RATIO, PROPORTION, INDICES & LOGARITHMS

(1)

CHAPTER # 1

RATIO, PROPORTION, INDICES & LOGARIT HMS

!

A ratio is a comparison of the sizes of two or more same kind quantities.

!

If a and b are two quantities of the same kind (in same units) then the fraction a/b is called ratio of a to b. It is written as a : b

!

The quantities a and b are called the terms of ratio.

a : b

First Terms Second Terms

(Antecedent) Consequent

« a & b must be same kind.

« The units of a & b must be same.

« Ratio always represent in simplest term.

« Importance of order.

for Example

- Find ratio between 150 gm & 2 kg.

150 gm 2 kg (Same kind quantities because both quantities belong to weight)

150 gm 2000gm (unit must be same, convert kg into gm)

150 : 2000 (Ratio always represent is simplest term)

15 : 200

3 : 40

So ratio of 150 gm to 2 kg is 3:40 we cannot write it 40 : 3 because the order of the term in a ratio is important.

Types of Ratio 1. Inverse Ratio

2. Compound Ratio

3. Duplicate Ratio

4. Triplicate Ratio a b

b a Inverse Ratio

a b

c d

e f

ace bdf , , C.R.

a b

a² b² D. R.

a T. R. a³

A - RATIO

(2)

5. Sub-Duplicate Ratio

6. Sub-Triplicate Ratio

Some Important Points :

1. A ratio a : b is said to be of greater inequality if a > b.

2. A ratio a : b is said to be of less inequality if a < b.

3. If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable.

4. If the ratio of two similar quantities can be expressed as a ratio of two non-integers, the quantities are said to be incommensurable.

5. Continued ratio is the relation between the magnitudes of three or more quantities.

Solved Example :

1. Find Ratio between 25 minute and 45 Seconds

25 min : 45 second (Same kind quantities)

(25 x 60) sec. : 45 sec. (same unit)

100 : 3 (simplify term)

so ratio is 100 : 3

2. Simplify the ratio : :

: :

8 : 3 : 4

24 so ratio is 8 : 3 : 4 3. Simplify ratio 2.5 : 5.45 : 4

2.5 : 5.45 : 4

2.50 : 5.45 : 4.00

250 : 545 : 400

50 : 109 : 80

4. If p : q is the sub duplicate ratio of p - x² : q - x² then find the value of x² ? Solution :

= (According the question)

a b

S.T.R.

= a b1/3 1/3

1 3

1 8

1 6 1

3

1 8

1 6

p - x² q - x²

√

p q

√a

√b

³

³ a

b

√a

√b S.D.R.

= a b1/2 1/2

(3)

= (

pq² - q²x² = p²q - p²x²

p²x² - q²x² = p²q - pq² x² (p² - q²) = pq (p - q)

x² =

x² = \ a² - b² = (a + b) (a - b)

x² =

5. If 2s : 3t is the duplicate ratio of 2s - p : 3t - p then find the value of p² ?

Solution :

= (According the question)

= [\ (A - B)² = A² + B² - 2AB]

12s²t + 3p²t - 12spt = 18st² + 2sp² - 12spt 12s²t + 3p²t = 18st² + 2sp² 3p²t - 2sp² = 18st² - 12s²t

p²(3t - 2s) = 6st (3t - 2s)

p² =

p² = 6st

by squaring both side)

(2s - p)² (3t - p)² 4 ² + p² - 2.2 .p

9t² + p² - 2.3t.p

s s

2s 3t 2 3t

s

6 t (3t - 2 ) (3t - 2s)

s s

pq (p - q) (p² - q²)

pq (p - q) (p + q) (p - q)

pq p + q p - x²

q - x²

p² q²

(4)

EXERCISE # 1 A

RATIO

1. The ratio of quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is :

(a) 5 (b) √5

(c) 7 (d) None

2. Ratio between 150 gm and 2 kg is :

(a) 40 : 3 (b) 3 : 40

(c) 9 : 20 (d) None of these

3. Ratio between 25 minutes and 45 seconds is :

(a) 100 : 3 (b) 50 : 3

(c) 25 : 45 (d) 5 : 9

4. The ratio of 4 kg and 500 gm is :

(a) 4 : 500 (b) 400 : 500

(c) 8 : 1 (d) 1 : 8

5. Simplify the ratio 1/3 : 1/8: 1/6

(a) 3 : 4 : 8 (b) 8 : 4 : 3

(c) 4 : 3 : 8 (d) 8 : 3 : 4

6. : : is equal to :

(a) 12 : 15 : 20 (b) 12 : 15 : 10

(c) 12 : 15 : 40 (d) None of these

7. Simplify the ratio 1.5 : 22.5 : 25

(a) 3 : 45 : 50 (b) 3 : 9 : 10

(c) 3 : 45 : 55 (d) None of these

8. 1.1 : 1.21 : 22 is equals to :

(a) 200 : 11 : 10 (b) 11 : 10 : 200

(c) 10 : 11 : 200 (d) None of these

9. Write each of the following ratios in the descending order 1 : 3, 2 : 5, 3 : 7 :

(a) 1 : 3, 2 : 5 and 3 : 7 (b) 3 : 7, 2 : 5 and 1 : 3

(c) 2 : 5, 3 : 7 and 1 : 3 (d) 3 : 7, 1 : 3 and 2 : 5

10. Which ratio is greater 2 : 3 ; 3.6 : 4.8 :

(a) 2 : 3 (b) 3.6 : 4.8

(c) Both are equal (d) None of these

11. Arrange the following ratio in descending order , , :

(a) < < (b) < <

(c) > > (d) None of these

1 5

1 4

2 3

1 3

1 3 1

3 1 3

2 3

4 5

5 2 7

3 4 5

5 7 4

5 5 7

2 3

5 7

4 5

2 3

(5)

12. The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is :

(a) 16 (b) 60

(c) 22 (d) 20

13. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 25, the antecedent is :

(a) 35 (b) √5

(c) 7 (d) None of these

14. The ratio of two quantities is 5 : 6. If the consequent is 180, then the antecedent is :

(a) 510 (b) 150

15. The inverse ratio of 11 : 15 is :

(a) 15 : 11 (b) √11 : √15

16. The inverse ratio of the ratio √48 : √64 is :

(a) √2 : 3 (b) √15 : √30

(c) 2 :√3 (d) None of these

17. The duplicate ratio of 3 : 4 is :

(a) √3 : 2 (b) 4 : 3

18. The sub duplicate ratio of 25 : 36 is :

(a) 6 : 5 (b) 36 : 25

(c) 50 : 72 (d) 5 : 6

19. The triplicate ratio of 2 : 3 is ______ :

(a) 8 : 27 (b) 6 : 9

20. The sub triplicate ratio of 8 : 27 is _________ :

(a) 27 : 8 (b) 24 : 81

21. The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is :

(a) 1 : 4 (b) 1 : 3

22. The ratio compounded of 2 : 3, 9 : 4, 5 : 6 and 8 : 10 is :

(a) 1 : 1 (b) 1 : 5

23. The ratio compound of two ratios 4:3 and 7:3 is

(a) 12:21 (b) 28:9

(c) 9:28 (d) None of these

24. The compounded ratio of duplicate ratio of 3:5 and the sub triplicate ratio of 1:27 and reciprocal of 1:5 is equal to

