Question: 11 -

If 3^4m+1 = 3^7m-5, solve for m.

Options:

2

3

1/3

1/2

Answer:

2

Solution:

3^4m+1 = 3^7m-5 equating powers of 3 on both sides, 4m+1 = 7m - 5 => 3m = 6 => m = 2.

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Question: 12 -

Simplify : 1 - {1 + (a² - 1)^-1}^-1 

Options:

-1/a²

-a²

1/a²

a²

Answer:

1/a²

Solution:

1 - {1 + (a² - 1)^-1}^-1 = 1 - {1 + 1/a² - 1}^-1 = 1 - {a² / a² - 1}^-1 = 1 - (a² - 1)/a² = 1/a².

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Question: 13 -

Solve for m if 36 (6^m) = (216)^3m+4

Options:

-9/4

-5/4

-11/4

-7/4

Answer:

-5/4

Solution:

36 (6^m) = (216)^3m+4 => 6² 6^m = (6³)^3m+4 => 6^2+m = 6^9m+12 Equating powers of 6 on both sides. 

m + 2 = 9m + 12 => -10 = 8m => m = -5/4

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Question: 14 -

Simplify : (y^4b-a . y^4c-b . y^4a-c) / (y^a . y^b . y^c)³

Options:

0

y

y²

1

Answer:

1

Solution:

(y^4b-a . y^4c-b . y^4a-c) / (y^a . y^b . y^c)³ 

= (y^4b-a . y^4c-b . y^4a-c) / (y^a+b+c)³ 

= y^{4b-a+4c-b+4a-c} / y^3(a+b+c) 

= y^3(a+b+c) / y^3(a+b+c) = 1.

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Question: 15 -

If 2^4m+2 = 4^6m-4, solve for m.

Options:

3

7/4

2

5/4

Answer:

5/4

Solution:

2^4m+2 = 4^6m-4 = 2^4m+2 = (2²)^6m-4 => 2^4m+2 = 22^12m-8 Equating powers of 2 both sides. 

4m + 2 = 12m - 8 => 10 = 8m => m = 5/4.

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