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

The m^th term of an A.P. is n and the n^th term is m. The (m + n)^th term would be

(a) m (b) n (c) (m + n) (d) (m – n) (e) Zero.

(1 mark)

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The sum of three numbers in an A.P. is 15. If 1, 4, 19 are added to them respectively, then they are in G.P. The possible highest number of the three is

(a) 18 (b) 21 (c) 26 (d) 29 (e) 31.

(1 mark)

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The third term of an A.P. is 7 and its seventh term is 2 more than thrice of its third term. Find the sum of its first 20 terms.

(a) 740 (b) 750 (c) 760 (d) 770 (e) 780.

(1 mark)

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The A.M. of two numbers exceeds their G.M. by 15 and H.M. by 27. The lowest of the two numbers is

(a) 20 (b) 25 (c) 30 (d) 40 (e) 50.

(2 marks)

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Arithmetic, geometric and harmonic means of two numbers X and Y, are A, G and H respectively. Which of the following is false?

(a) AH = XY

(b) X – A = A – Y

(c)

(d)

(e) G = XY.

(1 mark)

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The ratio of harmonic mean of two numbers to the geometric mean of the same numbers is 12:13. The actual numbers are in the ratio of

(a) 3:4 (b) 4:5 (c) 5:8 (d) 9:4 (e) 3:7.

(2 marks)

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The sum of an arithmetic progression (A.P.) consisting of n terms is zero. Which of the following is true? (t₁ represents the first term and t_n represents the n^thterm of the A.P.)

(a) t₁ = – t_n(b) t₁ = t_n(c) t₁ / t_n = 0 (d) t₁ / t_n > 1 (e) t_n / t₁ > 1.

(2 marks)

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If the first term in a geometric progression is greater than 1 and the common ratio is less than 1, then

(a) The consecutive terms will be in increasing order

(b) The consecutive terms will be in decreasing order

(d) All the consecutive terms will be less than 1

(e) All the consecutive terms will be greater than 1.

(1 mark)

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The reciprocals of the terms in a harmonic progression are

(a) In geometric progression (b) In harmonic progression

(e) Always in increasing order.

(1 mark)

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10.

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

(d) Either a geometric progression or a harmonic progression

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

(1 mark)

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11.

If the sum of three numbers in A.P. is 15 whereas the sum of their squares is 83. The smallest of the three numbers would be

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

(1 mark)

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12.

The n^th term of a series is given to be . Find the sum of 105 terms of this series.

(a) 1575 (b) 2730 (c) 1470 (d) 2835 (e) 5460.

(1 mark)

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13.

In a G.P. 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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14.

The maximum number of basic solutions for the system of simultaneous equations :

where

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

(1 mark)

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15.

If , then the value of is equal to

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

(2 marks)

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16.

In a G.P. sum of n terms is 364. The first term is 1 and the common ratio is 3. The value of n is

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

(1 mark)

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17.

, and

The value of xyz would be

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

(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.

Find out the values of A, B and C by solving the following equations:

2log₂A + 3log₃B + 4log₄C = 18

4log₂A + 5log₃B + 6log₄C = 30

6log₂A + 4log₃B + 10log₄C = 40

(a) A = 2, B = 3 and C = 4

(b) A = 2, B = 2 and C = 2

(d) A = 16, B = 4 and C = 9

(e) A = 4, B = 9 and C = 16.

(2 marks)

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20.

are in the arithmetic progression, the value of x would be

(a) 10 (b) 8 (c) 6 (d) 4 (e) 3.

(1 mark)

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21.

(a) 0 (b) 1 (c) 3 (d) 4 (e) 5.

(1 mark)

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22.

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

(1 mark)

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23.

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

(2 marks)

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24.

The value of the expression

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

(1 mark)

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25.

(a) - 4 (b) - 2 (c) 2 (d) 4 (e) 3.

(2 marks)

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26.

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

(1 mark)

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27.

The number of ways in which all the letters of the word INTEGRATION can be arranged so that all vowels are always in the beginning of the word is

(a) 10800 (b) 60480 (c) 120960 (d) 1260 (e) 720.

(1 mark)

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28.

If . The value of r is

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

(1 mark)

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29.

The number of words that can be made by rearranging the letters of the word CORRESPONDENCE beginning with C and ending with C is

(a) 14! (b) (c) (d) (e)

(1 mark)

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30.