(a) 3:5 (b) 1:2 (c) 3:25 (d) 9:25

(6)

25. The duplicate ratio of 2/3 : 3/5 is :

(a) 10 : 9 (b) 100 : 90

(c) 100 : 81 (d) 1000 : 810

26. The ratio compounded of 4 : 9, 9 : 7, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 is :

(a) 2 : 7 (b) 7 : 2

27. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is

(a) 18 : 25 (b) 8 : 25

28. If x : y = 3 : 4, the value of x²y + xy² : x³ + y³ is :

(a) 13 : 12 (b) 12 : 13

29. If p : q = 2 : 3 and x : y = 4 : 5, then the value of 5px + 3qy : 10px + 4qy is :

(a) 71 : 82 (b) 27 : 28

30. If 15(2x² - y²) = 7xy and x, y are positive, then find x : y is :

(a) 5 : 6 (b) 6 : 5

(c) 3 : 5 (d) 4 : 5

31. If the ratio of (5x - 3y) and (5y -3x) is 3 : 4, then the value of x:y is :

(a) 27 : 29 (b) 29 : 27

(c) 3 : 4 (d) 4 : 3

32. If a : b = 4 : 1 then + is :

(a) 5/2 (b) 4

33. If A : B = 3 : 2 and B : C = 3 : 5, then A : B : C is :

(a) 9 : 6 : 10 (b) 6 : 9 : 10

(c) 10 : 9 : 6 (d) None o these

34. If a : b = n : d and b : c = n : d then :1 1 2 2

(a) a : b : c = n d : n d : d d_{1 1} _{2 2} _{1 2} (b) a : b : c = n d : n d : d d_{1 2} _{2 1} _{2 1} (c) a : b : c = n n : d n : d d _{1 2} _{1 2} _{1 2} (d) None of these

35. The ratio of X : Y is 7 : 9 and Y : Z is 6 : 7. The ratio of X : Z is :

(a) 2 : 3 (b) 3 : 2

(c) 1 : 2 (d) 2 : 1

36. If p : q = 3 : 2, r : q = 4 : 3, then r : q : p is : (a) 6 : 8 : 9

(b) 8 : 6 : 9 (c) 9 : 6 : 8 (d) None of these

a b

ba

(7)

37. If A : B = 3 : 4, B : C = 8 : 9 and C : D = 15 : 16 then A : D will be :

(a) 3 : 16 (b) 1 : 2

(c) 5 : 8 (d) 9 : 16

38. If 2A = 3B = 4C then A : B : C is :

(a) 2 : 3 : 4 (b) 4 : 3 : 2

(c) 6 : 4 : 3 (d) 3 : 4 : 6

39. If A : B = 2 : 3, B : C = 4 : 5, C : D = 3 : 7, then A : B : C : D is :

(a) 24 : 36 : 45 : 105 (b) 12 : 36 : 45 : 105

(c) 24 : 36 : 65 : 105 (d) 24 : 48 : 45 : 105

40. A, B, C, D are four numbers so that A : B = 2 : 3, B : C = 4 : 5. C : D = 5 : 8, then A : D is :

(a) 2 : 3 (b) 3 : 2

(c) 1 : 3 (d) 3 : 1

41. If (7p+3q) : (3p-2q) = 43:2 , then p:q is

(a) 5:4 (b) 4:5

(c) 7:2 (d) 2:7

42. If a:b = 4:5, find the value of (5a - 3b): (6a + 3b)

(a) 7:39 (b) 15:39

(c) 1:3 (d) 5:39

43. If a² + b² = 45 & ab = 18, then + is

(a) (b)

(c) (d) None of these

44. If = - then the value of is :

(a) 1 (b) - 1/7

(c) 1/7 (d) 7

45 If = , then the value of = ?

(a) (b) 7

(c) 7 : 2 (d) 15

46. If A : B = 2 : 5 then (10 A + 3 B) : (5A + 2 B) is equal to :

(a) 7 : 4 (b) 7 : 3

(c) 6 : 5 (d) 7 : 9

47. An amount of Rs. 950 is distributed among A, B & C in the ratio of 5:11:3, what is the difference between the share of B and A.

(a) 300 (b) 340

1 a

1 b

1 3

2 3 1

2 p q

2 3

2p + q 2p - q

x 2y

3 2

2x + y x - 2y 1

7

(8)

48. The ages of two persons are in the ratio 5 : 7. Eighteen years ago their ages were in the ratio of 8 : 13, their present ages (in years) are :

(a) 50, 70 (b) 70, 50

(c) 40, 56 (d) none

49. If A, B and C started a business by investing Rs. 1,26,000, Rs. 84,000 and Rs. 2,10,000. If at the end of the year profit is Rs. 2,42,000 then the share of each is :

(a) 72,600, 48,400, 1,21,000 (b) 48,400, 1,21,000, 72,600

(c) 72,000, 49,000, 1,21,000 (d) 48,000, 1,21,400, 72,600

50. Find the two numbers which are in the ratio 7 : 15 and whose difference is 72 :

(a) 35, 75 (b) 49, 105

(c) 63, 135 (d) 56, 128

51. The number which when subtracted from each of the terms of the ratio 19 : 31 reducing it to 1 : 4 is ___ :

(a) 15 (b) 5

52. What must be added to each term of the ratio 49 : 68, so that it becomes 3: 4?

(a) 3 (b) 5

(c) 8 (d) 9

53. The ratio of two numbers is 4:1. If 5 is added to both the numbers, the ratio of the new numbers obtained becomes 3:1. Then the numbers are

(a) 30,20 (b) 40, 10

(c) 20, 10 (d) None of these 54. What must be subtracted from each term of the ratio 27:43 to make it equal to 7:15?

(a) 13 (b) 15

55. The ratio of the number of boys and girls in a school is 2:5. If there are 280 students in the school, find the number of girls in the school.

(a) 200 (b) 250

56. The ratio of the sum and the difference of two numbers is 7:1. Find the ratio of those two numbers.

(a) 5:3 (b) 4:3

57. Rs. 407 are to be divided among A, B and C so that their shares are in the ratio 1/4 : 1/5 : 1/6. The respective shares of A, B, C are:

(a) Rs. 165, Rs. 132, Rs. 110 (b) Rs. 165, Rs. 110, Rs. 132

(c) Rs. 132, Rs. 110, Rs. 165 (d) Rs. 110, Rs. 132, Rs. 165

58. The students of two classes were in the ratio 5 : 7, if 10 students left from each class, the remaining students are in the ration of 4 : 6 then now the number of students in each class is:

(a) 30, 40 (b) 25, 24.

