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Section A : Basic
Concepts (40 Points) |
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· This section consists of questions with serial number 1 - 40. · Answer all questions. · Each question carries one point. |
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If a straight line in X-Y plane has a negative slope, then a. For a given value of x the value of y will always be negative b. It falls from left to right as the values increase along the X-axis c. It always passes through the point of intersection of the X and Y axes d. It is always parallel to the Y-axis e. It is
always parallel to the X-axis. |
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a, m and n are integers. am × an = a. am + n b. am – n c. am × n d. am / n e.
(m × n)a |
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Which of the following is false with regard to the simplex method of solving a linear programming problem on profit maximization? a. The pivot element is located at the intersection of pivot column and pivot row b. The variable to leave solution can be identified by inspecting the values of Zj – Cj row c. The simplex method can be applied when there are more than two decision variables d. The values in the Zj – Cj row indicate whether the solution is optimal or not e.
The slack variables
can assume non-negative values only. |
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Which of the following is false with regard to the graphical method of solving linear programming problems? a. The decision variables are represented by the horizontal and vertical axes b. The graphical method can be applied when there are two decision variables c. The optimal solution, if it exists, occurs at one of the corner points of the feasible region d. The optimal solution to the problem always occurs at a point outside the feasible region e.
The feasible region is
determined by the structural constraints and non-negativity
constraints. |
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Which of the following is true with regard to the simplex method of solving a linear programming problem (LPP) on profit maximization? a. At the optimal solution the slack variables are always equal to zero b. The constraints which contain ‘£’ are converted into equations by adding slack variables c. There can be only one feasible solution to a LPP d. The slack variables make positive contributions to profit e. The slack
variables can only assume the values 0 or 1. |
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Which of the following conditions indicates the existence of multiple optimal solutions when a linear programming problem is solved by the graphical method? a. One of the constraints is parallel to the horizontal axis b. The objective function is parallel to the vertical axis c. The objective function is parallel to one of the edges of the feasible region which is in the direction of optimal movement of the objective function d. All the decision variables assume negative values e. One of the decision variables assumes negative values. |
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Which of the following indicates that the distribution of data is skewed to the right? a. Arithmetic Mean > Median > Mode b. Median = Mode = Arithmetic Mean c. Median = 0 d. Arithmetic Mean = 0 e. Standard
Deviation = 0. |
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Which of the following statements is true? a. The tallest rectangle in a histogram represents the modal class of the distribution b. In a symmetrical distribution the mean, median and mode are unequal c. The medians of two sets of data can be combined mathematically d. The median can not be determined graphically e. The mode is always uniquely defined. |
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Which of the following measures cannot be combined mathematically? a. Standard deviation b. Arithmetic mean c. Geometric mean d. Harmonic mean e. Median. |
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If a function, f(x), has a relative maxima at
a point x = c, then, which of the following is
true? a. The first order derivative of f(x) at x = c is positive b. The first order derivative of f(x) at x = c is negative c. The second order derivative of f(x) at x = c is zero d. The second order derivative of f(x) at x = c is positive e. The second order derivative of f(x) at x = c is negative. |
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logab + logac = 0 This implies that a. b = c b. b = -c c. b + c = 1 d. b - c = 1 e. b and c are reciprocals. |
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Which of the following statements is false? a. The set of natural numbers is an infinite set b. 2 is a natural number but not a whole number c. The set of integers includes positive as well as negative numbers d. A rational number can be expressed in decimal form e. The set of
real numbers does not include complex numbers. |
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Which of the following is true with regard to the classical approach to probability? a. 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. |
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If the average deviation from the arithmetic mean is used as a measure of dispersion then, it will always indicate that the dispersion of the data set is a. A positive value b. A negative value c. Either a positive or negative value d. Equal to zero e. A multiple of the arithmetic mean. |
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Which of the following measures can be determined graphically? a. Arithmetic mean b. Geometric mean c. Harmonic mean d. Standard deviation e. Median. |
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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. |
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The domain of a function contains a. Only negative numbers b. Only positive numbers c. The values that may be taken by the function d. The values that may be taken by the independent variable of the function e. The values
for which the function is not defined. |
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Which of the following statements is false? a. If the primal formulation of a linear programming problem is a minimizing problem then the dual formulation will be a maximizing problem b. If the primal formulation of a linear programming problem is a maximizing problem then the dual formulation will be a minimizing problem c. The dual formulation is an inverse of the primal formulation in every respect d. The dual formulation can not be solved for an optimal solution e. The dual
formulation can be derived from the same data from which the primal is
formulated. |
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The probability of the occurrence of an event is expressed as a number which lies between a. 0 and 1 b. 1 and 2 c. –1 and 0 d. –2 and
–1 e. 1 and infinity. |
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Which of the following can be used to represent graphically the type of events that may result from a random experiment? a. Histogram b. Frequency polygon c. Ogive d. Frequency distribution e. Venn diagram. |
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If every observation in a data set is increased by a constant, k, then the range of the resulting data set is equal to a. Range of the original data set b. Range of the original data set minus k c. Range of the original data set plus k d. k e.