If . The possible value of r is

(a) 4 (b) 5 (c) 6 (d) 7 (e) 8.

(1 mark)

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31.

A candidate is required to answer 7 out of 12 questions, which are divided in to two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many different ways can we choose the 7 questions?

(a) 180 (b) 210 (c) 600 (d) 780 (e) 792.

(1 mark)

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32.

A guard of 12 persons is to be formed from a group of n soldiers in all possible ways. If the number of times two particular soldiers are working together is 2m and the number of times three particular soldiers are working together is m, the value of n is

(a) 18 (b) 20 (c) 22 (d) 24 (e) 26.

(1 mark)

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33.

(a) x y z (b) x + y + z (c) (x + y + z)^-1(d) 1 (e) 0.

(2 marks)

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34.

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

(1 mark)

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35.

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

(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)

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36.

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

(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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37.

If x = 8 and y = 4, the value of would be

(a) 2 (b) 4 (c) 8 (d) 16 (e) 32.

(1 mark)

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38.

Differentiation of with respect to x is

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

(1 mark)

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39.

Differentiation of with respect to x is

(a) (b) (c) – (d) (e) .

(1 mark)

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40.

The geometric mean of 10 observations on a certain variable was calculated as 16.2. It was later discovered that one of the observations was wrongly recorded as 12.9 that was actually 21.9. What is the correct geometric mean?

(a) 16.58 (b) 16.83 (c) 17.08 (d) 17.33 (e) 17.58.

(1 mark)

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41.

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

(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)

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42.

If y = x² – 3x + 4, then which of the following statements is true for y at the point where it intersects the y axis?

(a) y is increasing at an increasing rate

(b) y is increasing at a decreasing rate

(d) y is decreasing at a decreasing rate

(e) y is decreasing at a constant rate.

(2 marks)

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43.

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

(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
Resource	STD	DEL	Quantity of Resources Available During Period
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 and a respective control unit. The objective is to maximize the total contribution.

At the optimum point how many hours would remain in balance for first phase?

(a) 0 (b) 50 (c) 100 (d) 150 (e) 200.

(2 marks)

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45.

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

(1 mark)

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46.

A marathon runner decides to spare his energy for the end of a race and to also not lag too far behind the other runners in the earlier parts of the race. For a while, he adjusts his speed so that half the runners overtake him and half lag behind him. Which average may be used to represent the above runner’s speed?

(a) Arithmetic mean (b) Geometric mean

(1 mark)

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47.

If by mistake you have added 29 to each of the biggest 200 numbers in a set of 401 numbers, how will the median of the numbers is affected?

(a) The median will increase by 29

(b) The median will increase by 200

(d) The median will increase by

(e) The median will increase by .

(1 mark)

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48.

Two students from the same class dropped out of a B-school. Their scores in Financial Management were equal to the mean score of the class. How does their leaving affect the measures characterizing the distribution of the class’s scores?

(a) The mean decreases and the variance decreases

(b) The mean decreases and the variance does not change

(d) The mean does not change and the variance increases

(e) The mean increases and the variance decreases.

(2 marks)

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49.

A data set includes some quantities. The sum of reciprocals of the quantities in the data set is. The harmonic mean of the data set is . The data set is expanded by including the quantities 5 and 10 into it.

What is the harmonic mean of the expanded data set?

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

(1 mark)

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50.

Let the sum of n terms of two A.P. i.e., S₁ and S₂ be in the ratio of 7n – 5: 5n + 17. The difference between the 6^th term of the two series is

(a) –1 (b) 0 (c) 1 (d) 2 (e) 12.

(1 mark)

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51.

The probability of getting a number less than four when a die is rolled is

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

(1 mark)

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52.

A single letter is selected at random from the word PROBABILITY. The probability that it is a vowel is

(a) 2/11 (b) 3/11 (c) 4/11 (d) 5/11 (e) 7/11.

(1 mark)

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53.

Two horses A and B run a race. If the probability of A’s win is twice that of B and there is no tie, then the probability of B’s win is

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

(1 mark)

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54.

If two cards are drawn from a pack of 52 cards, the probability that the two cards are aces is

(a) 1/220 (b) 1/221 (c) 1/222 (d) 1/223 (e) 1/224.

(1 mark)

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55.

The number of words that can be formed by using the letters of the word CUSHION without repetition so that vowels always occupy even places is

(a) 24 (b) 48 (c) 72 (d) 96 (e) 144.

(1 mark)

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56.

