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

1.

If all the terms of an arithmetic progression are multiplied by a constant quantity the resulting terms will always form

(a) A geometric progression

(b) A harmonic progression

(c) An arithmetic progression

(d) Either a geometric progression or a harmonic progression

(e) Either a geometric progression or an arithmetic progression.

(1 mark)

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2.

Which of the following is not a property of the arithmetic mean?

(a) The sum of the squared deviations of the items from the arithmetic mean is less than the sum of the squared deviations of the items from any other value

(b) If all the items in the data set are reduced by the same quantity then the arithmetic mean of the resulting set of values is same as that of the original data set

(c) If all the items in the data set are replaced by their arithmetic mean then their sum is equal to the sum of all the items in the original data set

(d) The arithmetic means of two or more sets of data can be algebraically combined to find out the arithmetic mean of all the sets of data taken together

(e) It is most suitable for further algebric treatments.

(1 mark)

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3.

In an arithmetical progression (A.P) j times the j^th term is equal to k times the k^th term. The first term is a and common difference is d. What is the (j + k)^th term of the A.P.?

(a) 0 (b) (j + k) d (c) (j + k –1) d (d) a (e) d.

(1 mark)

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4.

The sums of first n terms of three arithmetical progressions are . The first term of each of these A.Ps is unity and the common differences are 1, 2 and 3 respectively. Which of the following equations is true?

(a) (b) (c)

(d) (e)

(1 mark)

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5.

P and Q are the arithmetic mean and geometric mean respectively between the numbers M and N.

Which of the following is/are correct?

I. M = II. N =

III. M = P – IV. N = P +

V. M = P + VI. N = P –

VII. M = Q + VIII N = Q –

(a) Only (I) above

(b) Only (II) above

(c) Both (III) and (IV) above

(d) Both (V) and (VI) above

(e) Both (VII) and (VIII) above.

(1 mark)

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6.

The fifth term of an A.P. is 1 whereas its thirty-first term is –77. Find the sum of its first fifteen terms.

(a) 0 (b) -84.4 (c) -120 (d) -142.5 (e) -165.

(1 mark)

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7.

The sum of four numbers of a G.P is 85 and the product of the same is 4096. The highest of the three numbers is

(a) 56 (b) 64 (c) 81 (d) 128 (e) 144.

(1 mark)

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8.

The fifth term of a G.P 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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9.

The p^th term of an H.P is qr and q^th term is rp. The rth term of the H.P. is

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

(1 mark)

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10.

Find the number of terms in the series the sum of which is 300?

(a) 18 or 24 (b) 25 or 36 (c) 36 or 48 (d) 35 or 45 (e) 24 or 36.

(1 mark)

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11.

The ratio of the sum of first three terms of a G.P and the sum of the first six terms of the same G.P is 125:152. The common ratio of the series is

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

(1 mark)

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12.

The harmonic mean of two numbers is 4. The arithmetic and geometric mean of the same numbers are A, G respectively. The relationship between A and G are . One of the two numbers is

(a) 3 (b) 5 (c) 7 (d) 9 (e) 11.

(1 mark)

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13.

The sum of three numbers in an A.P 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.

(1 mark)

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14.

An arithmetic progression (AP) consists of the following terms :

9, 6, 3, 0, … , -75

How many terms are there in the AP?

(a) 10 (b) 19 (c) 29 (d) 35 (e) 39.

(1 mark)

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15.

Three numbers are in arithmetical progression. The sum of the three numbers is 18 and the sum of their squares is 140.

Which of the following statements are correct?

I. The first number is 6 or 10.

II. The first number is 2 or 10.

III. The first number is 6.

IV. The second number is 6.

V. The second number is 2 or 6.

VI. The second number is 6 or 10.

VII. The third number is 6 or 2.

VIII. The third number is 6.

IX. The third number is 10 or 2.

(a) Both (I) and (VI) above

(b) Both (V) and (VII) above

(c) Both (II) and (VIII) above

(d) Both (III) and (IX) above

(e) (II), (IV) and (IX) above.

(2 marks)

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16.

The A.M. and H.M. between two numbers are 27 and 12 respectively. The G.M of the two numbers would be

(a) 39 (b) 15 (c) 18 (d) 36 (e) 324.

(1 mark)

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17.

If a, m and n are integers.

Then a^m × aⁿ =

(a) a^{m + n} (b) a^{m – n} (c) a^{m × n} (d) a^{m / n} (e) (m × n)^a

(1 mark)

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18.

, and

The value of xyz is

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

(1 mark)

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19.

(a) 216 (b) 625 (c) 720 (d) 890 (e) 1184.

(1 mark)

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20.

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

(1 mark)

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21.

(a) (b)

(c) (d)

(e) .

(1 mark)

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22.

(a)

(b)

(c)

(d)

(e) .

(2 marks)

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23.

If

(a) 1 (b) ¾ (c) 5/4 (d) 7/4 (e) 9/4.

(1 mark)

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24.

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

(1 mark)

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25.

