sells for Rs.800.
The salesman at the counter is given a commission of Rs.40 on every trouser sold. The stock of this
brand of trousers is maintained for a period of 4 months. In the past the sales of this brand of trousers
during the same period of time have averaged 300 with a standard deviation of 40. The trousers,
which remain unsold at the end of the 4-month period are sold at 45% discount from the selling price.
Further, on each trouser sold at discount the salesman at the counter is given a commission of Rs.80.
The demand for the trousers is assumed to be normally distributed.
What is the optimal number of the “Boss” brand trousers which should be stocked by the company?
(Round off your answer to the nearest integer).
(a) 200 (b) 266 (c) 290 (d) 310 (e) 334.
(2 marks)
< Answer >
10
59. Orient Electricals Company manufactures water pumps. The general manager of the company wants to
compare the performance of the water pumps manufactured by the company with the industry
standards. He knows that 18% of all the water pumps of similar type sold in the industry require
repairs during the first year of sale. The company sampled 200 customers and found that in case of 44
customers the water pumps required repairs in the first year of sale. It is to be tested whether the
performance of the water pumps manufactured by the company is different from the industry
standards.
At a significance level of 5% which of the following can be concluded?
(a) The value of the test statistic is –1.471 and the performance of the water pumps manufactured by
the company is not significantly different from the industry standards
(b). The value of the test statistic is –54.201 and the performance of the water pumps manufactured
by the company is significantly different from the industry standards
(c) The value of the test statistic is 54.201 and the performance of the water pumps manufactured by
the company is not significantly different from the industry standards
(d) The value of the test statistic is 1.471 and the performance of the water pumps manufactured by
the company is not significantly different from the industry standards
(e) The value of the test statistic is 1.471 and the performance of the water pumps manufactured by
the company is significantly different from the industry standards.
(2 marks)
< Answer >
60. The linear programming formulation for a problem is given below:
Maximize : Z = 8X + 6Y
Subject to : 4X + 2Y £ 60
2X + 4Y £ 48
X, Y ³ 0
If the above problem is solved by the simplex method then which of the following statements is/are
correct?
I. According to the first tableau the solution is X = 15 and Y = 18
II. According to the first tableau the solution is X = 12 and Y = 6
III. According to the first tableau the solution is X = 0 and Y = 0
IV. According to the second tableau the solution is X = 12 and Y = 18
V. According to the second tableau the solution is X = 15 and Y = 12
VI. According to the second tableau the solution is X = 15 and Y = 0
VII. According to the third tableau the solution is X = 15 and Y = 0
VIII. According to the third tableau the solution is X = 12 and Y = 15
IX. According to the third tableau the solution is X = 12 and Y = 6
X. The maximum value of Z cannot be found
XI. The maximum value of Z is 120 which occurs at X = 15 and Y = 0
XII. The maximum value of Z is 132 which occurs at X = 12 and Y = 6.
(a) (I), (IV), (VII) and (X)
(b) (II), (V), (VIII) and (XII)
(c) (I), (VI), (VII) and (XI)
(d) (III), (IV), (VIII) and (X)
(e) (III), (VI), (IX) and (XII).
(2 marks)
< Answer >
61.
If
2 n1 2n 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 >
11
62. If S1, S2, S3,……, Sn are the sums of infinite geometric series whose first terms are 1, 2, 3,
…, n and whose common ratios are 1/2, 1/3, …., 1/(n+1) respectively.
Then S1+ S2 + S3+ …..+ Sn is equal to
(a) n(n+1)/2 (b) n(n+2)/2
(c) n(n+3)/2 (d) n(n+4)/2 (e)
n(n+5)/2.
(2 marks)
< Answer >
63. In an A.P. if S m : Sn = m2: n2, where S p denotes the sum of first p terms and tp denotes the pth term,
then the value of tm: tn is equal to
(a) (2m+1) : (2n+1) (b) (m+1) : (n+1)
(c) (2m–1) : (2n-1) (d) (m-1) : (n-1) (e) m : n.
(2 marks)
< Answer >
64. If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of the G.P. is
(a)
q+r
p+q (b)
q - r
p - q (c)
p - q
q - r (d)
p+q
q+r (e)
q - r
p+q .
(2 marks)
< Answer >
65. If (b + c), (c + a) and (a + b) are in H.P. then a2, b2 and c2 are in
(a) A. P. (b) G. P. (c) H. P. (d) A.P. and G.P. both
(e) An unknown progression.
(2 marks)
< Answer >
66. Atlas Sporting Goods has implemented a special trade promotion for its tennis balls and feels that the
promotion should result in a price benefit for the consumer. Atlas knows that before the promotion
began, the average retail price of the ball was Rs. 44.95, with a standard deviation of Rs.5.75. Atlas
samples 25 of its retailers after the promotion was launched and finds the mean price for the ball is
now Rs.42.95. Atlas wants to examine whether the price of balls has significantly reduced after the
trade promotion. Which of the following conclusions can be drawn on the basis of the above
information at a significance level of 2%?
(a) The average retail price to the consumers has increased significantly
(b) The average retail price to the consumers has not decreased significantly
(c) The average retail price to the consumer has decreased significantly
(d) We cannot draw any conclusion, as the sample size is too small
(e) We cannot draw any conclusion, as information is insufficient.
(2 marks)
< Answer >
67. A random sample of 300 loans was collected from the loans made during the recent five year period,
by a financial institution which finances small scale enterprises. This sample showed that 120 of the
loans were made to women entrepreneurs. A complete census of all the loans made by the financial
institution five years ago showed that 42 percent of the loans were made to women entrepreneurs. It is
to be tested whether the proportion of loans made to the women entrepreneurs by the financial
institution has reduced in the past five years.
At a significance level of 2 percent, what is the conclusion?
(a) The sample information is incorrect
(b) The population proportion is incorrect
(c) The proportion of loans made to the women entrepreneurs has reduced in the past five years
(d) The proportion of loans made to the women entrepreneurs has increased in the past five years
(e) The proportion of loans made to the women entrepreneurs has not reduced in the past five years.
(2 marks)
< Answer >
12
68. A commodity merchant knows that the mean retail price of a specific variety of rice three months ago
was Rs. 14.50 per kg. In the current month the merchant has collected the information on the price
charged for the same variety of rice by 16 randomly selected merchants in the same city. It was found
from the sample that the mean retail price was Rs. 15.00 per kg and the standard deviation was Rs.