The number of ways in which 6 boys and 6 girls can sit in a row so that no two boys sit together and always a row starts with the boy, is

(a) (6!)²(b) (c) 7!.5! (d) (e) (7!)².

(1 mark)

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57.

The average deviation of all items in the data from zero is equal to the

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

(d) Standard deviation (e) Variance.

(1 mark)

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58.

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

(a) The mean of the original data set

(b) C + Mean of the original data set

(d) Mean of the original data set ¸ C

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

(1 mark)

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59.

Which of the following is a relative measure of dispersion?

(a) Variance (b) Standard deviation (c) Range

(d) Coefficient of variation (e) Mean deviation.

(1 mark)

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60.

If every item in a data set is decreased by the same quantity then the standard deviation of the resulting data set

(a) Remains the same

(b) Increases by the same quantity by which every data item is decreased

(d) Increases by the square root of the same quantity by which every data item is decreased

(e) Decreases by the square root of the same quantity by which every data item is decreased.

(1 mark)

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61.

The appropriate mean for a set of ratios using the denominators of the ratios as weights is

(a) Geometric mean (b) Harmonic mean

(1 mark)

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62.

Which of the following measures cannot be combined mathematically?

(a) Standard deviation (b) Arithmetic mean

(1 mark)

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63.

From a city population, the probability of selecting a male or a smoker is 0.70, a male smoker is 0.40 and a male provided he/she is smoker is 0.67. What is the probability of selecting a non-smoker?

(a) 0.40 (b) 0.50 (c) 0.55 (d) 0.60 (e) 0.65.

(1 mark)

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64.

The coefficient of variation cannot be meaningfully used to compare the variability of two or more sets of data, when

(a) The standard deviation is zero for one or more sets of data

(b) The standard deviation is 1 for one or more sets of data

(d) The mean is 1 for one or more sets of data

(e) The mean and standard deviation are equal for one or more sets of data.

(1 mark)

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65.

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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66.

In a positively skewed distribution

(a) Majority of the observations are concentrated towards the higher end of the scale

(b) Majority of the observations are concentrated towards the lower end of the scale

(d) The observations have the same frequency

(e) The distribution of the data is symmetrical.

(1 mark)

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67.

A bucket contains 9 balls, two of which are red, three blue and four black. Three balls are drawn at random, what is the probability that these are of same colour?

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

(1 mark)

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68.

If events A and B are mutually exclusive then which of the following is true?

(a) P(A and B) = 0 (b) P(A) = 0 (c) P(B) = 0

(d) P(A or B) = 0 (e) P(A or B) = 1.

(1 mark)

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69.

The probability of occurrence of an event is expressed as a number which lies between

(a) 0 and 1 (b) 1 and 2 (c) –1 and 0

(d) –2 and –1 (e) 1 and infinity.

(1 mark)

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70.

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

(d) Marginal probability of event A

(e) Zero.

(1 mark)

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71.

Which one of the following is false with respect to classical probability of an event A?

(a) If A1, A2, …, Ak are k mutually exclusive and exhaustive events in the sample space () then

(b)

(d) If

(e) If , .

(1 mark)

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72.

The letters of the word SUCCESS are arranged in a row at random. The probability that all S’s come together is

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

(1 mark)

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73.

Which of the following number(s) is an/are irrational number?

(a) (b)

(1 mark)

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74.

From the following simultaneous equations, the value of x will be equal to

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

(1 mark)

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75.

The set of whole numbers includes the set of

(a) Natural numbers (b) Rational numbers

(e) Irrational numbers.

(1 mark)

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76.

If the initial cost of a machine is Rs.15,000 and it loses 10% of its value each year, the value of the machine after 10 years would be

(a) Rs.5230 (b) Rs.5811 (c) Rs.1500 (d) Rs.2530 (e) Rs.3520.

(1 mark)

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77.

The sum of the observations in a data set containing 10 observations is 120 and the standard deviation of the data set is 3. If 3 is added to every observation in the data set then, what will be the coefficient of variation of the resulting data set?

(a) 0 (b) 7.5% (c) 20% (d) 25% (e) 50%.

(1 mark)

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78.

The appropriate average for a set of ratios using the denominators of the ratio data as weights is

(a) Simple arithmetic mean (b) Weighted arithmetic mean

(e) Geometric mean.

(1 mark)

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79.

The sum of three numbers of a G.P. 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.

