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Question Paper Quantitative Methods – I (131): April 2006
	· Answer all questions. · Marks are indicated against each question.

Consider the three places Hyderabad, Nanded and Nagapur. If there are 3 routes from Hyderabad to Nanded and 2 from Nanded to Nagapur in how many different ways can a person travel from Hyderabad to Nagapur via Nanded?

(a) 1 (b) 3 (c) 2 (d) 5 (e) 6.

(1 mark)

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How many arrangements of the elements of {P, E, R, M, U, T, A, I, O, N, S} can be formed when the vowels come together?

(a) 5! (b) 6! (c) 7! (d) 5!.6! (e) 5!.7!.

(1 mark)

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A lock contains three rings, each ring containing four letters. If the lock opens in only one arrangement of three letters, how many unsuccessful events are possible?

(a) 144 (b) 63 (c) 24 (d) 121 (e) 17.

(1 mark)

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Find the number of ways in which 8 boys and 6 girls are to sit for a lunch at a round table so that no two girls are to sit together.

(a) 7!. (b) 8!.6! (c) 8!. (d) 7! (e) 8!.

(2 marks)

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Out of 8 men and 5 ladies a committee of 5 is to be formed. Find the number of ways in which this can be done so as to include at least 3 ladies.

(a) + (b) ++

(e) +.

(1 mark)

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In how many different ways can 8 different shirts be distributed among 4 different people so that each receives 2 shirts?

(a) 2850 (b) 2680 (c) 2520 (d) 1908 (e) 1800.

(1 mark)

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The value of y in the above simultaneous equations would be

(a) 3 (b) 6 (c) 9 (d) 12 (e) 24.

(1 mark)

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The sums of first 5 terms of three arithmetical progressions are . The first term of each of these A.P’s is unity and the common differences are 3, 4 and 5 respectively. Which of the following equations is/are true?

I. .

II. .

III. .

(a) Only (I) above (b) Only (II) above

(e) Both (II) and (III) above.

(2 marks)

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If . The value of n would be

(a) 8 (b) 9 (c) 10 (d) 11 (e) 12.

(2 marks)

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10.

If x²+ y²= 6xy, then 2 log(x + y) is equal to

(a) log x + log y + 3 log2 (b) log x – log y – log2

(e) log x + log y + log3.

(2 marks)

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11.

The value of is equal to

(a) (b) (c) (d) 0 (e) 1.

(1 mark)

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12.

The value of is equal to

(a) 0 (b) 1 (c) 2 (d) 16 (e) 1/16.

(1 mark)

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13.

If log_ex . log₅625 = log₁₀16. log_e10 then the value of x is

(a) 1 (b) 2 (c) 5 (d) 10 (e) 16.

(2 marks)

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14.

If is equal to

(a) f(x) (b) f(2x) (c) 2f(x) (d) f(1+x²) (e) f(1 – x²).

(1 mark)

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15.

The value ofis equal to

(a) 0 (b) 1 (c) –14 (d) –1/2 (e) 7.

(1 mark)

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16.

Which of the following is/are true?

I. The logarithms of the same numbers to different bases are different.

II. The logarithm of 1 to any base is zero.

III. The logarithm of any number (not equal to 1) itself as base is unity.

(a) Only (I) above (b) Only (II) above

(e) All (I), (II) and (III) above.

(1 mark)

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17.

If then the value of abc is

(a) 0 (b) 1 (c) b – c (d) c – a (e) a – b.

(2 marks)

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18.

What is the value of ?

(a) 0 (b) (c) 1 (d) 1/2 (e) 1/4.

(1 mark)

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19.

Which of the following is the value of ?

(a) 3/5 (b) 5/3 (c) (d) (e) 0.

(1 mark)

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20.

What is the value of ?

(a) 0 (b) 1 (c) 1/3 (d) 1/4 (e) .

(2 marks)

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21.

If f(x) = 2x²+ x – 3, then the value of is equal to

(a) 9 (b) 5 (c) 2 (d) 1 (e) does not exist.

(1 mark)

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22.

If x = ct and y = c/t, then is equal to

(a) –t (b) 1/t (c) –t² (d) –1/t² (e) t.

(1 mark)

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23.

Which of the following is the derivative of ?