–k. |
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The derivative of a constant is equal to a. 0 b. 1 c. –1 d. A positive number if the constant is positive e.
A negative number if the constant is negative. |
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Which of the following measures does not use every observation in the data set? a. Variance b. Coefficient of variation c. Mode d. Geometric mean e.
Arithmetic mean. |
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Which of the following is true, if there is no dispersion in a data set? a. All the mathematical and positional averages are equal b. All the mathematical averages are equal but the positional averages are not equal c. All the positional averages are equal but the mathematical averages are not equal d. All the mathematical averages are equal to zero e.
All the positional averages are equal to zero. |
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The reciprocal of the harmonic mean 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. |
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Which of the following statements is false? a. The tallest rectangle in a histogram indicates the location of the mean of the distribution b. For a multimodal distribution the mode is not a meaningful measure of central tendency c. In a symmetrical distribution the mean, median and mode are equal d. The median and the mode can be found out graphically e. The medians
of two data sets cannot be combined mathematically. |
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Which of the following is false with regard to the derivative of a function? a. It indicates the rate of change of the function at a given point b. The slope of the tangent to a function at a point is equal to the derivative of the function at the point c. The derivative may be a function of the independent variable d. The derivative of a linear function changes with the value of the independent variable e. If the
derivative of a function at a point is negative then it indicates that the
function is decreasing at that point. |
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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. |
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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. |
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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. |
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If events A and B are mutually exclusive then which of the following is true? a. P(A and B) = 0 b. P(A) = 0 c. P(B) = 0 d. P(A or B) = 0 e. P(A or B) =
1. |
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Which of the following is false with regard to histograms? a. The widths of the rectangles are proportional to the range of values falling within the classes b. The heights of the rectangles are proportional to the number of items falling within the classes c. The classes are marked along the horizontal axis d. The frequencies are marked along the vertical axis e. The tallest rectangle represents the class which has the least frequency. |
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Which of the following is a relative measure of variability? a. Range b. Mean absolute deviation c. Standard deviation d. Coefficient of variation e. Arithmetic
mean. |
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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. |
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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 arithmetic mean and the geometric mean between the two quantities e.
The ratio of the arithmetic mean to the geometric mean between the
two quantities. |
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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. |
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If every item in a data set is multiplied by the same quantity, k, then the standard deviation of the resulting data set a. Is equal to the variance of the original data set b. Is equal to the average of the original data set c. Is equal to the standard deviation of the original data set multiplied by k d. Is equal to the range of the original data set e.
Is always less than the standard deviation of the original data
set. |
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The average deviation of all items in a data set from zero is equal to a. Arithmetic mean b. Geometric mean c. Coefficient of variation d. Standard deviation e. Variance. |
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The logarithm of 1 with respect to any base is equal to a. –10 b. –1 c. 1 d. 0 e.
10. |
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If logab = k then which of the following is correct? a. a = bk b. b = ak c. k = ab d. k = ba e.