(c) 40, 60. (d) 50, 70

59. Anand earns Rs. 80 in 7 hours and Promod Rs. 90 in 12 hours. The ratio of their earnings is :

(a) 32 : 21 (b) 23 : 12

(9)

60. The ratio between the speeds of two trains is 7 : 8. If he second train runs 400 Kms. in 5 hours, the speed of the first train is :

(a) 10 Km/h (b) 50 Km/hr

(c) 70 Km/hr (d) None of these

61. A scooter covers a distance of 200 km in 2 hour 40 minutes, while a car covers the same distance in 2 hours.

The ratio of their speeds is :

(a) 2 : 3 (b) 5 : 4

(c) 4 : 5 (d) 3 : 4

62. The angles of a triangle are in ratio 2 : 7 : 11. The angles are :

(a) (20º, 70º, 90º) (b) (30º, 70º, 80º)

(c) (18º, 63º, 99º) (d) None of these

63. Division of Rs. 324 between X and Y is in the ratio 11 : 7. X and Y would get Rupees :

(a) (204, 120) (b) (200, 124)

(c) (180, 144) (d) (198, 126)

64. An amount of money is to be distributed among P, Q and R in the ratio 3 : 5 : 7. If Q's share is Rs. 1500, the differences between P's share and R's shares :

(a) Rs. 1200 (b) Rs. 1500

(c) Rs. 1600 (d) Rs. 1900

65. A boy has five rupee coins, two rupee coins and one rupee coins in the ratio 3 : 4 : 5. If he has an amount of Rs. 224, then find the number of one rupee coins with the boy :

(a) 20 (b) 30

(c) 5 (d) 40

66. An amount of Rs. 4686 is divided among A, B, C and D. The ratio of the shares of A and B is 3 : 2, while the ratio of the shares of B & C and C & D are 3 : 4 and 4 : 5 respectively. Find share of B :

(a) Rs. 952 (b) Rs. 1260

(c) Rs. 852 (d) Rs. 1540

67. Three numbers are in the ratio 2 : y : 5. The third number is greater than the first one by 21. Find the numbers, if the sum of numbers is 70 :

(a) 14, 21 and 35 (b) 7, 5 and 21

(c) 7, 14 and 21 (d) 14, 15 and 35

68. Daily earnings of two persons are in the ratio 4 : 5 and their daily expenses are in the ratio 7 : 9. If each saves Rs. 50 per day, their daily incomes in Rs. are :

(a) (400, 50) (b) (500, 450)

(c) (400, 500) (d) None of these

69. The sum of ages of 3 persons is 150 years. 10 years ago their ages were in the ratio 7 : 8 : 9. Their present ages are :

(a) (45, 50, 55) (b) (40, 60, 50)

(c) (35, 45, 70) (d) None of these

70. Two numbers are in the ratio 4 : 5. If 2 is subtracted from each of them, then their ratio becomes 3 : 4. The numbers are :

(a) 9, 15 (b) 8, 10

(c) 20, 25 (d) 12, 15

(10)

71. Two numbers are in ratio 7 : 16. If 10 is added to both the numbers, then the ratio will be 1 : 2. Find the numbers :

(a) 35 and 80 (b) 70 and 160

(c) 60 and 150 (d) 45 and 90

72. There are oranges and apples in the ratio 3 : 14. If 10 oranges are eaten, then the ratio will become 1 : 6. Find the number of apples :

(a) 120 (b) 60

(c) 210 (d) 160

73. If the length and breadth of a room measure 14 feet 3 inches and 11 feet 6 inches respectively, then the ratio of length to breadth is :

(a) 14 : 11 (b) 56 : 22

(c) 57 : 36 (d) 57 : 46

74. If 35% of A's income is equal to 25% of B's income, then the ratio of theirs income is :

(a) 4 : 3 (b) 5 : 7

(c) 7 : 5 (d) 4 : 7

75. If 15% of x is the same as 20% of y, then x : y is :

(a) 3 : 4 (b) 4 : 3

(c) 17 : 16 (d) 16 : 17

76. The ratio o two numbers is 7 : 10 and their difference is 105. The numbers are :

(a) (200, 305) (b) (185, 290)

(c) (245, 350) (d) None of these

77. Two numbers are in the ratio 2 : 3 and the difference of their squares is 320. The numbers are :

(a) 12, 18 (b) 16, 24

(c) 14, 26 (d) None of these

78. Two whole numbers whose sum is 64, cannot be in the ratio :

(a) 5 : 3 (b) 7 : 1

(c) 3 : 4 (d) 9 : 7

79. The ratio of the number of boys and girls in a school is 2 : 5. If there are 350 students in the school, the number of girls in the school are :

(a) 200 (b) 275

(c) 225 (d) 250

80. P, Q and R are three cities. The ratio of average temperature between P and Q is 11 : 12 and that between P and R is 9 : 8. The ratio between the average temperature of Q and R is :

(a) 22 : 27 (b) 27 : 22

81. The ages of two persons are in the ratio 5 : 7 eighteen years ago their ages were in the ratio of 8 : 13, their present ages (in years) are :

(a) 50, 70 (b) 70, 50

(c) 40, 56 (d) None of these

82. Sides of two squares are in the ratio of 3 : 4. Their perimeters are in the ratio of :

(a) 5 : 6 (b) 7 : 8

(c) 3 : 4 (d) 6 : 7

(11)

83. A bag contains Rs. 187 in the form of 1 Rupees, 50 Paise, and 10 Paise coins in the ratio of 3:4:5. Find the number of each type of coins?

(a) 30,40, 50 (b) 102, 136, 170

(c) 60,80, 100 (d) None of these

84. A bag contains one rupee, 50 paise and 25 paise coins in the ratio 10:14:18. If the total amount in the bag is Rs. 430, find the number of coins of each kind.

(a) 200, 280, 360 (b) 280, 200, 360

(c) 360, 280. 200 (d) None of these

85. The ratio of the incomes of two persons is 9:7 and the ratio of their expenditures is 4:3. If each of them saves Rs. 200 per month, then their monthly incomes.

(a) Rs. 1,800, Rs. 1,400 (b) Rs. 1,600, Rs. 1,200

(c) Rs. 1,400, Rs. 1,200 (d) None of these

86. The income of A and B are in the ratio 3 : 2 and their expenditure in the ratio 5 : 3. If each saves Rs. 1,500, then B's income is:

(a) Rs. 6,000 (b) Rs. 4,500

(c) Rs. 3,000 (d) Rs. 7,500

87. An employer reduces the number of employees in the ratio of 9:8 and increases their wages in the ratio of 14:15. In what ratio is the wages bill decreased?