(a) x^x(1+logx) (b) x(1+logx) (c) (1–logx) (d) x^xlogx (e) –logx.

(2 marks)

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24.

What is the derivative of

(a) (b)

(e) .

(1 mark)

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25.

(a) log x + c (b) log(logx) + c

(2 marks)

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26.

The value of is

(a) (b) (c) (d) (e) .

(2 marks)

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27.

If then the value of is equal to

(a) U (b) 2U (c) 3U (d) 4U (e) 5U.

(1 mark)

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28.

If Find the value of .

(a) (b) (c) (d) (e) .

(1 mark)

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29.

The sum of first n terms of an A.P., is 49 and common difference is 2. If the seventh term is 13, then what is the value of n?

(a) 6 (b) 7 (c) 8 (d) 9 (e) 10.

(1 mark)

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30.

How many terms of series 1 + 4 + 16 + ….. is required to make a sum of 341?

(a) n = 1 (b) n = 2 (c) n = 3 (d) n = 4 (e) n = 5.

(1 mark)

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31.

If the sum of first n terms of an A.P., is , then n^th term is

(a) n (b) n + 1 (c) 2n + 1 (d) 3n + 1 (e) 4n + 1.

(2 marks)

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32.

A negatively skewed distribution curve tails off towards the lower end and for such a curve

(a) A.M > Median > Mode (b) A.M < Median < Mode

(e) A.M > Median = Mode.

(1 mark)

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33.

Which of the following is a geographical classification:

(a) Classification according to place, area and region

(b) Classification according to lapse of time

(d) Classification according to magnitude of numerical values

(e) Classification according to attributes of subjects or items and magnitude of numerical values.

(1 mark)

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34.

Which of the following is not true in data array?

(a) We can observe the distance between succeeding values in the data

(b) We can easily display the large quantity of data

(d) We can quickly notice the lowest and highest values in the data

(e) We can see whether any values appear more than once in the array.

(1 mark)

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35.

A histogram is/are not suitable for

I. Frequency distribution with continuous classes.

II. Discrete grouped frequency distribution.

III. Individual series.

IV. Chronological data.

V. Geographical data.

(a) Only (IV) above (b) Both (I) and (II) above

(e) (II), (III) and (V) above.

(1 mark)

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36.

For equal class intervals the frequency polygon can be obtained by

(a) Joining the midpoints of lower sides of adjacent rectangles of the histogram by means of straight lines

(b) Joining the midpoints of lower sides of adjacent rectangles of the histogram by means of smooth curve

(d) Joining the midpoints of upper sides of adjacent rectangles of the histogram by means of smooth curve

(e) Joining the upper classes of adjacent rectangles of the histogram by means of smooth curve.

(1 mark)

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37.

A graph of cumulative frequency distribution is called as

(a) An ogive curve (b) A histogram

(e) Relative frequency histogram.

(1 mark)

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38.

Which one of the following is a commercial average?

(a) Moving average (b) Arithmetic mean (c) Median

(d) Quartiles (e) Harmonic mean.

(1 mark)

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39.

Which of the following is not true with respect to average in general?

(a) Average gives a value which is feasible always

(b) Often average is a value which does not exist in the set of data

(d) Average can help in taking decisions

(e) Average presents a simple picture of large and complicated set or data.

(1 mark)

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40.

Which of the following is most suitable average for index number

(a) Arithmetic mean (b) Geometric mean

(1 mark)

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41.

The relation ship between A.M., G.M., H.M., is:

(a) (b)

(e)

(1 mark)

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42.

A man travel’s from his house to his office at a speed of 25 mph and back from office to his house 30 mph. What is the average speed?

(a) 27.27 mph (b) 27.05 mph (c) 27.50 mph (d) 27.20 mph (e) 27.02 mph.

(2 marks)

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43.

If the data type is nominal which of the following average is/are appropriate?

I. Mean.

II. Median.

III. Mode.

(a) Only (I) above (b) Only (II) above

(e) Both (II) and (III) above.

(1 mark)

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44.

What is the variance of first n natural numbers?

(a) (b) (c) (d) (e) .

(2 marks)

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45.

What is the probability that a card drawn at random from the pack of playing cards may be either a king or ace?

(a) 1/13 (b) 2/13 (c) 3/13 (d) 4/13 (e) 5/13.

