Question Paper Quantitative Methods – I (131): July 2005
	· Answer all questions. · Marks are indicated against each question.

1.

The first and last term of an arithmetic mean are a and l respectively. If S is the sum of all the terms of the arithmetic mean, then the common difference of the A.P. is

(a) (b) (c)

(d) (e) .

(1 mark)

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2.

Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by 3.

(a) 156250 (b) 156500 (c) 312750 (d) 156000 (e) 156375.

(1 mark)

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3.

The mth term of an arithmetic mean is n and its nth term is m. Then its pth term will be

(a) m + n + p (b) m - n + p (c) m - n - p

(d) n - m - p (e) m + n - p.

(1 mark)

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4.

The arithmetic mean between two numbers exceeds their geometric mean by 2 and the geometric mean exceeds their harmonic mean by 8/5. The highest of the two numbers is

(a) 12 (b) 16 (c) 20 (d) 24 (e) 32.

(2 marks)

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5.

If the pth, qth and rth terms of an arithmetic progression are a, b and c respectively, then the value of the expression is equal to

(a) abc (b) pqr (c) ap + bq + cr

(d) abc( p + q + r) (e) Zero.

(1 mark)

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6.

The fifth term of a geometric progression is 81 whereas its second term is 24. The sum of the first four terms of the series is

(a) 110 (b) 120 (c) 130 (d) 140 (e) 150.

(1 mark)

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7.

The sum of three numbers of a geometric progression is 65 and the product of the same is 3375. The lowest of the three numbers is

(a) 5 (b) 10 (c) 15 (d) 20 (e) 25.

(2 marks)

< Answer >

8.

The sixth and eight terms of a harmonic progression (H.P.) are and respectively. The first five terms of the H.P are

(a) , , , and

(b) , , , and

(c) , , , and

(d) , , , and

(e) , , , and .

(1 mark)

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9.

The sum of three numbers in an arithmetic progression is 12 whereas the sum of their cubes is 288. The lowest of the three numbers is

(a) 0.5 (b) 1.0 (c) 1.5 (d) 2.0 (e) 2.5.

(2 marks)

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10.

In a geometric progression sum of n terms is 255, the last term is 128 and the common ratio is 2. The value of n is

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

(1 mark)

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11.

A man arranges to pay off a debt of Rs.3600 in 40 annual installments that form an arithmetic series. When 30 of the installments are paid he dies leaving one-third of the debt unpaid. Find the value of the first installment. (Ignore time value of money)

(a) Rs.11 (b) Rs.21 (c) Rs.31 (d) Rs.41 (e) Rs.51.

(1 mark)

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12.

Find the sum of n terms of the series: to n terms.

(a) (n/2) [n log ab - log (a/b)]

(b) (n/2) [log ab - n log (a/b)]

(c) (n/2) [n log ab + log (a/b)]

(d) (n/2) [log ab + n log (a/b)]

(e) (n/2) [log (a/b) – n log ab].

(1 mark)

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13.

In an arithmetical progression (A.P.) the first term is 4 and the last term is 24. The sum of all the terms in the A.P. is 154. What is the common difference of the A.P.?

(a) 10 (b) 6 (c) 4 (d) 2 (e) 11.

(1 mark)

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14.

Sum to n terms of the series 7, 77, 777, 7777, ….would be

(a) [7(10ⁿ⁺¹ – 10)/81] – [ 7n/9] (b) [7(10ⁿ⁺¹ – 10)/81] + [ 7n/9]

(c) [7(10ⁿ⁺¹ + 10)/81] – [ 7n/9] (d) [7(10ⁿ⁺¹ +10)/81] + [ 7n/9]

(e) [7(10ⁿ⁺¹ – 10)/81].

(2 marks)

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15.

If a = 2, b = 3, c = 6, The value of is

(a) 4 (b) 6 (c) 8 (d) 10 (e) 12.

(1 mark)

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16.

If a = 4, the value of would be

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

(1 mark)

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17.

?

(a) (b) (c)

(d) (e) .

(1 mark)

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18.

?

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

(1 mark)

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19.

?

(a) 4 (b) 3 (c) 2 (d) 1 (e) 0.

(1 mark)

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20.

?

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

(1 mark)

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21.

?

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

(1 mark)

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22.

?

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

(1 mark)

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23.

If ?

