Bits Pilani Goa Test - 5 Paper Expl Answers

BITS PILANI Test 5 – Answer Key & Explanatory Answers 1.A father can do job as fast as 2 sons working together. If one son does the job in 3 hours and the other in 6 hours, the number of hours taken by the father, to do the job alone is 1) 1 2) 2 3) 3 4) 4 2.The perimeter of a rhombus is 240 m and the distance between any two parallel sides is 20 m. The area of the rhombus in sq.m. is 1) 600 2) 1200 3) 2400 4) 4800 3.A man sold an article for Rs. 450, after allowing a discount of 16 2/3 % on the printed price. What is that printed price? 1) Rs.525 2) Rs.530 3) Rs.535 4) Rs.540 4.A sum of Rs. 770 has been divided among A, B, C in such a way that A receives 2/9th of what B and C together receive. Then A's share is 1) Rs.140 2) Rs.154 3) Rs.165 4) Rs.170 5.A man bought 4 dozen eggs at Rs. 24 per dozen and 2 dozen eggs at Rs. 32 per dozen. To gain 20% on the whole, he should sell the eggs at 1) 16 per dozen 2) 21 per dozen 3) 32 per dozen 4) 35 per dozen 6. P's salary is 25% higher than Q, what percentage is Q's salary lower than that of P? 1) 20 2) 29 3) 31 4) 331/3 7.A and B start running at the same time and from the same point around a circle. If A can complete one round in 40 seconds and B in50 seconds, how many seconds will they take to reach the starting point simultaneously? 1) 10 2) 200 3) 90 4) 2000 8.∆ABC is an isosceles triangle with AB = AC = 15 cm and altitude from A on BC is 12 cm. Length of side BC is 1) 9 cm 2) 12 cm 3) 18 cm 4) 20 cm 9.The mid points of AB and AC of a triangle ABC are respectively X & Y. If BC + XY = 12 units, then the value of BC XY is: 1) 2 units 2) 6 units 3) 8 units 4) 4 units 10.The difference between two numbers is 9 and the difference between their squares is 207. The numbers are 1) 17, 8 2) 16, 7 3) 15, 6 4) 23, 14 11.Find the sum of all two-digit numbers that leave a remainder, which is a prime number, when divided by 13? (1) 1942 (2) 2030 (3) 1592 (4) 2278 (5) None of these Solution: The possible prime number remainders when divided by 13 are 2, 3, 5, 7 and 11. When divided by 13, the two digit numbers which would give a remainder of: i) 2:- 2, 15, 28, 41, 54, 67, 80 and 93. (8 terms) Sum = 2 + 15 + 28 + … + 93 = 380. ii) 3:- 3, 16, 29, 42, 55, 68, 81 and 94. (8 terms) Sum = 3 + 16 + 29 + … + 94 = 388. iii) 5:- 5, 18, 31, 44, 57, 70, 83 and 96. (8 terms) Sum = 5 + 18 + 31 + … + 96 = 404. iv) 7:- 7, 20, 33, 46, 59, 72, 85 and 98. (8 terms) Sum = 7 + 20 + 33 + … + 98 = 420. v) 11:- 11, 24, 37, 50, 63, 76 and 89. (7 terms) Sum = 11 + 24 + 37 + … + 89 = 350. Hence, the total sum = 380 + 388 + 404 + 420 + 336 = 1942. Hence, the correct answer is option 1. 12.Set D is defined as follows: D = {1/n, where n is a two-digit number}. An operation C(x, y) is defined such that: C(x, y) = x + y + xy. This operation is performed on the elements of the set D taking 2 consecutive elements at a time until the set of the elements is reduced to a single value. What is the value thus obtained? (1) 11.1