1.25 per kg. It is to be tested whether the mean retail price of the rice in the current month is more than
Rs. 14.50 per kg.
At a significance level of 5 percent, what is the conclusion?
(a) The sample standard deviation is incorrect
(b) The sample mean is incorrect
(c) The mean retail price of the rice in the current month is more than Rs. 14.50 per kg
(d) The mean retail price of the rice in the current month is not more than Rs. 14.50 per kg
(e) The mean retail price of the rice in the current month is less than Rs. 14.50 per kg.
(2 marks)
< Answer >
69. The following data are collected from a wholesaler of commodities:
Year
Commodity 1997 2004 1999
Price
(Rs./kg.)
Price
(Rs./kg)
Price
(Rs./kg)
Quantity
(kg.)
Rice 11.00 14.00 13.00 2500
Wheat 8.50 11.50 9.75 1000
Salt 3.50 5.00 4.25 500
Sugar 12.50 14.00 13.00 750
Pulses 24.50 29.50 27.00 1200
What is the weighted average of relative price index for the year 2004, using the year 1999 for
weighting and the year 1997 for the base year?
(a) 123.33 (b) 124.27 (c) 127.57 (d) 90.54 (e) 111.67.
(2 marks)
< Answer >
70. Ten salesmen in a firm were put to a competency test. The test scores obtained by the salesmen and
the monthly sales made by them are given below:
Test score 30 60 40 50 70 40 80 30 50 50
Monthly sales (Rs. '000) 25 50 35 45 55 30 60 30 40 30
A regression equation has to be developed for estimating the amount of monthly sales from the test
scores. On the basis of this regression equation what is the estimated monthly sales for a salesman who
scores 90 in the competency test?
(a) Rs. 687.50 (b) Rs. 5,625 (c) Rs. 50,625 (d) Rs. 61,875 (e) Rs. 67,500.
(2 marks)
< Answer >
END OF QUESTION PAPER
13
Suggested Answers
Quantitative Methods (MB151): January 2005
1. Answer : (c)
Reason : If all the terms of an arithmetic progression are multiplied by a constant the resulting terms will
always form an arithmetic progression with the first term multiplied by the constant as well as the
common difference multiplied by the constant. The resulting series will neither be in a geometric
series or a harmonic series because nature of the resulting terms will not satisfy their requirements.
< TOP >
2. Answer : (b)
Reason : a. The consecutive terms of the G.P. will be in increasing order if the first term in a geometric
progression is greater than one and the common ratio is more than 1.
b. The consecutive terms of the G.P. will be in decreasing order if the first term in a geometric
progression is greater than one and the common ratio is less than 1.
c. If the first term in a geometric progression is greater than one and the common ratio is less
than 1, then the consecutive terms will not be the same.
d. All the consecutive terms will be less than 1 if the first term as well as the common ratio is
less than 1.
e. All the consecutive terms will be greater than 1 if the first term as well as the common ratio is
more than 1. However a decreasing G.P. may still have all the terms greater than 1; this
depends upon the number of terms in the G.P.
< TOP >
3. Answer : (e)
Reason : The following are true with regard to the graphical method of solving LPPs:
a. It is applicable when there are two decision variables.
b. The decision variables are represented by the horizontal and vertical axes.
c. Straight lines are used to demarcate the feasible region.
d. The feasible region shows the solutions that satisfy all the constraints.
e. The corner points of the feasible region may not include the origin.
< TOP >
4. Answer : (e)
Reason :
(a) b = c cannot be implied from the given condition.
(b) b = -c cannot be implied from the given condition.
(c) b + c = 1 cannot be implied from the given condition.
(d) b - c = 1 cannot be implied from the given condition.
(e) Let logab = k and logac = m.
Hence b = ak and c = am
loga b + logac = 0 implies that k + m = 0.
\k = -m
Þ ak = a-m Þ ak = 1/ am Þ ak. am = 1 Þ b.c = 1 Þ b = 1/c and vice versa.
Hence they are reciprocals.
< TOP >
5. Answer : (c)
Reason : (a) The classical approach to probability assumes that the outcomes are equally likely.
(b) In the relative frequency approach to probability the probability of an event is determined after
performing the experiment large number times.
(c) In the classical approach to probability the probability of an event is determined before
performing the experiment.
(d) The classical approach to probability assumes that all possible outcomes of the experiment are
known.
(e) The classical approach can be used to find out the probability of mutually exclusive events.
< TOP >
6. Answer : (e)
Reason : (a), (b) and (c) If every item of the data set is divided by a constant then the arithmetic mean,
geometric mean and mode will also be divided by the constant.
< TOP >
14
(d) The variance will be divided by the square of the constant.
(e) The standard deviation as well as mean will be divided by the constant. Hence coefficient of
variation ((Mean/Standard deviation)100) will remain unchanged.
7. Answer : (d)
Reason : Harmonic mean = Reciprocal of the average of reciprocals of the observations.
(a) The reciprocal of the harmonic mean is not equal to the arithmetic mean.
(b) The reciprocal of the harmonic mean is not equal to the sum of all the observations.
(c) The reciprocal of the harmonic mean is not equal to the sum of the reciprocals of the
observations.
(d) \ The reciprocal of the harmonic mean is equal to average of the reciprocals of the
observations.
(e) The reciprocal of the harmonic mean is not equal to the product of the reciprocals of the
observations.
< TOP >
8. Answer : (c)
Reason : Coefficient of variation = (standard deviation / mean) × 100
(a) A standard deviation 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.
(b) Even when the standard deviation is 1 the c.v. can be meaningfully used for comparison of
var iability provided mean is not equal to zero.
(c) Hence it cannot be meaningfully used for comparison of variability when mean of one or more
data sets is zero.
(d) When the mean is equal to 1, the c.v. can be meaningfully used for comparison of variabil ity.
(e) When the mean and standard deviation are equal for one or more sets of data, the c.v. can be
meaningfully used for comparison of variability.
< TOP >
9. Answer : (b)
Reason : (a) & (b) For two dependent events A and B, the joint probability of the events A and B is not equal
to the product of their marginal probabilities.