(a) 1 (b) 2 or 4 (c) 3 or 5 (d) 5 or 7 (e) 6.

(2 marks)

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24.

! =

?

(a) 153 (b) 816 (c) 252 (d) 306 (e) 4896.

(1 mark)

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25.

Everybody in a room shakes hand with everybody else. The total number of handshakes is 66. The total number of persons in the room is

(a) 10 (b) 11 (c) 12 (d) 13 (e) 14.

(2 marks)

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26.

In an examination a candidate is required to answer 6 out of 10 questions, which are divided in two groups each containing 5 questions. If the candidate is not permitted to answer more than four questions from each group, in how many ways he can make his choice?

(a) 120 (b) 200 (c) 330 (d) 460 (e) 680.

(1 mark)

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27.

If . The value of n would be

(a) 6 (b) 5 (c) 4 (d) 3 (e) 2.

(1 mark)

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28.

In the Drawing department of a switch-manufacturing multinational company 6 gentlemen and 4 ladies are working. For the proper upkeep of the old drawings the department is planning to form a team of 5 members. If the team is to include at least one lady, in how many ways the team can be formed?

(a) 60 (b) 120 (c) 180 (d) 240 (e) 246.

(2 marks)

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29.

If , the value of r would be

(a) 3 (b) 4 (c) 5 (d) 6 (e) 7.

(1 mark)

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30.

= ?

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

(1 mark)

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31.

Differentiation of with respect to x is

(a) (b) (c)

(d) (e) .

(2 marks)

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32.

Differentiation of with respect to x is

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

(2 marks)

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33.

Differentiation of with respect to x is

(a) (b) (c)

(d) (e) .

(1 mark)

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34.

Differentiation of with respect to x is

(a) (b) (c)

(d) (e) .

(1 mark)

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35.

Differentiation of with respect to x is

(a) (b) (c)

(d) (e)

(1 mark)

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36.

Differentiation of with respect to x is

(a) (b)

(c) (d) (e)

(1 mark)

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37.

Which of the following would have a value of zero?

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

(1 mark)

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38.

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

(1 mark)

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39.

Royce Electric Co., a manufacturer of gas dryers, produces two models—a standard (STD) model and a deluxe (DEL) model. Production consists of two major phases. In the first phase, stamping and painting (S & P), sheet metal is formed (stamped) into the appropriate components and painted. In the second phase, assembly and testing (A & T), the sheet metal components along with the motor and controls are assembled and tested. (Ignore any scheduling problems that might arise from the sequential nature of the operations.) Information concerning the resource requirements and availability is shown in the following table:

Quantity of Resources Required per Unit of Output

Resource	Dryer Type		Quantity of Resources Available During Period
	STD	DEL
S & P (hours)	1.0	2.0	2,000
Motors (units)	1.0	1.0	1,400
A & T (hours)	1/3	1.0	900
Profit contribution (Rs./unit)	100	125

Each dryer (STD or DEL) requires one motor. The objective is to maximize the total contribution towards profit.

The total number of dryers of DEL model to be produced is

(a) 600 (b) 650 (c) 700 (d) 750 (e) 800.

(2 marks)

< Answer >

40.

It is required to compute the dispersion 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:

Deviation of class mid point from assumed mean	Number of observations
-20	5
-10	10
0	20
10	30
20	20
30	15
Total number of observations	100

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.

(2 marks)

< Answer >

41.

The following details are available with regard to two groups of data, A and B:

Group	Number of observations	Mean	Standard deviation
A	15	20	4
B	25	16	2

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.

(2 marks)

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42.

In a textile factory there are 50 skilled workers, 125 semiskilled workers and 75 unskilled workers. It has been observed that on average a unit length of a particular fabric is woven by a skilled worker in 4 hours, by a semiskilled worker in 5 hours and by an unskilled worker in 6 hours. After three years of experience the semiskilled workers are expected to become skilled and the unskilled workers to become semiskilled. It is assumed that there will be no turnover of the workers within the next three years. What will be the change in time taken for weaving an unit length of the same fabric, after three years?

(a) No change (b) Reduction by 1.60 hour

(c) Reduction by 0.80 hour (d) Increase by 0.80 hour

(e) Increase by 1.60 hours.

(1 mark)

< Answer >

43.

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 mark)

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44.

A data set contains the following observations:

3, 8, 5, 6, 10, 7.