(1 mark)

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80.

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

(1 mark)

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81.

Which of the following is/are true?

I. The general LPP calls for optimizing a linear function for variables called the ‘constraints’ or ‘restrictions’.

II. A graphical method of solving LPP is applicable where two( or at most three) variables are involved.

III. In solving the graphical method of LPP, at least one corner of the region of feasible solutions would be an optimal solution whenever the maximum or minimum value of Z was finite.

IV. In solving the graphical method of LPP, the objective function could be represented by a line or a plane for any fixed value of Z.

(a) Only (I) above

(b) Only (II) above

(d) (II), (III) and (IV) above

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

(1 mark)

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82.

The value of (y + z) in the above simultaneous equations would be

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

(1 mark)

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Which of the following inequalities is/are not true when x, a and b are real numbers and a < b?

(a) If x < 0, then ax > bx (b) If x > 0, then ax < bx

(e) Both (a) and (d) above.

(1 mark)

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84.

If . The value of r would be

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

(1 mark)

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85.

(where b and c are constants)

(a) (b + c) (b) (b – c) (c) 2bc (d) (2b + c) (e) (b – 2c).

(1 mark)

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86.

The terms in a series are as given below:

4, 8, 16, 32, 64, ………

The ninth term of the series is

(a) 256 (b) 512 (c) 1024 (d) 2048 (e) 4096.

(1 mark)

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87.

(a) z (b) ab (c) abz (d) 2abz (e) 3abz.

(1 mark)

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88.

(a) 0 (b) 1 (c) -1 (d) 10 (e) -10.

(1 mark)

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89.

Which of the following statements is true with regard to a linear programming problem?

(a) The objective function is a statement of a constraint

(b) The coefficients of the decision variables in the constraints represent the per unit contribution of the decision variables to the value of the objective function

(d) The constraint inequations specify the consumption of resources and the amount of resources available

(e) The coefficients of the decision variables in the objective function represent the per unit consumption of resources by the decision variables.

(1 mark)

< Answer >

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

Answer : (e)

Reason : mth term in an AP would be S_m =a + (m –1)d = n

nth term in an AP would be S_n =a + (n – 1)d = m

By subtracting above two equations we get (m-n)d = n – m Or, d = -1

So, the value of a is n – (m – 1)(–1) = n + m – 1

T_m+n =a + (m + n – 1)d

Putting the values of a and d we get

S_m+n = m + n – 1 + (m + n – 1)(–1) = 0.

< TOP >

Answer : (c)

Reason :

< TOP >

Answer : (a)

Reason : Given that T₃ = 7 and T₇ = 2 + 3 ´ T₃ = 2 + 3 ´ 7 = 23

Therefore T₃ = a + 2d = 7 and T₇= a + 6d = 23

Solving the above two equations we have, a = - 1 and d = 4,

Therefore S₂₀= (20/2)[2 ´ (-1) + 19 ´ 4] = 740.

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Answer : (c)

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 15. So, A – G = 15; Or, (A –15)² = G² ……..(i)

As AM exceeds HM by 27. So, A – H = 27 Or, H = A – 27

G² = AH = A(A – 27) Putting this value of G² in equation (i) we get

(A –15)² = A(A – 27)

Or, – 30A + 225 = –27A

Or, 3A = 225

Or, A = 75. Or, (a+b) = 150

So, G = 60 Or, = 60 i.e ab = 3600

So, a + = 150

Or, a² –150a + 3600 = 0.

Or, a² –120a – 30a+ 3600 = 0.

Or, a(a – 120) – 30(a – 120) = 0

Or, (a – 30)(a – 120) = 0

So the value of the numbers would be (30, 120) or (120, 30)

So the lowest value is 30.

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Answer : (e)

Reason : The relationship between arithmetic mean (A), geometric mean (G) and harmonic mean (H) of two numbers if given by

G² = AH.

Also

And

From the above we can derive the identities stated in the alternatives.

Hence the answer is (e).

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Answer : (d)

Reason :

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Answer : (a)

Reason : The sum of n terms of an A.P., S_n = {2a + (n – 1)d}

= {t₁ + t_n}

S_n = 0 Þ {t₁ + t_n} = 0

Þ t₁ + t_n = 0

Þ t₁= –t_n.

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Answer : (b)

Reason : 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.

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.

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.

All the consecutive terms will be less than 1 if the first term as well as the common ratio is less than 1.