(c) For two dependent events A and B, the joint probability of the events A and B is not equal to
the sum of their marginal probabilities.
(d) For two dependent events A and B, the joint probability of the events A and B is not equal to
the difference between their marginal probabilities.
(e) For two dependent events A and B, the joint probability of the events A and B is not always
equal to 1.
< TOP >
10. Answer : (c)
Reason : The following are true for any LPP:
a. The contribution of each unit of the decision variables towards to the objective to be achieved
is known.
b. The consumption of resources by each unit of the decision variables is known.
c. The values of the decision variables in the optimal solution may be fractional numbers.
d. Linear programming is used either to maximize or to minimize the value of the objective
function.
e. The optimal solution to any linear programming problem is one of the possible feasible
solutions.
< TOP >
11. Answer : (d)
Reason : The following are true with regard to the simplex method of solving linear programming problems:
a. It involves an iterative procedure for arriving at the optimal solution.
b. Slack variables are used to represent the unused resources.
c. Slack variables make zero contribution towards the objective to be achieved.
d. The Zj – Cj values indicate the variable to enter solution.
e. The Zj – Cj values indicate whether the solution is optimal or not.
< TOP >
12. Answer : (d) < TOP >
15
Reason : If two events A and B are independent then, the conditional probability of event A given event B is
equal to marginal probability of event A because the occurrence of event B does not influence the
occurrence of event A.
13. 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 >
14. Answer : (c)
Reason : (c) The geometric mean between two given quantities is equal to the geometric mean of the
arithmetic mean and the harmonic mean between the two given quantities.
(a), (b), (d) and (e) are all incorrect with regard to the geometric mean between two given
quantities.
< TOP >
15. Answer : (b)
Reason : (a) Simple arithmetic mean is the appropriate average when the denominators are same and no
weighting is required.
(b) Weighted arithmetic mean is appropriate when the denominators of the ratio data are used as
weights.
(c) Simple harmonic mean is used when the numerators of the ratios are same.
(d) Weighted harmonic mean is appropriate when the numerators of the ratio data are used as
weights.
(e) Geometric mean is appropriate when the quantities vary over time.
< TOP >
16. Answer : (a)
Reason : The sum of n terms of an A.P., Sn =
n
2 {2a + (n – 1)d}
=
n
2 {t1 + tn}
Sn = 0 Þ
n
2 {t1 + t n} = 0
Þ t1 + tn = 0
Þ t1 = - tn.
< TOP >
17. Answer : (c)
Reason : The least common multiple of a group of quantities can be divided by each quantity in the group
without leaving any remainder. While, each number of the group can be divided by a factor, even
by the highest common factor also. But that condition is not satisfied, if each number is divided by
the average or the sum of all the quantities.
< TOP >
18. Answer : (e)
Reason : (e) is true because the value at the bottom of the solution column indicates profit. (b) 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 >
19. Answer : (d)
Reason : Since the two events A and B are mutually exclusive, the happening of A precludes the occurrence
of B and vice versa.
Hence P(A/B) = 0 and P(B/A) = 0
< TOP >
20. Answer : (c)
Reason : a. In a logarithmic function the logarithmic operation is applied on the independent variable.
b. In a logarithmic function the base of the logarithm may be any number other than 10.
c. In a logarithmic function the independent variable cannot be negative.
d. In a logarithmic function the base of the logarithm cannot be zero or negative numbers.
< TOP >
16
21. Answer : (c)
Reason : (a), (b), (d) and (e) are all assumptions underlying linear programming. However, (c) is not an
assumption because linear programming assumes that the amount of resources consumed by each
unit of the decision variables is certain.
< TOP >
22. Answer : (c)
Reason : The reciprocals of the terms in harmonic progression are in arithmetic progression.
< TOP >
23. Answer : (e)
Reason : The events for which the Baye’s theorem may be applied for computing posterior probabilities must
be mutually exclusive and collectively exhaustive.
< TOP >
24. Answer : (e)
Reason : (a) This is a wrong answer. The number of degrees of freedom does not depend on the sample
size.
(b) This is the wrong answer. The number of degrees of freedom does not depend on the ratio
of sample size and the population.
(c) This is the wrong answer. The number of degrees of freedom does not depend on the
number of rows in the contingency table only.
(d) This is the wrong answer. The number of degrees of freedom does not depend on the
number of columns in the contingency table only.
(e) This is the right answer. When the chi-square distribution is used as a test of independence, the
number of degrees of freedom is related to both the number of rows and the number of columns in
the contingency table.
< TOP >
25. Answer : (c)
Reason : (a) This is the wrong answer. The weighted average of price relatives using base values as
weights is not same as the unweighted aggregates price index.
(b) This is the wrong answer. The weighted average of price relatives using base values as
weights is not same as the unweighted aggregates quantity index.
(c) This is the right answer. The weighted average of price relatives using base values as weights
is same as the Laspeyres price index.
(d) This is the wrong answer. The weighted average of price relatives using base values as
weights is not same as the Laspeyres quantity index.
(e) This is the wrong answer. The weighted average of price relatives using base values as
weights is not same as the Paasche’s price index.
< TOP >
26. Answer : (d)
Reason : (a) This is a wrong answer. The standard error of the estimate of a regression line is not the slope
of the regression line.
(b) This is a wrong answer. The standard error of the estimate of a regression line is not The
square root of the coefficient of determination.
(c) This is a wrong answer. The standard error of the estimate of a regression line is not the Y
intercept of the regression line.
(d) This is the right answer. The standard error of the estimate of a regression line is the measure
of the variability of the observed values around the regression line.
(e) This is a wrong answer. The standard error of the estimate of a regression line is not the
coefficient of correlation between two variables.
< TOP >
27. Answer : (e)
Reason : (a) This is an advantage of sampling. The population may be too large to be studied in full. In
such situations it is always better to use sampling compared to the study of whole population.
(b) This is an advantage of sampling. A study of sample is usually cheaper than a study of the
population. As the number of observations required is less the cost incurred in obtaining and
processing these information is also less for sampling.
(c) This is an advantage of sampling. Sampling usually provides information quicker than a
census. Since the time required for studying the whole population is large, it is advisable to go
for sampling to reduce the time spent in the studying the whole population.