b = ka. |
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END OF PART A |
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Section B : Problems
(60 Points) |
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· This section consists of questions with serial number 41 - 71. · Answer all questions. · Points are indicated against each question. |
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A and B are two non-mutually exclusive and independent events. If P(A) = 0.60 and P(B) = 0.50 then, what is the probability of either event A or event B taking place? a. 0.10 b. 0.30 c. 0.50 d. 0.80 e. 0.90. (1 point) |
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Event B is dependent upon event A. P(B) = 0.60 and P(A and B) = 0.30, then P(A/B) is a. 0.18 b. 0.30 c. 0.40 d. 0.50 e. 0.90. (1 point) |
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The median of the series 7, 4, 9, 10, 6, 8 is a. 5.0 b. 6.0 c. 7.5 d. 10.0 e. 9.0. (1 point) |
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The harmonic mean of the numbers 2, 5 and 8 is a. 1/5 b. 5 c. 11/40 d. 40/11 e. 2/5. (1 point) |
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A geometric progression having infinite number of terms starts with the term 1/2. The progression is of the following form: 1/2, 1/4, 1/8, 1/16, …..infinite terms The sum of the terms in the geometric progression is a. 1/4 b. 1/2 c. 3/5 d. 3/4 e. 1. (1 point) |
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A bag contains 30 black balls and 20 red balls. If 14 balls are drawn from the bag without replacement then the probability of drawing 8 black balls and 6 red balls is approximately a. 0.125 b. 0.242 c. 0.625 d. 0.75 e. 0.81. (1 point) |
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The geometric mean of a set of five numbers is 3. If each of the five numbers is multiplied by 2, then the geometric mean of the resulting values will be a. 2 b. 3 c. 2/3 d. 3/2 e. 6. (1 point) |
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The first term and the fifth term of an arithmetic progression containing of five terms are 2 and 14 respectively. The first term and the fourth term of another arithmetic progression containing four terms are 3 and 9 respectively. If all the terms in both the series are added together then their sum will be equal to a. 16 b. 23 c. 24 d. 40 e. 64. (1 point) |
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The geometric mean of a set of five numbers is 2. The geometric mean of a set of four numbers is 4. What is the geometric mean of the numbers in both the sets considered together? a. 2 b. 2.72 c. 3 d. 3.54 e. 4. (1 point) |
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The roots of the quadratic equation x2 + 7x + 12 = 0 are a. –3 and 4 b. 3 and –4 c. –3 and –4 d. 3 and 4 e. 6 and 8. (1 point) |
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In a survey of 100 readers, it was found that 50 read magazine A, 30 read magazine B and 15 read both the magazines. What is the probability of finding a person in the group who reads neither magazine A nor magazine B? a. 0.15 b. 0.35 c. 0.65 d. 0.80 e. 0.85. (2 points) |
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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 past 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%. (3 points) |
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Three candidates, A, B and C, are being considered by the board of directors of a company for the post of the CEO. The probability that A will be appointed as the CEO is 0.20, the probability that B will be appointed is 0.30 and the probability that C will be appointed to the post is 0.50. If A is appointed as the CEO then the probability of successfully launching a new product is 30%. If B is appointed as the CEO then the probability of successfully launching a new product is 50% and if C is appointed then the probability of successfully launching a new product is 60%. What is the probability that a new product will be successfully launched? a. 6% b. 15% c. 30% d. 51% e. 60%. (2 points) |
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An arithmetic progression (AP) consists of the following terms : 9, 6, 3, 0, … , -75 How many terms are there in the AP? a. 10 b. 19 c. 29 d. 35 e. 39. (1 point) |
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The third term of an arithmetic progression (AP) is
11 and the sixth term is 20. What
is the tenth term of the AP? a. 8 b. 17 c. 20 d. 29 e. 32. (2 points) |
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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 points) |