(a) 20:22 (b) 20:33

88. Find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in he number of workers in the ratio 15 : 11 and an increment in their wages in the ratio 22 : 25 :

(a) decreased 6 : 5 (b) increased 6 : 5

(c) decreased 5 : 6 (d) None of these

89. The compounded ratio of 4:9 and sub-duplicate ratio of 9:16 is (a) 1:4 (b) 1:3

(c) 3:1 (d) None of these 90. The ratio Compounded of 4:5 and sub-duplicate of “a”:9 is 8:15. Then value of “a” is:

(a) 2 (b) 3

(c) 4 (d) 5

Answer Key

1 c 2 b 3 a 4 c 5 d 6 c 7 a 8 c 9 b 10 b 11 c 12 d 13 a

14 b 15 a 16 c 17 c 18 d 19 a 20 c 21 d 22 a 23 b 24 a 25 c 26 c

27 a 28 b 29 c 30 a 31 a 32 a 33 a 34 c 35 a 36 b 37 c 38 c 39 a

40 c 41 b 42 d 43 c 44 c 45 b 46 a 47 a 48 a 49 a 50 c 51 a 52 c

53 b 54 a 55 a 56 b 57 a 58 c 59 a 60 c 61 d 62 c 63 d 64 a 65 d

66 c 67 a 68 c 69 a 70 b 71 a 72 c 73 d 74 b 75 b 76 c 77 b 78 c

79 d 80 b 81 a 82 c 83 b 84 a 85 a 86 a 87 c 88 a 89 b 90 c

(12)

a b = c

(

d

(

b a = d

c a c = b

d a + b

b = c + d d

B - PROPORTION

!

An equality of two ratios is called a proportion.

!

Four quantities a, b, c, d are said to be in proportion if.

a: b = c : d

or

a : b :: c : d

I II III IV

Term Term Term Term

Middle Terms (Means)

Extremes

!

Law of proportion/cross product rule Product of means = product of extremes

bc = ad Continuous Proportion

Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion if a : b = b : c

by cross product rule b² = ac

b = √ac

« b is called mean proportional between a & c and it is equals to √ac

« a is the first proportional

« c is the third proportional

!

Properties of Proportion If a : b = c : d then

1. (Invertendo)

2. (Alternendo)

3. (Componendo)

(13)

4. (Dividendo)

5. (Componendo - Dividendo)

6. If = = ... then

= = ... = (Addendo)

7. If = = ... then

= = ... = (Subtrahendo) a + b

a - b = c + d c - d a

b c d

e f a

b c d

e f

a + c + e + ...

b + d + f + ...

a b

c d

e f a

b c d

e f

a - c - e - ...

b - d - f - ...

a - b

b = c - d d

(14)

1. In case of proportion product of means is always equal to :

(a) Avg. of Extremes (b) Avg. of means

(c) Product of Extremes (d) Product of Antecedents

2. If = then a, b, c, d are known as 1st, 2nd, 3rd and 4th proportional respectively :

(a) True (b) Partly True

(c) False (d) Can not say

3. The value of x if 10/3 : x : : 5/2 : 5/4 is

(a) 7/9 (b) 5/3

(c) 2/5 (d) None

4. The fourth proportional to 2/3, 3/7, 4 is

(a) 18/7 (b) 18/11

(c) 9/7 (d) None

5. The third proportion to 2.4 kg, 9.6 kg is

(a) 38.5 (b) 38

(c) 38.4 (d) 36

6. The mean proportion between 1.25 and 1.8 is

(a) 1.5 (b) 2.5

(c) 2 (d) 5

7. Find the mean proportion between 5.25 and 8.9

(a) 8.63 (b) 6.83

(c) 9.23 (d) None of these.

8. If a:b = c:d = 3: 5, then what are the values of ad : bc is

(a) 3:5 (b) 5:3

(c) 1:1 (d) None of these.

2 2

9. The mean proportional between 4p and 9q is

(a) 6 pq (b) 36 pq

2 2

(c) 6 p q (d) None of these 10. If 5:6 = 45:a, find the value of a

(a) 54 (b) 45

(c) 15 (d) 0

2 2 3 3

11. The fourth proportional to (a - ab + b ), (a + b ) and (a-b) is equal to

2 2 2 2

(a) a + b (b) a -b

12. Fourth proportional to x, 2x, (x+1) is :

(a) (x+2) (b) (x-2)

(c) (2x+2) (d) (2x-2)

EXERCISE # 1 B

PROPORTION

a b

c d

(15)

13. 24, 32, ___________,. 40 are in proportion.

(a) 25 (b) 20

(c) 30 (d) None of these 14. The fourth proportional to 4, 6, 8 is :

(a) 12 (b) 32

15. The third proportional to 12, 18 is :

(a) 24 (b) 27

16. The mean proportional between 25, 81 is :

(a) 40 (b) 50

17. The fourth proportional to 2a, a³, c is :

(a) ac/2 (b) ac

(c) 2/ac (d) None of these

18. If four numbers 1/2, 1/3, 1/5, 1/x are proportional then x is :

(a) 6/5 (b) 5/6

(c) 15/2 (d) None of these

19. The mean proportional between 12x² and 27y² is :

(a) 18xy (b) 81xy

(c) 8xy (d) None of these

20. The number which has the same ratio to 26 that 6 has to 13 is :

(a) 11 (b) 10

.

21 The number_____________ has same ratio to 7/33 as that 8/21 does to 4/9.

(a) 2/11 (b) 3/11

(c) 2/21 (d) None of these 22. The third proportional to (x² - y²) and (x - y) is :

(a) (b)

(c) x + y (d) x - y

23. First, third and fourth terms of a proportion are 3, 15, 35 respectively. The second term is :

(a) 5 (b) 7

(c) 25 (d) 45

24. The third proportional between (a² - b²) and (a + b)² is :

(a) (b)

x + y x - y

x - y x + y

a + b a - b (a - b)²

a - b a + b (a + b)³

(16)

25. If b is the mean proportional of a and c, then : (a + b + c) (a - b + c) = :

(a) a² + b² - c² (b) a² - b² + c²

(c) a² + b² + c² (d) a² - b² - c²

26. If = = , then find the value of

(a) 1 (b) 2

(c) 3 (d) 4

27. If A = B/2 = C/5, then A : B : C is

(a) 3 : 5 : 2 (b) 2 : 5 : 3

(c) 1 : 2 : 5 (d) None of these

28. If p/q = r/s = 2.5/1.5, the value of ps : qr is :

(a) 3/5 (b) 1

29. If x : y = z : w = 2.5 : 1.5, the value of (x + z) / (y + w) is

(a) 1 (b) 3/5

30. If (5x - 3y) / (5y - 3x) = 3/4, the value of x : y is

(a) 2 : 9 (b) 7 : 2

31. If A : B = 3 : 2 and B : C = 3 : 5, then A : B : C is :

(a) 9 : 6 : 10 (b) 6 : 9 : 10

32. If x/2 = y / 3 = z/7, then the value of (2x - 5y + 4z)/y is

(a) 6/23 (b) 23/6

33. If x : y = 2 : 3, y : z = 4 : 3 then x : y : z is

(a) 2 : 3 : 4 (b) 4 : 3 : 2

34. Division of Rs. 750 into 3 parts in the ratio 4 : 5 : 6 is

(a) (200, 250, 300) (b) (250, 250, 250)