(d) This is an advantage of sampling. In destructive testing, where the product becomes unusable
after the performance of test, we cannot test the whole population. In such cases sampling is
< TOP >
17
the only method of testing.
(e) This is not an advantage of sampling. In a sampling exercise we use a small number of
samples to study the whole population the degree of accuracy is not very high. For accurate
information census is always the preferable method compared to sampling.
28. Answer : (d)
Reason : (a) This is a wrong answer. If the covariance of two random variables X and Y is equal to zero
then it can be said that the random variables are independent of each other. If the variables are
dependent on each ot her then the covariance will assume any non-zero value.
(b) This is a wrong answer. If the coefficient of correlation is positive then we say that there is a
positive correlation between two variables.
(c) This is a wrong answer. If the coefficient of correlation is negative then we say that there is a
negative correlation between two variables.
(d) This is the correct answer. If the covariance of two random variables X and Y is equal to zero
then it can be said that the random variables are independent of each other.
(e) This is a wrong answer. If the covariance of the two random variables is zero it does not mean
that the variances of the individual variables will be zero.
< TOP >
29. Answer : (e)
Reason : a. Fisher’s ideal price index considers base year prices.
b. Fisher’s ideal price index considers base year quantities.
c. Fisher’s ideal price index considers current year prices.
d. Fisher’s ideal price index considers current year prices.
e. Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price indices
< TOP >
30. Answer : (a)
Reason : a. The graphical plot of the values of the dependent and independent variables, in the context of
regression analysis, is called scatter diagram.
b. A frequency polygon is a graphical representation of a frequency distribution which uses
straight lines to join the top mid points of the rectangles in a histogram.
c. A histogram is a graphical representation of a frequency distribution.
d. A p chart is a quality control chart.
e. An ogive is a graphical plot of a cumulative frequency distribution
< TOP >
31. Answer : (e)
Reason : (a) This is true for multiple correlation coefficient. It depends on the correlation coefficient
between Y and X1.
(b) This is true for multiple correlation coefficient. It depends on the correlation coefficient
between X1 and X2.
(c) This is true for multiple correlation coefficient. It depends on the correlation coefficient
between Y and X2.
(d) This is true for multiple correlation coefficient. It will take values between 0 and 1.
(e) This is false for multiple correlation coefficient. It will take values between 0 and 1. It does
not take negative values.
< TOP >
32. Answer : (c)
Reason : a. The Y intercept of the regression line does not represent the true value of Y
when X= 0.
b. The Y intercept of the regression line does not represent the change in average value of Y per
unit change in X.
c. The Y intercept of the regression line represents the mean value of Y when X = 0.
d. The Y intercept of the regression line does not represent the standard deviation of the values of
X.
e. The Y intercept of the regression line does not represent mean of the values of X
< TOP >
33. Answer : (c)
Reason : The acceptance of null hypothesis does not prove that it is true because there are situations when
the sample statistic falls within the acceptance region, though null hypothesis is false. That is why
type II error occurs.
< TOP >
18
34. Answer : (e)
Reason : In a binomial distribution the probability of getting zero or more number of successes
= P(x = 0) + P(x = 1) + … P(x = n) = 1.00.
< TOP >
35. Answer : (b)
Reason : In any right tailed test of hypothesis, the rejection region lies under the right tail. So the alternative
(b) is correct.
< TOP >
36. Answer : (c)
Reason : The standard error of estimate for any regression equation is given by:
Se =
(y yˆ )2
n 2
-
-
å
where
y = Observed values of the dependent variable
ˆy = Estimated values from the estimating equation that correspond to each Y value and
n = Number of data points.
< TOP >
37. Answer : (b)
Reason : The dependent variable decreases as the independent variable increases in an estimating equation
signifies that there is a negative nature of association between the two variables. Hence the
correlation co-efficient will be negative and will lie in the range of 0 to –1.00 (because correlation
coefficient cannot be less than –1).
< TOP >
38. Answer : (a)
Reason : Coefficient of determination represents the proportion of variation in the dependent variable that is
explained by the regression line. Coefficient of correlation signifies the nature of association
between two variables. Coefficient of variation states the extent of variability in a set of data points.
Standard error of estimate measures the variability of the observed values around the regression
line. So the alternative (a) is correct.
< TOP >
39. Answer : (b)
Reason : If the regression equation is a perfect estimator of the dependent variable, then the following are
possible:
i. The standard error of estimate is zero as å(y - yˆ )2 = 0.
ii. The coefficient of determination is 1.00 as the coefficient of correlation is either
+1.00 or –1.00.
iii. Naturally, all the data points must lie on the regression line.
Hence, the alternative (b) is correct.
< TOP >
40. Answer : (a)
Reason : a. The coefficient of determination is the square of coefficient of correlation (r). Hence it will
always be ³ 0 and £ 1.
b. From above we can see that the coefficient of determination cannot be less than zero.
c. The coefficient of determination is will be equal to 1, only if the coefficient of correlation is
equal to –1 or 1; in other cases it will be > 0 and < 1.
d. The coefficient of determination is always positive; The coefficient of correlation may be
negative.
e. Since -1 £ r £ 1, the coefficient of determination which is the square of coefficient of
correlation (r), will always have magnitude less than or equal to r
< TOP >
41. Answer : (e)
Reason : a2, b2 , c2 are in A.P.
By adding adding ad + ac + bc to each term, we see that
a2 + ab +ac + bc, b2 + ba + bc + ac, c2 + ca + cb + ab are in A.P
or (a + b) (a + c), (b + c)(b + a), (c + a)(b + c) are in A.P.
Dividing each term by (a + b)(b + c)(c + a) we get,
< TOP >
19
1 1 1
, ,
b + c c +a a + b are in A.P.
\ b + c , c + a, a + b are in H.P
\ c + a =
2(a b )(b c) 2(a b)(b c)
b c a b a 2b c
+ + + +
=
+ + + + +
or a + 2b + c =
( )( )
( )
2 a b b c
c a
+ +
+
42. Answer : (d)
Reason : 1st drawing
3 balls may be drawn from total 5 + 8 = 13 balls in
13
C3 ways,
3 red balls may be drawn in
5
C3 ways.
\ P (3 red balls in 1st drawing ) =
5
3
13
3
C 10 5
C 286 143
= =
.
