RATIO AND PROPORTION, INDICES, LOGARITHMS

Exercise 1(A)
Choose the most appropriate option (a) (b) (c) or (d)
1. The inverse ratio of 11 : 15 is
(a) 15 : 11 (b) √11 : √15 (c) 121 : 225 (d) none of these
2. The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is
(a) 16 (b) 60 (c) 22 (d) 20
3. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is
(a) 5 (b) √ 5 (c) 7 (d) none of these
4. 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
5. The duplicate ratio of 3 : 4 is
(a) √3 : 2 (b) 4 : 3 (c) 9 : 16 (d) none of these
6. The sub duplicate ratio of 25 : 36 is
(a) 6 : 5 (b) 36 : 25 (c) 50 : 72 (d) 5 : 6
7. The triplicate ratio of 2 : 3 is
(a) 8 : 27 (b) 6 : 9 (c) 3 : 2 (d) none of these
8. The sub triplicate ratio of 8 : 27 is
(a) 27 : 8 (b) 24 : 81 (c) 2 : 3 (d) none of these
9. The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4 is
(a) 1 : 4 (b) 1 : 3 (c) 3 : 1 (d) none of these
10. The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is
(a) 2 : 7 (b) 7 : 2 (c) 2 : 21 (d) none of these
11. The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio
of 81 : 256 and sub triplicate ratio of 125 : 512 is
(a) 4 : 512 (b) 3 : 32 (c) 1 : 12 (d) none of these
12. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is
(a) 54 : 25 (b) 8 : 25 (c) 17 : 24 (d) 18 : 25
13. Two numbers are in the ratio 2 : 3. If 4 be subtracted from each, they are in the ratio 3 : 5.
The numbers are
(a) (16,24) (b) (4,6) (c) (2,3) (d) none of these
14. The angles of a triangle are in ratio 2 : 7 : 11. The angles are
(a) (20o, 70o, 90o) (b) (30o, 70o, 80o) (c) (18 ° , 63 ° , 99 ° ) (d) none of these
15. Division of Rs. 324 between X and Y is in the ratio 11 : 7. X & Y would get Rupees
(a) (204, 120) (b) (200, 124) (c) (180, 144) (d) none of these

1.6 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 16. Anand earns Rs. 80 in 7 hours and Promode Rs. 90 in 12 hours. The ratio of their earnings is
(a) 32 : 21 (b) 23 : 12 (c) 8 : 9 (d) none of these
17. The ratio of 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
18. 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
19. 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
20. If p : q is the sub duplicate ratio of p–x2 : q–x2 then x2 is
p q pq
(a) (b) (c) (d) none of these
p+q p+q p+q

21. If 2s : 3t is the duplicate ratio of 2s – p : 3t – p then

(a) p2 = 6st (b) p = 6st (c) 2p = 3st (d) none of these
22. 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
23. 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
24. 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) (40, 50) (b) (50, 40) (c) (400, 500) (d) none of these
25. The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 Kms. in 5
hours, the speed of the first train is
(a) 10 Km/hr (b) 50 Km/hr (c) 70 Km/hr (d) none of these

1.2 PROPORTION
LEARNING OBJECTIVES
After reading this unit, a student will learn –
u What is proportion?
u Properties of proportion and how to use them.
If the income of a man is increased in the given ratio and if the increase in his income is given
then to find out his new income, Proportion problem is used.
Again if the ages of two men are in the given ratio and if the age of one man is given, we can find
out the age of the another man by 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 (also written as a : b :: c : d) i.e. if a/b = c/d i.e. if ad = bc.

MATHS 1.7

© The Institute of Chartered Accountants of India

 a b c a + b+c
Example 2: If = = , then prove that =2
3 4 7 c
a b c a+ b+c a+ b+c
Solution: We have = = = =
3 4 7 3+ 4+7 14
a+ b+c c a+ b+c 14
∴ = or =2 =
14 7 c 7
Example 3: A dealer mixes tea costing Rs. 6.92 per kg. with tea costing Rs. 7.77 per kg. and sells
the mixture at Rs. 8.80 per kg. and earns a profit of 17 1 2 % on his sale price. In what proportion
does he mix them?
Solution: Let us first find the cost price (C.P.) of the mixture. If S.P. is Rs. 100, profit is
17 1 2 Therefore C.P. = Rs. (100 - 17 1 2 ) = Rs. 82 1 2 = Rs. 165/2

If S.P. is Rs. 8.80, C.P. is (165 × 8.80)/(2 × 100) = Rs. 7.26

= C.P. of the mixture per kg = Rs. 7.26
2nd difference = Profit by selling 1 kg. of 2nd kind @ Rs. 7.26
= Rs. 7.77 – Rs. 7.26 = 51 paise
1st difference = Rs. 7.26 – Rs. 6.92 = 34 paise
We have to mix the two kinds in such a ratio that the amount of profit in the first case
must balance the amount of loss in the second case.
Hence, the required ratio = (2nd diff) : (1st diff.) = 51 : 34 = 3 : 2.

