If x + 1/x= 5/2 then what is the value of the following? {5x}{7x² - 3x + 7}

Correct Answer is : 102910/292910

Correct Answer is : 102910/292910

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UPSC CDS 1 2022 Math Paper

Question : 25 of 100

Marks: +1, -0

x + \frac{1}{x}= \frac{5}{2}

\frac{5x}{7x^{2} - 3x + 7}

Solution:

Concept:

Simplify the given expression by factoring

x

from the denominator, then substitute

x + \frac{1}{x}

from the given condition.

Explanation:

We are given

x + \frac{1}{x} = \frac{5}{2}

We need to find

\frac{5x}{7x^{2} - 3x + 7}

Take

x

common from the denominator:

7x^{2} - 3x + 7 = x\left(7x - 3 + \frac{7}{x}\right)

So the expression becomes

\frac{5x}{x\left(7x - 3 + \frac{7}{x}\right)} = \frac{5}{7x + \frac{7}{x} - 3}

Factor out

7

7x + \frac{7}{x} = 7\left(x + \frac{1}{x}\right)

Thus the expression is

\frac{5}{7\left(x + \frac{1}{x}\right) - 3}

Substitute

x + \frac{1}{x} = \frac{5}{2}

\frac{5}{7 \times \frac{5}{2} - 3} = \frac{5}{\frac{35}{2} - 3}

Simplify:

\frac{35}{2} - 3 = \frac{35 - 6}{2} = \frac{29}{2}

So the expression becomes

\frac{5}{\frac{29}{2}} = 5 \times \frac{2}{29} = \frac{10}{29}

Answer:

\frac{10}{29}

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