The expression

is equal to

(a) (b) (c)

(d) (e)

(1 mark)

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26.

(a)

(b)

(c)

(d)

(e) .

(1 mark)

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27.

If . The value of is

(a) 165 (b) 190 (c) 220 (d) 351 (e) 2925.

(1 mark)

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28.

If . The value of n would be

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

(1 mark)

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29.

How many different words can be formed by rearranging the letters of the word GANESHPURI when the vowels always occupy the even places?

(a) 5!6! (b) 4!6! (c) (d) (e) .

(1 mark)

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30.

There are 5 professors and 10 students out of whom a committee of 2 professors and 3 students is to be formed. When a particular student must be included in this committee, how many ways can such committee be formed?

(a) 220 (b) 330 (c) 360 (d) 420 (e) 480.

(1 mark)

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31.

A guard of 12 persons is to be formed from a group of 16 soldiers in all possible ways. How many times two particular soldiers are together on guard?

(a) 210 (b) 460 (c) 680 (d) 1001 (e) 1240.

(1 mark)

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32.

Mr. Khan has 7 relatives, 4 of them are ladies and 3 gentlemen, his wife has 7 relatives, 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of three ladies and three gentlemen, so that there are 3 of Mr. Khan’s relatives and 3 of his wife’s relatives?

(a) 16 (b) 144 (c) 324 (d) 468 (e) 485.

(1 mark)

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33.

(a)

(b)

(c)

(d)

(e)

(2 marks)

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34.

=

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

(1 mark)

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35.

If the derivative of a function f(x) is zero for all values of x, then it implies that

(a) f(x) has a constant value for all values of x

(b) f(x) increases at a constant rate as x increases

(c) f(x) decreases at a constant rate as x increases

(d) f(x) increases at a variable rate as x increases

(e) f(x) decreases at a variable rate as x increases.

(1 mark)

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36.

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)

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37.

A function is a rule or correspondence which always associates to

(a) Each number x in a set A a unique number f(x) in a set B

(b) All the numbers in a set A the same number in a set B

(c) All the numbers in a set A a fixed positive number

(d) All the numbers in a set A a fixed negative number

(e) Each number x in a set A two numbers in a set B.

(1 mark)

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38.

For any function f(x) the limit of f(x) as x approaches a value a is

(a) Always equal to a (b) Always a value greater than a

(c) Always a value less than a (d) Always less than f(a)

(e) A value L, such that f(x) approaches L as x approaches a.

(1 mark)

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39.

(a) (b) (c) 2 (d) 4 (e) .

(1 mark)

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40.

(a) ln 60 (b) (c) (d) (e) ln.

(1 mark)

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41.

Which of the following is/are correct?

(a)

(b)

(c)

(d)

(e) .

(2 marks)

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42.

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)

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43.

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)

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44.

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 STD model to be produced is

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

(2 marks)

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45.

Consider the following distribution:

X:	– 4	– 3	– 2	– 1	0	1	2	3	4
f:	2	4	5	7	10	7	5	4	2

Which of the following statement(s) is/are the property(ies) of the above distribution?

(a) The mean of the distribution is zero

(b) The distribution is symmetric

(c) The distribution is unimodal

(d) The median of the distribution is zero

(e) All (a), (b), (c) and (d) above.

(1 mark)

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46.

In a group of 30 girls, the median score in quantitative methods is 80 while for a group of 20 boys, the same is 70. Which of the following related to the median score M_e for combined group of 50 candidates is correct?

(a) M_e = 75

(b) M_e = 76

(c) M_e = 150

(d) M_e = 78

(e) As the 50 scores are not given, nothing can be concluded.

(1 mark)

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47.

The mean salary in a certain plant was Rs.1,500 and the standard deviation was Rs.400. A year later each employee got increment by Rs.100 while in the second year, it has been hiked by 20 percent. The mean and standard deviation of the salaries will be:

(a) Rs.1,920 and Rs.480 respectively

(b) Rs.1,920 and Rs.400 respectively

(c) Rs.1,600 and Rs.480 respectively

(d) Rs.1,920 and Rs.416 respectively

(e) Rs.1,900 and Rs.416 respectively.

(1 mark)

< Answer >

48.

The data below represents the amount of grams of carbohydrates in a serving of breakfast cereal:

11, 15, 23, 29, 19, 22, 21, 20, 15, 25, 17.

The sample variance of the carbohydrate amounts is ________ (grams squared).

(a) 5.10 (b) 23.65 (c) 26.02 (d) 31.54 (e) 32.50.

(2 marks)

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49.

Which of the following is true, if there is no dispersion in a data set?

(a) All the mathematical and positional averages are equal

(b) All the mathematical averages are equal but the positional averages are not equal

(c) All the positional averages are equal but the mathematical averages are not equal

(d) All the mathematical averages are equal to zero

(e) All the positional averages are equal to zero.

(1 mark)

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50.