(2) 10.0

(3) 9.9

(4) 9.0

(5) 99.0

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Solution: Set ‘D’ contains the terms: D = {1/10, 1/11, 1/12 … 1/99}. C(x, y) = x + y + xy. It is given that this operation is performed taking 2 elements of set D at a time. Consider the first 2 elements: x = 1/10, y = 1/11, then C(x, y) = x + y + xy = 1/10 + 1/11 + (1/10)(1/11) = (11 + 10 + 1)/(10 × 11) = 2/10. Now consider the above obtained term and the next available term, now x = 2/10, y = 1/12, then C(x, y) = x + y + xy = 2/10 + 1/12 + (2/10)(1/12) = (24 + 10 + 2)/(10 × 12) = 3/10. Now consider the above obtained term and the next available term, now x = 3/10, y = 1/13, then C(x, y) = x + y + xy = 3/10 + 1/13 + (3/10)(1/13) = (39 + 10 + 3)/(10 × 13) = 4/10. The similar pattern continues for another (99 – 13) = 86 terms, and the final value obtained would be (4/10) + {(4/10) – (3/10)} × (86) = (4/10) + (86/10) = 90/10 = 9.0 Hence, the correct answer is option 4. 2

2

2

2

13.A series S = 1 + 2 + 3 + …. n ; where: 1 < n ≤ 400. How many values of n exist such that S is exactly divisible by 8? (1) 8 (2) 24 (3) 16 (4) 32 (5) None of these Solution: 2 2 2 2 S = 1 + 2 + 3 + …. n ; where: 1 < n ≤ 400. i) Consider squares of consecutive odd numbers: 2 2 2 2 2 2 2 2 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + … 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + … When the terms of the above series are divided by 8, the remainders obtained are as follows: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8, which is exactly divisible by 8. ii) Consider squares of consecutive even numbers: 2 2 2 2 2 2 2 2 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + … 4 + 16 + 36 + 64 + 100 + 144 + 196 + 256 + … When the terms of the above series are divided by 8, the remainders obtained are as follows: 4 + 0 + 4 + 0 + 4 + 0 + 4 + 0 = 16, this is exactly divisible by 8. Thus, from above, it is clear that the first 16 terms of series S have to be added to get a number which is exactly divisible by 8. Similarly, every subsequent 16 terms added together would be exactly divisible by 8. Thus, there are a total of (400/16) = 25 such sets or 25 such values for n. Hence, the correct answer is option 5. 14.At a particular manufacturing industry, the price (P) is determined by the expression: P = (300 – 5x), where x is the 2 number of units sold. The manufacturing cost is given by: C(x) = 300 + 60x – x . Now, an additional tax of Rs 40 per unit is levied on the industry. If the number of units sold is such that it yields the maximum possible profit, then what is the corresponding profit per unit item sold? (1) 288 (2) 88 (3) 70 (4) 310 (5) None of these Solution: 2 Given: Price (P) = (300 – 5x); Cost = C(x) = 300 + 60x – x ; Tax = Rs 40/unit. 2 2 Total cost to the company = Given cost + tax = TC(x) = 300 + 60x – x + 40x = 300 + 100x – x . 2 Total selling price = P = (300 – 5x) × (x) = 300x – 5x . 2 2 2 Total Profit (Pr) = Selling Price – Cost to the company = (300x – 5x ) – (300 + 100x – x ) = (–4x + 200x – 300). To maximize the total profit, set dPr/dx to be equal to aero. dPr/dx = –8x + 200 = 0. → x = 200/8 = 25. 2 Thus, total maximum possible profit = –4x + 200x – 300 2 = –4(25) + 200(25) – 300 = –2500 + 5000 – 300 = 2200. Profit per unit item = 2200/25 = Rs 88. Hence, the correct answer is option 2. 15.A milk solution has milk and water in the ratio of 5:3. If 40% of this mixture is replaced with only water, then the concentration of milk in that solution is reduced by? (1) 37.5 (2) 62.5 (3) 40.0 (4) 50.0 (5) 66.7 © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Solution: Let the amount of the milk solution be 8x Litres (8 L). The ratio of milk and the water in the solution is 5:3, hence the amount of milk in the solution is 5x L and the amount of water in the solution is 3x L. Thus, concentration of milk in the solution = (5x/8x) × 100 = 62.5%. 40% of the above solution = 40% of 8x L = 3.2x L. In this, amount of milk = 40% of 5x L = 2x L and the amount of water = 40% of 3x L = 1.2x L. This portion of the solution is replaced with water. Thus, out of the 5x L of milk, 2x L is removed, hence only 3x L of milk is remaining and also to the 3x L of water, 2x L is added, hence now 5x L of water is present. Thus, the new ratio of milk and water in the new solution is 3:5. Thus, concentration of milk in the solution = (3x/8x) × 100 = 37.5%. Thus, the reduction in the concentration of milk in the solution is given by: 100 × (62.5 – 37.5)/(62.5) = 40%. Hence, the correct answer is option 3. 16.If ‘A’ is the first of n Arithmetic Means (n > 1) and ‘H’ is the first of n Harmonic Means (n > 1) between two positive integers x and y, then which of the following is [(A/H) – 1] equal to? 2 2 (1) [(n)/(n+1) ][{√(x/y)} – {√(y/x)}] 2 2 (2) [(n)/(n+1) ][{√(x/y)} + {√(y/x)}] 2 2 (3) [(n)/(n+1) ][{√(xy)} – {√(1/xy)}] 2 2 (4) [(n)/(n+1) ][{√(xy)} + {√(1/xy)}] (5) None of these Solution: Given 2 positive numbers x and y; ‘n’ Arithmetic and Harmonic means. st 1 Arithmetic Mean = A = x + {(y – x)/(n + 1)} = (xn + y)/(n + 1) st 1 Harmonic Mean = H = (n + 1)(xy)/(x + ny) Let R = [(A/H) – 1] → R = {[(xn + y)/(n + 1)] / [(n + 1)(xy)/(x + ny)]} – 1 2 → R = {[(xn + y) (x + ny)] / [(n + 1) (xy)]} – 1 2 2 → R = {[(xn + y) (x + ny)] – [(n + 1) (xy)]} / [(n + 1) (xy)] 2 2 2 2 → R = {(x n + n xy + xy + ny ) – (n xy + 2nxy + xy)} 2 / [(n + 1) (xy)] 2 2 2 → R = (x n + ny – 2nxy) / [(n + 1) (xy)] 2 2 2 → R = {(n)/ [(n + 1) ]} × [(x + y – 2xy) / (xy)] 2 → R = {(n)/ [(n + 1) ]} × [(x/y) + (y/x) – 2] 2 2 → R = {(n)/ [(n + 1) ]} × [{√(x/y)} – {√(y/x)}] . Hence, the correct answer is option 1. Directions for Questions 17 and 18: Refer the following data to answer the questions given below. Port City Aloha is located 1800 kms directly to the east of another port city Bailamos. They lie in two different time zones. Ships between these two port cities travel at 80 km/hr in still water but their actual running speeds are altered due to ocean currents flowing in the direction of Aloha to Bailamos. The arrival and departure time schedule of ships is given in the table below. Note that the timings listed are in local time (of the corresponding city). Departure Starting City Aloha Bailamos

