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UPSC CDS 2 2023 Math Paper

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Question : 15 of 100

Marks: +1, -0

If

3^{x - 1} + 3^{3 - x} = 6

2^{x - 1} + 2^{3 - x}

equal to ?

4
3
2
1

Solution:

Concept:

The sum of two powers with a constant product can be solved by equating each term to the same value.

Explanation:

We have

3^{x-1} + 3^{3-x} = 6

.

The product of the two terms is

(3^{x-1})(3^{3-x}) = 3^{(x-1)+(3-x)} = 3^2 = 9

.

Their sum is

6

and product is

9

.

The only pair of positive numbers with sum

6

and product

9

is

3

and

3

.

Thus,

3

and

3^{x-1} = 3

.

Since

3^{3-x} = 3

, we get

3^1 = 3

and

x-1 = 1

.

Both give

3-x = 1

.

Now compute

x = 2

with

2^{x-1} + 2^{3-x}

:

x=2

.

Answer:

2^{2-1} + 2^{3-2} = 2^1 + 2^1 = 2 + 2 = 4

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