2nd drawing :
The balls taken in the first drawing are not replaced. If 3 red balls were drawn in the 1st drawing
then the balls present before the 2nd drawing are 2 red and 8 black balls.
\At the second trial 3 balls may be drawn from 2 + 8 = 10 balls in
10
C3 ways
3 black balls may be drawn in
8
C3 ways.
\ P(3 black balls in 2nd drawing ) =
8 10
3 3
56 7
C C
120 15
¸ = =
P(3 red balls in 1st draw and 3 black balls in 2nd draw)
=
5 7 7
143 15 429
´ =
.
< TOP >
43. Answer : (d)
Reason : There are 4 equally likely possibilities which reference to the remaining balls:
A. All the 3 remaining balls are green P(A) =
1
4
B. 2 balls are green P(B) =
1
4
C. 1 ball is green P(C) =
1
4
D. No ball is green P(D)=
1
4
If A was true then, probability of drawing three green balls in the first three draws
=
6
3
6
3
C
1
C
=
= P( 3 green balls drawn | A)
If B was true then, probability of drawing three green balls in the first three draws
=
5
3
6
3
C 10 1
C 20 2
= =
= P(3 green balls drawn | B)
If C was true then, P (3 green balls drawn | C) =
4
3
6
3
C 4 1
C 20 5
= =
If D was true then, P(3 green balls drawn | D) =
3
3
6
3
C 1
C 20
=
\P(3 green balls drawn) =
< TOP >
20
P(A). P(3 green balls drawn | A) + P(B). P(3 green balls drawn | B) + P(C). P(3 green balls drawn |
C) + P(D). P (3 green balls drawn | D)
=
1 1 1 1 1 1 1 1 7 7
144 2
4
5
4
20 4
4
16
æ ´ ö +æ ´ ö +æ ´ ö + æ ´ ö = ´ = ç ÷ ç ÷ ç ÷ ç ÷
è ø è ø è ø è ø
P(D | 3 green balls drawn) =
( )
( )
P D and 3 green balls drawn
P 3 green balls drawn
=
1 1
4 20 1 16 1
7 80 7 35
16
æ ö
ç ´ ÷
è ø= ´ =
44. Answer : (a)
Reason : Let T be the event of obtaining 15 in 3 throws.
If 15 should be obtained in three throws then the number that should be obtained in first throws must be
3, 4, 5, or 6.
Let A, B, C, D and E respectively be the events that 3, 4, 5, 6 and any other number(1 or 2) is
obtained in the 1st throw.
\ P(A) = P(B) = P(C) = P(D) =
1
6
P(E) =
2 1
6 3
=
If A happens then in the remaining two throws the total must be 12 and this will happen as : (6 ,6)
\P(T | A) = P (6 in 2nd throw) ´ P (6 in 3rd throw) =
1 1 1
6 6 36
´ =
If B happens then in the remaining two throws the total must be 11 and this will happen as : (5,6) or
(6,5)
\P(T | B) = P (5 in 2nd throw and 6 in 3rd throw) + P(6 in 2nd throw and 5 in 3rd throw)
=
1 1 1 1 2 1
6 6 6 6 36 18
æ ´ ö +æ ´ ö = = ç ÷ ç ÷
è ø è ø
If C happens then in the remaining two throws the total must be 10 and this will happen as : (4,6),
(6,4) or (5,5)
\ P (T | C) =
1 1 1 1 1 1 3 1
6 6 6 6 6 6 36 12
æ ´ ö +æ ´ ö +æ ´ ö = = ç ÷ ç ÷ ç ÷
è ø è ø è ø
If D happens then in the remaining two throws the total must be 9 and this will happen as :
(3,6 ), (4,5), (5,4) or (6,3)
\ P(T | D) =
1 1
6 6
æ ´ ö ç ÷
è ø+
1 1
6 6
æ ´ ö ç ÷
è ø+
1 1
6 6
æ ´ ö ç ÷
è ø+
1 1
6 6
æ ´ ö ç ÷
è ø=
4 1
36 9
=
If E happens then in the remaining two throws the total must be 14 or 13 (in order to get a total of
15 in 3 throws). But this is impossible because maximum that can be obtained in the 2nd and 3rd
throws is 6 + 6 = 12.
\ P (T | E) = 0
Now, P(T) = P(T and A) + P(T and B) + P(T and C) + P(T and D) + P(T and E)
= P(A) . P(T | A) + P(B).P(T | B) + P(C).P(T | C)+ P(D).P(T | D)+ P(E).P(T | E)
=
1 1 1 1 1 1 1 1 1 5
0
6 36 6 18 6 12 6 9 3 108
æ ´ ö + æ ´ ö + æ ´ ö + æ ´ ö+ æ ´ ö = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷
è ø è ø è ø è ø è ø
\ P(B|T) =
( )
1 1
P(BandT) 6 18 1 5 1
P T 5 108 108 5
108
æ ö
ç ´ ÷
= è ø = ¸ =
< TOP >
45. Answer : (b)
Reason : Let the following notations be used:
< TOP >
21
A: Sunil has committed a crime A¢ : Sunil has not committed a crime
B: Ramesh will lie in the court. B¢ : Ramesh will speak truth in court
(i.e. will not lie in the court)
Given : P(A) = 0.65
P (A¢) = 1- 0.65 = 0.35
P (B | A) = 0.25
P (B|A¢) = 0
P(B) = P(A and B) + P( A¢ and B)
= P(A). P(B | A) + P( A¢ ).P(B | A¢ )
= 0.65 ´ 0.25 + 0.35 ´ 0
= 0.1625
P (B¢) = 1-P (B) = 0.8375.
46. Answer : (d)
Reason : The number is odd and is divisible by 3
No. of values which are odd and divisible by 3 =
No. of values divisible by 3 – No of values which are even and divisible by 3
=
9999 9996
3333 1666 1667
3 6
- = - =
P(Number is divisible by 3 | Number is odd) =
( )
( )
P Number is odd and divisible by 3
P Number of is odd
No. of values which are odd = 10,000 –
10,000
5,000
2
=
\P(Number is divisible by 3 | Number is odd ) =
1667 10000 1667
1 1 1
5000 10000 5000
1 1 1
C C C
0.3334
C C C
¸
= =
¸
< TOP >
47. Answer : (d)
Reason : Sum of 1st term through 10th term,
( ) 10
10
S 2a 10 1d 50
2
= éë + - ùû =
Or 5(2a+9d) = 50
Or 2a+9d = 10 …….(A)
Sum of 1st term though 20th term , S20 = S10 + Sum of 11th term through 20th term
Or
20 ( )
2a 20 1d 50 250
2
éë + - ûù = +
Or 10[2a + 19d] = 300
Or 2a + 19d = 30 ………(B)
Subtracting (A) from (B) …….