Local Time 8 PM 12 PM

Arrival (Next Day) Destination City Local Time Bailamos 12 PM Aloha 8 PM

Solutions to questions 17 and 18: Let the speed of the current and the time gap between Aloha and Bailamos be denoted by s and t respectively. i) Consider journey from Aloha to Bailamos: Journey time = 8 PM to 12 PM (next day) + Time gap = (16 + t) hrs (Since Aloha lies to the east of Bailamos). Boat Speed = (80 + x) km/hr (Since the current flows from Aloha to Bailamos). Distance = 1800 = (16 + t) (80 + x) (1) ii) Consider journey from Bailamos to Aloha: © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Journey time = 12 PM to 8 PM (next day) – Time gap = (32 – t) hrs (Since Bailamos lies to the west of Aloha). Boat Speed = (80 – x) km/hr (Since the current flows from Aloha to Bailamos). Distance = 1800 = (32 – t) (80 – x) (2) Solving (1) and (2), we get t = 2 hrs, x = 20 km/hr. 17.What is the speed of the ocean current flowing between Aloha and Bailamos (in km/hr)? (1) 10 (2) 20 (3) 25 (4) 15

(5) None of these

Solution: Speed of the ocean current (from Aloha to Bailamos) = x = 20 km/hr. Hence, the correct answer is option 2. 18.Another city Chow lies exactly halfway between Aloha and Bailamos, but Chow is in the same time zone as Bailamos. The water current flows at twice the speed between Aloha and Chow as it did between Aloha and Bailamos. If a boat, which travels at 60 km/hr in still water, leaves Aloha at 12 PM, then what is the time at Chow when the boat reaches there? (1) 11 PM (2) 9 PM (3) 7 PM (4) 6 PM (5) None of these Solution: Distance between Aloha and Chow = 1800/2 = 900 kms. Time gap between Aloha and Chow = t = 2 hrs. Speed of the current = 2x = 40 km/hr. Boat Speed (in still water) = 60 km/hr. Total speed = 60 + 40 = 100 km/hr. So, the journey time from Aloha to Chow = 900/100 = 9 hrs. Departure time at Aloha = 12 PM. Thus, the arrival time at Chow = 12 + 9 – 2 = 7 PM. Hence, the correct answer is option 3. 1