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The following details are available with regard to a data set: Σx = 33 Σx2 = 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. (35/3)1/2 b. 35/3 c. 3 d. 25 e. 35. (2 points) |
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The following distribution shows the ages of 100 persons in a group:
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. 45.2 years. (2 points) |
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The following distribution shows the distribution of wages of 500 workers in a factory:
What is the median wage earned by the workers in the factory? (Select the nearest figure) a. Rs. 325.70 b. Rs. 350.25 c. Rs. 374.84 d. Rs. 395.30 e. Rs. 412.50. (3 points) |
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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 points) |
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The observations from a population are presented in the following frequency distribution:
What is the standard deviation of the population? a. 14.25 b. 17.23 c. 20.18 d. 22.16 e. 24.36. (3 points) |
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It is required to compute the standard deviation of a population. The observations from the population were grouped into class intervals and, the deviations of the class mid points from an assumed mean were computed. The following details are available:
The width of the class interval is 10. What is the variance of the population? a. 184.75 b. 1847.5 c. 18475 d. 18550 e. 19450. (3 points) |
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The following details are available with regard to two groups of data, A and B:
What is the combined standard deviation for both the groups? a. 2.50 b. 3.50 c. 6.25 d. 7.00 e. 12.25. (3 points) |
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The following details are available with regard to four sets of data, A, B, C and D:
Which data set is the most consistent? a. A b. B c. C d. D e. All are equally consistent. (2 points) |
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Which of the following is correct? a. b. c. d. e. (2 points) |
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Which of the following is correct? a. g(x) is a linear function of x and, g(1) = 5 and g(2) = 8. The slope of the function is 2 b. g(x) is a linear function of x and, g(2) = 13 and g(4) = 23. \ g(7) = 30 c. y is
a linear function of x and d. y is
a linear function of x and e. None of the above. (3 points) |
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Which of the following is/are correct? a. b. c. d. e. None of the above. (3 points) |
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Which of the following is/are correct? a. b. c. d. e. None of the above. (3 points) |
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A firm has the following revenue function for its product: R = 8Q Where, R = Revenue earned in a day Q = Number of units of the product produced and sold in a day
The total cost of producing and selling the product on any day is given by the following function:
C = 1,26,000 + 60 Where, C = The total cost of producing and selling the product in a day Q = Number of units of the product produced and sold in a day
What is the number of units of the product that must be produced and sold by the firm in a day in order to maximize its daily profit? a. 6,750 units b. 12,500 units c. 40,000 units d. 54,000 units e. 1,26,000 units. (3 points) |
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f(x) = 50x2 – 400x + 1000 f(x) is minimum at x = a. 1 b. 4 c. 40 d. 100 e. 400. (1 point) |
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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). (3 points) |
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is defined at x = 3
is defined at x = 5
is defined at x = – 5
is not defined at x = 3
is not defined at x = – 6.
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Section A : Basic Concepts 1. Answer : (b) < TOP > Reason : (a) A straight line with a negative slope does not necessarily mean that the value of y will always be negative for any given value of x. (b) If a straight line has a negative slope then it falls from left to right as the values increase along the X-axis because for increase in the x – values will be accompanied with decrease in the y-values. (c) A straight line passes through the origin when the y intercept is zero; a negative slope will not pass through the origin if the y intercept is not zero. (d) A straight line is parallel to the Y-axis when the slope is infinite. (e) A straight line is parallel to the X-axis when the slope is zero. 2. Answer : (a) < TOP > Reason : (a) am × an = am + n (b) am – n = am / an (c) am × n = (am) n = (an) m (d) am / n = (am) 1/n (e) (m × n)a = ma ´ na 3. Answer : (b) < TOP > Reason : (a) The pivot element is located at the intersection of the pivot column and pivot row. (b) The variable to enter solution can be identified by inspecting the values of the Zj – Cj row. (c) The simplex method can be applied when there are more than two decision variables. (d) The values in the Zj – Cj indicate whether the solution is optimal or not.