(c) (350, 250, 150) (d) None of these

35. The sum of the ages of 3 persons is 150 years. 10 years ago their ages were in he ratio 7 : 8 : 9. Their present ages are

(a) (45, 50, 55) (b) ((40, 60, 50)

(c) (35, 45, 70) (d) None of these

36. The number 14, 16, 35, 42 are not in proportion. The fourth term for which they will be in proportion is

(a) 45 (b) 40

37. If x / y = z / w, implies y / x = w / z, then the process is called

(a) Dividendo (b) Componendo

(c) Altertendo (d) None of these

a 3

b 4

c 7

a + b + c c

(17)

38. If p / q = r / s = p - r / q - s, the process is called

(a) subtrahendo (b) Addendo

(c) Invertendo (d) None of these

39. If a / b = c / d, implies (a + b) / (a - b) = (c + d) / (c - d), the process is called

(a) Componenod (b) Dividendo

(c) Componendo and Dividendo (d) None of these

40. If u / v = w/p, then (u - v) / (u + v) = (w - p) / (w + p). The process is called

(a) Invertendo (b) Alternendo

(c) Addendo (d) None of these

41. 12, 16, *, 20 are in proportion. Then * is

(a) 25 (b) 14

42. 4, *, 9, 13½ are in proportion. Then * is

(a) 6 (b) 8

43. The mean proportional between 1.4 gms and 5.6 gms is :

(a) 28 gms (b) 2.8 gms

(c) 3.2 gms (d) None of these

44. If = = then is

(a) 4 (b) 2

45. Two numbers are in the ratio 3 : 4, if 6 be added to each terms of the ratio, then the new ratio will be 4 : 5, then the numbers are

(a) 14, 20 (b) 17, 19

(c) 18 and 24 (d) None of these

46. If = then :

(a) = (b) =

(c) = (d) None of these

47. If a : b = 4 : 1 then + is

(a) 5/2

(b) 4

(c) 5

(d) None of these a

4 b 5

c 9

a + b + c c

a 4

b 5 a + 4

a - 4

b + 4 b - 4

a + 4 a - 4

b + 5 b - 5 a - 4

a + 4

b + 5 b - 5

a b

b a

(18)

48. If = = then is

(a) 8 (b) 5

(c) 4 (d) None of these.

49. What must be added to each of the numbers 6, 15, 20 and 43 to make them proportional.

(a) 5 (b) 4

(c) 3 (d) 2

50. Find two numbers such that mean proportional between them is 18 and third proportional to them is 144.

(a) 9, 36 (b) 29, 56

(c) 18, 72 (d) None of these

51. What must be added to each of the four numbers 10, 18, 22, 38. So that they become in proportion?

(a) 2 (b) 5

52. An alloy is to contain copper and zinc in the ratio 9 : 4. The zinc required to melt with 24 kg of copper is:

(a) (b)

(c) (d) 9kg

55. In what ratio should tea worth Rs.10 per kg be with tea worth Rs. 14 per kg, so that the average prices of the mixture may be Rs. 11 per kg?

(a) 2 : 1 (b) 3 : 1

(c) 3 : 2 (d) 4 : 3

53. If = = then (b - c) x + (c - a) y + (a - b) z is :

(a) 1 (b) 0

54. If a : b = c : d, then = :

(a) c (b) c + d

(c) c - d (d) a + b

56. In what proportion must rice @ Rs. 3.10/ kg be mixed with rice @ 3.60/ k to make the mixture worth Rs. 3.25/

kg :

(a) 3 : 5 (b) 5 : 3

(c) 3 : 7 (d) 7 : 3

57. In 40 liters mixture of glycerine and water the ratio of glycerine and water is 3 : 1. The quantity of water added in the mixture in order to make this ratio 2 : 1 is :

(a) 15 liters (b) 10 liters

(c) 8 liters (d) 5 liters

58. In a mixture of 35 liters, the ratio of milk and water is 4 : 1. Now, 7 liters of water is added to the mixture. Find the ratio of milk and water in the new mixture :

(a) 2 : 1 (b) 1 : 2

(c) 3 : 2 (d) 2 : 3

p 7

q 6

r 11

p + q + r p

10 2 kg 3 9 2 kg

3

10 1 kg 3

x b + c - a

y c + a - b

z a + b - c

a + b b

? d

(19)

59. In a mixture of 42 liters, the ratio of the milk to water is 6 : 1. Another 12 liters of water is added to the mixture.

Then, the ratio of the milk and water in the resultant mixture is :

(a) 2 : 1 (b) 1 : 2

(c) 7 : 1 (d) 3 : 4

x y z

60. If 4 = 5 = 20 then z is equal to:

(a) xy (b) x+y/xy

(c) 1/xy (d) xy/x+y

Answer Key

1 c 2 a 3 b 4 a 5 c 6 a 7 b 8 c 9 a 10 a 11 b 12 c 13 c

14 a 15 b 16 c 17 d 18 c 19 a 20 d 21 a 22 b 23 b 24 d 25 c 26 b

27 c 28 b 29 c 30 d 31 a 32 d 33 d 34 a 35 a 36 b 37 d 38 a 39 c

40 d 41 c 42 a 43 b 44 b 45 c 46 b 47 a 48 d 49 c 50 a 51 a 52 a

53 b 54 b 55 b 56 d 57 d 58 a 59 a 60 d

(20)

!

If a is a non - zero real number and n is a natural number then a n = a x a x a x ... x a

n times here, a is called base and n is called the index or power

!

Laws of Indices

m n m+n

(a) a x a = a

m n m-n

(b) a ÷ a = = a

m n mn

(c) (a ) = a

n n n

(d) (ab) = a b

(e) =

(f) a = 1 0

(g) a = -m

(h) = am

am

an

a b

( (

n ^abⁿ

n

1 am

1 a-m

C - INDICES

(21)

1/2 - 1

1. Simply 2x 3x if x = 4 :

(a) 2 (b) 3

(c) 4 (d) - 3

- 2 - 3

2. Simplify 6ab²c³ x 4b c d :

(a) 24ad (b) 24

(c) 24a/d (d) None of these

3. Find The value of :

(a) (b)

(c) (d) None of these

4. Simplify if a = 4 :

(a) 3 (b)

(c) (d)

a - b 3 3 2 - a

5. Simplify (x . y ) . (x y ) :

(a) (b)

a + b 3b + 2a

(c) y (d) y

4b 6 2/3 - 1 - b

6. a x . (a x ) :6

1 - b 1 + b

(a) x (b) x

(c) x (d) 0

- 7 - 5 k

7. Find the value of k from (√9) x (√3) = 3 :

(a) (b)

(c) (d)