(1 mark)

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46.

A junior analyst X completes on average 35 analyses with a standard deviation of 7, Y completes on average 120 analyses with a standard deviation of 10 and Z completes on average 55 analyses with a standard deviation 5. Which employee (junior analyst) shows less variability?

(a) Junior analyst X (b) Junior analyst Y

(e) Both Junior analyst X and Junior analyst Z.

(2 marks)

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47.

The probability of a person A winning a race is 1/6 and the probability of a person B winning the same race is 1/7. What is the probability that none of them will win?

(a) 11/42 (b) 12/21 (c) 13/42 (d) 14/21 (e) 29/42.

(2 marks)

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48.

From a well-shuffled pack of cards, four cards are drawn at random. Find the probability that all are spades.

(a) 0.3611 (b) 0.1606 (c) 0.0160 (d) 0.0261 (e) 0.0026.

(1 mark)

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49.

What is the probability of occurrence of a number greater than 4 in a throw of a die if it is known that only even number can occur?

(a) 0 (b) 1/3 (c) 2/3 (d) 1/6 (e) 1.

(1 mark)

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50.

If P(A) = 1/2, P(B) = 11/36 and P(A Ç B) = 1/6 then P() is

(a) 29/36 (b) 11/72 (c) 13/36 (d) 5/6 (e) 1/36.

(2 marks)

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51.

An analysis of monthly wages paid to the workers of two firms X and Y belonging to the same industry gives the following results

	Firm X	Firm Y
Number of workers	500	600
Average daily wage	Rs.186.00	Rs.175.00
Standard deviation of distribution of wages	9	10

The variance of the distribution of wages of all the workers in the firms X and Y taken together is

(a) 120.63 (b) 119.63 (c) 121.36 (d) 121.63 (e) 120.00.

(2 marks)

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52.

A bag contains 10 red and 15 black balls and a second bag contains 8 blue and 6 green balls. A ball is taken out from each bag. What is the probability that one ball is red and the other is blue?

(a) 2/5 (b) 5/7 (c) 8/35 (d) 4/7 (e) 1/4.

(2 marks)

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53.

There are two packs of cards; one card is drawn at random from each pack. What is the probability that both of them are red?

(a) 1 (b) 1/2 (c) 1/3 (d) 1/4 (e) 1/13.

(1 mark)

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54.

A four-digit number of distinct digits is written down at random by using digits 1, 2, 3, 4. What is the probability that the number begins with the digit 4?

(a) 1 (b) 1/2 (c) 1/3 (d) 1/4 (e) 1/5.

(1 mark)

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55.

A football match is played from 9A.M to 11 A.M. A boy arrives to see the match (not before the match starts). What is the probability that he will miss the only goal of the match, which takes place at the 21^st minute of the match?

(a) 0.126 (b) 0.935 (c) 0.736 (d) 0.825 (e) 0.901.

(1 mark)

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56.

Linear programming cannot be used in

(a) Selection of a product mix

(b) Determination of a capital budget

(d) Determination of NPV of a project

(e) Problems where constraints are linear.

(1 mark)

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57.

Which of the following statements is false?

(a) A dual of a LPP is also a LPP

(b) Every LPP has a unique dual

(d) The dual cannot be considered as inverse of the primal

(e) The directions of the equalities are reversed in the dual.

(1 mark)

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58.

A finance company offers to pay a lump sum of Rs.16,000 at the end of 6 years to investors who deposit Rs.2,000 annually for 6 years. What is the implicit interest rate?

(a) 10.00% (b) 10.58% (c) 11.00% (d) 11.42% (e) 12.00%.

(1 mark)

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59.

A linear programming problem cannot be solved by the simplex method if

I. The number of basic variables are lesser than the number of constraints.

II. The number of constraints are lesser than the number of basic variables.

III. Some of the right hand side values of the inequalities are negative.

(a) Only (I) above (b) Only (II) above

(e) Both (I) and (III) above.

(1 mark)

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60.

The feasible region in a graphical representation of a linear programming problem is/are indicated by the

I. Objective function.

II. Structural constraints.

III. Non-negativity constraints.

(a) Only (I) above (b) Only (II) above

(e) Both (II) and (III) above.

(1 mark)

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61.

