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Ratio & Proportion, Indices & Logarithm

1. The fourth proportional to (𝒂𝟐 − 𝐚𝐛 + 𝒃𝟐 ), (𝒂𝟑 + 𝒃𝟑 ) and (𝐚 − 𝐛) is equal to

(a) 𝒂𝟐 + 𝒃𝟐
(c) 1
(b) 𝒂𝟐 − 𝒃𝟐
(d) None of these

2. The third proportional to 15 and 20 is

𝟖𝟎
(a) 𝟑
𝟖𝟎
(c) 𝟕
(b) 80
(d) None of these

3. What must be added to each of the four numbers 𝟏𝟎, 𝟏𝟖, 𝟐𝟐, 𝟑𝟖. So that they become in proportion?
(a) 2
(b) 5
(d) None of these

4. The age of two persons is in the ratio 𝟓: 𝟕. Eighteen years ago their ages were in the ratio of 8:13,
their present age (in years) are:
(a) 50,70
(c) 40,56
(b) 70,50
(d) None
5. 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) Rs. 72,600 , Rs. 48,400 , Rs. 𝟏, 𝟐𝟏, 𝟎𝟎𝟎
(b) Rs. 48,400 , Rs. 𝟏, 𝟐𝟏, 𝟎𝟎𝟎, Rs. 72,600
(c) Rs. 72,000 , Rs. 49,000 , Rs. 𝟏, 𝟐𝟏, 𝟎𝟎𝟎
(d) Rs. 48,000 , Rs. 𝟏, 𝟐𝟏, 𝟒𝟎𝟎, Rs. 72,600
𝒑 −𝟐 𝟐𝒑+𝒒
6. If 𝒒 = then the value of 𝟐𝒑−𝒒 is:
𝟑
(a) 1

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 (c) 𝟏/𝟕
(b) −𝟏/𝟕
(d) 7
7. The mean proportional between 2 and 1.25 is:
(a) 0.625
(c) 2.5
(b) 25
(d) 𝟏. 𝟓𝟖𝟏
8. If (𝒙 + 𝟓) is the mean proportional between (𝒙 + 𝟐) and (𝒙 + 𝟗) then the value of ' 𝒙 ' is
(a) 4
(c) 7
(b) 5
(d) 8
9. The students of two classes are in the ratio 𝟓: 𝟕, if 10 students left from each class, the remaining
students are in the ratio of 𝟒: 𝟔 then the number of students initially in each class was:
(a) 30,40
(c) 40,60
(b) 25,24
(d) 50,70
10. If 𝑨: 𝑩 = 𝟐: 𝟓 then (𝟏𝟎𝑨 + 𝟑𝑩): (𝟓𝑨 + 𝟐𝑩) is equal to:
(a) 𝟕: 𝟒
(c) 𝟔: 𝟓
(b) 𝟕: 𝟑
(d) 𝟕: 𝟗
𝟏
𝟏𝒏+𝟏
11. {(𝒙𝒏 )𝒏−𝒏 } is equal to
(a) 𝒙𝒏
(c) 𝒙𝒏−𝟏
(b) 𝒙𝒏+𝟏
(d) none of these
𝒍𝟐 +𝒍𝒎+𝒎𝟐 𝟐 𝟐
𝒙𝒍 𝒙𝒎 𝒎 +𝒎𝒏+𝒏 𝒙𝒏 𝒏
12. 23. [𝒙𝒎 ] × [ 𝒙𝒏 ] × [ 𝒙𝒍 ]
(a) 0
(b) 1
(c) 𝒙
(d) none of these

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 13. Tick the correct of these when 𝒙 = 𝒑𝟏/𝟑 − 𝒑−𝟏/𝟑
(a) 𝒙𝟑 + 𝟑𝒙 = 𝒑 + 𝟏/𝒑
(c) 𝒙𝟑 + 𝟑𝒙 = 𝒑 + 𝟏
(b) 𝒙𝟑 + 𝟑𝒙 = 𝒑 − 𝟏/𝒑
(d) none of these

14. On simplification, 𝟏/(𝟏 + 𝒂𝒎−𝒏 + 𝒂𝒎−𝒑 ) + 𝟏/(𝟏 + 𝒂𝒏−𝒎 + 𝒂𝒏−𝒑 ) + 𝟏/(𝟏 + 𝒂𝒑−𝒎 + 𝒂𝒑−𝒏 ) is equal
to
(a)
(c) 1
(b) 𝒂
(d) 𝟏/𝒂