1.2.2 LAWS ON PROPORTION AS DERIVED EARLIER

(i) p : q = r : s => q : p = s : r (Invertendo)
(p/q = r/s) => (q/p = s/r)
(ii) a : b = c : d => a : c = b : d (Alternendo)
(a/b = c/d) => (a/c = b/d)
(iii) a : b = c : d => a+b : b = c+d : d (Componendo)
(a/b = c/d) => (a+b)/b = (c+d)/d
(iv) a : b = c : d => a–b : b = c–d : d (Dividendo)
(a/b = c/d) => (a–b)/b = (c–d)/d
(v) a : b = c : d => a+b : a–b = c+d : c–d (Componendo & Dividendo)
(a+b)/(a–b) = (c+d)/(c–d)
(vi) a : b = c : d = a+c : b+d (Addendo)
(a/b = c/d = a+c/b+d)

MATHS 1.11

© The Institute of Chartered Accountants of India

 RATIO AND PROPORTION, INDICES, LOGARITHMS

(vii) a : b = c : d = a–c : b–d (Subtrahendo)

(a/b = c/d = a–c/b–d)
(viii) If a : b = c : d = e : f = ............ then each of these ratios = (a – c – e – .......) : (b – d – f – .....)
Proof: The reader may try it as an exercise (Subtrahendo) as the proof is similar to that
derival in 7 above
Exercise 1(B)
Choose the most appropriate option (a) (b) (c) or (d)
1. The fourth proportional to 4, 6, 8 is
(a) 12 (b) 32 (c) 48 (d) none of these
2. The third proportional to 12, 18 is
(a) 24 (b) 27 (c) 36 (d) none of these
3. The mean proportional between 25, 81 is
(a) 40 (b) 50 (c) 45 (d) none of these
4. 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
5. The fourth proportional to 2a, a2, c is
(a) ac/2 (b) ac (c) 2/ac (d) none of these
6. 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
7. The mean proportional between 12x2 and 27y2 is
(a) 18xy (b) 81xy (c) 8xy (d) none of these
(Hint: Let z be the mean proportional and z = (12x 2 x 27y 2 )
8. 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
9. If a/3 = b/4 = c/7, then a+b+c/c is
(a) 1 (b) 3 (c) 2 (d) none of these
10. If p/q = r/s = 2.5/1.5, the value of ps:qr is
(a) 3/5 (b) 1:1 (c) 5/3 (d) none of these
11. 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
12. 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
13. 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

1.12 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 14. If x/2 = y/3 = z/7, then the value of (2x–5y+4z)/2y is
(a) 6/23 (b) 23/6 (c) 3/2 (d) 17/6
15. 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
16. 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) 8 : 12 : 9
17. The sum of the 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
18. The numbers 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
19. If x/y = z/w, implies y/x = w/z, then the process is called
(a) Dividendo (b) Componendo (c) Alternendo (d) none of these
20. If p/q = r/s = p–r/q–s, the process is called
(a) Subtrahendo (b) Addendo (c) Invertendo (d) none of these
21. If a/b = c/d, implies (a+b)/(a–b) = (c+d)/(c–d), the process is called
(a) Componendo (b) Dividendo (c) Componendo (d) none of these
and Dividendo
22. 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
23. 12, 16, *, 20 are in proportion. Then * is
(a) 25 (b) 14 (c) 15 (d) none of these
24. 4, *, 9, 13½ are in proportion. Then * is
(a) 6 (b) 8 (c) 9 (d) none of these
25. 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

a b c a+ b+c
26. If = = then is
4 5 9 c
(a) 4 (b) 2 (c) 7 (d) none of these.
27. 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

a b
28. If = then
4 5
a+4 b-5 a+4 b+5 a-4 b+5
(a) = (b) = (c) = (d) none of these
a-4 b+5 a-4 b-5 a+4 b-5
MATHS 1.13

© The Institute of Chartered Accountants of India

 RATIO AND PROPORTION, INDICES, LOGARITHMS

a b
29. If a : b = 4 : 1 then + is
b a
(a) 5/2 (b) 4 (c) 5 (d) none of these

x y z
30. If = = then
b+c −a c+a − b a+ b−c
(b – c)x + (c – a)y + (a – b)z is
(a) 1 (b) 0 (c) 5 (d) none of these