10d = 20
or d =
20
2
10
=
Putting d = 2 , in (A) we get
2a + 9 ´ 2 = 10
or a=
10 18
4
2
-
= -
\ a = –4 , d=2
\ Sum of twenty sixth term through thirtieth term = S30 – S25
< TOP >
22
=
30 ( ) 25 ( )
2 4 30 12 2 4 25 1 2
2 2
éë ´ - + - ùû - éë ´ - + - ùû
= 15 [–8 + 58] – 25 [ ]
8 48
2
- +
= 750 – 500 = 250.
48. Answer : (d)
Reason : S = logx + log2 + logx2 + log4 + logx3 + log8 + …………………+ logx10 + log1024
= (logx + log2) + (logx2 + log22)+ (logx3 + log23 )+ ……………+(logx10 + log210)
= (logx+log2 )+ (2logx + 2log2) + (3logx + 3log2) + ………….+ (10logx + 10log2)
= (logx + log2) + 2(logx + log2) + 3(logx + log2) ………………+ 10(logx + log2)
= log(2x) + 2log(2x) + 3log(2x) + ………….+10log(2x)
This is an A.P with
First term , a = log(2x)
Common difference, d = log(2x)
Number of terms, n = 10
\Sum , S =
( ) [ ] n
n n
2a n 1d a t
2 2
éë + - ùû = +
=
10 ( ) ( )
log 2x 10log 2x
2
éë + ùû
= 5 éë11log (2x )ùû
= 55 log(2x)
< TOP >
49. Answer : (d)
Reason :
Marks Frequency Cumulative Frequency
5 – 25 7 7
25 – 45 15 22
45 – 65 18 40
65 – 85 12 52
85 – 105 6 58
105 – 125 2 60
60
Median position :
60 1
30.5
2
+
= th value
This falls in the class 45 – 65.
Median =
( )
m
m
N 1
(F 1)
2 W L
f
é + ù
ê - + ú
ê ú +
ê ú
êë úû
Now, Lm = 45, N = 60, F = 22, W = 20, fm =18
\ Median =
60 1
(22 1)
2 30.5 23
20 45 20 45 53.3
18 18
éæ + ö ù êç ÷ - + ú êè ø ú æ - ö ´ + = ç ÷ ´ + = ê ú è ø
êë úû (approx.).
< TOP >
50. Answer : (d)
Reason : nA = 30 mA =10 sA = 4
nB = 20 mB =15 sB = 6
mc = Combined mean =
( ) A A B B
A B
n . n 30 10 (20 15) 600
12
n n 30 20 50
m + m ´ + ´
= = =
+ +
< TOP >
23
dA = mA - mc = 10-12 = -2
dB = mB - mc = 15-12 = 3
C s = Combined standard deviation
=
1
2 2 2 2 2
A A B B A A B B
A B
n n n d n d
n n
é s + s + + ù
ê + ú ë û
=
( ) ( ) ( ) ( )
1
30 42 20 62 30 2 2 20 32 2
30 20
é ´ + ´ + é - ù + ´ ù ê ë û ú
ê + ú
êë úû
=
1
480 720 120 180 2
50
é + + + ù
êë úû
= 30
= 5.48
51. Answer : (c)
Reason : Subgroup A : NA =
2
150
2 5 3
´
+ + = 30
A x = 8 years
Subgroup B : NB =
5
150
2 5 3
´
+ + = 75
B x = 10 years
Subgroup C : NC =
3
150
2 5 3
´
+ + = 45
C x = 12 years
\ Average age of the entire group =
( ) ( ) ( ) A A B B C C
A B C
N .x N .x N .x
N N N
+ +
+ +
=
(30 8) (75 10) (45 12)
30 75 45
´ + ´ + ´
+ + =
1530
150 = 10.20 years.
< TOP >
52. Answer : (b)
Reason : Let the following notations be used:
A : Pipe is produced by Plant A.
B : Pipe produced by Plant B.
C : Pipe is produced by Plant C.
D : Pipe is defective.
Given: P (A) =
1000 1000 1
1000 2000 4000 7000 7
= =
+ +
P (B) =
2000 2000 2
1000 2000 4000 7000 7
= =
+ +
P (C) =
4000 4000 4
1000 2000 4000 7000 7
= =
+ +
P (D/A) = 2% = 0.02
P (D/B) = 3% = 0.03
< TOP >
24
P (D/C) = 5% = 0.05
P (A and D) = P (A) . P (D/A) =
1 0.02
0.02
7 7
´ =
P (B and D) = P (B) . P (D/B) =
2 0.06
0.03
7 7
´ =
P (C and D) = P (C) . P (D/C) =
4 0.20
0.05
7 7
´ =
\ P (D) = P (A and D) + P (B and D) + P (C and D) =
0.28
7
P (B/D) =
P(B and D) (0.06/7)
P (D) (0.28/7)
=
= 0.2143. i.e. 21.43%
53. Answer : (b)
Reason : a, b and c are in A.P.
\ b – a = c – b Þ 2b = a + c
p is the G.M. between a and b Þ p = ab Þ p2 = ab
q is the G.M. between b and c Þ q = bc Þ q2 = bc
\ p2 + q2 = ab + bc
or p2 + q 2 = b(a + c)
or p2 + q 2 = b(2b)
or p2 + q 2 = 2b2
or b2 =
p2 q2
2
+
\ b2 is the A.M. between p2 and q2.
< TOP >
54. Answer : (e)
Reason : The probability that a randomly selected person supports party A is 0.55. Then the probability that
he/she does not support party A is 1- 0.55 = 0.45
\The event that a randomly selected person does not support party A, follows a binomial
distribution with probability of success = 0.45.