The standard deviation of a data set

(a) Is expressed in the same unit as the observations in the data set

(b) Is expressed in the square of the unit of the observations in the data set

(c) Is expressed in the square root of the unit of the observations in the data set

(d) Is expressed in a different unit from the unit in which the observations in the data set are expressed

(e) Is always expressed as a percentage of the mean of the data set.

(1 mark)

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51.

Because a new medical procedure has been shown to be effective in early detection of an illness, a medical screening of the population is proposed. The probability that the test correctly identifies someone with the illness as positive is 0.99 and the probability that the test correctly identifies some one without the illness as negative is 0.95. The incidence of illness in the general population is 0.0001. You take the test and the result is positive. What is the probability that you have the illness? (Round off your answer upto the third decimal)

(a) 0.002 (b) 0.005 (c) 0.995 (d) 0.998 (e) 0.949.

(1 mark)

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52.

Two computers A and B are to be sold. A salesman who is assigned the job of selling these has the chances of 60 percent and 40 percent respectively to get success. The two computers may be sold independently given that he is able to sell at least one computer. The salesman has been able to sell atleast one computer. What is the probability that computer A has been sold? (Round off your answer up to the second decimal)

(a) 0.40 (b) 0.64 (c) 0.79 (d) 0.90 (e) 0.24.

(1 mark)

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53.

Events S and T are independent with P(S) < P(T), P(S and T) = 0.24 and P(S/T) + P(T/S) =1, then P(S) is

(a) 0.04 (b) 0.20 (c) 0.24 (d) 0.40 (e) 0.60.

(1 mark)

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54.

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)

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55.

Events A and B are dependent. The joint probability of the events A and B is

(a) Equal to the product of the marginal probabilities of the events A and B

(b) Not equal to the product of the marginal probabilities of the events A and B

(c) Equal to the sum of the marginal probabilities of the events A and B

(d) Equal to the difference between the marginal probabilities of the events A and B

(e) Is always equal to 1.

(1 mark)

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56.

a, b and c are three different real numbers. They are related as a < c < b. If the value of b is negative which of the following statements would be true for the value of the expression ?

(a) It is greater than zero (b) It is less than zero

(c) It is greater than one (d) It is less than one

(e) It falls between zero and one.

(1 mark)

< Answer >

57.

The following distribution shows the ages of 100 persons in a group:

Age group (in years)	Number of persons
20 - 25	3
25 - 30	16
30 - 35	22
35 - 40	18
40 - 45	14
45 - 50	10
50 - 55	7
55 - 60	6
60 - 65	4

What is the average age of the persons in the group?

(a) 25.5 years (b) 28.4 years (c) 32.6 years
(d) 39.3 years (e) 45.2 years.

(1 mark)

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58.

A data set contains the following observations:

7, 8, 6, 11, 9, 5, 7, 6, 4, 1, 5, 7, 2, 7.

What is mode of the data set?

(a) 4 (b) 5 (c) 6 (d) 7 (e) 11.

(1 mark)

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59.

The variance of a set of data is 9 and the arithmetic mean is 5. What is the coefficient of variation?

(a) 6% (b) 60% (c) 80% (d) 150% (e) 167%.

(1 mark)

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60.

Which of the following measures of dispersion is also called ‘root mean square deviation’?

(a) Range (b) Average deviation

(c) Mean absolute deviation (d) Standard deviation

(e) Median absolute deviation.

(1 mark)

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61.

Which of the following is true about median?

(a) Median is not useful as a measure of central tendency when the data set contains extreme values

(b) Median is not useful when the data set contains open ended classes

(c) Median can be calculated when the data set is not arranged in ascending or descending order

(d) Median is the middle item of a data set arranged in ascending or descending order

(e) Median is the value that has the highest frequency in a data set.

(1 mark)

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62.

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)

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63.

Which of the following measures will remain unchanged when every observation in the data set is divided by the same quantity?

(a) Range (b) Quartile deviation

(c) Standard deviation (d) Coefficient of variation (e) Mode.

(1 mark)

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64.

If every observation in a data set is increased by a constant quantity then the coefficient of variation of the resulting set of values will be

(a) Less than the coefficient of variation of the original data set

(b) Greater than the coefficient of variation of the original data set

(c) Equal to the coefficient of variation of the original data set

(d) Equal to the coefficient of variation of the original data set plus the square root of the constant quantity

(e) Equal to the coefficient of variation of the original data set multiplied by the square root of the constant quantity.

(1 mark)

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65.

Which of the following measures of central tendency may assume multiple values for the same set of data?

(a) Arithmetic mean (b) Geometric mean

(c) Harmonic mean (d) Median (e) Mode.

(1 mark)

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66.

Which of the following is false with regard to histograms?

(a) The class intervals are represented by the base of the rectangles

(b) The frequencies are represented by the heights of the rectangles

(c) If the class intervals are of equal width then the bases of the rectangles will be equal in length

(d) The tallest rectangle in a histogram represents the class interval with the lowest frequency

(e) If a grouped frequency distribution is not continuous, first it is to be converted into continuous distribution and then the histogram is drawn.