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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Answer : (c)

Reason : The reciprocals of the terms in a harmonic progression are in arithmetic progression. And vice versa. Therefore the reciprocals of a H.P. cannot be in G.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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10.

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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11.

Answer : (b)

Reason : Lets the middle number be m and the common difference of the A.P is d

So, (m-d) + m + (m +d) = 15 Or, m = 5

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12.

Answer : (c)

Reason : Putting n = 1, 2, 3, …. in T_n = (3 + n)/4, we get the series as 1, 5/4, 3/2, …

\a = 1, d = 1/4.

S₁₀₅ = (105/2)[2´1 + (104) ´ (1/4)] = (105/2) ´ 28 = 1470.

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13.

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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14.

Answer : (e)

Reason : The maximum number of basic solutions will be

Therefore (e) is the correct answer.

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15.

Answer : (a)

Reason :

Therefore,

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16.

Answer : (b)

Reason :

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17.

Answer : (c)

Reason :

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18.

Answer : (c)

Reason :

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19.

Answer : (e)

Reason :

Given: 2log₂A + 3log₃B + 4log₄C = 18 . . (1)

4log₂A + 5log₃B + 6log₄C = 30 . . . . (2)

6log₂A + 4log₃B + 10log₄C = 40 . . . . (3)

Multiplying equation (1) by 2 and subtracting equation (2) from the modified equation (1) we get

4log₂A + 6log₃B + 8log₄C = 36

4log₂A + 5log₃B + 6log₄C = 30 (3)

– – – –

log₃B + 2log₄C = 6. . . . (4)

Multiplying equation (2) by 3 and equation (3) by 2, and subtracting the modified equation (3) from the modified equation (2) we get

12log₂A + 15log₃B + 18log₄C = 90

12log₂A + 8log₃B + 20log₄C = 80

– – – –

7log₃B – 2log₄C = 10. . . . (5)

Adding equations (4) and (5) we get

log₃B + 2log₄C = 6

7log₃B – 2log₄C = 10

8log₃B = 16

\ log₃B = 2

Substituting the value of log₃B in equation (4) we get

2 + 2log₄C = 6

2log₄C = 4

\log₄C = 2

Substituting the values of log₃B and log₄C in equation (1) we get

2log₂A + 3 ´ 2 + 4 ´ 2 = 18

or 2log₂A + 6 + 8 =18

or 2log₂A = 18 – 14 = 4

\log₂A = 2

From above we get the following:

log₂A = 2 log₃B = 2 log₄C = 2

\A = 2² \B = 3² \ C = 4²

or A = 4 or B = 9 or C =16.

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20.

Answer : (e)

Reason :

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21.

Answer : (c)

Reason : According to the given equation, we can say that

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22.

Answer : (d)

Reason : According to the given equation we can say that

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23.

Answer : (e)

Reason : The given equation can be written as ,

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24.

Answer : (d)

Reason : The given expression can be written as,

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25.

Answer : (a)

Reason : The given equality can be written as,

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26.

Answer : (b)

Reason : Given that

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27.

Answer : (a)

Reason : In the word “INTEGRATION” number of vowels are : I, E, A, I, O = 5 and number of consonants are : N, T, G, R, T and N = 6.

Keeping five vowels together as one unit, the remaining six consonants can be arranged in ways and the five vowels can be arranged as ways.

Total number of ways = ´ = 10800.

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28.

Answer : (c)

Reason :

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29.

Answer : (d)

Reason : With two Cs occupying the first and the last place, the total number of places to be filled is 12 with 12 letters which has 2 O’s, 2 R’s, 3 E’s and 2 N’s. Therefore the no. of arrangements =

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30.

Answer : (b)

Reason :

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31.

Answer : (d)

Reason : The number of ways the candidate can choose 7 questions is

(5A, 2B), (4A, 3B), (3A, 4B) and (2A, 5B), where 5A, 2B means 5 questions from part A and 2 question from part B.

= 90 + 300 + 300 + 90 = 780.

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32.

Answer : (c)

Reason : When two particular persons are together, the remaining 10 soldiers are to be selected from (n–2) in ways. So, 2m =

When three particular persons are together, the remaining 9 soldiers are to be selected from (n–3) in ways. So, m =

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33.

Answer : (d)

Reason : Given that

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34.

Answer : (a)

Reason : Given that , this can be rewritten as,

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35.

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.

(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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36.

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 independent 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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37.

