Post on 24-Dec-2021
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.1
CHAPTER # 1
RATIO, PROPORTION, INDICES & LOGARITHMS
! 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 : 2003 : 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
b
a³
b³
T. R.
A - RATIO
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.2
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 : 424
so ratio is 8 : 3 : 4
3. Simplify ratio 2.5 : 5.45 : 4
2.5 : 5.45 : 42.50 : 5.45 : 4.00250 : 545 : 40050 : 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.=
a1/3b
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.=
a1/2b
1/2
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.3
= (
pq² - q²x² = p²q - p²x²p²x² - q²x² = p²q - pq²x² (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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.4
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
15
14
23
13
13
13
13
23
45
57
23
45
57
45
57
23
57
45
23
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.5
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(c) 30 (d) None of these
15. The inverse ratio of 11 : 15 is :(a) 15 : 11 (b) √11 : √15(c) 121 : 225 (d) None of these
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(c) 9 : 16 (d) None of these
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(c) 3 : 2 (d) None of these
20. The sub triplicate ratio of 8 : 27 is _________ : (a) 27 : 8 (b) 24 : 81(c) 2 : 3 (d) None of these
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(c) 3 : 1 (d) None of these
22. The ratio compounded of 2 : 3, 9 : 4, 5 : 6 and 8 : 10 is :(a) 1 : 1 (b) 1 : 5(c) 3 : 8 (d) None of these
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.6
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(c) 2 : 21 (d) None of these
27. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is(a) 18 : 25 (b) 8 : 25(c) 17 : 24 (d) None of these
28. If x : y = 3 : 4, the value of x²y + xy² : x³ + y³ is :(a) 13 : 12 (b) 12 : 13(c) 21 : 31 (d) None of these
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(c) 17 : 28 (d) None of these
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(c) 5 (d) None of these
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 (b) a : b : c = n d : n d : d d1 1 2 2 1 2 1 2 2 1 2 1
(c) a : b : c = n n : d n : d d (d) None of these1 2 1 2 1 2
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
ab
ba
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.7
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(c) 500 (d) None of these
1
a
1
b
13
23
12
p
q
2
32p + q
2p - q
x
2y
3
2
2x + y
x - 2y1
7
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.8
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(c) 1 (d) None of these
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(c) 17 (d) None of these
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(c) 150 (d) None of these
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(c) 4:5 (d) None of these
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(c) 8 : 9 (d) None of these
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.9
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.10
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(c) 32 : 33 (d) None of these
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.11
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(c) 21:20 (d) None of these
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
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a
b=
c
d( (b
a=
d
c
a
c=
b
d
a + b
b=
c + d
d
B - PROPORTION
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.12
! 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 : dor
a : b :: c : d
I II III IVTerm 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 ifa : 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)
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.13
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.14
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 pq2 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 3 311. 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
(c) 1 (d) None of these
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.15
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
(c) 48 (d) None of these
15. The third proportional to 12, 18 is :
(a) 24 (b) 27
(c) 36 (d) None of these
16. The mean proportional between 25, 81 is :
(a) 40 (b) 50
(c) 45 (d) None of these
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
(c) 21 (d) None of these
.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)
(c) (d)
x + y
x - y
x - y
x + y
a + b
a - b
(a - b)²
a + b
a - b
a + b
(a + b)³
a - b
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.16
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
(c) 5/3 (d) None of these
29. If x : y = z : w = 2.5 : 1.5, the value of (x + z) / (y + w) is
(a) 1 (b) 3/5
(c) 5/3 (d) None of these
30. If (5x - 3y) / (5y - 3x) = 3/4, the value of x : y is
(a) 2 : 9 (b) 7 : 2
(c) 7 : 9 (d) None of these
31. 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 of these
32. If x/2 = y / 3 = z/7, then the value of (2x - 5y + 4z)/y is
(a) 6/23 (b) 23/6
(c) 3/2 (d) None of these
33. If x : y = 2 : 3, y : z = 4 : 3 then x : y : z is
(a) 2 : 3 : 4 (b) 4 : 3 : 2
(c) 3 : 2 : 4 (d) None of these
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
(c) 32 (d) None of these
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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.17
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
(c) 15 (d) None of these
42. 4, *, 9, 13½ are in proportion. Then * is
(a) 6 (b) 8
(c) 9 (d) None of these
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
(c) 7 (d) None of these
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
ba
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.18
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
(c) 7 (d) None of these
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
(c) 5 (d) None of these
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 kg3
9 2 kg3
10 1 kg3
x
b + c - a
y
c + a - b
z
a + b - c
a + b
b
?