A possible increase in profits in a maximization problem is/are indicated by

I. A negative value of Z_j– C_j for a slack variable.

II. A negative value of Z_j– C_j for a structural variable.

III. A positive value of Z_j– C_j for a slack variable, but you learn that more of the resource corresponding to that slack variable can be obtained.

(a) Only (I) above (b) Both (I) and (II) above

(e) All (I), (II) and (III) above.

(1 mark)

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62.

The vertices of the feasible region for a profit maximizing linear programming problem is bounded by the vertices (0,0), (0,10), (13,0) and (10,5). The profit function is Z = 7X + 12Y. The maximum value of Z is

(a) 90 (b) 98 (c) 110 (d) 121 (e) 125.

(1 mark)

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63.

Which of the following techniques/methods can be used when confronted with more than one objective function in a real life problem?

(a) Linear programming (b) Graphical method

(e) Goal programming.

(1 mark)

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64.

Which of the following is true with regard to the graphical method of solving linear programming problems?

(a) The horizontal and vertical axis does not represent the decision variables

(b) The graphical method can be applied when there are two or more than two decision variables

(d) The optimal solution to the problem always occurs at a point outside the feasible region

(e) The feasible region is not determined by the structural constraints and non-negative constraints.

(1 mark)

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65.

Which of the following describes a linear function?

(a) (b)

(e) .

(1 mark)

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66.

The primal linear programming problem formulation is

Maximize : 10X₁ + 12X₂

Subject to : 2X₁ + 3X₂ £ 60

5X₂+ 7X₂ £ 50

X₁ ³ 0, X₂ ³ 0

Which of the following would be false with regard to its dual formulation?

(a) Objective Function : Minimize 60Y₁ + 50Y₂

(b) Subject to : 2Y₁ + 5Y₂ ³ 10

(d) Subject to : Y₁, Y₂ £ 0

(e) Both (b) and (c) above.

(1 mark)

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67.

Which of the following is not amenable to mathematical manipulation?

(a) Arithmetic Mean (b) Geometric Mean

(1 mark)

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68.

For an asymmetric frequency distribution the mean and empirical mode are 10 and 7 respectively. The value of the median is

(a) 5 (b) 6 (c) 7 (d) 8 (e) 9.

(1 mark)

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69.

Which of the following is a relative measure of dispersion?

(a) Range (b) Mean absolute deviation

(1 mark)

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70.

The mean of squares of first 23 natural numbers is

(a) 4324 (b) 188 (c) 1128 (d) 104 (e) 3424.

(1 mark)

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71.

Which of the following is used as denominator when calculating standard deviation of a sample?

(a) N (b) n (c) n –1 (d) N – 1 (e) n + 1.

(1 mark)

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72.

For a class consisting of 60 students, a test on Mathematics was conducted. The marks obtained by the students are given below:

Marks obtained out of 120	Numbers of students
0 - 20	6
20 - 40	14
40 - 60	18
60 - 80	12
80 -100	8
100 -120	2

What is the median marks obtained by the students in the class?

(a) 35.5 (b) 45.8 (c) 50.6 (d) 55.4 (e) 65.4.

(2 marks)

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73.

If every item in a data set is increased by a constant K, then the arithmetic mean of the resulting data set will be equal to

(a) The mean of the original data set

(b) K + Mean of the original data set

(d) Mean of the original data set ¸ K

(e) Mean of the original data set ´ K.

(1 mark)

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74.

For which of the following, interpolation method cannot be used?

(a) Finding out the optimal solution to problem

(b) Making insertions in a series of data

(d) Finding out the internal rate of return

(e) Finding out the yield to maturity of a bond.

(1 mark)

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75.

A sum of money deposited today gets doubled in 6 years. Find the annual rate of interest.

(a) 12.00% (b) 12.23% (c) 13.18% (d) 13.64% (e) 14.00%.

(1 mark)

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76.

At K = 16% Present value of cash flow of a project is equal to Rs.64.30 and at K = 17% the present value of cash flow is Rs.29.82 at what percent the present value of cash flow is Rs.30?

(a) 16.18% (b) 16.32% (c) 16.73% (d) 16.81% (e) 16.95%.

(1 mark)

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77.

Which of the following is false?

