Test Index

UPSC CDS 1 2020 Math Paper

© examsnet.com

Question : 19 of 100

Marks: +1, -0

What is

\frac{1}{{{{\rm{a}}^{{\rm{m}} - {\rm{n}}}} - 1}} + \frac{1}{{{{\rm{a}}^{{\rm{n}} - {\rm{m}}}} - 1}}

equal to?

1
$-1$
0
$2a^{m-n}$

Solution:

Concept:

Use substitution

x = a^{m-n}

and the property

a^{-k} = \frac{1}{a^k}

to simplify the expression.

Explanation:

Let

x = a^{m-n}

.

Then

a^{n-m} = a^{-(m-n)} = \frac{1}{x}

.

Rewrite the given sum:

\frac{1}{x - 1} + \frac{1}{\frac{1}{x} - 1}

.

Simplify the second term:

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

.

Now the sum is

\frac{1}{x-1} + \frac{x}{1-x}

.

Note that

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

. So the sum becomes

-\frac{1}{1-x} + \frac{x}{1-x} = \frac{x-1}{1-x}

.

Factor

-1

:

\frac{-(1-x)}{1-x} = -1

.

Thus the value is

-1

.

Answer:

\boxed{-1}

© examsnet.com

Go to Question:

Prev Question

Next Question