d
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.19
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 z60. 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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.20
! If a is a non - zero real number and n is a natural number then
na = 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) =
0(f) a = 1
-m(g) a =
m(h) = a
ma
na
a
b
n
( ( an
b
n
1m
a
1-m
a
C - INDICES
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.21
1/2 - 11. Simply 2x 3x if x = 4 :(a) 2 (b) 3(c) 4 (d) - 3
- 2 - 32. 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 - b66. a x . (a x ) :
1 - b 1 + b(a) x (b) x(c) x (d) 0
- 7 - 5 k7. Find the value of k from (√9) x (√3) = 3 :
(a) (b)
(c) (d)
EXERCISE # 1 C
INDICES
-14x-1/3x
4x1/3x
2x2/3x
42/3x
2a x a x 6a
9a x a
12
23
73
- 53
32
2332
13
13b + 2ay
1
y
192
57
192
75
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.22
- 1/48. 4x is expressed as :1/4 - 1(a) - 4x (b) x
1/4(c) 4/x (d) None of these
1/39. The value of 8 is :(a) 3√2 (b) 4(c) 2 (d) None of these
1/510. The value of 2 x (32) is :(a) 2 (b) 10(c) 4 (d) None of these
1/511. The value of 4/(32) is :(a) 8 (b) 2(c) 4 (d) None of these
1/312. The value of (8/27) is : (a) 2/3 (b) 3/2(c) 2/9 (d) None of these
- 1/813. The value of 2(256) is :(a) 1 (b) 2(c) 1/2 (d) None of these
1/2 3/414. 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 - a16. x . x . x is equal to :(a) x (b) 1(c) 0 (d) None of these
17. The value of is equal to :
(a) 0 (b) 2/3(c) 1 (d) None of these
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
481x - 8y[ [
14
2p²q³3xy(
0
(
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.23
a - b b - c c - a - a - b20. 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 2/3³(a) x = √x² (b) x = √x³2/3 2/3³ ³(c) x > √x² (d) x < √x²
- 3 2 - 1 3 - 222. Simplified value of 16x y x 8 x y is : (a) 2xy (b) xy/2(c) 2 (d) None of these
- 1/3 - 1/523. The value of (8/27) x (32 / 243) is : (a) 9/4 (b) 4/9(c) 2/3 (d) None of these
.2/3 3/2 3 624. 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 1/2³25. Simplified value of (125) x √25 x √5 x 5 is : (a) 5 (b) 1/5(c) 1 (d) None of these
1/2 )3/4 5/6 7/8 9/10 4 3/2526. [{(2) . (4 . (8) . (16) . (32) } ] is : (a) A fraction (b) An integer(c) 1 (d) None of these
n n-1/n 1/n + 127. {(X ) } is equal to : n n + 1(a) X (b) Xn - 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 P29. 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(c) 0 (d) None of these
1/4 3/4 5/431. 3 x 6 x 2 is equal to(a) 10 (b) 12 (c) 14 (d) 0
p q r32. If 1176 = 2 .3 .7 Find the value of p, q, r.(a) 1, 2, 3 (b) 1,3, 2(c) 3, 1, 2 (d) 1,3, 5
827
1/3
pq( (
pq
- ppq - 1
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.24
a - m m -n n - a33. 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 4/535. The value of + (64) + (- 32)
(a) 17 (b) 17
(c) 11 (d) None of these
3 8 n36. If (4) x ( 2) =2 , then n is(a) 10 (b) 12(c) 13 (d) None of these
37. Simplification of is: m -m(a) x (b) xn -n(c) x (d) x
x x-1 x38. If 2 -2 =4, then the value of x is:(a) 2 (b) 1(c) 64 (d) 27
X y z39. If 3 = 2, 5 =3 and 2 = 5, find the value of multiply of x. y. z(a) 0 (b) 1(c) 2 (d) None of these
a b c40. If x=y , y=z and z = x then abc is:(a) 2 (b) 1(c) 3 (d) 4
x y z41. If a = b, b = c, c = a, then xyz is :(a) 1 (b) 2(c) 3 (d) None of these
n/m n/m42. If m and n are two natural numbers then a x a ..................m terms =n(a) a (b) a
m(c) a (d) m
-1-1-1 43. [1/n] :
(a) 1/n (b) n - 3 - 3(c) n (d) 1/n
m - n 2n k44. If a . a = a then k = :(a) n (b) m + n(c) m - 3n (d) None of these
1
64( (0
18
78
38
m+3n 4m-9nx .x6m-6nx
[[ [ [
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.25
x ³45. If √a = √a² then x = :(a) 2/3 (b) 4/3(c) 1/3 (d) 5/3
x46. Find x, if 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 ?