The probability that less than 8 out of 10 person do not support party A
=1- P(At least 8 persons do not support party A)
=1– [P(8) + P(9) + P(10)]
=1– ( ) ( ) ( ) ( ) ( ) ( ) 10 8 2 10 9 1 0 10 0
8 9 10 éë C 0.45 1- 0.45 + C 0.45 1- 0.45 + C 0.45 1- 0.45 ùû
=1– [0.02289 + 0.00416 + 0.00034]
= 0.97261
< TOP >
55. Answer : (b)
Reason : Let X be the number of cricket matches played and Y be the number of accidents during weekends.
åX = 132, åY= 41, n = 7
åXY = 873,
åX2 = 2994,
X = 18.86 , Y = 5.86
The coefficient of regression b = ( )2 2
n XY X Y
n X X
å -å å
å - å =
( ) ( )
( ) ( 2 )
7 873 132 41
7 2994 132
´ - ´
´ -
=
699
3534 = 0.198
The Y intercept a = Y- b X = 5.86 – (0.198 ´ 18.86) = 2.126
Therefore the estimating equation is ˆY= 2.126+ 0.198 X
< TOP >
25
The estimate of Y for X = 25, is ˆY = 2.126 + 0.198 ´ 25 = 7.08 (Approx)
56. Answer : (e)
Reason : Proportion of variations in Y that is explained by variations in X
= r2 = 1 –
2
2
(Y Y)
) Yˆ (Y
S -
S -
From above, we have the following:
r2 = 0.84
S(Y -Y)2 = 1369 (given)
\ 0.84 = 1 –
(Y Yˆ )2
1369
S -
or
(Y Yˆ )2
1369
S -
= 1 – 0.84 = 0.16
\
2 ) Y ˆ S(Y - = 1369 ´ 0.16 = 219.04
Standard error of estimate = n 2
) Yˆ (Y 2
-
S -
=
219.04
6- 2 = 7.40
< TOP >
57. Answer : (c)
Reason : This is an A.P with ……
a = p2
d = 2p 2 + q
No. of terms = n + 1
Now, Sn =
( ) [ ] n
n n
2a n 1d a t
2 2
éë + - ùû = +
\ Here, Sn+1 =
(n+1) 2 ( ) 2
p 2n 1p nq
2
é + + + ù ë û
=
(n 1) 2 2
2p 2np nq
2
+
é + + ù ë û
=
(n 1) 2 ( 2 )
2p n 2p q
2
+ é ù + +
ë û
< TOP >
58. Answer : (c)
Reason : Marginal profit (MP) = Selling price – Cost – Commission
= 800 – 600 – 40 = Rs.160
Marginal loss (ML):
Discounted price of the Trousers = 800 (1 – 0.45) = Rs.440
Less: Commission payable = Rs. 80
Amount receivable from discount sale = Rs. 360
Marginal loss (ML) = Cost of the T -shirt – Amount receivable from the discount sale
= 600 – 360 = Rs.240
Minimum required probability of selling an additional T-shirt
(p*) = ML MP
ML
+ =
240
240 +160 = 0.60 i.e. 60%.
< TOP >
26
Since 0.50 of the area under the normal distribution is located between the mean and the right tail
0.10 of the shaded area in the diagram must be to the left of the mean, (0.60 – 0.50 = 0.10). From
the standard normal distribution table we find by interpolation the Z-value which corresponds to an
area of 0.10 from the mean i.e. = 0.25 +
( )
( ) ( ) 0.26 0.25
0.10 0.0987
0.1026 0.0987
-
-
- = 0.253.
Let the stock level be denoted X.
\
X -m
s = –0.253 ( –ve because it is on the left side of the mean).
Given, m = 300, s = 40
\
X 300
40
-
= –0.253
or X = 300 + (–0.253 ´ 40) = 289.88 @ 290
\ The company should stock 290 Trousers of the brand.
59. Answer : (d)
Reason : Sample size, n = 200
Sample proportion, p=
44
200 = 0.22
H0 : p = 0.18 (= p0)
H1 : p ¹ 0.18
a = 0.05
Standard error of proportion, sp = n
p0.q0
=
(0.18)(1 0.18)
200
-
= 0.0272
Standardizing the sample proportion:
Z = p
p p 0
s
-
=
0.22 0.18
0.0272
-
= 1.471
From the normal distribution curve we can find out that the test statistic falls in the acceptance
region. So we accept the null hypothesis. Hence, we infer that the performance of the water pumps
manufactured by the company does not significantly differ from the industry standards.
< TOP >
60. Answer : (e)
Reason : Rewriting the formulation using the slack variables:
< TOP >
27
Maximize: Z = 8X + 6Y + 0S1 + 0S2
Subject to: 4X + 2Y + S1 + 0S2 = 60
2X + 4Y + 0S1 + S2 = 48
All variables ³ 0
Tableau I
Profit 8 6 0 0
Variables X Y S1 S2 Ratio
Profit Variables Solution
0 S1 60 ? 2 1 0 60/4 = 15
0 S2 48 2 4 0 1 48/2 = 24
(Zj – Cj) 0 –8 –6 0 0
Enter : X (Most negative value in (Zj – Cj) row)
Leave : S1 (Minimum positive ratio)
Tableau II
Profit 8 6 0 0
Variables X Y S1 S2 Ratio
Profit Variables Solution
8 X 15 1 1/2 1/4 0 15 ¸1/2 = 30
0 S2 18 0 ? –1/2 1 18/3 = 6
(Z j – Cj) 120 0 –2 2 0
Enter : Y
Leave : S2
Tableau III
Profit 8 6 0 0
Variables X Y S1 S2
Profit Variables Solution
8 X 12 1 0 1/3 –1/6
6 Y 6 0 1 –1/6 1/3
(Z j – Cj) 132 0 0 5/3 2/3
In tablean III we find that all the Zj – Cj values are non -negative. This indicates that the
optimal solution has been reached.
\ At the optimal solution:
X = 12
Y = 6
Z = 132 (maximum value).