(1 mark)

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67.

In which of the following conditions, the graphical solution to a linear programming problem will not have multiple optimal solutions?

(a) One of the edges of the feasible region coincides with the horizontal axis

(b) One of the edges of the feasible region is parallel to the horizontal axis

(c) One of the edges of the feasible region coincides with the vertical axis

(d) One of the edges of the feasible region is parallel to the vertical axis

(e) None of the edges of the feasible region is parallel to the objective function.

(1 mark)

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68.

If A and B are any two events, the probability that exactly one of them occurs is given by

(a) P(A) + P(B) – 2 P(AB) (b) P(A) + P(B) – P(AB)

(c) (d)

(e) .

(1 mark)

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69.

The events B, C and D are mutually exclusive and collectively exhaustive; A is another event, which can jointly occur with B, C or D.

P(A and B) + P(A and C) + P(A and D) =

(a) 1.00 (b) P(D) (c) P(C) (d) P(B) (e) P(A).

(1 mark)

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70.

A firm manufactures steel pipes in three plants namely A, B and C. The daily production volumes from the three firms A, B and C respectively are 1000 units, 2000 units and 4000 units respectively. It is known from past experience that 2% of the output from plant A, 3% of the output from plant B and 5% of the output from plant C are defective. A pipe is selected from a days total production and found to be defective. What is the probability that the pipe is manufactured by plant B?

(a) 14.29% (b) 21.43% (c) 28.57% (d) 4% (e) 57.14%.

(2 marks)

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71.

Five students namely P, Q, R, S and T are independently trying to solve a problem in mathematics. The probabilities that they will be able to solve the problem are , , , and respectively.

What is the probability that the problem will be solved?

(a) 5.2% (b) 0.83% (c) 25.5% (d) 56.7% (e) 94.8%.

(1 mark)

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72.

An NGO has been studying the literacy level among the male and female adults in a certain village. 3 out of every 5 adults in the village are males. It is observed that 1 out of every 2 adult males is illiterate and 4 out of every 5 adult females are illiterates.

If an adult is randomly selected and found to be illiterate then what is the probability that the adult is a female?

(a) 32% (b) 25.8% (c) 51.6% (d) 12.9% (e) 62%.

(2 marks)

< Answer >

73.

Which of the following is true?

(a) In probability theory, the result of an experiment is known as activity

(b) The probability of two or more statistically independent events occurring together is equal to the sum of their marginal probabilities

(c) Bayes’ theorem is normally used to develop probabilities according to the classical approach

(d) The classical probability approach enables us to determine revised probabilities or posterior probabilities

(e) The set of all possible outcomes of an experiment is called the sample space of the experiment.

(1 mark)

< Answer >

74.

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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75.

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 >

76.

Which of the following indicates that the distribution curve of the data is negatively skewed?

(a) Arithmetic Mean < Median < Mode

(b) Median < Mode < Arithmetic Mean

(c) Mode < Arithmetic Mean < Median

(d) Mode < Median < Arithmetic Mean

(e) Median < Arithmetic Mean < Mode.

(1 mark)

< Answer >

77.

Which of the following is not an unbounded interval?

(a) The set of all real numbers greater than a specified real number

(b) The set of all real numbers less than a specified real number

(c) The set of all real numbers greater than or equal to a specified real number

(d) The set of all real numbers less than or equal to a specified real number

(e) The set of all real numbers greater than one real number and less than another real number.

(1 mark)

< Answer >

78.

Which of the following statements is false?

(a) The set of natural numbers is an infinite set

(b) 2 is a natural number but not a whole number

(c) The set of integers includes positive as well as negative numbers

(d) A rational number can be expressed in decimal form

(e) The set of real numbers does not include complex numbers.

(1 mark)

< Answer >

79.

The first term of an A.P. is a, second term is b and the last term is c, then the sum of all the terms of the series is

(a)

(b)

(c)

(d)

(e) Zero.

(1 mark)

< Answer >

80.

The ratio between the sum of n terms of two A.P.’s is (3n + 8) : (7n + 15). Find the ratio between their twelfth terms.

(a) 6 : 17 (b) 7 : 15 (c) 8 : 17 (d) 7 : 16 (e) 9 : 19.

(2 marks)

< Answer >

81.

Three simultaneous equations involving three variables can be solved if the three equations

(a) Can be satisfied by the three variables assuming equal value in all the equations

(b) Can be satisfied when two out of the three variables assume equal value in all the equations

(c) Can be satisfied when each of the three variables assumes a different value in each of the three equations

(d) Can be satisfied when each of the variables are set to zero in all the equations

(e) Can be satisfied when the three variables respectively assume the same values in the three equations.

(1 mark)

< Answer >

82.

Which of the following is true?

(a) In a logarithmic function the logarithmic operation is applied on the dependent variable

(b) In a logarithmic function the base of the logarithm is always 10

(c) In a logarithmic function the independent variable cannot be negative

(d) In a logarithmic function the base of the logarithm can be any real number

(e) In a logarithmic function the dependent variable cannot be negative.