1.3 INDICES
LEARNING OBJECTIVES
After reading this unit, a student will learn –
u A meaning of indices and their applications;
u Laws of indices which facilitates their easy applications.
We are aware of certain operations of addition and multiplication and now we take up certain
higher order operations with powers and roots under the respective heads of indices.
We know that the result of a repeated addition can be held by multiplication e.g.
4 + 4 + 4 + 4 + 4 = 5(4) = 20
a + a + a + a + a = 5(a) = 5a
Now, 4 × 4 × 4 × 4 × 4 = 45;
a × a × a × a × a = a5 .
It may be noticed that in the first case 4 is multiplied 5 times and in the second case ‘a’ is multiplied
5 times. In all such cases a factor which multiplies is called the “base” and the number of times it is
multiplied is called the “power” or the “index”. Therefore, “4” and “a” are the bases and “5” is the
index for both. Any base raised to the power zero is defined to be 1; i.e. ao = 1. We also define
1
r
a =a r .
If n is a positive integer, and ‘a’ is a real number, i.e. n ∈ N and a ∈ R (where N is the set of
positive integers and R is the set of real numbers), ‘a’ is used to denote the continued product of
n factors each equal to ‘a’ as shown below:
an = a × a × a ………….. to n factors.
Here an is a power of “a“ whose base is “a“ and the index or power is “n“.
For example, in 3 × 3 × 3 × 3 = 34 , 3 is base and 4 is index or power.

1.14 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 Example 8: Find the value of k from (√9)–7 × (√3)–5 = 3k
Solution: (√9)–7 × (√3)–5 = 3k
or, (32 × 1/2) –7 × (3½) –5 = 3k
or, 3 −7 − 5/2 = 3k
or, 3 –19/2 = 3k or, k = –19/2

1.3.1 LAWS OF INDICES

(i) am × an = am+n (base must be same)
Ex. 23 × 22 = 23+2 = 25
(ii) am ÷ an = am–n
Ex. 25 ÷ 23 = 25–3 = 22
(iii) (am)n = amn
Ex. (25)2 = 25×2 = 210
(iv) ao = 1
Example : 20 = 1, 30 = 1
(v) a–m = 1/am and 1/a–m = am
Example: 2–3 = 1/23 and 1/2–5 = 25
(vi) If ax = ay, then x=y
(vii) If xa = ya, then x=y

a = a1/m , √x = x½ , √4 = (2 )
2 1/2
(viii) m = 21/2 × 2 = 2

3
Example: 8 = 81/3 = (23)1/3 = 2 3×1/3 = 2

MATHS 1.19

© The Institute of Chartered Accountants of India

 RATIO AND PROPORTION, INDICES, LOGARITHMS

Exercise 1(C)
Choose the most appropriate option (a) (b) (c) or (d)
1. 4x–1/4 is expressed as
(a) –4x 1/4 (b) x–1 (c) 4/x1/4 (d) none of these
2. The value of 81/3 is
(a) 3√2 (b) 4 (c) 2 (d) none of these
1/5
3. The value of 2 × (32) is
(a) 2 (b) 10 (c) 4 (d) none of these
1/5
4. The value of 4/(32) is
(a) 8 (b) 2 (c) 4 (d) none of these
1/3
5. The value of (8/27) is
(a) 2/3 (b) 3/2 (c) 2/9 (d) none of these
6. The value of 2(256)–1/8 is
(a) 1 (b) 2 (c) 1/2 (d) none of these
½ ¾
7. 2 .4 is equal to
(a) a fraction (b) a positive integer (c) a negative integer (d) none of these
1
4
¨ 81x · 4
8. © 8 ¸ has simplified value equal to
ª y ¹

(a) xy2 (b) x2y (c) 9xy2 (d) none of these

9. xa–b × xb–c × xc–a is equal to
(a) x (b) 1 (c) 0 (d) none of these
0
⎛ 2p2q 3 ⎞
10. The value of ⎜ ⎟ where p, q, x, y ≠ 0 is equal to
⎝ 3xy ⎠
(a) 0 (b) 2/3 (c) 1 (d) none of these
11. {(33)2 × (42)3 × (53)2} / {(32)3 × (43)2 × (52)3} is
(a) 3/4 (b) 4/5 (c) 4/7 (d) 1
12. Which is True ?
(a) 20 > (1/2)0 (b) 20 < (1/2)0 (c) 20 = (1/2)0 (d) none of these
13. If x1/p = y1/q = z1/r and xyz = 1, then the value of p+q+r is
(a) 1 (b) 0 (c) 1/2 (d) none of these
14. The value of ya–b × yb–c × yc–a × y–a–b is
(a) ya+b (b) y (c) 1 (d) 1/ya+b