(e)
The slack variables can assume non-negative values only. 4. Answer : (d) < TOP > Reason : (a) In the graphical method of solving LP problems the decision variables are represented by the horizontal and vertical axes. (b) This method can be applied when there are two decision variables. (c) If the optimal solution exists then it occurs at one of the corner points. (d) The optimal solution to the LPP can not occur at a point outside the feasible region because a point outside the feasible region means that the constraints are violated. (e) The feasible region is determined by the structural constraints as well as the non-negativity constraints. 5. Answer : (b) < TOP > Reason : (a) At the optimal solution the slack variables need not be equal to zero. (b) The constraints which contain the £ sign are converted into equations by adding slack variables. (c) There can be more than one feasible solution to a LPP. (d) The slack variables make zero contribution towards profit. (e) The slack variables can only assume any non-negative value. 6. Answer : (c) < TOP > Reason : (a) and (b) are not the indicators of multioptimality. (c) In the graphical method of solving a LPP the situation of multioptimality arises when the objective function is parallel to one of the edges of the feasible region which is in the direction of the optimal movement of the objective function. (d) and (e) are not correct because they only assume non-negative values. 7. Answer : (a) < TOP > Reason : (a) indicates that the distribution is skewed to the right. (b), (c) and (d) are incorrect indicators of skewness to the right. (e) indicates that there is no variability in data. 8. Answer : (a) < TOP > Reason : (a) The tallest rectangle in a histogram represents the modal class. (b) In a symmetrical distribution mean, median and mode are equal. (c) The median can not be mathematically manipulated; hence medians of two sets of data can not be combined. (d) The median can be determined graphically. (e) The mode is not uniquely determined when more than one observations have the highest frequency. 9. Answer : (e) < TOP > Reason : (a) Standard deviations of two or more data sets can be mathematically combined (b) Arithmetic means of two or more data sets can be mathematically combined (c) Geometric means of two or more data sets can be mathematically combined (d) Harmonic means of two or more data sets can be mathematically combined. (e) Medians of two or more data sets cannot be mathematically combined 10. Answer : (e) < TOP > Reason : (a) implies that the function is f(x) is increasing at x = c. (b) implies that the function is f(x) is decreasing at x = c. (c) implies that a function is changing at a constant rate. (d) is true when the function has a relative minima at x = c. (e) is true when the function has a relative maxima at x = c. 11. Answer : (e) < TOP > 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 logab + 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. 12. Answer : (b) < TOP > Reason : (a) The set of natural numbers is an infinite set. (b) 2 is both a natural number and a whole number. (c) The set of integers includes positive as well as negative numbers. (d) A rational number can be expressed in decimal form. (e) The set of real numbers does not include complex numbers. 13. Answer : (c) < TOP > 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. 14. Answer : (d) < TOP > Reason : (a) The average deviation from the arithmetic mean will never be greater than zero. (b) The average deviation from the arithmetic mean will never be less than zero. (c) The average deviation from the arithmetic mean can never be less than or greater than zero. (d) The sum of the deviations from the arithmetic mean is equal to zero. Hence the average deviation from the arithmetic mean will always indicate that the dispersion of the data set is equal to zero. (e) There is no reason why the average deviation from the arithmetic mean will be a multiple of the arithmetic mean. 15. Answer : (e) < TOP > Reason : (a) Arithmetic mean cannot be determined graphically. (b) Geometric mean cannot be determined graphically. (c) Harmonic mean cannot be determined graphically. (d) Standard deviation cannot be determined graphically. (e) Median can be determined graphically. 