(1 mark)

< Answer >

83

The least common multiple of a group of numbers is

(a) The smallest number in the group

(b) A multiple of only the smallest number in the group

(c) A multiple of only the largest number in the group

(d) The smallest multiple of only the smallest number in the group

(e) The smallest number that can be divided by each number in a group of numbers without leaving a remainder.

(1 mark)

< Answer >

84.

Find out the sum of the following series:

S = 4.+ 4. + 4.+ 4. + .......

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

(1 mark)

< Answer >

85.

The mth term of an A.P. is n and its nth term is m. Then its (m + n)th term will be

(a) m + n (b) m - n (c) m n (d) n - m (e) zero.

(1 mark)

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86.

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)

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87.

Which of the following is not true with regard to a logarithmic function y = log_b x?

(a) The function is not defined for zero or negative values of x

(b) The value of the function is zero when x = 1

(c) The value of the function is negative when x lies between 0 and 1, and b > 1

(d) The value of the function decreases as the value of x increases for b > 1

(e) The value of the function is positive when x > 1.

(1 mark)

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88.

u = ln

(a) t +

(b)

(c)

(d)

(e) t² + 4.

(2 marks)

< Answer >

89.

If f (x)=, what is the value of the expression

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

(1 mark)

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90.

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 >

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

1.

Answer : (c)

Reason : If all the terms of an arithmetic progression are multiplied by a constant the resulting terms will always form an arithmetic progression with the first term multiplied by the constant as well as the common difference multiplied by the constant. The resulting series will neither be in a geometric series or a harmonic series because nature of the resulting terms will not satisfy their requirements.

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2.

Answer : (b)

Reason : All other alternatives are properties of the arithmetic mean. Refer to pages 69 and 70 of text book.

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3.

Answer : (a)

Reason : Let ‘a’ be the first term and ‘d’ be the common difference in the A.P.

\ jth term, t_j = a + (j – 1) d

kth term, t_k = a + (k – 1) d

Given: j [a + (j – 1)d] = k [a + (k – 1) d]

or ja + j (j – 1)d = ka + k (k – 1) d

or a (j – k) = [k (k – 1) – j (j – 1)] d

or a (j – k) = [k² – k – j² + j] d

or a (j – k) = [k² – j² – (k – j)] d

or a (j – k) = [(k – j) (k + j) – (k – j)] d

or a (j – k) = [(k – j) (k + j – 1)] d

or a =

or a = – (j + k – 1) d ……… (1)

(j + k)^th term, t_(j+k) = a + (j + k – 1) d

From (1) above,

t_(j+k) = – (j + k – 1) d + (j + k – 1) d = 0

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4.

Answer : (b)

Reason :

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5.

Answer : (d)

Reason : M, P and N are in A.P

\ P – M = N – P Þ 2P = M + N Þ P = ……….(1)

\ P² = ……………….(2)

M , Q and N are in G.P

\ \ P² – Q² = ……………….(3)

= \ ……………….(4)

Adding equation (1) and (4):

P + [P² – Q²]=

i.e. M = P + Subtracting equation (4) from equation (1):

P – i.e. N = P

\ V and VI are correct

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6.

Answer : (c)

Reason : The fifth term of an A.P., = 1,

The thirty first term of the A.P. is

Solving the above equations we get a = 13 and d = -3.

The sum of its first fifteen terms is

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7.

Answer : (b)

Reason:

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8.

Answer : (c)

Reason :

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9.

Answer : (b)

Reason : The pth term of the A.P. is i.e. A + (p – 1)d =

The qth term of the A.P. is i.e. A + (q – 1)d =

By subtracting the two equations we get

(p – q)d = – =

Or, d =

The rth term would exceed the pth term by (r – p)d. i.e.

So the value of rth term is

So the rth term of the H.P. is pq.

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10.

Answer : (b)

Reason : The given series is an A.P. with first term a = 20 and common difference d = -2/3

Given that the sum S_n = 300.

\(n/2)[2´20 + (n -1) (-2/3)] = 300.

Simplifying n²- 61n + 900 = 0 or (n – 25) (n – 36) = 0

n = 25 or 36.

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11.

Answer : (c)

Reason :

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12.

Answer : (a)

Reason :

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13.

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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14.

Answer : (c)

Reason :

For the A.P. : 9,6,3,0 … – 75

a = 9

d = 6 – 9 = –3

Let there be ‘n’ terms

\ t_n = –75

\ –75 = a + (n – 1)d = 9 + (n – 1) (–3)

or –75 = 9 – 3n + 3

or 3n = 75 + 9 + 3 = 87

or n = = 29.

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15.