222(a) 2 (b)
2x3(c) 22x2x2 (d) 2
x y49. 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(c) 0 (d) None of these
1/p 1/q 1/r51. If x = y = z and xyz = 1, then the value of p + q = r is :(a) 1 (b) 0(c) 1/2 (d) None of these
x y - z52. If 2 = 3 = 6 , + + is :
(a) 1 (b) 0(c) 2 (d) None of these
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(c) 2 (d) None of these
[ { { [
22[( (2 [
2
1a - b a - c1 + x + xΣ
1x
1y
1z
lxmx[ [
2 2l + lm + m
xmxnx[ [
2 2m + mn + n nxlx[ [
2 2l + ln + n
x
aXbX( (
a + b
xbXcX( (
b + c
xcXaX( (
c + a
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.26
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
58. Show that reduces to :
(a) 1 (b) 3(c) - 1 (d) None of these
59. Show that reduces to :
(a) 1 (b) 3(c) 0 (d) 2
60. Show that reduces to :
(a) - 1 (b) 0(c) 1 (d) None of these
61. Show that is reduces to :
(a) 1 (b) 0(c) - 1 (d) None of these
m - n m - p n - m n - p p - m p - n62. 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
aXbX( (
(a² + ab + b²)
xbXcX( (
(b² + bc + c²)
xcXaX( (
(c² + ca + a²)
x( (1
a - b
1a - c
x x( (1
b - c
1b - a
x x( (1
c - a
1c - b
2bX2cX
(b+c) x2cX2aX
(c +a)
2aX2bX
(a+b) x
bXcX( (
a
xcXaX( (
b
xaXbX( (
c
x( (b + cc - a
1a - b
x x( (c + aa - b
1b - c
x x( (a + bb - c
1c - a
bXcX( (
1/bc
xcXaX( (
1/ca
xaXbX( (
1/ab
bXcX( (
b + c - a
xcXaX( (
c + a - b
xaXbX( (
a + b - c
1a - b a - c1 + z + z
1b - c b - a1 + z + z
1c - a c - b1 + z + z
12(a + b + c)z
1(a + b + c)z
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.27
64. The value of + is given by :
(a) - 1 (b) 0(c) 1 (d) None of these
1/3 - 1/365. 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 -1/3 366. If x = 3 + 3 , then 3x - 9x is : (a) 15 (b) 10(c) 12 (d) None of these
1/3 - 1/3 367. 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 -1/270. 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
1y - x1+ a
1x - y1+ a
√93
5/29
3√3
7/2
x 9
3
9√3
n n - 12 + 2n + 1 n2 - 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
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.28
! 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
xIf a = b then
log b = xa
! Properties of logarithms
(a) log (m x n) = log m + log na a a
(b) log (m/n) = log m - log na a a
n(c) log m = nlog ma a
(d) log m = log mb aa
(e) log a = 1a
(f) log 1 = 0a
(g) log a = b
(h) log a = b
(i) log a x log b = 1b a
(j) log a x log b x log c = log ab c d d
Note :
1. log x = log y x = ya a
2. a > 1 then x > y log x > log ya a
3. If 0 < a < 1 then x > y log x < log ya a
x [ ; ]
1
b
log am
log bm
1
log ba
D - LOGARITHMS
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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(c) 3 (d) None of these
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(c) 2 (d) None of these
7. log 64 is equal to :√2
(a) 12 (b) 6(c) 1 (d) None of these
8. log 1728 is equal to :2√3
(a) 2√3 (b) 2(c) 6 (d) None of these
9. log (1/81) to the base 9 is equal to :(a) 2 (b) ½(c) - 2 (d) None of these
10. log 0.0625 to the base 2 is equal to :(a) 4 (b) 5(c) 1 (d) None of these
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
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EXERCISE # 1 D
LOGARITHMS
Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.29