16. Answer : (e) < TOP > 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. (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. 17. Answer : (d) < TOP > Reason : (a) and (b) The domain of the function is determined by definiti(a) and (b) The domain of the function is determined by definition of the function. Hence it can not be said that the domain will contain only negative values or positive values. (c) The range and not the domain includes the values which will be taken by the function. (d) The domain of the function consists of the values that may be taken by the independent variable of the function. (e) The domain does not contain the values for which the function is not defined. 18. Answer : (d) < TOP > Reason : (a) and (b) If the primal formulation of the LPP is a minimizing problem then the dual will be a maximizing problem and vice versa. (c) The dual formulation is an inverse of the primal in every respect. (d) The dual formulation can also be solved for the optimal solution. (e) The dual formulation can be derived from the same data from which the primal is formulated. 19. Answer : (a) < TOP > Reason : (a) The probability of the occurrence of an event is expressed as a number which lies between 0 and 1. 20. Answer : (e) < TOP > Reason : (a) Histogram cannot be used to graphically represent the type of events that may result from an experiment. (b) Frequency polygon cannot be used to graphically represent the type of events that may result from an experiment. (c) Ogive cannot be used to graphically represent the type of events that may result from an experiment. (d) Frequency distribution cannot be used to graphically represent the type of events that may result from an experiment. (e) Venn diagrams can be used to graphically represent the type of events that may result from an experiment. 21. Answer : (a) < TOP > Reason : Range = Highest observation – Lowest observation (a) \ If every observation in a data set is increased by a constant k , then the range of the resulting data set is equal to the range of the original data set. (b) If every observation in a data set is increased by a constant k , then the range of the resulting data set is not equal to range of the original data set minus k. (c) If every observation in a data set is increased by a constant k , then the range of the resulting data set is not equal to range of the original data set plus k. (d) If every observation in a data set is increased by a constant k , then the range of the resulting data set is not equal to k. (e) If every observation in a data set is increased by a constant k , then the range of the resulting data set is not equal to -k. 22. Answer : (a) < TOP > Reason : (a) The derivative of a constant is equal to 0. (b), (c), (d) and (e) are all incorrect conclusions with regard to the derivative of a constant. 23. Answer : (c) < TOP > Reason : (a) Variance uses every observation in the data set. (b) Coefficient of variation uses every observation in the data set. (c) Mode does not use every observation in the data set. (d) Geometric mean uses every observation in the data set. (e) Arithmetic mean uses every observation in the data set. 24. Answer : (a) < TOP > Reason : (a) If there is no dispersion in a data set then all the mathematical and positional averages are equal. (b) Hence b. is false. (c) Hence c. is false. (d) Hence d. is false. (e) Hence e. is false. 25. Answer : (d) < TOP > 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. 26. Answer : (a) < TOP > Reason : (a) The tallest rectangle in a histogram indicates the location of the mode. (b) A multimodal distribution has more than one mode. Hence one can not say which of the values of the mode is the correct figure for central tendency. (c) In a symmetrical distribution the mean, median and mode are all equal. (d) The median and the mode can be found out graphically. (e) The medians of two data sets can not be combined mathematically because mode is a positional average. 27. Answer : (d) < TOP > Reason : (a) The derivative of a function indicates the rate of change of the function. (b) The slope of the tangent to a function at a point is equal to the derivative of the function at that point. (c) The derivative of a function can be said to be a function of the independent variable if the expression of the derivative contains the independent variable. (d) The derivative of a linear function is the slope of the linear function, which is a constant value for all values of the dependent variable. (e) If the derivative of any function at a point is negative then it indicates that the function is decreasing at that point. 28. Answer : (c) < TOP > 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 decision variables for which the objective is accomplished with in the given constraints.