15. If 𝒂𝒙 = 𝒃, 𝒃𝒚 = 𝒄, 𝒄𝒛 = 𝒂, then 𝒙𝒚𝒛 is

(a) 1
(c) 3
(b) 2
(d) none of these
𝟐 𝟐 (𝒃𝟐+𝒃𝒄+𝒄𝟐 ) 𝟐 𝟐
𝒙𝒂 (𝒂 +𝒂𝒃+𝒃 ) 𝒙𝒃 𝒙𝒄 (𝒄 +𝒄𝒂+𝒂 )
16. The value of (𝒙𝒃 ) × ( 𝒙𝒄 ) × (𝒙𝒂 )
(a) 1
(c) -1
(b) 0
(d) none of these
𝟏 𝟏 𝟏
17. If 𝟐𝐱 = 𝟑𝐲 = 𝟔−𝐳 , 𝒙 + 𝒚 + 𝒛 =
(a) 1
(c) 2
(b) 0
(d) none of these
𝟏 𝟏
18. If 𝒙 + 𝒙 = 𝟐, then the value of 𝒙𝟐 + 𝒙𝟐 =
(a) 2
(c) 8
(b) 4
(d) 16

19. 𝟑𝟏/𝟒 × 𝟔𝟑/𝟒 × 𝟐𝟓/𝟒 is equal to

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

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

(a) 𝟏, 𝟐, 𝟑
(c) 𝟑, 𝟏, 𝟐
(b) 𝟏, 𝟑, 𝟐
(d) 𝟏, 𝟑, 𝟓

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

(a) 0
(c) 2
(b) 1
(d) None of these
𝟏 𝟏
22. If 𝒙 + 𝒙 = 𝟑, then the value of 𝒙𝟑 + 𝒙𝟑 =
(a) 16
(c) 18
(b) 14
(d) None of these

𝟐𝟏𝟔 ×𝟑𝟏𝟎 ×𝟓𝟒

23. The value of is equal to
𝟐𝟏𝟐 ×𝟑𝟔 ×𝟓𝟑
(a) 2160
(c) 648
(b) 6480
(d) 3240
𝟒
24. Evaluate √𝟎. 𝟓𝟏𝟕𝟑
(a) 0.8480
(c) 0.6480
(b) 0.8210
(d) None of theses

𝟑 𝟎.𝟕𝟐𝟏𝟒×𝟐𝟎.𝟑𝟕
25. Evaluate √ 𝟔𝟗.𝟖
(a) 1.5948
(c) 0.2348
(b) 0.5948
(d) None of these

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 26. If (𝟒)𝟑 × (√𝟐)𝟖 = 𝟐𝒏, then 𝐧 is
(a) 10
(c) 13
(b) 12
(d) None of these

𝟑𝒏+𝟏+𝟑𝒏
27. The value of 𝟑𝒏+𝟑 −𝟑𝒏+𝟏
(a) 𝟏/𝟑
(c) 𝟏/𝟒
(b) 𝟏/𝟔
(d) 𝟏/𝟗
𝒙
28. Value of 𝒙, if 𝒙, 𝒙𝟏/𝟑 = (𝒙𝟏/𝟑 ) :
(a) 3
(c) 2
(b) 4
(d) 6
𝟐 𝟐 𝟐
29. If 𝟐𝒙 = 𝟑𝒚 = 𝟏𝟐𝒛
𝟏 𝟏 𝟏
(a) 𝒙𝟐 + 𝒚𝟐 = 𝒛𝟐
𝟏 𝟐 𝟏
(b) 𝒙𝟐 + 𝒚𝟐 = 𝒛𝟐
𝟐 𝟏 𝟏
(c) 𝒙𝟐 + 𝒚𝟐 = 𝒛𝟐
(d) None
30. Find the value of ' 𝒂 ' from the following
(√𝟗)−𝟓 × (√𝟑)−𝟕 = (√𝟑)−𝒂
(a) 13
(c) 15
(b) 11
(d) 17

𝟑𝒕−𝟏
31. Find the value of 𝒕−𝟏/𝟑
𝟑
(a) 𝒕𝟐/𝟑
𝟑
(b)
𝒕𝟑/𝟐
𝟑
(c) 𝒕𝟏/𝟑
𝟑
(d) 𝒕𝟐

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 𝒂𝟐 𝒃𝟐 𝒄𝟐
32. The value of 𝐥𝐨𝐠 𝒃𝒄 + 𝐥𝐨𝐠 𝒄𝒂 + 𝐥𝐨𝐠 𝒂𝒃 is equal to
(a) 0
(b) 1
(c) -1
(d) None of these
𝟏
33. The value of 𝐥𝐨𝐠 𝟑 𝟖𝟏 is
(a) 4
(c) 2
(b) -4
(d) -2
𝟏
34. The value of 𝐥𝐨𝐠 𝟐√𝟐 𝟐𝟓𝟔
𝟏𝟔
(a) 𝟑
(b) -4
(c) 3
−𝟏𝟔
(d) 𝟑

