BITS PILANI Test - 7 – Answer Key & Explanatory Answers

BITS PILANI Test - 7 – Answer Key & Explanatory Answers 1.A, B and C can work together for Rs. 550/- A and B together are to do 7/11 of the work. The share of C should be? 1) Rs. 200 2) Rs. 300 3) Rs. 400 4) Rs. 450 2.In Δ ABC, the height CD intersects AB at D. The midpoints of AB and BC are P and Q respectively. If AD = 8 cm and CD = 6 cm, then the length of PQ is? 1) 3 cm 2) 7 cm 3) 9 cm 4) 5 cm 3.The price of a chair is Rs. 500. It has been sold at two succesive discounts of 10% each. What is its selling price? 1) Rs. 400 2) Rs. 405 3) Rs. 415 4) Rs. 425 4.The percent profit made when an article is sold for Rs. 78 is twice as much as when it is sold for Rs. 69, the cost price of the article is? 1) Rs. 60 2) Rs. 51 3) Rs. 55.50 4) Rs. 70 5.In a Village panchayat society 574 names are enlisted as 'below poverty level'. If 14% of the villagers are below poverty level, the total number of villagers is? 1) 4100 2) 4200 3) 4000 4) 3800 6.A train 240 meters in length crosses a telegraph post in 16 seconds. The speed of the train is? 1) 50 Km/hr 2) 52 Km/hr 3) 54 Km/hr 4) 56 Km/hr 7.If a2+1 = a, then the value of a3 is 1) 0 2) 1 2

3) -1

4) 2

3) 1/4

4) 4

2

8.Ifx+3y+y, then x /2y is equal to 1) 1/8 2) 1/2

9.From an external point two tangents to a circle are drawn. The chord passing through the points of contact subtends an angle72° at the centre. The angle between the tangents is? 1) 36° 2) 72° 3) 108° 4) 144° 10.Length of three line segments isgiven. Is construction of a triangle possible with the segments in the given cases? 1) 8 cm, 7 cm, 18 cm 2) 8 cm, 15 cm, 17 cm 3) 10 cm, 6 cm, 4 cm 4) 8 cm, 10 cm, 20 cm

11. If n is a positive integer and product of all integers from 1 to n is a multiple of (450 × 990), then find the least possible value of n? (1) 15 (2) 30 (3) 10 (4) 45 (5) None of these Solution: Given M = (450 × 990) = (2 × 3 × 3 × 5 × 5) × (2 × 3 × 3 × 5 × 11) Product of 1 to n should include all the above factors. Upon observation, it can be noticed that the product of 1 to 4 includes the factors 2 × 2. Similarly, it can be noticed that the product of 1 to 9 includes the factors 3 × 3 × 3 × 3. Similarly, it can be noticed that the product of 1 to 15 includes the factors 5 × 5 × 5. Similarly, it can be noticed that the product of 1 to 11 includes the factors 11. Thus, it can be seen that the product of 1 to 15 includes all the factors (2 × 3 × 3 × 5 × 5) × (2 × 3 × 3 × 5 × 11). Hence, the least possible value of n is 15. Hence, the correct answer is option 1. 1000

12.What is the remainder of 2000 divided by 7? (1) 6 (2) 4

(3) 3

(4) 2

(5) 1

Solution: 1000 Given: 2000 divided by 7. 2000/7 gives a remainder of 5. 1000 1000 Thus, remainder of 2000 by 7 = remainder of 5 by 7. 1000 500 Now, 5 = 25 ; 25/7 gives a remainder of 4. © 2017 ETHNUS

www.eguru.ooo

facebook.com/ethnus

BITS PILANI Test - 7 – Answer Key & Explanatory Answers 500

500

1000

Thus, remainder of 25 by 7 = remainder of 4 = remainder of 2 by 7. By using the property of power cycle for 2, 1 2 = 2 gives a remainder of 2 when divided by 7. 2 2 = 4 gives a remainder of 4 when divided by 7. 3 2 = 1 gives a remainder of 1 when divided by 7. 4 2 = 16 gives a remainder of 2 when divided by 7. And so on where the remainder repeats after every 3 terms. 1000 (1000 mod 3) 1 Thus, remainder of 2 by 7 would be equal to the remainder of 2 by 7 = remainder of 2 by 7 = 2. Hence, the correct answer is option 4. 13.Find the maximum value of the expression ([1/{(p – 1)(q – 1)}] + [1/{(p – 1)(r – 1)}] + [1/{(q – 1)(r – 1)]}), given that p, q and r are integers (not equal to one) and (p + q + r) is equal to 4? (1) 3 (2) 4 (3) 1 (4) 2 (5) 5 Solution: Given: p + q + r = 4. Let the given expression: E = ([1/{(p – 1)(q – 1)}] + [1/{(p – 1)(r – 1)}] + [1/{(q – 1)(r – 1)]}) Also, let F = (p – 1)(q – 1)(r – 1). Then, E = {(r – 1) + (q – 1) + (p – 1)}/(F) = (p + q + r – 3)/F. → E = 1/F. Thus, ‘E’ would take the maximum value when F has the minimum value. F (minimum) cannot be negative as then E also would be negative, and F ≠ 0 since p or q or r ≠ 1. Hence, the minimum value F can take is 1. Thus, maximum value of E = 1. Hence, the correct answer is option 3. 2

2

14.What is the value of loge (3x + y) if 9x + y = 21xy? (1) loge 3 + loge x + loge y (3) (3loge 3) + (1/2) × (loge x + loge y) (5) None of these

(2) (1/2) × (3loge 3 + loge x + loge y) (4) loge x + loge y – (1/2) × loge 3

Solution: 2

2

Given: 9x + y = 21xy. 2 2 9x + y + 6xy = 21xy + 6xy = 27xy. 2 (3x + y) = 27xy. Taking log on both sides, 3 2 log (3x + y) = log (27xy) = log 27 + log xy = log 3 + log xy → log (3x + y) = (½) × (3log 3 + log x + log y). Hence, the correct answer is option 2. 15.If (1/ log3 p) + (1/ log9 p) + (1/ log27 p) + ... (1/ log59049 p) = 11, then find the value of p, given that p is a positive integer? (1) 729 (2) 333 (3) 81 (4) 27 (5) 243 Solution: Given: (1/ log3 p) + (1/ log9 p) + (1/ log27 p) + ... (1/ log59049 p) = 11. → (log 3/ log p) + (log 9/ log p) + … + (log 59049/ log p) = 11. → (log 3 + log 9 + … log 59049)/ (log p) = 11. 2 10 → (log 3 + log 3 + … log 3 )/ (log p) = 11. → (log 3 + 2 log 3 + … 10 log 3)/ (log p) = 11. → (55 log 3)/ (log p) = 11. 5 → (log p) = 5 (log 3) = log 3 . 5 → p = 3 = 243. Hence, the correct answer is option 5.

© 2017 ETHNUS

www.eguru.ooo

facebook.com/ethnus

BITS PILANI Test - 7 – Answer Key & Explanatory Answers 16.In a triangle ABC, angles A, B and C are two digit integers such that 13 times the angle A equals 11 times the angle B. Find the minimum possible positive value of angle (A + B – C)? (1) 12 (2) 6 (3) 24 (4) 30 (5) 18 Solution: Given: A + B + C = 180° (Sum of angles of a triangle). → A + B – C = A + B – (180 – A – B) = 2(A + B) – 180. It is given that A and B are both two digit numbers and also 13A = 11B. Thus, if A = 11, then B = 13. If A = 22, then B = 26 and so on as shown in the following table: A

B

A+B

2(A + B)

2(A + B) – 180

11

13

24

48