2

1

n

n-1

19.The function f(x) is defined as follows: f(x) = x + (1/x), when x > 0, f (x) = f(f(x)), f (x) = f(f (x)), … f (x) = f(f (x)). If x = 1 2 10 1, what is the value of f(x) × f (x) × f (x) × … f (x)? (1) 55 (2) 377 (3) 89 (4) 233 (5) 144 Solution: Given: x = 1, f(x) = x + (1/x) = 1 + (1/1) = 2. 1 f (x) = f(f(x)) = f(2) = 1 + (1/2) = 3/2. 2 1 f (x) = f(f (x)) = f(3/2) = 1 + (1/(3/2)) = 5/3. 3 2 f (x) = f(f (x)) = f(5/3) = 1 + (1/(5/3)) = 8/5. 4 3 f (x) = f(f (x)) = f(8/5) = 1 + (1/(8/5)) = 13/8. 5 4 f (x) = f(f (x)) = f(13/8) = 1 + (1/(13/8)) = 21/13. 6 5 f (x) = f(f (x)) = f(21/13) = 1 + (1/(21/13)) = 34/21. 7 6 f (x) = f(f (x)) = f(34/21) = 1 + (1/(34/21)) = 55/34. 8 7 f (x) = f(f (x)) = f(55/34) = 1 + (1/(55/34)) = 89/55. 9 8 f (x) = f(f (x)) = f(89/55) = 1 + (1/(89/55)) = 144/89. 10 9 f (x) = f(f (x)) = f(144/89) = 1 + (1/(144/89)) = 233/144. 1 2 10 Thus, f(x) × f (x) × f (x) × … f (x) = 2 × 3/2 × 5/3 × 8/5 × … × 144/89 × 233/144 = 233. Hence, the correct answer is option 4. 20.A cube is cut into 27 equal cubes. How many distinct straight lines can be drawn in all such that they pass the centre of any three of the smaller cubes? (1) 27 (2) 39 (3) 45 (4) 48 (5) None of these Solution: A cube is cut into 27 cubes, so it has 3 cubes along the length, breadth and the height axes. Along the length axis, there are 9 sets of 3 adjacent cubes each, through each set, a straight line can be drawn passing through the centres of 3 cubes, thus 9 lines can be drawn. Similarly, along the breadth axis, there are 9 sets of 3 adjacent cubes each, through each set, a straight line can be drawn passing through the centres of 3 cubes, thus 9 lines can be drawn. © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Similarly, along the height axis, there are 9 sets of 3 adjacent cubes each, through each set, a straight line can be drawn passing through the centres of 3 cubes, thus 9 lines can be drawn. Also, the cube can be split into 3 sets of a block of 9 cubes (3 by 3) each, along each axis. Along each of this 3 by 3 block, the 2 diagonals connect the centres of 3 cubes each. Thus, along the 3 different axes, there are 3 × 3 = 9 blocks and thus the total number of such diagonals = (2 × 3 × 3) = 18 such diagonals (on the face). Also, in the complete cube, there are 4 diagonals which connect the opposite vertices, which also join the centres of 3 cubes. Thus, in all, the number of straight lines can be drawn such that they pass the centre of any three of the smaller cubes = 9 + 9 + 9 + 18 + 4 = 49 lines. Hence, the correct answer is option 5. 21.Select the related word/letters/number from the given alternatives. Head : Human Body :: ? 1) Arc : Circle 2) Cube : Circle 3) Square : Circle 22.Select the related word/letters/number from the given alternatives. PZQW : NXOU :: FISK : ? 1) EFPJ 2) FERI

4) Triangle : Circle

3) DGQI

23.Select the related word/letters/number from the given alternatives. 7 : 19 :: 10 : ? 1) 25 2) 30 3) 21