61. 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 >
62. Answer : (c)
Reason : For the G .P. with first term n and common ratio 1/(n+1)
< TOP >
28
Sn=
n
n 1
1
1
n 1
= +
-
+
\ S1+ S2 + S3+ …..+ Sn= i
n(n 1) n(n 3)
S (i 1) i n n
2 2
+ +
å =å + = å += + =
63. Answer : (c)
Reason : Let a be the first term and d be the common difference of the A.P.
m
mS[2a (
m
1)d]
2
\ = + -
and n
nS[2a (
n
1)d]
2 =
+ -
2
2 2
m n 2
m 2a (m 1)d m
S :S m :n 2a(n m) d(n m) 2a d
n 2a (n 1)d n
é + - ù
\ = Þ ê ú = Þ - = - Þ = ë + - û
Now, tm : t n = [a + (m-1)d]:{a + (n-1)d} = (a + 2am- 2a):(a + 2an - 2a)
= (2m - 1) : (2n - 1)
< TOP >
64. Answer : (b)
Reason : We are given that Tp, Tq and Tr of an A.P. are in G.P.
\ = = q r
p q
T T
R
T T (common ratio of G.P.)
or ,
( 1) ( 1)
( 1) ( 1)
+ - + -
=
+ - + -
a q d a r d
a p d a q d
From the property of ratio we know that if
=
x z
y w, then the ratio is equal to
- +
- +
x z x z
or
y w y w
Using the property we have,
[ ( 1) ] [ ( 1) ] ( )
[ ( 1) ] [ ( 1) ] ( )
+ - - + - - -
= = =
+ - - + - - -
a q d a r d q rd q r
R
a p d a q d p qd p q.
< TOP >
65. Answer : (a)
Reason : Taking the reciprocals
1 1
,
b + c c+ a and
1
a + b are in A.P.
1 1 1 1
c a b c a b c + a
\ - = -
+ + +
or, ( )( ) ( )( )
b a c b
c a b c a b c a
- -
=
+ + + +
Canceling (c + a) and cross multiplying, we get,
b2 - a2 = c 2 - b2
\a2,b2 and c2 are in A.P.
< TOP >
66. Answer : (b)
Reason : We want to test the hypothesis
H0 : m = 44.95 ( Null Hypothesis: Average retail price has not changed)
H1: m < 44.95 (Alternative Hypothesis: Average retail price has reduced after sales promotion)
The standard error of mean = n
s
=
5.75
25 = 1.15
The standard normal value of the mean retail price of the sample =
X
/ n
-m
s
< TOP >
29
=
42.95 44.95
1.15
-
=
2
1.15
-
= -1.739
Looking at the table for the critical value corresponding the area under the normal curve as 2% in
the left tail i.e. –2.05. We find that the sample statistic falls in the acceptance region therefore we
accept null hypothesis and at significance level of 2 % we can conclude that the price of the tennis
balls has not reduced significantly after the trade promotions.
67. Answer : (e)
Reason : 0 H: p = 0.42
1 H
:p < 0.42
a = 0.02
( ) Ho HO
p
p q 0.42 1 0.42
0.0285
n 300
´ -
s = = =
Sample proportion,
120
p 0.40
300
= =
Standardized value of sample proportion,
Ho
p
p p 0.40 0.42
Z
0.0285
- -
= =
s
= – 0.702
Since the sample size is large we shall use the normal distribution.
Critical Z –value = – 2.05
Thus we can see that the sample statistic falls in the acceptance region. Hence we accept the null
hypothesis at a significance level of 2 percent. It can be concluded that the proportion of loans made
to the women entrepreneurs has not reduced in the past five years.
< TOP >
68. Answer : (d)
Reason : 0 H: m = 14.50
1 H
:m > 14.50
a = 0.05
Estimated standard error of mean X
ˆ s 1.25 0.3125
n 16
s = = =
Since the population standard deviation is not known and the sample size is less than 30, the tdistribution
will be used.
Degrees of freedom = n – 1 = 16 – 1 = 15
Since this is a right tailed test with a = 0.05 , the rejection area is 0.05 under the right tail. Hence
we should look under the 0.10 column in the t-distribution (0.10 area in both tails combined).
\Critical t-value = 1.753
Standardized sample statistic, t =
H0
X
X 15.00 14.50
1.60
ˆ 0.3125
-m -
= =
s
< TOP >
30
We find that the sample statistic falls in the acceptance region.
Hence we accept the null hypothesis.
It is concluded that the mean price of the rice is not more than Rs.14.50 per kg.
69. Answer : (b)
Reason : Weighted average of relatives price index =
1 ( )
n n
0
n n
P
100 P Q
P
P Q
éæ ö ù
êç ´ ÷ ú
ëè ø û
å
P0 = Base year (1997) price
P1 = Current year (2004) price
Pn = Price in the year 1999 to be used in weights
Qn= Quantity in the year 1999 to be used in weights
\ Weighted average of relatives price index for 2004
=
14 ( ) 11.50 ( ) 5.00 ( ) 14.00 ( ) 29.50 ( )
100 13 2500 100 9.75 1000 100 4.25 500 100 13 750 100 27 1200
11 8.50 3.50 12.50 24.50
(13 2500)+(9.75 1000) + (4.25 500) +(13 750) +(27
éæ ö æ ö æ ö æ ö æ ö ù êç ´ ÷ ´ ´ +ç ´ ÷ ´ ´ + ç ´ ÷´ ´ + ç ´ ÷ ´ ´ +ç ´ ÷ ´ ú
ëè ø è ø è ø è ø è ø û
[ ´ ´ ´ ´ ´ 1200) ]
=
4136363.64 + 1319117.65 + 303571.43 + 1092000 + 3901224.49
124.27
86525
=
< TOP >
70. Answer : (e)
Reason : Let the following notations be used:
X : Test score
Y : Monthly sales (Rs. ‘000)
Yµ = a + bX
b = ( )2 2
n XY X. Y
n X X
-
-
å å å
å å
a =Y - bX
n = 10
åXY = 21650 åX = 500 åY = 400 åX2 = 27400
\
( ) ( )
( ) ( )
10 21650 500 400 216500 200000
b 0.6875
10 27400 500 500 274000 250000
´ - ´ -
= = =
´ - ´ -
a =
400 500
Y bX 0.6875 5.625
10 10
- = æ ö - æ ö = çè ÷ø çè ÷ø
\ Y = 5.625+ 0.6875X
)
< TOP >
31
For X = 90, Y = 5.625+ (0.6875´90) = 67.5(inRs.'000)
)
\Estimated monthly sales = Rs.67500