Answer : (e)

Reason : Let the three numbers be (a – d), a and (a + d)

Given: a–d + a + a+d = 18 Þ 3a = 18 or a = = 6

(a – d)² + a² + (a + d)² = 140

or a² – 2ad + d² + a² + a² + 2ad + d² = 140

or 3a² + 2d² = 140

or 3(6²) + 2d² = 140

or 108 + 2d² = 140

or d² = = 16

\ d = = ± 4

\ The numbers are: 2, 6 and 10 or

10, 6 and 2

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16.

Answer : (c)

Reason : We know G² = A.H = 27.12 Or, G = 18.

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17.

Answer : (a)

Reason : (a) a^m × aⁿ = a^{m + n}

(b) a^{m – n} = a^m / aⁿ

(c) a^{m × n} = (a^m) ⁿ = (aⁿ) ^m

(d) a^{m / n} = (a^m) ^1/n

(e) (m × n)^a = m^a ´ n^a

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18.

Answer : (c)

Reason :

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19.

Answer : (d)

Reason :

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20.

Answer : (e)

Reason :

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21.

Answer : (b)

Reason :

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22.

Answer : (a)

Reason : Given that,

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23.

Answer : (e)

Reason :

Given that,

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24.

Answer : (a)

Reason : The given expression can be rewritten as,

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25.

Answer : (b)

Reason : The given sum is

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26.

Answer : (c)

Reason : Given that,

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27.

Answer : (d)

Reason :

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28.

Answer : (c)

Reason :

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29.

Answer : (c)

Reason : The word GANESHPURI has 10 distinct letters. There are five odd positions and five even positions. As four vowels – A, E, U and I – can take only even positions so for them choice is five. The number of ways by which 4 vowels can be placed in five even positions is . Remaining six positions are to be filled by remaining six letters. This can be done in 6! ways. The total number of ways by which words can be formed so that vowels are placed in even places is .

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30.

Answer : (c)

Reason : 2 professors can be selected from 5 professors in ways. As one particular student must be the part of the committee so, effectively, 2 students are to be selected from remaining 9 students in ways. So, the ways this committee can be formed in the given condition.

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31.

Answer : (d)

Reason : The size of the guard group is 12. When two particular persons are together, the remaining 10 soldiers are to be selected from (16 – 2) or 14 soldiers in or 1001 times.

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32.

Answer : (e)

Reason : There are four possibilities:

3 ladies from Mr. Khan’s side and 3 gentlemen from his wife’s side: No. of ways in this case =

3 gentlemen from Mr. Khan’s side and 3 ladies from his wife’s side: No. of ways in this case =

2 ladies and one gentleman from Mr. Khan’s side and one lady and 2 gentlemen from his wife’s side: No. of ways in this case =

One lady and 2 gentlemen from Mr. Khan’s side and 2 ladies and one gentleman from his wife’s side: No. of ways in this case =

So, total no. of ways = (16 + 1 + 324 + 144) or 485

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33.

Answer : (d)

Reason :

Given that log₆15 = , log₁₂18 =

Therefore, Or, Or,

Taking ‘5’ as the base,

or, 1 + log3 = (log2 + log3)

or, a log2 + (a –1) log3 = 1 ……………. (1)

Now, log₁₂18 = Or, Or,

Or,

Or, log2 + 2log3 = (log3 + 2 log 2)

Or, (1 –2 ) log2 + (2–) log3 = 0 …………. (ii)

Log2 + (a – 1) log3 =1 ………………x (1-2 )

(1 – 2) log2 + (2 – ) log3 = 0 ………………..x

(–1)(1–2) log3 – (2 – ) log3 = 1–2

or, Log3 = =

= log2 = = =

=

Now, log₂₅24 = = =

with base ‘5’ = (log 3 + 3 log 2)

=

= =

=

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34.

Answer : (a)

Reason :

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35.

Answer : (a)

Reason : If the derivative of a function f(x) is zero, then it implies that f(x) has a constant value for all values of x.

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36.

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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37.

Answer : (a)

Reason : A function is a rule or correspondence which always associates to each number x in a set A, a unique number f(x) in a set B.

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38.

Answer : (e)

Reason : For any function f(x) the limit of f(x) as x approaches a value a is a value L, such that f(x) approaches L as x approaches a.

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39.

Answer : (a)

Reason :

=

=

=

=

=

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40.

Answer : (d)

Reason :

If f(x) = ln g(x), then

\

=

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41.

Answer : (d)

Reason :

(a)

= = =

= = =

(b)

= =

(c)

(d)

= =

= =

=

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42.

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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43.

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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44.

Answer : (a)

Reason :

Let X₁ and X₂ be the number of STD and DEL dryers manufactured per period, respectively

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) and

X₁, X₂ 0

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 no. of dryers of STD model to be produced is 800.

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45.

Answer : (e)

Reason : The weighted arithmetic mean of the distribution is zero as all the values are in opposite sign in the same weighted proportion. Therefore, the distribution is symmetric in nature. As the observation zero occurs with highest frequency, it may be considered to be mode of the distribution. Hence, the option (e) is the answer.

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46.