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.30
13. The value of log 0.333 to the base 9 is :(a) - ½ (b) ½(c) 1 (d) None of these
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(c) 1 (d) None of these
16. log 32/4 is equal to___________(a) log 32/log4 (b) log 32 - log4
3 (c) 2 (d) None of these
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
5 22. The value of (log a x log b x log c) is equal tob c a
(a) 5 (b) 0(c) 1 (d) None of these
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
610
log10
log 610
13
1log (ab)a
1log (ab)b
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.31
25. The value of log 216√6 is equal to6
(a) 7/2 (b) 5/2(c) 1 (d) None of these
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 2 230. 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
231. log (3x5 x7) is equal to_________(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(c) 10000 (d) None of these
35. If log 3 = find the value x x
(a) 9 (b) 27(c) 18 (d) None of these
log 5
log 7
1
81( (
1
256 ( (16
3
- 16
3
√16
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.32
36. The value of log 0.00001 = - 5, then x isx
2(a) 10 (b) 10(c) 10° (d) None of these
n
37. The value of Iog √ Aa
(a) log A (b) a log Aa 1/n
(c) A log (d) None of these a
38. The value of
(a) (b)
(c) (d) None of these
If log X=-1/4 , then x is given by:10000
(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
342. The value of log [log {log (log 27 )}] is equal to :2 2 3 3
(a) 1 (b) 2(c) 0 (d) None of these
43. If log x + log x + log x = 21/4, these x is equal to :2 4 16
(a) 8 (b) 4(c) 16 (d) None of these
44. The simplified value of 2 log 5 + log 8 - 1/2 log 4 is : 10 10 10
(a) 1/2 (b) 4(c) 2 (d) None of these
2 - 1 - 1 - 1/245. 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 - 4/36 46. The simplified value of log 729 9 .27 is :(a) log 3 (b) log 2(c) log 1/2 (d) None of these
39.
40.
log 4log 8
10
10
1
3
4
3
2
3
1
n
1
n( (
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.33
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) 8ab2 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 =
251. log (x +x)-log (x+1) = 2. Find x4 4
(a) 16 (b) 0(c) - 1 (d) None of these
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 3(c) 10 (d) 10
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(c) 6.6738 (d) None of these
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 + b2
12
nn - 1
nn + 1
n + 1n
n + 1n - 1
log x - 310
2 11 -
3log x10+ = 2
2315
0.03
0.7
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.34
458. Evaluate √0.5173
(a) 0.8480 (b) 0.8210(c) 0.6480 (d) None of these
59. Evaluate
(a) 1.5948 (b) 0.5948(c) 0.2348 (d) None of these
log xa60. The value of a is (a) x (b) log xa
2 (c) x (d) None of these
2-log 6361. The value of 3 is (a) 9/5 (b) 3 /2(c) 9/4 (d) None of these
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)
10064. If log 2 = 0.3010, find the number of digits in 2(a) 36 (b) 31(c) 38 (d) None of these
64Number of digits in the numeral for 2 . [Given log 2 = 0.30103] :(a) 18 digits (b) 19 digits(c) 20 digits (d) 21 digits
The value is:
(a) 3 log 2 (b) 7 log 310 10
(c) 3 log z (d) None.e
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 20.3769.8√³