4) HKVM

4) 23

24. For the following questions Find the odd word/letters/number pair from the given alternatives. 1) Bigger 2) Faster

3) Greater

4) Taller

25. For the following questions Find the odd word/letters/number pair from the given alternatives. 1) AEIM 2) BFJN

3) CGKO

4) FDKN

26. For the following questions Find the odd word/letters/number pair from the given alternatives. 1) 140 → 45 2) 110 → 35

3) 100 → 30

4) 80 → 25

27.Arrange the following words as per order in the dictionary 1. Forest 2. Fascinating 3. Fantastic 4. Fabulous 1) 2, 4, 3, 1 2) 4, 2, 3, 1

3) 4, 3, 2, 1

4) 2, 3, 4, 1

28.Which one set of letters when sequentially placed at the gaps in the given letter series shall complete it ? a_ ca_c_dc_d_ad_ 1) ddacdc 2) daadca 3) dadaac 4) ddaacc 29.Which number comes next in the series? 1, 5, 2, 6, 3, 7, ? 1) 6 2) 5

3) 4

4) 3

30.P and Q are sisters. R and S are brothers. P's daughter is R's sister. What is Q's relation to S? 1) Mother 2) Grandmother 3) Sister 4) Aunt Directions for questions 31 to 35: Refer the following data to answer the questions given below. The four members of a Chinese family: Mother, Father, Son and Daughter-in-law – who are sitting around a circular table at equidistant seats, in an Indian restaurant. They ordered only Idlies and nothing else. The number of Idlies consumed by each one in the family is a prime number and the total number of idlies consumed by this family of four is not more than 44. The middle names of the four Chinese family members are Hong, Leeong, Huang and Liang, not necessarily in the same order. Liang is not sitting adjacent to Hong. The person on the right of Leeong is not a female. The person sitting to the left of Liang ate 11 Idlies, but it is not Leeong. Huang is a blood relative of a person among these four, who ate the highest number of Idlies and sits opposite to Huang. No two males sit opposite to each other and © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers the son sits to the right of the person who ate the second highest number of Idlies. Eleven Idlies are neither the maximum nor the minimum number of Idlies ate by anyone. Solutions for 31 to 35: 1. Hong and Liang are not sitting adjacent to each other, so they must be sitting opposite to each other. Therefore Huang and Leeong must be sitting opposite to each other. 2. Huang and his/her blood relative is Leeong, who sitting opposite to Huang. Since no two males sit opposite to each other, Huang and Leeong must be the mother and the Son in the family, not necessarily in the same order. 3. The person sitting to the left of Liang is not Leeong. So, it must be Huang and Huang ate 11 Idlies. 4. The person sitting to the right of Leeong is a male. So Hong, who is sitting to the right of Leeong is the father in the family and Liang is the daughter in law of the family. 5. Huang ate 11 Idlies. Leeong ate the highest number of Idlies. Huang or Leeong could be the son in the family. The son in the family is sitting to the right of the person, who ate the second highest number of Idlies. 6. Leeong must have consumed 17 Idlies. Either Hong or Liang must have consumed 13 Idlies. 17 + 13 + 11 = 41, One of the four in the family could have consumed either 3 or 2 Idlies. The two possible orders are Case I: Hong (Father) 2 or 3 Huang (Mother) 11

Leeong (Son) 17 Liang (Daughter in law) 13

Case II: Hong (Father) 13 Huang(Son) 11

Leeong (Mother) 17 Liang (Daughter in law) 2 or 3

31.Which of the following could not be the total number of Idlies consumed by the ladies in the family? (1) 20 (2) 24 (3) 19 (4) 14

(5) None of these

ANSWER:Hence, the correct answer is option 4. 32.Which of the following could be the total number of Idlies consumed by the Father in the family? (1) 17 (2) 7 (3) 5 (4) 13

(5) None of these

ANSWER:Hence, the correct answer is option 4. 33.What is the relation of Huang with Leeong? (1) Mother (2) Father

(3) Daughter

(4) Wife

(5) Indeterminate

(3) Hong

(4) Huang

(5) Indeterminate

(3) 19

(4) 3

(5) None of these

ANSWER:Hence, the correct answer is option 5. 34.Who is sitting opposite to the daughter-in-law? (1) Leeong (2) Liang ANSWER:Hence, the correct answer is option 3. 35.How many Idlies did Leeong consume? (1) 13 (2) 17 ANSWER:Hence, the correct answer is option 2.