Answer : (e)

Reason : The median score of the group of girls is 80 while that of the boys is 70. For the composite group, neither can we take the average of these two (that results 75) nor can we apply the formulae of composite median, as we used to do for the mean (that results 76) nor we can sum the two numbers. As the 50 scores are not given, nothing can be concluded.

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47.

Answer : (a)

Reason : The mean is affected by the change in scale and origin. Hence, the mean salary will be = (1500 + 100) ´ 1.20 = 1920 while the standard deviation of salary is affected only by the change in origin, not by the scale. So, the increment of Rs.100 will not affect the standard deviation while the increment of 20 percent will increase it to Rs.480. Hence, the option (a) is correct.

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48.

Answer : (c)

Reason :

Sample variance (S2) =

= 19.73, S² = 26.02

n = 11

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49.

Answer : (a)

Reason : (a) If there is no dispersion in a data set then all the mathematical and positional averages are equal.

(b) Hence b. is false.

(c) Hence c. is false.

(d) Hence d. is false.

(e) Hence e. is false.

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50.

Answer : (a)

Reason : (a) The standard deviation of a data set is expressed in the same unit as the observations in the data set.

(b), (c), (d) and (e) are incorrect with regard to the standard deviation.

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51.

Answer : (a)

Reason : Let PI = the event that the test is positive and NI = the event that the test is negative.

Also, let D denote the event that subject has the illness and D^c the complementary event that the subject does not have the illness. The following information has been provided in this problem:

P(PI I D) = 0.99 and therefore P(NI I D) = 0.01

P(NI I D^C) = 0.95 and therefore P(PI I D^c) = 0.05

P(D) = 0.0001 and therefore P(D^c) = 0.9999

Our goal is to find P(D I PI). By the Bayes’ theorem

P(D I PI) = {P(PI I D)P(D)}/{ P(PI I D)P(D) + P(PI I D^c)P(D^c)} = {0.99 x 0.0001}/{0.99 x 0.0001 + 0.05 x 0.9999} = 0.002 (approx)

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52.

Answer : (c)

Reason : Let the events A and B denote that the computers A and B respectively be sold. As per the problem, P(A) = 0.60 and P(B) = 0.40 and hence, and

The probability of selling at least one computer is given by P(A or B). Hence, the required probability may be obtained as: P{A/(A or B)}. Therefore, from the multiplication theorem of probability, we get,

P{A/(A or B)} ==

=

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53.

Answer : (d)

Reason : We have, the events S and T are independent while P(S) < P(T) and P(S/T) + P(T/S) = 1 while P(ST) = 0.24

or, or, = P(T) ´ P(S) (as the events S and T are independent) or, P(S) + P(T) = 1 Therefore, by solving the given relationships, we get P(S) = 0.4

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54.

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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55.

Answer : (b)

Reason : (a) & (b) For two dependent events A and B, the joint probability of the events A and B is not equal to the product of their marginal probabilities.

(c) For two dependent events A and B, the joint probability of the events A and B is not equal to the sum of their marginal probabilities.

(d) For two dependent events A and B, the joint probability of the events A and B is not equal to the difference between their marginal probabilities.

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

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56.

Answer : (b)

Reason :

As, a < c < b, and b is negative so a and c would also be negative. Highest negative number would always have lowest absolute number. So . Moreover as both a and b are negative numbers so ab is a positive number. On the whole it would be a negative number i.e. less than zero.

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57.

Answer : (d)

Reason:

Class	Mid-value (m)	Frequency (f)	f ´ m
20-25	22.5	3	67.5
25-30	27.5	16	440
30-35	32.5	22	715
35-40	37.5	18	675
40-45	42.5	14	595
45-50	47.5	10	475
50-55	52.5	7	367.5
55-60	57.5	6	345
60-65	62.5	4	250
		Sf = 100	Sf ´ m = 3930

\ Mean = = 39.3 years

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58.

Answer : (d)

Reason : Mode is the observation which has the highest frequency.

\ Mode of the given data set is 7, because 7 has the highest frequency (4).

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59.

Answer : (b)

Reason : Standard deviation = = = 3

Coefficient of variation = =

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60.

Answer : (d)

Reason : Standard deviation is also called root mean square deviation.

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61.

Answer : (d)

Reason : All other alternatives are false with regard to median except alternative (d). Refer to pages 71 and 72 of text book.

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62.

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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63.

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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64.

Answer : (a)

Reason : If every observation in a data set is increased by a constant then the coefficient of variation of the resulting set of values will be less than the coefficient of variation of the original data set because the numerator remains the same whereas the denominator increases.

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65.

Answer : (e)

Reason : Mode can assume multiple values for the same data set if there are multiple values which occur the maximum number of times.

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66.

Answer : (d)

Reason : The following are true with regard to histograms:

a. The classes are represented by the base of the rectangles.

b. The frequencies are represented by the heights of the rectangles.

c. If the classes are of equal width then the bases of the rectangles will be of equal length.

d. The tallest rectangle in a histogram represents the class with the highest frequency.

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67.