35. If 𝐥𝐨𝐠 𝟒 [𝐥𝐨𝐠 𝟑 (𝐥𝐨𝐠 𝟐 𝐱)] = 𝟎; then value of 𝐱 is

(a) 16
(b) 32
(c) 4
(d) None of these

36. The value of 𝐥𝐨𝐠 𝒙 (𝟎. 𝟎𝟎𝟎𝟎𝟏) = −𝟓, then 𝐱 is

(a) 10
(b) 𝟏𝟎𝟐
(c) 𝟏𝟎∘
(d) None of these
𝒏
37. The value of 𝐥𝐨𝐠 𝒂 √𝑨
𝟏
(a) 𝒏 𝐥𝐨𝐠 𝒂 𝑨
(b) 𝒂𝐥𝐨𝐠 𝟏/𝒏 𝑨
𝟏
(c) 𝒂𝐥𝐨𝐠 𝒂 (𝒏)
(d) None of these
𝐥𝐨𝐠 𝟒
38. The value of 𝐥𝐨𝐠 𝟏𝟎 𝟖
𝟏𝟎
𝟏
(a) 𝟑
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 𝟒
(b) 𝟑
𝟐
(c) 𝟑
(d) None of these

39. If 𝐥𝐨𝐠 𝟏𝟎 𝟏𝟐. 𝟒𝟓 = 𝟏. 𝟎𝟗𝟓𝟐 and 𝐥𝐨𝐠 𝟏𝟎 𝟑. 𝟕𝟗 = 𝟎. 𝟓𝟕𝟖𝟔, Find the value of 𝐥𝐨𝐠 𝟏𝟎 𝟏𝟐𝟒. 𝟓 + 𝐥𝐨𝐠 𝟏𝟎 𝟑𝟕𝟗
(a) 5.6738
(c) 6.6738
(b) 4.6738
(d) None of these

40. If 𝐥𝐨𝐠 𝟏𝟎 𝟓 + 𝐥𝐨𝐠 𝟏𝟎 (𝟓𝒙 + 𝟏) = 𝐥𝐨𝐠 𝟏𝟎 (𝒙 + 𝟓) + 𝟏, Then 𝒙 =

(a) 5
(c) 1
(b) 3
(d) None
𝟐
41. If (𝐥𝐨𝐠 √𝒙 𝟐) = 𝐥𝐨𝐠 𝒙 𝟐 then 𝐱 =
(a) 16
(c) 8
(b) 32
(d) 4
𝟑
42. Find value of [𝐥𝐨𝐠 𝒚 𝒙 𝐥𝐨𝐠 𝒛 𝒚
⋅ 𝐥𝐨𝐠 𝒙 𝒛 ] =
(a) 0
(c) 1
(b) -1
(d) 3

43. Find the value of 𝐥𝐨𝐠 𝟒 𝟗 ⋅ 𝐥𝐨𝐠 𝟑 𝟐 is

(a) 3
(b) 9
(c) 2
(d) 1

44. The value of 𝐥𝐨𝐠 𝟓 𝟑 × 𝐥𝐨𝐠 𝟑 𝟒 × 𝐥𝐨𝐠 𝟐 𝟓.

(a) 0
(c) 2
(b) 1
(d) 𝟏/𝟐

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 45. If 𝐥𝐨𝐠 𝒙 𝒚 = 𝟏𝟎𝟎 and 𝐥𝐨𝐠 𝟐 𝒙 = 𝟏𝟎 then the value of 𝒚
(a) 𝟐𝟏𝟎
(c) 𝟐𝟏𝟎𝟎𝟎
b) 𝟐𝟏𝟎𝟎
d) 𝟐𝟏𝟎,𝟎𝟎𝟎
𝟏 𝟏 𝟏
46. 𝐥𝐨𝐠 𝟓 (𝟏 + 𝟓) + 𝐥𝐨𝐠 𝟓 (𝟏 + 𝟔) + ⋯ … … … … + 𝐥𝐨𝐠 𝟓 (𝟏 + 𝟔𝟐𝟒) =
(a) 2
(c) 5
(b) 3
(d) 0

47. 𝐥𝐨𝐠 𝟎.𝟎𝟏 (𝟏𝟎, 𝟎𝟎𝟎) = 𝒙; Find the value of ?

(a) 1
(c) -4
(b) -2
(d) 2

48. If 𝐥𝐨𝐠 𝒙𝒚𝟐 − 𝐥𝐨𝐠 𝒚 = 𝐥𝐨𝐠(𝒙 + 𝒚) Find the value of 𝒚 in term of 𝒙

(a) 𝒙 − 𝟏
𝒙
(c) 𝒙−𝟏
𝒙
(b) 𝒙+𝟏
(d) 𝒙 + 𝟏

49. 𝐥𝐨𝐠 𝟗 + 𝐥𝐨𝐠 𝟓 is expressed as

(a) 𝐥𝐨𝐠(𝟗/𝟓)
(b) 𝐥𝐨𝐠 𝟒
(c) 𝐥𝐨𝐠(𝟓/𝟗)
(d) 𝐥𝐨𝐠 𝟒𝟓

𝒑𝟐 𝒒𝟐 𝒓𝟐
50. 𝐥𝐨𝐠 𝒒𝒓 + 𝐥𝐨𝐠 𝒑𝒓 + 𝐥𝐨𝐠 𝒑𝒒 =
(a) pqr
(c) 1
𝟏
(b) 𝒑𝒒𝒓
(d) 0

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1.

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