Consider the following expressions: 1) x+x²-1/x 2) {ax²+bx+x-c+d/x-{e}{x²}} 3) 3x2 - 5x + ab 4) {2}{x²-ax+b³} 5) 1/x-2/x+5 Which of the above are rational expressions?

Correct Answer is : 1, 3, 4 and 5 only

Correct Answer is : 1, 3, 4 and 5 only

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UPSC NDA II 2018 Mathematics Solved Paper

Question : 3 of 120

Marks: +1, -0

x+x^{2}-\frac{1}{x}

\sqrt{ax^{2}+bx+x-c+\frac{d}{x}-\frac{e}{x^{2}}}

\frac{2}{x^{2}-ax+b^{3}}

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

Which of the above are rational expressions?

Solution:

Concept:

Rational expressions are ratios of two polynomials, where the denominator is not zero.

Explanation:

Rewrite each expression to check if it is a ratio of polynomials.

Expression 1:

x + x^2 - \frac{1}{x} = \frac{x^2 + x^3 - 1}{x}

. Numerator and denominator are polynomials. So it is rational.

Expression 2:

\sqrt{ax^2 + bx + x - c + \frac{d}{x} - \frac{e}{x^2}}

contains a square root and is not a ratio of polynomials. So it is not rational.

Expression 3:

3x^2 - 5x + ab = \frac{3x^2 - 5x + ab}{1}

. Denominator is the polynomial

1

. So it is rational.

Expression 4:

\frac{2}{x^2 - ax + b^3}

is already a ratio of polynomials. So it is rational.

Expression 5:

\frac{1}{x} - \frac{2}{x+5} = \frac{(x+5) - 2x}{x(x+5)} = \frac{5 - x}{x^2 + 5x}

. Both numerator and denominator are polynomials. So it is rational.

Thus expressions 1, 3, 4, and 5 are rational expressions.

Answer:

Option B: 1, 3, 4 and 5 only

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