Answer : (b)

Reason :

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38.

Answer : (c)

Reason :

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39.

Answer : (d)

Reason :

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40.

Answer : (c)

Reason : Here, the geometric mean of 10 observations is 16.2 and hence, the product of all the observations will be = 16.2¹⁰_. Now, one of the observation was wrongly noted as 12.9 though it is actually 21.9. Hence, the actual product is given by: and correspondingly, the actual geometric mean is obtained as: .

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41.

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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42.

Answer : (c)

Reason : At y-axis the x – co-ordinate is always zero. Here = 2x – 3 = –3 and ; Fox x = 0, = –3 (a negative value) and = 2(a positive value).

So, we can say that y is decreasing at an increasing rate.

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43.

Answer : (c)

Reason : The given function is

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44.

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.

So, the optimum point is where the contribution is maximum i.e. (800, 600).

The total time in the phase 1 was 2000 hours.

The time consumed in production as per optimum schedule is .

So, the balance is 2000 – 2000 = 0 hr.

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45.

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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46.

Answer : (d)

Reason : Median is the middle most point of a set of observations. Here, the runner has maintained such a speed that he can maintain his position just at the middle of all the participants.

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47.

Answer : (a)

Reason : Here, 29 is being added to all the numbers 200 numbers in a set of 401 numbers. The median of these numbers is the 200^th one. As it is being increased by 29, the median will also increase by 29. Hence, the option (a) is correct.

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48.

Answer : (d)

Reason : As two students left the class whose scores were the same as that of the mean score of the entire class, the mean does not change but the variance increases.

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49.

Answer : (e)

Reason : Harmonic mean of a set of ‘n’ quantities, H =

Given: = H = =

From above, n = H.

n = = 3

Harmonic mean after the expansion = = =

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50.

Answer : (b)

Reason : Let the two A.P. s’ be

a, a + d, a + 2d, …, a + (n – 1)d

a₁, a₁ + d₁, a₁ + 2d₁, …, a₁ + (n – 1)d₁

We are given that

Þ …………. (1)

We want to find the ratio of the 6^th term, i.e., we want to find which can be obtained from (1) above by writing it as

i.e., by putting n = 11 in equation (1) above therefore,

= = = 1

Þ a + 5d = a₁ + 5d₁

\ The 6^th terms is equal in both the series.

\ The difference between the sixth term of the series is zero.

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51.

Answer : (e)

Reason : The sample space for this experiment is S ={1, 2, 3, 4, 5, 6}

The set of favorable event E = {1, 2, 3}

Therefore required probability P(E) =

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52.

Answer : (c)

Reason : The total number of ways one letter can be chosen from 11 letters of the word ‘PROBABILITY’ is .

The total number of ways a vowel is chosen =

Therefore the required probability is = 4/11.

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53.

Answer : (d)

Reason : If the race is between A and B only and there is no tie, then P(A) + P(B) = 1.

Given that P(A) =2 P(B)

\P(B) = 1/3

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54.

Answer : (b)

Reason : When the cards are first drawn and not replaced, the sample space is n(S) = 52 ´ 51

The number of ways two cards can be chosen such that both are aces is n(E)

The probability of the favorable event =

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55.

Answer : (e)

Reason : There are 7 letters in this word and there are three even positions namely 2^nd, 4^th and 6^th which is filled by vowels. So the number of ways these palaces can be filled by vowels: U, O and I is (3!) and the remaining 4 places can be filled by consonants C, S, H and N in (4!) ways.

\ Total number of arrangements = (3!) (4!) = 6 ´ 24 = 144.

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56.

Answer : (a)

Reason : The sequence in which 6 boys and six girls can sit with row starting with the boy is as follows:

B G B G B G B G B G B G.

\The number of ways 6 boys can occupy six positions is (6!), and the number of ways 6 girls can occupy six positions is (6!).

Total number of ways = (6!) (6!) = (6!)².

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57.

Answer : (a)

Reason : If there are n items in data then the average deviation from zero can be written as

= = = Mean.

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58.

Answer : (b)

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

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59.

Answer : (d)

Reason : Co-efficient of variation is a relative measure of dispersion based upon standard deviation and mean.

Co-efficient of variation = .

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60.

Answer : (a)

Reason : Standard deviation is independent of change of origin i.e., it remains unchanged even if all the items in the data set are increased or decreased by the same quantity.

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61.