(a) Two events A and B are said to be independent, if the occurrence of one does not affect the occurrence of the other

(b) If A and B are independent, their compliments are also independent

(c) If two events A and B are independent, the probability of their product is not equal to the product of their individual probabilities

(d) An event having only one sample point is called simple event

(e) Events are said to be mutually exclusive if the occurrence of one precludes the occurrence of others.

(1 mark)

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78.

A consignment of 20 mobiles contains 5 defectives. The mobiles are taken out one by one at random and examined. The ones examined are not put back. What is the probability that the thirteenth one examined is the last defective?

(a) 0.3699 (b) 0.0699 (c) 0.0319 (d) 0.1194 (e) 0.0496.

(2 marks)

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79.

The reciprocal of the harmonic mean of a data set is equal to

(a) Arithmetic mean

(b) Sum of all the observations

(d) Average of the reciprocals of the observations

(e) Product of the reciprocals of the observations.

(1 mark)

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80.

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

(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.

(1 mark)

< Answer >

Suggested Answers
Quantitative Methods – I (131): April 2006

Answer : (e)

Reason : A person can choose to travel by any one of the 3 routes from Hyderabad to Nanded and this can be done in 3 ways. He can choose to travel by any one of the 2 routes from Nanded to Nagapur and this can be done in 2 ways. Hence the number of different ways of traveling from Hyderabad to Nagapur via Nanded is 3 x 2 = 6.

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Answer : (d)

Reason : {P, E, R, M, U, T, A, I, O, N, S}

Since the vowels E, U, A, I, O are together, treat them as one entity. Now there are six elements and these can be arranged in 6! Ways. Also the vowels can be arranged themselves in 5! Ways. Since the two operations are independent, the total number of arrangements is 6!.5!=86400.

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Answer : (b)

Reason : Each ring contains four letters. Therefore we have for each ring 4 different ways of bringing a letter to the opening position.

Therefore the number of ways in which the 3 rings can combine = 4x4x4=64,

But of these attempts to open the lock, only one will be successful.

Hence the number of unsuccessful events =64–1=63

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Answer : (a)

Reason : Since no two girls are to sit together the arrangement of the seats is as fallows

Where B – Boys

G– Girls

Eight boys can be arranged among themselves round the table in

(8–1)! =7! Ways.

As a girl can sit, with the given restriction, between any 2 boys and as there are 8 seats for 6 girls, the girls can be arranged with respect to boys in ways.

Since the two arrangements are independent the number of arrangements required is 7!..

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Answer : (b)

Reason : i. 3 ladies from 5 and 2 men from 8, can be selected in

ii. 4 ladies from 5 and 1 men from 8, can be selected in

iii. 5 ladies from 5 can be selected in

Therefore total ways of selecting a committee of 5 with at least 3 ladies

=++.

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Answer : (c)

Reason: 2 shirts can be selected from 8 shirts in ways for giving them to the first person. Then six shirts remain. From these two can be selected in ways to give them to the second person. Then four shirts remain. From these two can be selected in ways to given them to the third person. Then two shirts remain and these can be given to the fourth person in ways. Therefore the required number of ways

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Answer : (c)

Reason :

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Answer : (b)

Reason :

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Answer : (d)

Reason :

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10.

Answer : (a)

Reason : Given x²+y²=6xy

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11.

Answer : (e)

Reason :

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12.

Answer : (b)

Reason :

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13.

Answer : (b)

Reason :

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14.

Answer : (c)

Reason :

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15.

Answer : (c)

Reason :

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16.

Answer : (e)

Reason : The logarithms of the same numbers to different bases are different.

The logarithm of 1 to any base is zero.

The logarithm of any number (not equal to 1)itself as base is unity.

Therefore (e) is the correct answer.

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17.

Answer : (b)

Reason :

Now add 1, 2 and 3 above

log a + log b + log c = k(b – c) + k(c – a) + k(a – b)

log a + log b + log c = k(b – c + c –a + a – b)

log a + log b + log c = k(0)

log a + log b + log c = 0

Þ log (abc) = 0

Þ log (abc) = log1

Þ abc = 1

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18.

Answer : (e)

Reason :

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19.

Answer : (c)

Reason :

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20.

Answer : (c)

Reason :

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21.

Answer : (a)

Reason :

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22.