What is the arithmetic mean of the data set?

(a) 3.5 (b) 4.5 (c) 5.5 (d) 6.5 (e) 7.5.

(1 mark)

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45.

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 mark)

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46.

A box contains 3 white balls and 2 black balls. What is the probability that both balls drawn are black if balls are not replaced after being drawn?

(a) 1/4 (b) 1/5 (c) 2/5 (d) 3/5 (e) 1/10.

(1 mark)

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47.

A fair die is thrown twice. What is the probability of getting a 4, 5 or 6 on the first throw and a 1, 2, 3 or 4 on the second throw?

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

(1 mark)

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48.

Sixty percent of the employees of a company are college graduates out of which ten percent are in sales. Of the employees who did not graduate from college, eighty percent are in sales. What is the probability that an employee selected at random is neither in sales nor a college graduate?

(a) 0.04 (b) 0.06 (c) 0.08 (d) 0.10 (e) 0.12.

(1 mark)

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49.

A and B are mutually exclusive and collectively exhaustive events. Both A and B are dependent on event C.

P (A and C) = 0.48, P (B and C ) = 0.32.

What is the probability of event B happening if event C happens?

(a) 0.32 (b) 0.40 (c) 0.48 (d) 0.60 (e) 0.80.

(1 mark)

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50.

A die is loaded so that probability of face x is proportional to x. The probability of an even number occurring when the die is rolled would be

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

(1 mark)

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51.

If the probability that student A will pass a certain computer science exam is 0.7, the probability that student B will pass the same exam is 0.6, and the probability that both student A and B will pass the exam is 0.5, what is the probability that at least one of these two students will pass the examination?

(a) 0.2 (b) 0.5 (c) 0.6 (d) 0.7 (e) 0.8.

(1 mark)

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52.

Which of the following is influenced the least by the occurrence of extreme values in the samples?

(a) Arithmetic mean (b) Median (c) Harmonic mean

(d) Geometric mean (e) Weighted arithmetic mean.

(1 mark)

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53.

Which of the following measures represents the scatter of the values in a data set?

(a) Arithmetic mean (b) Geometric mean (c) Harmonic mean

(d) Median (e) Standard deviation.

(1 mark)

< Answer >

54.

If a constant quantity is subtracted from every observation in a data set then the range of the resulting set of values will be equal to the

(a) Range of the original data set plus the constant quantity

(b) Range of the original data set minus the constant quantity

(c) Range of the original data set

(d) Range of the original data set multiplied by the constant quantity

(e) Range of the original data set divided by the constant quantity.

(1 mark)

< Answer >

55.

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.

(1 mark)

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56.

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.

(1 mark)

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57.

Which of the following is true?

(a) The tallest rectangle in a histogram represents the median class of the distribution

(b) In a symmetrical distribution the mean, median and mode are unequal

(c) The modes of two sets of data can be combined mathematically

(d) The median can be determined graphically

(e) The mode cannot be determined graphically.

(1 mark)

< Answer >

58.

Which of the following measures is based only on two observations in a data set?

(a) Arithmetic mean (b) Harmonic mean (c) Range

(d) Mean absolute deviation (e) Standard deviation.

(1 mark)

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59.

Which of the following measures cannot be combined mathematically?

(a) Standard deviation (b) Arithmetic mean (c) Geometric mean

(d) Harmonic mean (e) Median.

(1 mark)

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60.

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.

(1 mark)

< Answer >

61.

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.

(1 mark)

< Answer >

62.

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.

(1 mark)

< Answer >

63.

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) Always equal to 1.

(1 mark)

< Answer >

64.

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)

(d) P(A/B) = 0 and P(B/A) = 0 (e) 0 < P(A and B) < 1.

(1 mark)

< Answer >

65.

The probability of an outcome of an experiment is ‘p’ (0 < p < 1). If the experiment is repeated ‘n’ times then the probability of getting the same outcome every time is

(a) Equal to p (b) Less than p

(c) More than p (d) Zero (e) Approximately equal to 1.

(1 mark)

< Answer >

66.

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 >

67.

Bayes’ theorem helps the statistician to calculate

(a) Subjective probability (b) Classical probability (c) Revised probability

(d) Central tendency (e) Dispersion.

(1 mark)

< Answer >

68.

