x + 1/x Type Questions Practice Questions with Detailed Solutions – Level 2

x + 1 by x Type Questions Level 2: Higher Powers and Surd Forms

Aptitude360's Level 2 set of x + 1 by x type questions is an intermediate algebra quiz on reciprocal identities. It moves beyond basic squares to fourth, fifth and sixth powers, reverse problems and surd forms of x. Values taken from quadratics, square roots of x plus its reciprocal and special cases with odd and mixed powers also feature.

Question types in this Level 2 reciprocal set

• Fourth, fifth and sixth powers of x plus its reciprocal
• Reverse problems where a higher power is given and the base sum or difference is found
• Values from quadratics with a constant term of one, with either sign on the middle term
• Surd forms of x and the square root of x plus its reciprocal
• Special cases where x equals its own reciprocal, including sums of consecutive powers

Key formulas for higher-power reciprocal identities

• Fourth power: \( x^{4}+\dfrac{1}{x^{4}}=\left(x^{2}+\dfrac{1}{x^{2}}\right)^{2}-2 \)
• Sixth power: \( x^{6}+\dfrac{1}{x^{6}}=\left(x^{3}+\dfrac{1}{x^{3}}\right)^{2}-2 \)
• With \( k=x+\dfrac{1}{x} \): \( x^{5}+\dfrac{1}{x^{5}}=k^{5}-5k^{3}+5k \)
• Root form: \( \left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)^{2}=k+2 \)
• With \( m=x-\dfrac{1}{x} \): \( x^{2}-\dfrac{1}{x^{2}}=km \)

How to revise for these intermediate problems

• Build higher powers in a chain: square first, then cube, then combine them
• In reverse problems, add two and take square roots step by step, using the sign condition given on x
• When x is a sum with a root, check whether it is the perfect square of a simpler surd

Who should take this Level 2 set

Students who are comfortable with basic reciprocal identities and are preparing algebra for SSC, banking, CAT and similar aptitude exams.