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Directions for questions 36 to39: Each question is followed by two statements A and B. Indicate your responses based on the following directives: Mark (1) if the question can be answered using A alone but not using B alone. Mark (2) if the question can be answered using B alone but not using A alone. Mark (3) if the question can be answered using either statement alone. Mark (4) if the question can be answered using A and B together, but not using either A or B alone Mark (5) if the question cannot be answered even using A and B together. 36.There were five questions in a test. Each wrong answer caries negative marking of 0.5 and each right answer carries 1 mark. If Ramesh attempts all the questions, then is the marks scored by Ramesh greater than one? A. The square of the marks scored by Ramesh is greater than one. B. The square root of marks scored by Ramesh is greater than the marks scored by him. (1) if the question can be answered using A alone but not using B alone. (2) if the question can be answered using B alone but not using A alone. (3) if the question can be answered using either statement alone. (4) if the question can be answered using A and B together, but not using either A or B alone (5) if the question cannot be answered even using A and B together. Solutions: Let the marks scored by Ramesh be ‘x’. We know that – 2.5 ≥ x ≥ 2.5 From (A), 2 2 x > 1. i.e., x – 1 > 0 ⇒ (x – 1) (x + 1) > 0 i.e., (x – 1) [x –(–1)] >0 ∴ x does not lie between –1 and 1. It cannot be said if x > 1 From (B), Sqrt. (x) > x ⇒ x – Sqrt. (x) < 0 ⇒ Sqrt. (x) {Sqrt. (x) – 1} < 0 Therefore, Sqrt. (x) lies between 0 and 1. The second statement alone is sufficient to answer the question. Hence, the correct answer is option 2. 37.How many ordered pairs of positive integers (a, b) is possible? 4 4 A. a – b = 12652 B. Both ‘a’ and ‘b’ are odd. (1) if the question can be answered using A alone but not using B alone. (2) if the question can be answered using B alone but not using A alone. (3) if the question can be answered using either statement alone. (4) if the question can be answered using A and B together, but not using either A or B alone (5) if the question cannot be answered even using A and B together. Solutions: Statement A alone is sufficient to answer the given question. 4 4 a – b = 12652 4 4 i.e., a – b is even ⇒ both a, b are even (or) both a, b are odd. 4 4 2 2 a – b = (a + b) (a – b) (a + b ) Case 1: a and b are even, then a + b = even a – b = even 2 2 a + b = even 4 4 ∴ a – b is divisible by 8. Case 2: a and b are odd, then a + b = even a – b = even © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers 2

2

a + b = even 4 4 ∴ a – b is divisible by 8. But 12652 is not divisible by 8. Hence there is no such pair is possible. Statement B alone is not sufficient to answer the given question uniquely. Hence, the correct answer is option 1. 38.If a gathering of a certain number of families consists of people belonging to two generations only, then what is the minimum number of families in the gathering? A. The number of families is less than the number of boys, the number of boys is less than the number of girls and that the number of girls is less than the number of parents. B. The minimum number of single parent families is three. (1) if the question can be answered using A alone but not using B alone. (2) if the question can be answered using B alone but not using A alone. (3) if the question can be answered using either statement alone. (4) if the question can be answered using A and B together, but not using either A or B alone (5) if the question cannot be answered even using A and B together. Solutions: Both A and B are needed to answer the given question. The given data can be represented as follows: f < b < g < p (1) where the letters f, g, b and g represent the number of families, girls, boys and parents respectively Keeping the difference between the consecutive groups mentioned above, to the minimum it can be written, b = f + 1; g = b + 1 = f + 2; p = g + 1 = f + 3 (2) ‘p’, the number of parents is related to the number of families, by the question p = 2f – s, where ‘s’ is the number of single parent families. Minimum value of ‘s’ is given as 3. p = 2f – 3 (3) Combining (2) and (3) f + 3 = 2f – 3 ⇒ f = 6 Before concluding about the minimum number of families, let us check whether the condition about the number of children is satisfied or not. n = number of children ≤ 3f, because each family has less than or equal to 3 children. When f = 6, b = 7, g = 8 and the number of children is = 15, which is < (3 × 6), i.e. 18. Hence the condition is satisfied. Hence, the minimum number of families = 6. Hence, the correct answer is option 4. 39.What is the value of | 𝑎 # + 𝑏 # − 2𝑎𝑏 |? A. √a = 3. 2 2 B. a + b – 6a + 12b + 45 = 0 (1) if the question can be answered using A alone but not using B alone. (2) if the question can be answered using B alone but not using A alone. (3) if the question can be answered using either statement alone. (4) if the question can be answered using A and B together, but not using either A or B alone (5) if the question cannot be answered even using A and B together. Solutions: Statement A alone is not sufficient to answer the question. From statement B alone, 2 2 a + b – 6a + 12b + 45 = 0 2 2 ⇒ a – 6a + 9 + b + 12b + 36 = 0 2 2 ⇒ (a – 3) + (b + 6) = 0 a = 3 and b = – 6 2 2 a + b – 2ab = 9 + 36 + 36 = 81 2 2 | Sqrt. [a + b – 2ab] | = | Sqrt. [81] | = 9 Hence, statement B alone is sufficient to answer the question. Hence, the correct answer is option 2. Directions for questions 40 : Refer the following data to answer the questions given below. © 2017 ETHNUS