The totality of all possible outcomes of an experiment is called

(a) Sample space (b) Population (c) Event

(d) Data (e) Sample.

(1 mark)

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69.

If a, b and c are real numbers then, which of the following is the identity property of addition?

(a) a + b = b + a (b) (a + b) + c = a+ (b + c)

(c) a + 0 = 0 + a = a (d) a + (–a) = (–a) + a = 0 (e) a (b + c) = ab + ac.

(1 mark)

< Answer >

70.

If a is a fixed positive real number on the number line and x denotes any real number on the number line then which of the following represents the set of all real numbers x such that a < x?

(a) (a, +) (b) (–, a) (c) [a, +) (d) (–a, a) (e) (–, +).

(1 mark)

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71.

According to the inverse property of addition

(a) For every real number there exists another real number such that the sum of the two real numbers is equal to 1

(b) For every real number there exists another number such that the sum of the two real numbers is equal to 0

(c) The addition of zero to any real number is equal to that real number

(d) For every real number there exists another real number such that the product of the two numbers is equal to 1

(e) The product of any real number with 1 is equal to that real number.

(1 mark)

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72.

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.

(1 mark)

< Answer >

73.

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 >

74.

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)

< Answer >

75.

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 >

76.

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.

(1 mark)

< Answer >

77.

The logarithm of a number

(a) Is always expressed with respect to base 10

(b) Is always expressed with respect to base 1

(c) Is always expressed with respect to base e

(d) Is equal to the base which must be raised to a given exponent in order to get the number

(e) Is equal to the exponent to which a given base must be raised in order to get the number.

(1 mark)

< Answer >

78.

A quadratic equation is of the form ax² + bx + c = 0, where a, b and c are constants. If c is equal to zero, then the roots of the equation are

(a) Only positive values

(b) Only negative values

(c) All equal to zero

(d) or

(e) 0 or .

(1 mark)

< Answer >

79.

The set of whole numbers includes the set of

(a) Natural numbers (b) Rational numbers (c) Negative numbers

(d) Complex numbers (e) Irrational numbers.

(1 mark)

< Answer >

80.

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 >

81.

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.

(1 mark)

< Answer >

82.

The derivative of a function f(x) at x = a is

(a) Always equal to f(a)

(b) Always equal to a

(c) The rate of change in the value of the function at x = a

(d) The limit of f(x) as x approaches a

(e) The ratio of f(a) to (a).

(1 mark)

< Answer >

83

A function f(x) is said to be monotonically increasing if

(a) The first derivative of f(x) is a constant for all values of x

(b) The first derivative of f(x) is negative for all values of x

(c) The first derivative of f(x) is positive for all values of x

(d) The first derivative of f(x) is zero for all values of x

(e) The first derivative of f(x) is equal to 1 for all values of x.

(1 mark)

< Answer >

84.

For a function f(x) the first derivative is positive and the second derivative is negative at x = a. This means that

(a) f(x) is minimum at x = a

(b) f(x) is maximum at x = a

(c) f(x) is increasing at an increasing rate at x = a

(d) f(x) is increasing at a decreasing rate at x = a

(e) f(x) is decreasing at an increasing rate at x = a.

(1 mark)

< Answer >

85.

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.

(1 mark)

< Answer >

86.

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 >

87.

In the graphical method of solving linear programming problems the feasible region is the set of all points

(a) Which do not satisfy any of the constraints

(b) Which satisfy exactly one of the constraints

(c) Which satisfy all the constraints

(d) At which the objective function has the same value

(e) At which the objective function is equal to zero.

(1 mark)

< Answer >

88.

In the graphical method of solving linear programming problems if there is a unique optimal solution, then the optimal solution

(a) Is always found at the center of the feasible region

(b) Is always at the origin

(c) Lies outside the feasible region

(d) Is located at one of the corner points of the feasible region

(e) Always lies on one of the two axes.

(1 mark)

< Answer >

Suggested Answers
Quantitative Methods – I (131): July 2005

1.

Answer : (a)

Reason : Given that S = (n/2)(a + l) or = n …….(i)

l = a + (n – 1)d so that d = ……..(ii)

Now putting the value of n from equation (i) in equation (ii), we get

.

< TOP >

2.

Answer : (e)

Reason : Clearly the numbers between 250 and 1000 which are exactly divisible by 3 are 252, 255, 258, …, 999.