EXERCISE # 1 C

INDICES

4x -1

x-1/3

4x x1/3

2x x2/3

4 x2/3

2a x a x 6a 9a x a

1

2 2

3 7

3 - 5

3 3

2

2 3 3 2 1

3

1

3b + 2a

y

1 y

19 2 5 7

19 2 7 5

(22)

- 1/4

8. 4x is expressed as :

1/4 - 1

(a) - 4x (b) x

(c) 4/x1/4 (d) None of these

9. The value of 8 is :1/3

(a) 3√2 (b) 4

10. The value of 2 x (32) is :1/5

(a) 2 (b) 10

11. The value of 4/(32) is :1/5

(a) 8 (b) 2

12. The value of (8/27) is : 1/3

(a) 2/3 (b) 3/2

- 1/8

13. The value of 2(256) is :

(a) 1 (b) 2

1/2 3/4

14. 2 . 4 is equal to :

(a) A fraction (b) A positive integer

(c) A negative integer (d) None of these

15. has simplified value equal to :

(a) xy² (b) x²y

(c) 9xy² (d) None of these

a - b b - c c - a

16. x . x . x is equal to :

(a) x (b) 1

17. The value of is equal to :

(a) 0 (b) 2/3

18. {(3³)² x (4²)³ x (5³)²} / {(3²)³ x (4³)² x (5²)³} is :

(a) 3/4 (b) 4/5

(c) 4/7 (d) 1

19. Which is True ?

0 0 0 0

(a) 2 > (1/2) (b) 2 < (1/2)

0 0

(c) 2 = (1/2) (d) None of these

81x4

y - 8

[ [

¹⁴

2p²q³

(

3xy ⁰

(

(23)

a - b b - c c - a - a - b

20. The value of y x y x y x y is :

a + b

(a) y (b) y

a + b

(c) 1 (d) 1/y

21. The True option is :

2/3 ³ 2/3

(a) x = √x² (b) x = √x³

2/3 ³ 2/3 ³

(c) x > √x² (d) x < √x²

- 3 2 - 1 3 - 2

22. Simplified value of 16x y x 8 x y is :

(a) 2xy (b) xy/2

- 1/3 - 1/5

23. The value of (8/27) x (32 / 243) is :

(a) 9/4 (b) 4/9

.

2/3 3/2 3 6

24. The value of {(x + y) (x - y) /√x + y x √(x - y) } is :

(a) (x + y)² (b) (x - y)

(c) x + y (d) None of these

2/3 ³ 3 1/2

25. Simplified value of (125) x √25 x √5 x 5 is :

(a) 5 (b) 1/5

1/2 )3/4 5/6 7/8 9/10 4 3/25

26. [{(2) . (4 . (8) . (16) . (32) } ] is :

(a) A fraction (b) An integer

n n-1/n 1/n + 1

27. {(X ) } is equal to :

n n + 1

(a) X (b) X

n - 1

(c) X (d) None of these

28. The value of is

(a) 2/3 (b) 3/2

(c) 2/9 (d) None of these.

- 6 - 2 P

29. Find the value of p from (√4) x (√2) = 2

(a) 16 (b) 8 (c) -7 (d) 4

q p

30. If p = q then value of reduces to

(a) P (b) q

1/4 3/4 5/4

31. 3 x 6 x 2 is equal to

(a) 10 (b) 12 (c) 14 (d) 0

p q r

32. If 1176 = 2 .3 .7 Find the value of p, q, r.

(a) 1, 2, 3 (b) 1,3, 2

8 27

1/3

p

( (

q ^p^q^{- p}^p^{q - 1}

(24)

a - m m -n n - a

33. The value of y x y x y is equal to

(a) 1 (b) 0

(c) -1 (d) y

1/5 1/7

34. The value of (243) (128) is_________

(a) 1 (b) 6

(c) 2 (d) 3

-1/2 4/5

35. The value of + (64) + (- 32)

(a) 17 (b) 17

3 8 n

36. If (4) x ( 2) =2 , then n is

(a) 10 (b) 12

37. Simplification of is:

m -m

(a) x (b) x

n -n

(c) x (d) x

x x-1 x

38. If 2 -2 =4, then the value of x is:

(a) 2 (b) 1

(c) 64 (d) 27

X y z

39. If 3 = 2, 5 =3 and 2 = 5, find the value of multiply of x. y. z

(a) 0 (b) 1

a b c

40. If x=y , y=z and z = x then abc is:

(a) 2 (b) 1

(c) 3 (d) 4

x y z

41. If a = b, b = c, c = a, then xyz is :

(a) 1 (b) 2

n/m n/m

42. If m and n are two natural numbers then a x a ...m terms =

(a) a (b) an

(c) am (d) m

-1 -1 -1

43. [1/n] :

(a) 1/n (b) n

- 3 - 3

(c) n (d) 1/n

m - n 2n k

44. If a . a = a then k = :

(a) n (b) m + n

(c) m - 3n (d) None of these

1

( (

64 ⁰ 1

8 7 8

3 8

m+3n 4m-9n

x .x

6m-6n

x

[ ^[ ^[ [

(25)

x ³

45. If √a = √a² then x = :

(a) 2/3 (b) 4/3

(c) 1/3 (d) 5/3

46. Find x, if x √x = (x√x) : x

(a) 2 (b) 1

(c) 3 (d) 5

- 1/3 -1 -1

47. 1 - 1 - (1 - x³) = :

(a) 0 (b) 1

(c) x (d) x²

48. Which of the following is bigger ?

22

(a) 22 (b)

(c) 22x2x2 (d) 22x3

x y

49. 3 .9 = 81 and 2x - y = 3 then x, y = :

(a) 2,1 (b) 3,1

(c) 2,3 (d) None of these

50. = :

(a) 1 (b) - 1

1/p 1/q 1/r

51. If x = y = z and xyz = 1, then the value of p + q = r is :

(a) 1 (b) 0

x y - z

52. If 2 = 3 = 6 , + + is :

(a) 1 (b) 0

53. If a³ - b³ = (a - b) (a² + ab + b²), then the simplified form of

(a) 0 (b) 1

(c) x (d) None of these

54. The value of

(a) 1 (b) 0

[ ^{ ^{ [

22

[ ^{( (}

²

[

²

1

a - b a - c

1 + x + x

Σ

1 x

1 y

1 z

xl

xm

[ [

l + lm + m² ²

x x^m xn

[ [

m + mn + n² ² xn

xl

[ [

l + ln + n² ²

x

Xa

Xb

( (

^{a + b}^x

( (

^X_X^b^c ^{b + c}^x

( (

^X_X^c^a ^{c + a}

(26)

55. The value of

(a) 1 (b) 0

(c) - 1 (d) None of these

56. Show that is given by :

(a) 1 (b) - 1

(c) 3 (d) 0

57. Show that reduces to :

(a) 1 (b) 0

(c) -1 (d) None of these

(a) 1 (b) 3

(c) 0 (d) 2

(a) - 1 (b) 0

(c) 1 (d) None of these

61. Show that is reduces to :

(a) 1 (b) 0

m - n m - p n - m n - p p - m p - n

62. On simplification, 1/(1 + a + a ) + 1/(1 + a + a ) + 1/(1 + a + a ) is equal to :

(a) 0 (b) a

(c) 1 (d) 1/a

63. On simplification + + would reduces to :

(a) (b)