1.20 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 15. The True option is
(a) x2/3 = 3√x2 (b) x2/3 = √x3 (c) x2/3 > 3√x2 (d) x2/3 < 3√x2
16. The simplified value of 16x–3y2 × 8–1x3y–2 is
(a) 2xy (b) xy/2 (c) 2 (d) none of these
17. The value of (8/27)–1/3 × (32/243)–1/5 is
(a) 9/4 (b) 4/9 (c) 2/3 (d) none of these
18. The value of {(x+y)2/3 (x–y)3/2/√x+y × √ (x–y)3}6 is
(a) (x+y)2 (b) (x–y) (c) x+y (d) none of these
19. Simplified value of (125) 2/3
× √25 × √5 × 5 3 3 1/2
is
(a) 5 (b) 1/5 (c) 1 (d) none of these
1/2 3/4 5/6 7/8 9/10 4 3/25
20. [{(2) . (4) . (8) . (16) . (32) }] is
(a) A fraction (b) an integer (c) 1 (d) none of these
21. [1–{1–(1–x2)–1}–1]–1/2 is equal to
(a) x (b) 1/x (c) 1 (d) none of these
n (n–1)/n 1/n+1
22. {(x ) } is equal to
(a) xn (b) xn+1 (c) xn–1 (d) none of these
3 3 2 2
23. If a –b = (a–b) (a + ab + b ), then the simplified form of
⎡ x l ⎤ l 2 +lm+m 2 ⎡ x m ⎤ m 2 +mn+n 2 ⎡ x n ⎤ l 2 + ln+n 2
⎢xm ⎥ ×⎢ n⎥ ×⎢ l ⎥
⎣ ⎦ ⎣x ⎦ ⎣x ⎦
(a) 0 (b) 1 (c) x (d) none of these
24. Using (a–b)3 = a3–b3–3ab(a–b) tick the correct of these when x = p1/3 – p–1/3
(a) x3+3x = p + 1/p (b) x3 + 3x = p – 1/p (c) x3 + 3x = p + 1 (d) none of these
25. On simplification, 1/(1+am–n+am–p) + 1/(1+an–m+an–p) + 1/(1+ap–m+ap–n) is equal to
(a) 0 (b) a (c) 1 (d) 1/a
a+b b+c c+a
⎛ xa ⎞ ⎛ xb ⎞ ⎛ xc ⎞
26. The value of ⎜ b ⎟ ×⎜ c ⎟ ×⎜ a ⎟
⎝x ⎠ ⎝x ⎠ ⎝x ⎠
(a) 1 (b) 0 (c) 2 (d) none of these
1 1
-
27. If x = 3 +3 3 3
, then 3x 3 -9x is
(a) 15 (b) 10 (c) 12 (d) none of these
28. If ax = b, by = c, cz = a, then xyz is
(a) 1 (b) 2 (c) 3 (d) none of these

MATHS 1.21

© The Institute of Chartered Accountants of India

 RATIO AND PROPORTION, INDICES, LOGARITHMS

(a 2 +ab+b2 ) (b2 +bc+c 2 ) (c 2 +ca+a 2 )

⎛ xa ⎞ ⎛ xb ⎞ ⎛ xc ⎞
29. The value of ⎜ b ⎟ ×⎜ c ⎟ ×⎜ a ⎟
⎝x ⎠ ⎝x ⎠ ⎝x ⎠
(a) 1 (b) 0 (c) –1 (d) none of these

1 1 1
30. If 2x = 3y = 6-z, + + is
x y z

(a) 1 (b) 0 (c) 2 (d) none of these

1.4 LOGARITHM
LEARNING OBJECTIVE
u After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
The logarithm of a number to a given base is the index or the power to which the base must be
raised to produce the number, i.e. to make it equal to the given number. If there are three quantities
indicated by say a, x and n, they are related as follows:
If ax = n, where n > 0, a > 0 and a ≠ 1
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed
as follows:
logan = x
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i) 24 = 16 ⇒ log 216 = 4
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii) 103 = 1000 ⇒ log101000 = 3
i.e. the logarithm of 1000 to the base 10 is 3