\T_n = 999 = a + (n –1) d = 252 + ( n – 1) 3

Solving the above we get n = 250.

\ S = (n/2) [a + l] = (250/2) [252 + 999] = 156375.

< TOP >

3.

Answer : (e)

Reason : T_m = a + (m – 1) d = n

And T_n = a + (n – 1)d = m.

Solving both equations we get d = -1 and a = m + n – 1

Now T_p = a + (p –1) d = (m + n –1) + (p –1)(-1) = m + n – p.

< TOP >

4.

Answer : (b)

Reason : Let the numbers be a, b. Let the A.M be A, GM be G and HM be H

As A.M. exceeds GM by 2. So, A – G = 2; Or, (A –2)² = G² ……..(i)

As GM exceeds HM by 8/5. So, G – H = 8/5 Or, A – 2 – H = 8/5 Or, A – H =

G² = AH = A(A – ) Putting this value of G² in equation (i) we get

(A –2)² = A(A – ) Or, – 4A + 4 = –A

Or, 2A = 20 Or, A = 10. Or, (a+b) = 20

So, G = 8 Or, = 8 i.e. ab = 64

So, a + = 20

Or, a² –20a + 64 = 0.

Or, a² –16a – 4a+ 64 = 0

Or, a(a – 16) – 4(a – 16) = 0

Or, (a – 4)(a – 16) = 0

So the vale of the numbers would be (4, 16) or (16, 4)

So the highest value is 16.

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5.

Answer : (e)

Reason : Let A be the first term and D the common difference of A.P.

T_p = a = A + (p – 1) D = (A – D) + pD …… (i)

T_q = b = A + (q – 1) D = (A – D) + qD ……. (ii)

T_r = c = A + (r – 1) D = (A – D) + rD ……. (iii)

Multiplying (q - r), (r – p) and (p – q) with the equations (i), (ii) and (iii) respectively. We have,

a(q - r) + b(r – p) + c (p – q) = (A – D) [q – r + r – p + p – q] + D[p(q - r) + q(r – p) + r (p – q)] = 0.

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6.

Answer : (c)

Reason : ar⁴ = 81

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7.

Answer : (a)

Reason : Let the number be

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8.

Answer : (a)

Reason : The 6^th and 8^th term of the HP are and respectively. Therefore, the 6^th and 8^th term of the corresponding AP are 26 and 34 respectively.

Let the first term in the AP be a and the common difference be d.

So, the sixth term = a + 5d = 26 ………(1)

and the eight term = a + 7d = 34 ………(2)

Subtracting the equation (1) from the equation (2).

a + 7d = 34

a + 5d = 26

- – –

2d = 8 or, d = 4

Putting the value of d = 4 in the equation (1):

a + 5 x 4 = 26

or a = 26 – 20 = 6

\The AP is: 6, 10, 14, 18, 22.

\ The corresponding HP is: , , , and

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9.

Answer : (d)

Reason : Let the numbers be (m-d), m, (m+d)

So, (m-d) + m + (m+d) = 3m = 12

Or, m = 4

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10.

Answer : (c)

Reason : Let the first term be a

So, the nth term is a x 2^n-1 = 128 Or, a = 128/2^n-1

The sum is

Putting the value of a we get

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11.

Answer : (e)

Reason : Given that S₄₀ = 3600, and S₃₀ = 3600 – (3600/3) = 2400,

Because after paying 30 installments one third of the debt is still left. The above two equations give,

2a + 39d = 180 and 2a + 29d =160.

Solving them we get d = 2 and a = 51.

\The value of the first installment is Rs. 51.

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12.

Answer : (d)

Reason : We find that the given series is in A.P. with first term as log a and with a common difference of log (a/b)

Now sum of the n terms of the series is

S_n = (n/2) [2 log a + (n – 1) log (a/b)]

= (n/2) [log a² + n log (a/b) – log (a/b)]

= (n/2) [ log ab + n log (a/b)]

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13.

Answer : (d)

Reason : Given: t₁ = a = 4

Last term, t_n = 24

Sum of the terms, S_n = 154

In an A.P. sum of ‘n’ terms starting from the first term,

S_n =

\ From above

154 = = 14 n

or n = = 11

t_n = a + (n – 1) d

\ t₁₁ = 4 + (11 – 1) d = 24

or 4 + 10 d = 24

or d = = 2

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14.