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BITS PILANI Test 5 – Answer Key & Explanatory Answers Six teams viz. Manchester United, Bayern Munich, Chelsea, Real Madrid, Arsenal and Liverpool participated in a tournament. None of the matches scheduled were abandoned. Each team has to play exactly one match with each of the other teams. If a team wins a match, it is awarded two points and gets no points for losing a match. In case a match is a tie, then each of the two teams playing the match would be awarded one point. It is known that there were no more than two ties in the tournament. After the tournament was over, it was known that the total sum of the points scored by (Manchester United & Chelsea), (Chelsea & Bayern Munich), (Real Madrid & Chelsea), (Chelsea & Liverpool) and (Arsenal & Chelsea) are respectively 16, 15, 14, 12 and 9 points. 40. If it is known that Bayern Munich won a match against Manchester United, then Manchester United tied their match with which of the following teams? (1) Manchester United (2) Chelsea (3) Liverpool (4) Arsenal (5) Indeterminate Solutions: Let us take the first letter of the name of each team to represent its name. Now, it is given that MU + C = 16 C + BM = 15 RM + C = 14 C + L = 12 A+C=9 Adding the above five equations, we get MU + 5C + BM + RM + L + A = 66 (1) The total number of games played is 5 + 4 + 3 + 2 + 1 = 15 Each game is worth of two points, hence total points is 15 × 2 = 30 Or MU + C + BM + RM + L + A = 30 (2) Comparing with equation (1) and (2), we get; 4C = 36 C = 9. The maximum point that C can score is 10 points. Now it is 9 points, so it has to be of (4 wins and 1 tie) MU= 7 (3 wins, 1 tie, 1 loss) BM = 6 (3 wins, 2 loss) L = 3 (1 win, 1 tie. 3 loss) RM = 5 (2 wins, 1 tie, 2 loss) A = 0. It is clearly seen that A lost all the five matches. Team L wins only 1 match and that is against A. As there are two ties they must be between MU, RM, C and L, but not BM & A. Now we can have the following possibilities of ties:

Case I: C ties with MU then RM ties with L. Team MU

Win (BM or RM), L, A

Lose (BM or RM)

Tie C

BM

(MU or RM), L, A

(MU or RM), C

-

C

BM, RM, L, A

-

MU

RM

(MU or BM), A

(MU or BM), C

L

L

A

MU, BM, C

RM

A

-

MU, BM, C, L, RM

-

Team

Win

Lose

Case II: C ties with RM then MU ties with L.

Tie

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BITS PILANI Test 5 – Answer Key & Explanatory Answers MU BM C

BM, RM, A RM, L, A MU, BM, L, A

C MU, C -

L RM

RM L

L, A A

MU, BM RM, BM, C

C MU

A

-

MU, BM, C, L, RM

-

Case III: C ties with L then MU ties with RM. Team

Win

Lose

Tie

MU

BM, L, A

C

RM

BM C

RM, L, A MU, BM, RM, A

MU, C -

L

RM

L, A

BM, C

MU

L A

A -

MU, BM, RM MU, BM, C, L, RM

C -

Possibility of BM winning against MU is only in case I, in which RM ties with L and MU ties with C. Hence, the correct answer is option 2.



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