-3 1 ⎛ 1 ⎞
(iii) 5 = ⇒ log 5 ⎜ ⎟ = -3
125 ⎝ 125 ⎠
1
i.e. the logarithm of to the base 5 is –3
125
(iv) 23 = 8 ⇒ log28 = 3
i.e. the logarithm of 8 to the base 2 is 3
1. The two equations ax = n and x = logan are only transformations of each other and should
be remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power
zero is one.
Since a0 = 1 , loga1 = 0

1.22 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 1 1 1 1 1 1
Therefore 
x +1 y +1 z +1 log a abc log b abc log c abc

= logabca +logabcb + logabcc

= log abcabc = 1 (proved)
Example 3: If a=log2412, b=log3624, and c=log4836 then prove that
1+abc = 2bc
Solution: 1+abc = 1+ log2412 × log3624 × log4836
= 1+ log3612 × log4836
= 1 + log4812
= log4848 + log4812
= log4848×12
= log48 (2×12)2
= 2 log4824
= 2 log3624 x log4836
= 2bc
Exercise 1(D)
Choose the most appropriate option. (a) (b) (c) and (d)
1. log 6 + log 5 is expressed as
(a) log 11 (b) log 30 (c) log 5/6 (d) none of these
2. log28 is equal to
(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 – log 4 (c) 23 (d) none of these
4. log (1 × 2 × 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, the x is equal to
(a) 3 (b) 9 (c) 2 (d) none of these
7. log √2 64 is equal to
(a) 12 (b) 6 (c) 1 (d) none of these
8. log 2√3 1728 is equal to
(a) 2√3 (b) 2 (c) 6 (d) none of these

MATHS 1.31

© The Institute of Chartered Accountants of India

 RATIO AND PROPORTION, INDICES, LOGARITHMS

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 log2 log2 log2 16
(a) 0 (b) 2 (c) 1 (d) none of these
1
13. The value of log 3 to the base 9 is
(a) – ½ (b) ½ (c) 1 (d) none of these
14. 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
3
15. The value of log2 [log2 {log3 (log327 )}] is equal to
(a) 1 (b) 2 (c) 0 (d) none of these
16. If log2x + log4x + log16x = 21/4, then x is equal to
(a) 8 (b) 4 (c) 16 (d) none of these
17. Given that log102 = x and log103 = y, the value of log1060 is expressed as
(a) x – y + 1 (b) x + y + 1 (c) x – y – 1 (d) none of these
18. Given that log102 = x, log103 = y, then log101.2 is expressed in terms of x and y as
(a) x + 2y – 1 (b) x + y – 1 (c) 2x + y – 1 (d) none of these
19. 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
20. The simplified value of 2 log105 + log108 – ½ log104 is
(a) ½ (b) 4 (c) 2 (d) none of these
21. log [1 – {1 – (1 – x2)–1}–1]–1/2 can be written as
(a) log x2 (b) log x (c) log 1/x (d) none of these

22. The simplified value of log 4 729 3 9-1 .27 -4/3 is

(a) log 3 (b) log 2 (c) log ½ (d) none of these
23. The value of (logba × logcb × logac)3 is equal to
(a) 3 (b) 0 (c) 1 (d) none of these
24. The logarithm of 64 to the base 2√2 is
(a) 2 (b) √2 (c) ½ (d) none of these
25. The value of log825 given log 2 = 0.3010 is
(a) 1 (b) 2 (c) 1.5482 (d) none of these

1.32 COMMON PROFICIENCY TEST

© The Institute of Chartered Accountants of India

 ANSWERS
Exercise 1(A)
1. a 2. d 3. c 4. a 5. c 6. d 7. a 8. c
9. a 10. c 11. d 12. d 13. a 14. c 15. d 16. a
17. c 18. b 19. b 20. c 21. a 22. c 23. a 24. c
25. c
Exercise 1(B)
1. a 2. b 3. c 4. d 5. a 6. c 7. a 8. c
9. c 10. b 11. c 12. d 13. a 14. d 15. d 16. a
17. a 18. b 19. d 20. a 21. c 22. d 23. c 24. a
25. b 26. b 27. c 28. b 29. a 30. b
Exercise 1(C)
1. c 2. c 3. c 4. b 5. a 6. a 7. b 8. d
9. b 10. c 11. d 12. c 13. b 14. d 15. a 16. c
17. a 18. c 19. d 20. b 21. a 22. c 23. b 24. b
25. c 26. a 27. b 28. a 29. a 30 d
Exercise 1(D)
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. c 15. c 16. a
17. b 18. c 19. a 20. c 21. b 22. a 23. c 24. d
25. c

MATHS 1.33

© The Institute of Chartered Accountants of India