Answer : (a)

Reason : S_n = 7 + 77 + 777 + … to n terms

= 7 (1 + 11 + 111 + … to n terms)

= 7/9 ( 9 + 99 + 999 + … to n terms)

= 7/9 [(10-1) + (10² – 1) + (10³ – 1) + … to n terms]

= 7/9 [(10 + 10² + 10³ + … to n terms) – ( 1 + 1 + 1 + … to n terms)]

= 7/9 {[10 (10^n-1)/(10-1)]- n}

= [7(10ⁿ⁺¹ – 10)/81] – [ 7n/9].

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15.

Answer : (c)

Reason : If a =2, b=3, c=6, The value of is

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16

Answer : (e)

Reason :

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17.

Answer : (d)

Reason :

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18.

Answer : (b)

Reason :

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19.

Answer : (e)

Reason :

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20.

Answer : (b)

Reason : The Given expression,

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21.

Answer : (b)

Reason :

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22.

Answer : (d)

Reason :

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23.

Answer : (b)

Reason : Given that,

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24.

Answer : (b)

Reason :

Or

Or,

Or,

Þ n – 10 = 8

\ n = 10 + 8 = 18

\ = = 816

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25.

Answer : (c)

Reason : Let there be n number of people. Number of handshakes would be . In this case

Or,

Or,

Or,

Or, (n – 12)(n + 11) = 0

Or, n = 12.

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26.

Answer : (b)

Reason : The candidate can take three strategies:

4 questions from first group and 2 questions from second group: The number of ways he can do this is .

3 questions from first group and 3 questions from second group:

3 questions from first group and 3 questions from second group: .

So, total number ways by which the candidate can make his choice is (50+100+50) or 200.

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27.

Answer : (d)

Reason :

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28.

Answer : (e)

Reason : As the desired number of lady members in the committee would be at least one – i.e. there would be four types of groupings –

(i) 1 lady member and 4 gentlemen : This could be done in ways

(ii) 2 lady member and 3 gentlemen : ways.

(iii) 3 lady member and 2 gentlemen: ways.

(iv) 4 lady member and 1 gentleman:

So, total (60 + 120 + 60 + 6) or 246 ways the team can be formed in the give conditions.

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29.

Answer : (c)

Reason :

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30.

Answer : (b)

Reason :

=

=

Because as , and in the numerator tend to zero, and in the denominator tends to zero.

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31.

Answer : (b)

Reason :

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32.

Answer : (a)

Reason :

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33.

Answer : (d)

Reason :

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34.

Answer : (b)

Reason :

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35.

Answer : (b)

Reason :

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36.

Answer : (e)

Reason :

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37.

Answer : (c)

Reason : . , are rational number. is irrational number so is zero.

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38.

Answer : (e)

Reason :

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39.

Answer : (a)

Reason : If the program is formulated we get it as follows:

Objective Function:

Constraints: So, at the most ……………….(i)

So, at the most ……………….(ii)

So, at the most ………………….(iii)

Plotting the three constraints on a graph. We get the feasible region as OA BCD. Then we find the value of the objective function at the corners of the feasible region.

Points	(X1, X2)	Z = 100X1 + 125X2
O	(0, 0)	0
A	(0, 900)	Rs. 1,12,500
B	(600, 700)	Rs. 1,47,500
C	(800, 600)	Rs. 1,55,000
D	(1400, 0)	Rs.1,40,000)

Since the objective function has the maximum value for X₁ = 800 and X₂ = 600. This is the optimum point.

Hence total number of dryers of DEL model to be produced is 600.

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40.

Answer : (a)

Reason : The variance of the population using assumed mean can be calculated as:

s² =

Deviations	d =	Frequency	d^2	fd^2	fd
–20	–2	5	4	20	–10
–10	–1	10	1	10	–10
0	0	20	0	0	0
10	1	30	1	30	30
20	2	20	4	80	40
30	3	15	9	135	45
		100		275	95

Variance of the population =

=

= 184.75.

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41.

Answer : (b)

Reason : N_A = 15 m_A = 20 s_A = 4

N_B = 25 m_B = 16 s_B = 2

m = = = = 17.5.

d_A = m_A – m = 20 – 17.5 = 2.5

d_B = m_B – m = 16 – 17.5 = –1.5

s₁₂ =

=

= = 3.50.