(e)
The resources are in limited supply. 29. Answer : (c) < TOP > 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 variability 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 variability. (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. 30. Answer : (b) < TOP > 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. 31. Answer : (a) < TOP > Reason : (a) If events A and B are mutually exclusive then P(A and B) = 0 because A and B can not occur at the same time. (b) and (c) Mutual exclusiveness of two events A and B does not imply that both of them have zero probabilities of occurrence. (d) and (e) Mutual exclusiveness does not mean that the probability of any one of them occurring is either 0 or 1. 32. Answer : (e) < TOP > Reason : (a) The widths of the rectangles are proportional to the range of values falling within the classes. (b) The heights of the rectangles are proportional to the number of values falling within the classes. (c) The classes are marked along the horizontal axis. (d) The frequencies are marked along the vertical axis. (e) The tallest rectangle represents the class which has the maximum frequency. 33. Answer : (d) < TOP > Reason : (a), (b) and (c) are all absolute measures of dispersion. (d) is a relative measure of dispersion. (e) is a measure of central tendency. 34. Answer : (a) < TOP > 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. 35. Answer : (c) < TOP > 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. 36. Answer : (b) < TOP > 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. 37. Answer : (c) < TOP > Reason : (c) If every item is multiplied by the same quantity, k , then the standard deviation of the resulting data set is equal to the standard deviation of the original data set multiplied by k. (a), (b), (d) and (e) are incorrect conclusions with regard to the given condition. 38. Answer : (a) < TOP > Reason : (a) The average deviation of all the items from zero is equal to the arithmetic mean. (b), (c), (d) and (e). The geometric mean, coefficient of variation, standard deviation, and variance cannot be calculated on the basis of average deviation from zero. 39. Answer : (d) < TOP > Reason : (d) The logarithm of 1 with respect to any base is equal to 0 , because any quantity raised to the exponent of 0 is equal to 1. (a), (b), (c) and (e) are incorrect conclusions because raising any quantity to these exponents will not give 1. 40. Answer : (b) < TOP > Reason : (b) If logab = k then k is the exponent to which a must be raised in order to get b. Therefore logab = k implies that b = ak . (a), (c), (d) and (e) are incorrect interpretations of the logarithmic expression. Section B : Problems 41. Answer : (d) < TOP > Reason : A and B are non-mutually exclusive events P(A) = 0.60 P(B) = 0.50 \P(A or B) = P(A) + P(B) – P(A and B) = 0.60 + 0.50 – (0.60 ´ 0.50) = 0.80. 42. Answer : (d) < TOP > Reason : P(A and B) = P(B) ´ P(A/B) Or 0.30 = 0.60 ´ P(A/B) Or
P(A/B)
= 43. Answer : (c) < TOP > Reason : Median of the series (arranged in ascending order): 4, 6, 7, 8, 9, 10 Median
position = \
Median = 44. Answer : (d) < TOP >
Reason :
Harmonic mean of 2,5 and 8
= 45. Answer : (e) < TOP >
Reason :
Sum of an infinite geometric progression = a = \
Sum of the series = 46. Answer : (b) < TOP > Reason : Total number of balls = 30 + 20 = 50 Number of black balls = 30 Number of red balls = 20 Number
of ways in which 8 black balls can be selected from 30 = Number
of ways in which 6 red balls can be selected from 20 =
Number
of ways in which 8 black and 6 red balls can be selected = Number
of ways in which 14 balls can be selected from 50 = \
P(8 black balls and 6 red balls)
= 47. Answer : (e) < TOP > Reason : Geometric mean = (Product of the numbers)1/n Where n = Number of numbers \ (Geometric mean)n = Product of the numbers (before multiplication by 2) = 35 Each number is multiplied by 2. \ Product of the resulting values = 25 ´ 35 \ Geometric mean of the resulting values = (25 ´ 35)1/5 = 2 ´ 3 = 6. 48. Answer : (e) < TOP >
Reason :
Sum of the first A.P. containing 5 terms = Sum
of the second A.P. containing 4 terms
= \ Sum of all the terms in both the A.P.s = 40 + 24 = 64. 49. Answer : (b) < TOP > Reason : Geometric mean of the set of five numbers = 2 \ Product of the five numbers = 25 = 32 Geometric mean of the set of four numbers = 4 \ Product of the four numbers = 44 = 256 Product of the combined set of numbers = 32 ´ 256 = 8192. \ Geometric mean of the combined set of numbers = (8192)1/9 = 2.72 (approx.) 50. Answer : (c) < TOP > Reason : x2 + 7x + 12 = 0 x = 51. Answer : (b) < TOP >
Reason :