(c) 1 (d) 0

Xa

Xb

( (

(a² + ab + b²)

x Xb

Xc

( (

(b² + bc + c²)

x Xc

Xa

( (

(c² + ca + a²)

( (

x^{a - b}¹ ^{a - c}¹ ^{x x}

( (

^{b - c}¹ ^{b - a}¹ ^{x x}

( (

^{c - a}¹ ^{c - b}¹

b2

X

c2

X

(b+c) x

c2

X

a2

X

(c +a) a2

X

b2

X

(a+b) x

Xb

Xc

( (

^a^x

( (

^X_X^c^a ^b^x

( (

^X_X^a^b ^c

( (

x^{b + c}^{c - a} ^{a - b}¹ ^{x x}

( (

^{c + a}^{a - b} ^{b - c}¹ ^{x x}

( (

^{a + b}^{b - c} ^{c - a}¹

Xb

Xc

( (

^1/bc^x

( (

^X_X^c^a ^1/ca^x

( (

^X_X^a^b ^1/ab

Xb

Xc

( (

^{b + c - a}^x

( (

^X_X^c^a ^{c + a - b}^x

( (

^X_X^a^b ^{a + b - c}

1

a - b a - c

1 + z + z

1

b - c b - a

1 + z + z

1

c - a c - b

1 + z + z

1

2(a + b + c)

z

1

(a + b + c)

z

(27)

64. The value of + is given by :

(a) - 1 (b) 0

1/3 - 1/3

65. Using (a - b)³ = a³ - b³ - 3ab(a - b) tick the correct of these when x = p - p :

(a) x³ + 3x = p + 1/p (b) x³ + 3x = p - 1/p

(c) x³ + 3x = p + 1 (d) None of these

1/3 -1/3 3

66. If x = 3 + 3 , then 3x - 9x is :

(a) 15 (b) 10

1/3 - 1/3 3

67. If x = 4 + 4 prove that 4x - 12 x is given by :

(a) 12 (b) 13

(c) 15 (d) 17

68. is equal to

%

(a) 1 (b) √3

(c) 3√3 (d)

69.

(a) 1/2 (b) 3/2

(c) 2/3 (d) 1/3

1/8 -1/8 1/8 -1/8 1/4 -1/4 1/2 -1/2

70. Value of (a +a )(a - a )(a +a ) (a +a ) is:

(a) a + 1/a (b) a - 1/a

2 2

(c) a + 1/a² (d) a - 1/a²

1

y - x

1+ a

1

x - y

1+ a

√ 9

3 ^5/2 9 3√3

7/2

x 9

3 9√3

n n - 1

2 + 2

n + 1 n

2 - 2

Answer Key

1 b 2 a 3 b 4 c 5 a 6 b 7 a 8 c 9 c 10 c 11 b 12 a 13 a

14 b 15 d 16 b 17 c 18 d 19 c 20 d 21 a 22 c 23 a 24 c 25 d 26 b

27 d 28 a 29 c 30 c 31 b 32 c 33 a 34 b 35 a 36 d 37 b 38 d 39 b

40 b 41 a 42 b 43 b 44 b 45 b 46 b 47 c 48 a 49 a 50 a 51 b 52 b

53 b 54 a 55 a 56 a 57 a 58 a 59 a 60 c 61 a 62 c 63 c 64 c 65 b

66 b 67 d 68 a 69 b 70 b

(28)

!

^{If a = b} where b > 0 a > 0 (but a ¹ 1)

then the exponent x is called the logarithm of b to the base a If a x = b then

log b_a = x

!

Properties of logarithms

(a) log (m x n) = log m + log n_a _a _a (b) log (m/n) = log m - log n_a _a _a (c) log m = nlog m_a n _a

(d) log m = log m_b _a

a

(e) log a = 1_a

(f) log 1 = 0_a (g) log a = _b

(h) log a = _b

(i) log a x log b = 1_b _a

(j) log a x log b x log c = log a_b _c _d _d Note :

1. log x = log y_a _a x = y

2. a > 1 then x > y log x > log y_a _a 3. If 0 < a < 1 then x > y log x < log y_a _a

x [ ; ]

1 b

log a_m log b_m

1 log b_a

D - LOGARITHMS

(29)

1. log 6 + log 5 is expressed as :

(a) log 11 (b) log 30

(c) log 5/6 (d) None of these

2. log 8 is equal to :2

(a) 2 (b) 8

3. log 32/4 is equal to :

(a) log 32 / log 4 (b) log 32 - log4

(c) 2³ (d) None of these

4. log (1 x 2 x 3) is equal to :

(a) log 1 + log 2 + log 3 (b) log 3

(c) log 2 (d) None of these

5. The value of log 0.0001 to the base 0.1 is :

(a) - 4 (b) 4

(c) ¼ (d) None of these

6. If 2 log x = 4 log 3, then x is equal to :

(a) 3 (b) 9

7. log 64 is equal to :_√2

(a) 12 (b) 6

8. log 1728 is equal to :2√3

(a) 2√3 (b) 2

9. log (1/81) to the base 9 is equal to :

(a) 2 (b) ½

10. log 0.0625 to the base 2 is equal to :

(a) 4 (b) 5

11. Given log2 = 0.3010 and log3 = 0.4771 the value of log 6 is :

(a) 0.9030 (b) 0.9542

(c) 0.7781 (d) None of these

12. The value of log 2 is :2

(a) 0 (b) 2 (c) 1

(d) None of these

EXERCISE # 1 D

LOGARITHMS

(30)

13. The value of log 0.333 to the base 9 is :

(a) - ½ (b) ½

14. The logarithm of 64 to the base 2√2 is :

(a) 2 (b) √2

(c) ½ (d) None of these

15. The value of (log a x log b x log c)³ is equal to :_b _c _a

(a) 3 (b) 0

16. log 32/4 is equal to___________

(a) log 32/log4 (b) log 32 - log4

3

17. The logarithm of a number consists of two parts, the whole part or the integral part is called the_______and the decimal part is called the_________.

(a) Characteristic, Number (b) Characteristic, Mantissa

(c) Mantissa, Characteristic (d) Number, Mantissa

18. log (1 x 2 x 3) is equal to

(a) log 1 + log 2 + log 3 (b) log 3

(c) log 2 (d) None of these

19. The base of log 13 into the common logarithmic base is6

(a) (b)

(c) (d) None of these

20. The value of + is

(a) 0 (b) 1

(c) -1 (d) None of these.