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42.

Answer : (c)

Reason : Average time taken now = hours

Average time taken after three years = hours

Change in time taken = 5.1 – 4.3 = 0.80 hours reduction.

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43.

Answer : (b)

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.)

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44.

Answer : (d)

Reason : Arithmetic mean = =

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45.

Answer : (c)

Reason : Median of the series (arranged in ascending order): 4, 6, 7, 8, 9, 10

Median position = = 3.5

\ Median = = 7.5.

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46.

Answer : (e)

Reason : Probability that the first ball drawn is black = 2/(3+2) = 2/5

Probability that the second ball drawn is black = 1/(3+1) = ¼

Probability that both the balls drawn are black = P{E₁ E₂ } = P{ E₁} P{ E₂I E₁ } = 2/5 ´ 1/4 = 1/10.

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47.

Answer : (b)

Reason : Let E₁ be the event 4, 5 or 6 on the first toss, and let E₂ be the event 1, 2, 3 or 4 on the second toss. Each of the six ways a die can fall on the first to can be associated with each of the six ways in which it can fall on the second toss, a total of 36 ways, all equally likely event. Each of the three ways in which E₁ can occur can be associated with each of the four ways in which E₂ can occur, to give 12 ways. Thus P{ E₁ E₂ } = 12/36 =1/3.

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48.

Answer : (c)

Reason : Let A be the event that an employee is a college graduate and B be the event that an employee is in sales. Hence, P(A) = 0.60, P(B/A) = 0.10 and = 0.80.

=

= 0.60 ´ 0.10 + (1– 0.60) ´ 0.80 = 0.06 + 0.32 = 0.38

Now, the probability that a employee selected at random is neither in sales nor a college graduate is given by: = 1 – P(A or B) = 1 – {P(A) + P(B) – P(A and B)} = 1 – {0.60 + 0.38 – 0.60 ´ 0.10} = 1 – 0.92 = 0.08.

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49.

Answer : (b)

Reason : P(A and C) = 0.48

P (B and C) = 0.32

\ P(C) = P(A and C) + P(B and C) = 0.80

P (B and C) = P(C). P(B/C)

Or, P(B/C) =

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50.

Answer : (d)

Reason : There are six faces in a die and they are named as 1, 2, 3, 4, 5 and 6. So, the corresponding probability would be 1k, 2k, 3k, 4k, 5k and 6k respectively. Since one of the face must appear the sum of all these probabilities should be one i.e. k + 2k + 3k + 4k + 5k + 6k = 21k = 1 or k =

The probability of even number is 2k+4k+6k = 12k = = .

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51.

Answer : (e)

Reason : P(At least one student will pass) = P(A or B). Clearly,

P(A or B) = P(A) + P(B) – P(A and B)

= 0.7 + 0.6 – 0.5 = 0.8.

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52.

Answer : (b)

Reason : Median is least influenced by the occurrence of extreme values in the samples.

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53.

Answer : (e)

Reason : The standard deviation represents the scatter of the values in a data set.

Arithmetic mean, geometric mean, harmonic mean and median are measures of central tendency.

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54.

Answer : (c)

Reason : If a constant is subtracted from every observation in a data set then the range of the resulting set of values will be equal to the range of the original data set.

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55.

Answer : (d)

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.

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56.

Answer : (d)

Reason : Coefficient of variation is a relative measure of dispersion. Range, quartile deviation, standard deviation and mode are absolute measures of dispersion. When every observation in the data set is divided by a constant both the standard deviation and the mean of the resulting data set will divided by the constant. Since coefficient of variation is the ratio standard deviation to the mean, the coefficient of variation of the resulting data set will be same as the original data set. Since the other measures are absolute measures, they will permanently change due to the modification.

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57.

Answer : (d)

Reason : The tallest rectangle in a histogram represents the modal class. In a symmetrical distribution the mean, median and mode are equal. The modes of two sets of data cannot be combined mathematically. The mode can be determined graphically. Hence (a), (b), (c) and (e) are false. (d) is true because median can be determined graphically.

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58.

Answer : (c)

Reason : Range (= Highest value – Lowest value) is based only on two observations in a data set.

Arithmetic mean, harmonic mean, mean absolute deviation and standard deviation are based on all the observations.

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59.