Answer : (d)

Reason : given x=ct, y=ct^-1

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23.

Answer : (a)

Reason : let y=x^x

Now taking log on both sides

logy= log x^x

Now differentiating on both sides with respect to x

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24.

Answer : (e)

Reason :

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25.

Answer : (b)

Reason : let log x =t

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26.

Answer : (a)

Reason :

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27.

Answer : (c)

Reason : Given

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28.

Answer : (b)

Reason :

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29.

Answer : (b)

Reason :

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30.

Answer : (e)

Reason : a = 1 r = 4

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31.

Answer : (e)

Reason :

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32.

Answer : (b)

Reason : A positively skewed distribution curve tails off towards the higher end and for such a curve A.M>Median>Mode.

A negatively skewed distribution curve tails off towards the lower end and for such a curve A.M<Median<Mode.

A distribution curve for which mean, median, mode are equal is called symmetric curve i.e. A.M=Median=Mode.

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33.

Answer : (a)

Reason : The classification according to place, area or region comes under geographical classification. Therefore (a) is the correct answer

The classification according to lapse of time, e.g., weekly, monthly, yearly, etc., comes under chronological classification.

The classification according to attributes of the subjects or items, e.g., designation, qualification, knowledge etc., comes under qualitative classification.

The classification according to the magnitude of the numerical values, e.g., income, expenditure, weight, height etc., comes under quantitative classification.

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34.

Answer : (b)

Reason : It is cumbersome for displaying large quantities of data. We have to compress the information to use it for interpretation.

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35.

Answer : (e)

Reason : A histogram is suitable for Frequency distribution with continuous class and also for chronological data. It is not suitable for discrete grouped frequency distribution, individual series and geographical data. Bar diagrams are suitable for geographical data

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36.

Answer : (c)

Reason : For equal class intervals the frequency polygon can be obtained by joining the midpoints of upper sides of adjacent rectangles of the histogram by means of a straight line. If we join the midpoints of upper sides of adjacent rectangles of the histogram by means of a smooth curve then it called as frequency curve.

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37.

Answer : (a)

Reason : (a) A graph of cumulative frequency polygon is called as an ogive curve. Therefore (a) is the correct answer.

(b) A histogram is a series of rectangles, each proportional in width to the range of values within a class and proportional in height or the number of items falling in the class.

(c) A frequency polygon is simply a line graph that connects the midpoints of all the bars in a histogram by means of a straight line.

(d) A frequency polygon smoothed by adding classes and data points to a data set is called frequency curve.

(e) A histogram that uses the relative frequency of data points in each of the classes rather than the actual number of points is called a relative frequency histogram.

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38.

Answer : (a)

Reason : Moving average, progressive average and composite average are commercial averages. Therefore (a) is the correct answer.

Arithmetic mean and Harmonic mean are mathematical averages.

Median and quartiles are positional averages.

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39.

Answer : (a)

Reason : Average sometimes gives a value which is not feasible. The options (b), (c), (d) and (e) are correct.

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40.

Answer : (b)

Reason : Geometric mean is considered to be the most suitable average for index numbers

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41.

Answer : (a)

Reason : The relation ship between A.M., G.M., H.M., is

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42.

Answer : (a)

Reason : Since both the distances are same the average speed is calculated by means of simple harmonic mean.

Let the distance from the house to office be x miles. So from house to office the distance x miles is covered in hours.

And from office to house, the distance is covered in hours. The total distance () is covered in hours.

Average speed =

=== 27.27 mph.

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43.

Answer : (c)

Reason : If the data type is of nominal (no inherent order in categories e.g., hair color, ethnicity etc.), then the suitable average is mode.

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44.

Answer : (d)

Reason : The formula for variance

The above equation can also be written as

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45.

Answer : (b)

Reason : Let n(S) be the total sample space = n(S) = = 52

Let A be the event of being a king then n(A) = = 4

Let B be the event of being an ace, then n(B) = = 4

Therefore P(A) = = 4/52

P(B) = = 4/52

Clearly A, B are mutually exclusive

Therefore

= 4/52 + 4/52

= 8/52 = 2/13.

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46.

Answer : (b)

Reason : Coefficient of variation=

==20% for junior analyst X

Coefficient of variation =

== 8.3% for junior analyst Y

Coefficient of variation=

==9.09% for junior analyst Z

Therefore junior analyst Y shows less relative variation when compared to X and Z.

