Question 1
The ratio of 12√35 : 13√140 is equal to the ratio _________________
A
4:3✓✗
B
2:1✓✗
C
5:4✓✗
D
3:4✓✗
Explanation
First simplify √140: 140 = 4 × 35, so √140 = √4 × √35 = 2√35.

So the second term becomes 13 × 2√35 = 23√35.

Ratio = 12√35 : 23√35 = 12 : 23 = 12 × 32 = 34, i.e. 3 : 4.
Question 2
The value of
A
78✓✗
B
1516✓✗
C
158✓✗
D
34✓✗
Explanation
Work from the innermost radical outward. Each square root halves the running exponent of a, and each "a ×" adds a full power of a before the next root is taken:

√a = a1/2
a × a1/2 = a3/2, so √a × a1/2 = a3/4
a × a3/4 = a7/4, so √that = a7/8
a × a7/8 = a15/8, so √that = a15/16

So the expression inside the log simplifies to a15/16. Now, log√a(a15/16) = loga1/2(a15/16) = 1516 ÷ 12 = 158.
Question 3
If 3X x b5 logb x = 192, then the value of x is:
A
8✓✗
B
4✓✗
C
2✓✗
D
2√2✓✗
Explanation
Using the identity blogb x = x, we get b5 logb x = x5.

So the equation becomes 3x · x5 = 192, i.e. 3x6 = 192, so x6 = 64.

Since 26 = 64, x = 2.
Question 4
In India, an examination is conducted in two sessions. In the first session the ratio of boys to girls among 455 students is 8:5. If 50 new girls are admitted in the second session, how many new boys must be admitted so that the ratio of girls to boys becomes 3:4?
A
20✓✗
B
30✓✗
C
40✓✗
D
50✓✗
Explanation
Boys = 813 × 455 = 280, Girls = 513 × 455 = 175.

After 50 new girls join: Girls = 175 + 50 = 225. Let b new boys be admitted, so Boys = 280 + b.

Required ratio Girls:Boys = 3:4, so 225280 + b = 34.

900 = 3(280 + b) = 840 + 3b, so 3b = 60, b = 20.
Question 5
The simplified value of
A
x1+2b✓✗
B
x2b✓✗
C
1✓✗
D
a3b/2xb✓✗
Explanation
4√a6bx4 = (a6bx4)1/4 = a6b/4x4/4 = a3b/2x.

√(a3x-4)-b = [(a3x-4)-b]1/2 = a-3b/2x2b.

Multiplying: a3b/2x × a-3b/2x2b = a3b/2 - 3b/2x1+2b = a0x1+2b = x1+2b.
Question 6
If a : b = 3 : 4, then the value of 2a + 3b3a + 2b is:
A
18/17✓✗
B
17/18✓✗
C
6/7✓✗
D
7/6✓✗
Explanation
Let a = 3k, b = 4k.

2a + 3b3a + 2b = 2(3k) + 3(4k)3(3k) + 2(4k) = 6k + 12k9k + 8k = 18k17k = 1817.
Question 7
The product of two numbers is 7644 and their ratio is 12:13. Then, the smaller of the two numbers is __________.
A
91✓✗
B
84✓✗
C
82✓✗
D
90✓✗
Explanation
Let the numbers be 12k and 13k. Product = 12k × 13k = 156k2 = 7644, so k2 = 49, k = 7.

Smaller number = 12k = 12 × 7 = 84.
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