21. log log 27 is equal to_______3 3

(a) 0 (b) -1 (c) 3 (d) 1 614 22. The value of (log a x log b x log c) is equal tob c a 5

(a) 5 (b) 0

23. If log (x+1) = 2 then x is equal to______√3

(a) 1 (b) 3 (c) 2 (d) 0 24. The value of log log 81 is equal to4 3

(a) 1 (b) 0

(c) 3 (d) None of these log 10₆

log 136

log log 1310

106

log₁₀ log 610

13

1 log (ab)_a

1 log (ab)_b

(31)

25. The value of log 2166 √6 is equal to

(a) 7/2 (b) 5/2

26. log 512 is equal to64

(a) 3 (b) 2

(c) 1 (d) 3/2

27. log (3 x 5 x 7) is equal to

(a) log 3 x log 5 x log 7 (b) log 3 + log 5 + log 7 (c) log 3 - log 5 - log 7 (d) 0 28. log (5/7) is equal to

(a) log 5 x log 7 (b) log 5 + log 7

(c) log 5 - log 7 (d) None of these

29. The value of log (1 + 2 + 3 + ...+ n) is equal to

(a) log 1+ log 2+ .... + log n (b) log n + log (n+1) - log 2

(c) 0 (d) 1

2 2 2

30. log (1 + 2 + 3 ) is equal to_________

2 2 2

(a) log l + log 2 + log 3 (b) log 2 + log 7

(c) log 2 - log 7 (d) None of these

31. log (3x5 x7) is equal to_________2

(a) 2(log 3 + log 5 + log 7) (b) log (2x3x5x7)

(c) 2(log3-log5-log7) (d) None of these

32. The value of log is3

(a) 4 (b) - 4

(c) 2 (d) - 2

33. The value of log is2√2

(a) (b) - 4

(c) (d)

34. If Iog x = 4, then the value of x is₁₀

(a) 100 (b) 1000

35. If log 3 = find the value x x

(a) 9 (b) 27

log 5 log 7

1

( (

81

1

( (

256 16

3

- 16 3

√ ¹₆

(32)

36. The value of log 0.00001 = - 5, then x isx

(a) 10 (b) 102

(c) 10° (d) None of these

37. The value of Ioga √n A

(a) log Aa (b) a log A1/n

(c) A log _a (d) None of these

38. The value of

(a) (b)

(c) (d) None of these

If log₁₀₀₀₀ X=-1/4 , then x is given by:

(a) 1/100 (b) 1/10

(c) 1/20 (d) None

log 144 is equal to:

(a) 2 log 4 + 2 log 2 (b) 4 log 2 + 2 log 3

(c) 3 log 2 + 4 log 3 (d) 3 log 2 4 log 3

41. If log x + log y = log (x + y), y can be expressed as :

(a) x - 1 (b) x

(c) x/x - 1 (d) None of these

42. The value of log [log {log (log 27 )}] is equal to :₂ ₂ ₃ ₃ 3

(a) 1 (b) 2

43. If log x + log x + log x = 21/4, these x is equal to :2 4 16

(a) 8 (b) 4

44. The simplified value of 2 log 5 + log 8 - 1/2 log 4 is : 10 10 10

(a) 1/2 (b) 4

2 - 1 - 1 - 1/2

45. log [1 - {1 - (1 - x ) } ] can be written as :

(a) log x² (b) log x

(c) log 1/x (d) None of these

3 - 1 - 4/3

46. The simplified value of log 729 9 .27 is :6

(a) log 3 (b) log 2

(c) log 1/2 (d) None of these

39.

40.

log 4 log 8

10 10

1 3

4 3 2

3 1 n

1

( (

n

(33)

47. Iflog [log (log x)] = 0 ; then value of x is3 4 2

(a) 16 (c) 32

(c) 4 (d) None of these

48. If log = (log a + log b), the value of a² + b² is

(a) 6ab (b) 8ab

2 2

(c) 6a 6 (d) None of these

If log [log (log x)] = 1, then x equals:₂ ₃ ₂

(a) 128 (b) 256

(c) 512 (d) None

50. log (m + n) = log m + log n, m can be expressed as:

(a) m = (b) m =

(c) m = (d) m =

51. log (x +x)-log (x+1) = 2. Find x₄ 2 ₄

(a) 16 (b) 0

If log b + log c = 0 then._a _a

(a) b = c (b) b = -c

(c) b = c = 1 (d) b and c are reciprocals.

Solve :

-1 2

(a) 10 (b) 10

(c) 10 (d) 103

If log x + log x = 6 then the Value of x is2 4

(a) 16 (b) 32

(c) 64 (d) 128

55. If Iog x + Iog x + log x = then the value of x is2 8 32

(a) 8 (b) 5

(c) 2 (d) None of these

56. If log 12.45 = 1.0952 and log 3.79 = 0.5786, Find the value of log 124.5 + Iog 37910 10 10 10

(a) 5.6738 (b) 4.6738

57. If log3 = 0.48 and log 7 = 0.84, then the value of log is

(a) - 2.26 (b) - 3.26

(c) - 1.36 (d) None of these

49.

52.

53.

54.

a + b 2

1 2

n n - 1

n n + 1

n + 1 n n + 1

n - 1

log x - 310

2

11 - 3 log x10

+ = 2

23 15

0.03 0.7

(34)

58. Evaluate √0.5173 4

(a) 0.8480 (b) 0.8210

59. Evaluate

(a) 1.5948 (b) 0.5948

log xa

60. The value of a is

(a) x (b) log x_a

2

(c) x (d) None of these

2-log 63

61. The value of 3 is

(a) 9/5 (b) 3 /2

62. If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, there log 30

(a) 2.5717 (b) 2.4771

(c) 1.4771 (d) None of these

If log (2a - 3b) = Log a - Log b, then a = :

(a) (b)

(c) (d)

64. If log 2 = 0.3010, find the number of digits in 2100

(a) 36 (b) 31

Number of digits in the numeral for 2 . [Given log 2 = 0.30103] :64

(a) 18 digits (b) 19 digits

(c) 20 digits (d) 21 digits

The value is:

(a) 3 log 2₁₀ (b) 7 log 3₁₀

(c) 3 log ze (d) None.

67. Given that log 2 = x and log 3 = y, the value of log 60 is expressed as :10 10 10

(a) x - y + 1 (b) x + y + 1

(c) x - y - 1 (d) None of these

68. Given that log 2 = x, log 3 = y, then log 1.2 is expressed in terms of x and y as :10 10 10

(a) x + 2y - 1 (b) x + y - 1

(c) 2x + y - 1 (d) None of these

69. Given that log x = m + n and log y = m - n, the value of log 10x/y² is expressed in terms of m and n as :

(a) 1 - m + 3n (b) m - 1 + 3n

(c) m + 3n +1 (d) None of these

63.

65.

66.

0.7214 x 20.37

√

^³ 69.8

3b2

2b-1

3b2

2b+1 b2

2b+1

3b 2b-1

log 83

log 16.log 109 4

RATIO, PROPORTION, INDICES & LOGARITHMS