Answer : (c)

Reason : The appropriate mean for a set of ratios using the denominators of the ratios as weights is weighted arithmetic mean.

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62.

Answer : (e)

Reason : Standard deviations of two or more data sets can be mathematically combined

Arithmetic means of two or more data sets can be mathematically combined

Geometric means of two or more data sets can be mathematically combined

Harmonic means of two or more data sets can be mathematically combined.

Medians of two or more data sets cannot be mathematically combined

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63.

Answer : (a)

Reason : Let the events are: A: A male is selected and B: A smoker is selected. Now, we have, P(A or B) = 0.7, P(A and B) = 0.4 and P(A/B) = 0.67. The probability of selecting a non-smoker is given by:

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64.

Answer : (c)

Reason : Coefficient of variation = (standard deviation / mean) × 100

A standard deviation equal to zero implies that there is no deviation in the data set. The same will be reflected by the c.v. provided mean is not equal to zero.

Even when the standard deviation is 1 the c.v. can be meaningfully used for comparison of variability provided mean is not equal to zero.

Hence it cannot be meaningfully used for comparison of variability when mean of one or more data sets is zero.

When the mean is equal to 1, the c.v. can be meaningfully used for comparison of variability.

When the mean and standard deviation are equal for one or more sets of data, the c.v. can be meaningfully used for comparison of variability.

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65.

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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66.

Answer : (b)

Reason : In a positively skewed distribution the majority of the observations are concentrated towards the lower end of the scale. In a negatively skewed distribution majority of the observations are concentrated towards the higher end of the scale. In a skewed distribution the distribution of the data is not symmetrical.

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67.

Answer : (a)

Reason : Three balls may be selected at random by ⁹C₃ ways = 84 ways. The balls thus drawn may be of same colour, if either three blue balls or three black balls are drawn. That may be done by ³C₃ + ⁴C₃ ways = 1 + 4 = 5 ways (as the events are mutually exclusive). Therefore, the required probability will be = 5/84.

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68.

Answer : (a)

Reason : If events A and B are mutually exclusive then P(A and B) = 0 because A and B can not occur at the same time.

Mutual exclusiveness of two events A and B does not imply that both of them have zero probabilities of occurrence.

Mutual exclusiveness does not mean that the probability of any one of them occurring is either 0 or 1.

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69.

Answer : (a)

Reason : The probability of the occurrence of an event is expressed as a number which lies between 0 and 1.

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70.

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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71.

Answer : (e)

Reason : The options (a), (b), (c) and (d) are properties of classical probability. Therefore they are true. Option (e) is false because if , .

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72.

Answer : (a)

Reason : Total number of ways the letters of the word ‘SUCCESS’ can be rearranged n(S) = .

If all S’s have to come together, we assume them to be as a single letter, then the total number of possible ways the letters can be rearranged n(E) = .

The probability of the favorable event =

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73.

Answer : (c)

Reason : Numbers whose decimals are not terminating and non-repetitive are included in a set of numbers called as irrational numbers.

1/3 = 0.333333333……

65/67 = 0.970149253 …..

We see decimals for and 65/67 are not terminating and non-repetitive.

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74.

Answer : (b)

Reason :

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75.

Answer : (a)

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

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

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

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

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

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76.

Answer : (b)

Reason : The sequence is: 15000, 13500, 12150, 10935,…

a = 15000 and r = 9/10

Therefore, T₁₀ = ar^n-1 = 15000 (9/10)⁹ = 5811.

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77.

Answer : (c)

Reason : Given : s = 3 n = 10

After 3 is added to every observation, mean of the resulting data set =

Standard deviation of the resulting data set will be unchanged, i.e. = 3.

\ Coefficient of variation of the resulting data set = .

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78.

Answer : (b)

Reason : (a) Simple arithmetic mean is the appropriate average when the denominators are same and no weighting is required.

(b) Weighted arithmetic mean is appropriate when the denominators of the ratio data are used as weights.

(d) Weighted harmonic mean is appropriate when the numerators of the ratio data are used as weights.

(e) Geometric mean is appropriate when the quantities vary over time.

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79.

Answer : (a)

Reason :

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80.

Answer : (d)

Reason : The function given is,

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81.

Answer : (d)

Reason : The general LPP calls for optimizing a linear function for variables called the ‘objective function’ subject to a set of linear equations and/or inequalities called the ‘constraints’ or ‘restrictions’.

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82.

Answer : (e)

Reason :