Answer : (e)

Reason : A graphical solution to a linear programming problem will have multiple optimal solutions if the objective function is parallel to an edge of the feasible region which is in the direction of the optimal movement of the objective function. If none of the edges of the feasible region is parallel to the objective function then, the possibility of multiple optimal solution does not arise.

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68.

Answer : (e)

Reason : indicates the occurrence of at least any one of the two events. The first component indicate the probability of the occurrence of B but not A while the second component indicate the occurrence of A not B. Hence, the option (e) is happened to be the right choice.

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69.

Answer : (e)

Reason : If B, C and D are mutually exclusive and collectively exhaustive, and A is another event which can jointly occur with B, C or D, then

P (A) = P(A and B) + P(A and C) + P(A and D).

This means that P(A) is equal to the sum of all the joint probabilities which include event A.

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70.

Answer : (b)

Reason : Let the following notations be used:

A : Pipe is produced by Plant A.

B : Pipe produced by Plant B.

C : Pipe is produced by Plant C.

D : Pipe is defective.

Given: P (A) =

P (B) =

P (C) =

P (D/A) = 2% = 0.02

P (D/B) = 3% = 0.03

P (D/C) = 5% = 0.05

P (A and D) = P (A) . P (D/A) =

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

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

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

P (B/D) = = 0.2143. i.e. 21.43%

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71.

Answer : (e)

Reason : Let the probability that the students P, Q, R, S and T will be able to solve the problem be denoted by P(P), P(Q), P(R), P(S), and P(T) respectively.

Given:

P(P) = P(Q) = P(R) = P(S) = P(T) =

Probability that none of the students will be able to solve the problem

=

\ Probability that the problem will be solved = 1 – = = 0.948 i.e. 94.8%.

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72.

Answer : (c)

Reason : According to Bayes’ theorem:

P(Female/Illiterate) =

P(Female and Illiterate) = P(Female) . P(Illiterate / Female)

= = = .

P(Illiterate) = P(Female and Illiterate) + P(Male and Illiterate)

=

= = =

\ P(Female / Illiterate) = = 0.516 i.e. 51.6% approximately.

(In the above calculations, ‘males’ imply adult males and ‘females’ imply adult females).

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73.

Answer : (e)

Reason : The first four alternatives are clearly false. The fifth alternative is true. (Self explanatory.)

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74.

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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75.

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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76.

Answer : (a)

Reason : In case of a negatively skewed distribution arithmetic mean < median < mode. Hence (a) is the answer. The relationships stated in (b), (c), (d) and (e) are not the features of negatively skewed distributions.

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77.

Answer : (e)

Reason : a. This interval is bounded on the lower side but unbounded on the upper side.

b. This interval is bounded on the upper side but unbounded on the lower side.

c. This interval is bounded on the lower side but unbounded on the upper side.

d. This interval is bounded on the upper side but unbounded on the lower side.

e. This interval is bounded on the upper as well as lower sides.

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78.

Answer : (b)

Reason : (a) The set of natural numbers is an infinite set.

(b) 2 is both a natural number and a whole number.

(c) The set of integers includes positive as well as negative numbers.

(d) A rational number can be expressed in decimal form.

(e) The set of real numbers does not include complex numbers.

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79.

Answer : (b)

Reason : Given that T₁= a and T₂ = a + d = b \d = b – a

T_n = a + (n – 1) d = c

n – 1 =

n = …….(i)

S_n = (n / 2) [a + T_n] = …..(using the relation (i))

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80.

Answer : (d)

Reason : Given that

Or, ……….(i)

We have to find

Choosing (n-1)/2 = 11 or n = 23 in relation (i), we get

.

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81.

Answer : (e)

Reason : Three simultaneous equations involving three variables can be solved if the three equations can be satisfied when the three variables respectively assume the same values in each of the three equations.

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82.

Answer : (c)

Reason : a. In a logarithmic function the logarithmic operation is applied on the independent variable.

b. In a logarithmic function the base of the logarithm may be any number other than 10.

c. In a logarithmic function the independent variable cannot be negative.

d. In a logarithmic function the base of the logarithm cannot be zero or negative numbers.

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83.

Answer : (e)

Reason : The least common multiple of a group of numbers is the smallest number that can be divided by each number of the group without leaving a remainder.

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84.

Answer : (d)

Reason : The given series,

is of the form, a + ar + ar² + ar³ + ar⁴ + ....

The sum of a series of this form,

S =

a = ; r =

\ S = = = 2 ´ 2 = 4

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85.

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

Then T_m+n = a + (m + n –1) d

= (m + n – 1) + ( m + n –1) (-1)

= 0 .

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86.

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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87.

Answer : (d)

Reason : According to the properties of logarithmic functions all the alternatives (a) through (d) are true. Alternative (d) is not true because the value of the logarithmic function increases as the value of x increases, for b > 1.

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88.

Answer : (c)

Reason :

u = ln

=

=

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89.

Answer: (b)

Reason:

=

=

=

=

=

=

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90.

Answer : (c)

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

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