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UPSC CDS 1 2020 Math Paper

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Question : 15 of 100

Marks: +1, -0

If

5^{x+1} - 5^{x-1} = 600

, then what is the value of

10^{2x}

?

1
1000
100000
1000000

Solution:

Concept:

Use exponent rules:

a^{b+c} = a^b \cdot a^c

and

a^{b-c} = \frac{a^b}{a^c}

to simplify and solve for

x

.

Explanation:

Given:

5^{x+1} - 5^{x-1} = 600

Rewrite the terms:

5^{x+1} = 5^x \cdot 5

and

5^{x-1} = \frac{5^x}{5}

.

So the equation becomes:

5^x \cdot 5 - \frac{5^x}{5} = 600

Factor out

5^x

:

5^x \left(5 - \frac{1}{5}\right) = 600

Simplify inside brackets:

5 - \frac{1}{5} = \frac{25-1}{5} = \frac{24}{5}

Thus:

5^x \cdot \frac{24}{5} = 600

Multiply both sides by

\frac{5}{24}

:

5^x = 600 \cdot \frac{5}{24} = \frac{3000}{24} = 125

Since

125 = 5^3

, we get

x = 3

.

Now compute

10^{2x} = 10^{2 \cdot 3} = 10^6 = 1\,000\,000

.

Answer:

10^{2x} = 1\,000\,000

(Option D).

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