Answer : (e)

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

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60.

Answer : (c)

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.

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61.

Answer : (c)

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.

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62.

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.

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63.

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.

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.

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.

For two dependent events A and B, the joint probability of the events A and B is not always equal to 1.

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64.

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,

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65.

Answer : (b)

Reason : Probability of the outcome = p

The experiment is repeated n times.

The probability of getting the same outcome every time = pⁿ

Since 0 < p < 1, pⁿ is less than p. Hence the answer is (b).

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66.

Answer : (d)

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.

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67.

Answer : (c)

Reason : Bayes’ theorem helps the statistician to calculate posterior (or revised) probability.

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68.

Answer : (a)

Reason : The totality of all possible outcomes of an experiment is called sample space.

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69.

Answer : (c)

Reason : The chosen option (c) is the identity property of addition. The options (a) and (b) represent the commutative and associative properties of addition respectively. While the option (d) stands for the inverse property of addition.

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70.

Answer : (a)

Reason : (a, +¥) contains all values of ‘x’ such that a < x.

(–¥, a) contains all values of ‘x’ such that x < a.

[a, +¥) contains all values of ‘x’ such that x ³ a.

(–a, a) contains all values of ‘x’ such that –a < x < a.

Hence the answer is (a).

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71.

Answer : (b)

Reason : a. This is not a property of the real numbers

b. According to the inverse property of addition for every real number there exists another number such that the sum of the two real numbers is equal to 0.

c. This is the identity property of addition.

d. This is the inverse property of multiplication.

e. This is the identity property of multiplication.

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72.

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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73.

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.

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74.

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.

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75.

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.

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76.

Answer : (b)

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.

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77.

Answer : (b)

Reason : (b) If log_ab = k then k is the exponent to which a must be raised in order to get b. Therefore log_ab = k implies that b = a^k .

(a), (c), (d) and (e) are incorrect interpretations of the logarithmic expression.

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78.

Answer : (e)

Reason : ax² + bx + c = 0 x =

\ For c = 0 = 0 or –

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79.

Answer : (a)

Reason : a. The set of whole numbers includes the set of natural numbers.

b. The set of whole numbers does not include the set of rational numbers.

c. The set of whole numbers does not include the set of negative numbers.

d. The set of whole numbers does not include the set of complex numbers.

e. The set of whole numbers does not include the set of irrational numbers.

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80.

Answer : (c)

Reason : The reciprocal of the terms in a harmonic progression are in arithmetic progression. And vice versa. Therefore the reciprocals of a H.P. cannot be in H.P. There is no such connection between the H.P. and the G.P. If the terms in the corresponding A.P. are in the increasing order then the terms of the H.P. will be in the decreasing order and vice versa. Hence there is no reason why terms of a H.P. will always be in increasing or decreasing order.

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81.

Answer : (d)

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.

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82.

Answer : (c)

Reason : a. The derivative of a function f(x) at x = a may not be equal to f(a).

b. The derivative of a function f(x) at x = a may not be equal to a.

c. The derivative of f(x) at x = a is the rate of change in the value of f(x) at x = a.

d. The derivative of a function f(x) at x = a is not the limit of f(x) as x approaches a.

e. The derivative of a function f(x) at x = a is not the ratio of f(a) to a.

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83.

Answer : (c)

Reason : a. If the first derivative of f(x) is a constant for all values of x then it indicates that the function changes at a constant rate.

b. A function f(x) is said to be monotonically decreasing if the first derivative of f(x) is negative for all values of x.

c. A function f(x) is said to be monotonically increasing if the first derivative of f(x) is positive for all values of x.

d. If the first derivative of f(x) is zero for all values of x then it indicates that the function has a constant value.

e. This is not the condition for a monotonically increasing function.

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84.

Answer : (d)

Reason : For a function f(x) the first derivative is positive and the second derivative is negative at x = a. This means that f(x) is increasing at a decreasing rate at x = a. For a function f(x) to be minimum or maximum the first derivative should be equal to zero.

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85.

Answer : (e)

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.

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86.

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.

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87.

Answer : (c)

Reason : The feasible region is the set off all points which satisfy all the constraints in the LPP.

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88.

Answer : (d)

Reason : In the graphical method of solving linear programming problems if there is a unique optimal solution, then the optimal solution is located at one of the corner points of the feasible region.

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