23b2b-1
23b2b+1
2b2b+1
3b2b-1
log 83
log 16.log 109 4
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.35
70. log(1 + 2 + 3) is exactly equal to : (a) log 1 + log 2 + log 3 (b) log (1 x 2 x 3) (c) Both the above (d) None of these
logb - logc logc - loga loga - logb71. a x b x c has a value of :(a) 1 (b) 0(c) - 1 (d) None of these
72. + + is equal to :
(a) 0 (b) 1(c) 2 (d) - 1
73. + + is equal to :
(a) 0 (b) 1(c) 3 (d) - 1
74. + + is equal to :
(a) 0 (b) 1(c) 3 (d) - 1
75. log (a).log (b).log (c) is equal to :b c a
(a) 0 (b) 1(c) - 1 (d) None of these
1/2 3 2/376. log (a ).log (b ).log (c ) is equal to :b c a
(a) 0 (b) 1(c) - 1 (d) None of these
logb/c logc/a loga/b77. The value of a .b .c : (a) 0 (b) 1(c) - 1 (d) None of these
78. The value of log + log + log is :
(a) 0 (b) 1(c) - 1 (d) None of these
979. log (a ) + log a = 10 if the value of a is given by :(a) 0 (b) 10(c) - 1 (d) None of these
80. If + + = then the value of z is given by :
(a) abc b) a + b + c(c) a (b+ c) (d) (a + b) c
1log (abc)ab
1log (abc)bc
1log (abc)ca
11 + log (bc)a
11 + log (ca)b
11 + log (ab)c
1log (x)a/b
1log (x)b/c
1log (x)c/a
nanb
nbnc
ncna
1log ta
1log tb
1log tc
1log tz
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.36
2 3 n81. The sum of the series log b + log b + log b + ............log b is given by : a 2 3 na a an n(a) log b (b) log ba na
n(c) nlog b (d) (a) & (c) bothna
82. has a value of :
(a) a (b) b(c) (a + b) (d) None of these
log b.log c.log d.log ta b c d83. The value of the following expression a is given by :(a) t (b) abcdt(c) (a + b + c + d + t) (d) None of these
84. On solving the equation logt + log(t - 3) = 1 we get the value of t as : (a) 5 (b) 2(c) 3 (d) 0
85. On solving the equation log [log (log t)] = 1 we get the value of t as : 3 2 3
(a) 8 (b) 18(c) 81 (d) 6561
86. log [log (log 256)] =3 2 2
(a) 0 (b) 1(c) - 1 (d) None of these
87. Given log (1 + 2 + 3) = log1 + log2 + log3. Usin the given rule calculate log (3 + 4 + 5) : (a) log 3 + log 4 (b) log 3 + log 4 + lo 5(c) (a) or (b) (d) Can't be determined
log 5 log 23 388. 2 - 5 = :
(a) 0 (b) 2(c) 3 (d) 5
89. The characteristic of log 529.32 is .................. : (a) 1 (b) 2(c) 3 (d) 4
90. If the characteristic of a number is 5 then the number of digits in the integral part of the number is______ :(a) 5 (b) 4(c) 6 (d) 0
91. If the characteristic of a number is - 2 then the first non-zero number appears in which decimal place ? (a) 1st (b) 2nd(c) 3rd (d) 4th
3 2 x 92. Value of [log √25 - log (2) + log (4) ] %10 10 10
(a) x (b) 10(c) 1 (d) None of these
1log aba
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Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.37
Answer Key
1 b 2 c 3 b 4 a 5 b 6 b 7 a 8 c 9 c 10 d 11 c 12 c 13 a
14 d 15 c 16 b 17 b 18 a 19 c 20 b 21 d 22 c 23 c 24 a 25 a 26 d
27 b 28 c 29 b 30 b 31 a 32 b 33 d 34 c 35 b 36 a 37 a 38 c 39 b
40 b 41 c 42 c 43 a 44 c 45 b 46 d 47 a 48 d 49 c 50 a 51 a 52 d
53 a 54 a 55 c 56 b 57 c 58 a 59 b 60 a 61 b 62 c 63 a 64 b 65 c
66 a 67 b 68 c 69 a 70 c 71 a 72 c 73 b 74 a 75 b 76 b 77 b 78 a
79 b 80 a 81 d 82 b 83 a 84 a 85 d 86 b 87 a 88 a 89 b 90 c 91 b
92 c
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