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47.

Answer : (e)

Reason : Let the event of the person A winning =E₁

and the event of the person B winning =E₂

the event of none of the persons winning =

so, P[] = 1–

(but = = ).

= 1– =.

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48.

Answer : (e)

Reason : Exhaustive number of case =

Favorable number of cases=

Probability that all the four cards drawn are spades = = 0.0026.

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49.

Answer : (b)

Reason : Let A = the event of occurrence of even number ={ 2,4,6 } =3 numbers

Let B= the event of occurrence of even number greater than 4 ={ 6 }=1 i.e., only one number.

= 1

n(A) = 3

Therefore P(B|A) = =.

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50.

Answer : (c)

Reason : P() = P[]

= 1 –

but = P(A) + P(B) – P(AB)

= 1/2 +11/36-1/6

= 23/36

therefore P() = 1 –

= 1 – 23/36

= 13/36.

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51.

Answer : (c)

Reason : The average daily wages of all the workers in the two firms x and Y taken together is given by

==Rs.180

The combined variance is given by

d₁=186-180=6 and d₂=175-180=–5

=121.36

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52.

Answer : (c)

Reason : Let A be the event of getting red ball from the first bag, then n(A)==10

Let B be the event of getting blue ball from the second bag, then n(B)==8

For A, n(S) = 10+15=25

For B, n(S) = 8+6=14

Therefore P (A)= and P (B)=

Since A and B are independent =

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53.

Answer : (d)

Reason : Let A be the event of drawing a red card from the first pack and B be the event of drawing a black card from the second pack.

Since A and B are independent, in a pack there are 26 red and 26 black cards,

So, n(A)= = 26 and n(B) = = 26

For A, n(S) = 10+15=25

For B, n(S) = 8+6=14

Now P(A)= = = and

P(B)= = = .

Since A and B are independent, required probability is

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54.

Answer : (d)

Reason : Here n(S) = total number of four distinct digits that cam be made with 1, 2, 3, 4 =

n(E)= the number of numbers of four distinct digits beginning with 4 that can be made with 1, 2, 3, 4 =

Therefore the required probability===

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55.

Answer : (d)

Reason : A football match is played from 9 A.M to 11 A.M. so, the total possible length of time equal to 120 minutes. He will miss the goal if he arrives between 9.21 A.M to 11 A.M. So, the favorable length of time is equal to 1 hour 39 minutes =99 minutes

P(E)= favorable length / total length.

Therefore the probability of his missing the goal =99/120 = 0.825

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56.

Answer : (d)

Reason : Linear programming is a method for selecting an optimum combination of factors from a series of interrelated alternatives, each subjective to limitation. These models are concerned with the problem of allocating limited resources in such a way that the defined objectives are optimized. Hence, the answer is (d)

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57.

Answer : (d)

Reason : Every LP problem whether maximization or minimization has a mirror image associated with it. The original problem is the primal and the reverse of it or the inverse of it is the dual. If the primal has an objective function of maximization then the dual has an objective function of minimization.

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58.

Answer : (d)

Reason : We know that 16,000=2,000 x FVIFA(K,6)

Where FVIFA is the future value interest factor of an annuity, and K is the implicit rate.

FVIFA(K,6)=

If we refer to the FVIFA tables,

At k= 10%, FVIFA(10,6)=7.7156

At k= 12%, FVIFA(12,6)=8.1152

So K must be greater than 10% and less than12%.

For an ascent of (8.1152–7.7156), the ascent in rate is 2. for a required ascent of (8.1152–7.7156), the ascent in rate is

the implicit rate of interest==11.42%.

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59.

Answer : (c)

Reason : A linear programming problem cannot be solved by the simplex method if we give negative values for the resources available.

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60.

Answer : (e)

Reason : The feasible region in the graphical representation of a linear programming problem can be identified only by the structural and non-negativity constraints.

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61.

Answer : (e)

Reason : A possible increase in profits in a maximization problem is indicated by

I. A negative value of Z_j– C_j for a slack variable.

II. A negative value of Z_j– C_j for a structural variable.

III. A positive value of Z_j– C_j for a slack variable, but you learn that more of the resource corresponding to that slack variable can be obtained.