P(A) = P(A
and B) = P(A or B) = P(A) + P(B) – P(A and B) = 0.50 + 0.30 – 0.15 = 0.65 \ P(Neither A nor B) = 1 – P (A or B) = 1 – 0.65 = 0.35. 52. Answer : (b) < TOP > 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)
=
P (B) =
P (C) = P (D/A) = 2% = 0.02 P (D/B) = 3% = 0.03 P (D/C) = 5% = 0.05
P (A and D)
= P (A)
. P (D/A) =
P (B and D)
= P (B)
. P (D/B)
=
P (C and D)
= P (C)
. P (D/C)
=
\
P (D)
= P (A
and D) + P (B and D) + P (C and D) =
P (B/D)
= 53. Answer : (d) < TOP > Reason : Let the following notations be used, A : A is appointed as CEO. B : B is appointed as CEO. C : C is appointed as CEO. N : New product is launched. Given: P (A) = 0.20 P (B) = 0.30 P (C) = 0.50 P (N/A) = 30% = 0.30 P (N/B) = 50% = 0.50 P (N/C) = 60% = 0.60 \ P (N) = P (A and N) + P (B and N) + P (C and N) = (0.20 ´ 0.30) + (0.30 ´ 0.50) + (0.50 ´ 0.60) = 0.51 i.e. 51%. 54. Answer : (c) < TOP > Reason : For the A.P. : 9,6,3,0 … – 75 a = 9 d = 6 – 9 = –3 Let there be ‘n’ terms \ tn = –75 \ –75 = a + (n – 1)d = 9 + (n – 1) (–3) or –75 = 9 – 3n + 3 or 3n = 75 + 9 + 3 = 87 or
n
= 55. Answer : (e) < TOP > Reason : t3 = a + (3 – 1)d i.e., 11 = a + 2d (A) t6 = a + (6 – 1)d i.e., 20 = a + 5d (B) Subtracting equation (A) from equation (B) (20 – 11) = (a + 5d) – (a + 2d) or 9 = 3d or d = 9/3 = 3 \ a = 11 – (2 ´ 3) = 5 \ t10 = a + 9d = 5 + (9 ´ 3) = 32. 56. Answer : (b) < TOP > 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 = q
is the G.M. between b and c
Þ
q = \ p2 + q2 = ab + bc or p2 + q2 = b(a + c) or p2 + q2 = b(2b) or p2 + q2 = 2b2 or
b2
=
\ b2 is the A.M. between p2 and q2. 57. Answer : (a) < TOP > Reason : Original data set: Sx = 33 Sx2 = 199 N = 6 After modification: Sx = 2 ´ 33 = 66 Sx2 = 4 ´ 199 = 796 \Standard
deviation of the resulting data set
= = = 58. Answer : (d) < TOP > Reason :
\
Mean = 59. Answer : (c) < TOP >
Reason :
Median = The
position of the median is the
\ Median class is 350-400. Lm = 350 N = 500 F = 170 fm = 160 W = 50 \
Median = 350 +
= Rs.374.84 60. Answer : (c) < TOP >
Reason :
Subgroup A : NA =
Subgroup
B :
NB =
Subgroup
C :
NC
=
\
Average age of the entire group
= = 61. Answer : (b) < TOP >
Reason :
Standard deviation of population, s =
Mean,
Standard
Deviation = 62. Answer : (a) < TOP > Reason : The variance of the population using assumed mean can be calculated as: s2 =
Variance
of the population
=
= = 184.75. 63. Answer : (b) < TOP > Reason : NA = 15 mA = 20 sA = 4 NB = 25 mB = 16 sB = 2 m = dA = mA – m = 20 – 17.5 = 2.5 dB = mB – m = 16 – 17.5 = –1.5 s12 =
=
= 64. Answer : (a) < TOP > Reason : Data Set A: m
= s
= Coefficient
of variation = Data
Set B: m
= s
= Coefficient
of variation = Data
Set C: m
= s
= Coefficient of
variation = Data Set
D: m
= s
= Coefficient of
variation = The data set which has the least coefficient of variation is the most consistent data set. \From above we can see that data set A is the most consistent. 65. Answer : (e) < TOP >
Reason :
(a)
f(x) = \ f(x) is not defined at x = –1 and x = 3 (b) f(x) = \ f(x) is not defined at x = 3 and x = 5 (c) f(x) = \f(x) is not defined at x = –5 and x = 6 (d) f(x) = \f(x) is not defined at x = –3 and x = 1. f(x) is defined at x = 3 (e) f(x) = \ f(x) is not defined at x = –6 and x = 5. 66. Answer : (c) < TOP > Reason : (a) g(x) is a linear function. \g(x) = a + bx Given : g(1) = 5 Þ 5 = a + b(1) or 5 = a + b \ g(2) = 8 Þ 8 = a + b(2) or 8 = a + 2b \ Slope, b can be obtained as follows: 8 – 5 = (a + 2b) – (a + b) or 3 = b \ b = 3. (b) g(x) is a linear function. \g(x) = a + bx Given : g(2) = 13 Þ a + 2b = 13 g(4) = 23 Þ a + 4b = 23 \ (23 – 13) = (a + 4b) – (a + 2b) or 10 = 2b
or
b
= \ a = 13 – 2 ´ 5 = 3 \ g (7) = 3 + 5(7) = 38. (c) y = a + bx
Given:
\ Slope, b = –5 (d) y = a + bx
Given: \ Intercept on the y axis = 3. 67. Answer : (d) < TOP >
Reason :
(a)
=
=
=
=
=
(b)
=
(c)
(d)
=
=
=
=
=
68. Answer : (d) < TOP >
Reason :
(a)
Let y = f(t) =
(b) Let y =
f(t) =
(c) Let y = f(t) = (ln t)3
(d) Let y =
f(t) =
= 69. Answer : (d) < TOP > Reason : Profit function, F = Revenue – Total cost = R – C
=
8Q –
F =
8Q –
or
or
or Q = 8 x 6,750 = 54,000
\
At Q = 54,000 , \ The profit is maximum at Q = 54,000 units. 70. Answer : (b) < TOP > Reason : f(x) = 50x2 – 400x + 1000
At minima,
\ f(x) is minimum at x = 4. 71. Answer : (e) < TOP > Reason : Rewriting the formulation using the slack variables: Maximize: Z = 8X + 6Y + 0S1 + 0S2 Subject to: 4X + 2Y + S1 + 0S2 = 60 2X + 4Y + 0S1 + S2 = 48 All variables ³ 0 Tableau I
Enter : X (Most negative value in (Zj – Cj) row) Leave : S1 (Minimum positive ratio) Tableau II
Enter : Y Leave : S2 Tableau III
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). |
= 0.2143. i.e. 21.43%
item i.e., 
=
=
or
= 0 and 
< 0
(x) = 50(2x) – 400 = 100x – 400
and 