Therefore (e) is the correct answer.

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62.

Answer : (c)

Reason : Z=7X+12Y (0,0), (0,10), (13,0) and (10,5).

Substitute the different values of X and Y of the vertices of the feasible region and find the maximum value of Z as follows:

Z=7X+12Y

For (0,0), Z = 0

For (0,10), Z = 7(0) +8(10) = 80

For (13,0), Z = 7(13) +8(0) = 91

For (10,5), Z = 7(10) +8(5) = 110.

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63.

Answer : (e)

Reason : Goal programming is a modification and extension of linear programming. The goal programming approach allows a simultaneous solution of a system of complex objectives other than a single objective.

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64.

Answer : (a)

Reason : The horizontal and vertical axes represent the decision variables.

(b) The graphical method can be applied when there are two decision variables

(d) The optimal solution to the problem cannot occurs 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 and non-negative constraints.

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65.

Answer : (c)

Reason : Linear function is of the form

So (c) is the correct answer.

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66.

Answer : (d)

Reason : The dual is considered as the inverse of the primal. The direction of the inequalities are reversed for resource constraints. If the primal objective function is a ‘Maximization’ function then the dual objective function is a ‘Minimization’ function and vice versa. But the direction of the inequalities remains unchanged for the non-negativity constraints. Hence (d) is false.

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67.

Answer : (e)

Reason : Mode is a positional average; it is not capable of mathematical manipulations. All others are mathematical averages.

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68.

Answer : (e)

Reason : Mode = 3 ´ Median – 2 ´ Mean

Let the median be x

\ 7 = 3x – 2 ´ 10

or 3x = 27

or x = 9

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69.

Answer : (e)

Reason : Co-efficient of variation is a relative measure of dispersion (standard deviation expressed as a percentage of the mean). Other alternatives in questions are absolute measures of dispersion.

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70.

Answer : (b)

Reason : The sum of squares of first n natural numbers is

1²+2²+3² +...+ n² =.

Here n=23 =4324

mean of squares of first 23 natural numbers is =4324 / 23=188.

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71.

Answer : (c)

Reason : While calculating sample standard deviation we use n-1 ( no. of observations in the sample minus 1) as denominator. Therefore option (C) is the correct answer

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72.

Answer : (c)

Reason : Median marks =

Position of median =

Class	Frequency	Cumulative frequency
0-20	6	6
20-40	14	20
40-60	18	38
60-80	12	50
80-100	8	58
100-120	2	60
	60

Median class is 40 – 60

N = 60

L_M = 40

F = 20

f_m = 18

W = 20

\ Median =

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73.

Answer : (b)

Reason : If every item in a data set is increased by a constant K, then the arithmetic mean of the resulting data set will be equal to the mean of the original data set plus K, i.e., K + mean of the original data set.

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74.

Answer : (a)

Reason : Interpolation cannot be used to find out the optimal solution to a problem. For finding out optimal solution to a problem, linear programming is used.

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75.

Answer : (b)

Reason : Let the sum of money deposited be X. In 6 years it gets boubled to 2X.

We get X x FVIF_{(K, 6)} = 2x

Or FVIF_{(K, 6)} = 2

If we refer to FVIF tables,

At k=12.0%, FVIF_{(12, 6)} = 1.9738

At k=14.0%, FVIF_{(14, 6)} = 2.1950

So k should be greater than 12 but less than 14

By interpolation we get

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76.

Answer : (e)

Reason : at K = 16%, present value of cash flow is equal to 33.42

at K = 17%, present value of cash flow is equal to 29.82

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77.

Answer : (c)

Reason : Options (a), (b), (d) and (e) are true.

If two events A and B are independent, the probability of their product is equal to the product of their individual probabilities.

Therefore (c) is the correct answer.

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78.

Answer : (c)

Reason : The thirteenth one will be the last defective if the first 12 mobiles examined contains 4 defectives and 8 non-defectives, and the thirteenth mobile examined is defective.

Now the probability of drawing 4 defectives and 8 non-defectives in the first 12 examinations is

Now the probability of drawing 1 defective from the remaining 1 defective and 7 non-defectives in the thirteenth examination is

Therefore the required probability is x =

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79.

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.

(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.

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80.

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.

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