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

x + 1 by x Type Questions Level 1: Reciprocal Identity Basics

This is the Level 1 set of x + 1 by x type questions on Aptitude360, a beginner algebra quiz on reciprocal identities. It covers finding squares and cubes of x plus or minus its reciprocal from a given value. It also includes quadratics rewritten in reciprocal form and conjugate surd values of x.

What this x + 1 by x Level 1 set covers

• Square of x plus its reciprocal from a given sum or difference
• Cubes and fourth powers built step by step from the base value
• Quadratics whose constant term equals the leading coefficient, divided through by x
• Conjugate surd values of x and their reciprocals
• Special cases where x equals its own reciprocal, which make very high powers easy

Key formulas for x + 1 by x questions

• Square from sum: \( x^{2}+\dfrac{1}{x^{2}}=\left(x+\dfrac{1}{x}\right)^{2}-2 \)
• Square from difference: \( x^{2}+\dfrac{1}{x^{2}}=\left(x-\dfrac{1}{x}\right)^{2}+2 \)
• With \( k=x+\dfrac{1}{x} \): \( x^{3}+\dfrac{1}{x^{3}}=k^{3}-3k \)
• With \( m=x-\dfrac{1}{x} \): \( x^{3}-\dfrac{1}{x^{3}}=m^{3}+3m \)
• If \( x^{2}-kx+1=0 \): \( x+\dfrac{1}{x}=k \)

Preparation tips for reciprocal identity problems

• Memorise the square and cube results for both the sum and the difference, since the sign of the correction term changes
• For a surd value of x, rationalise to find the reciprocal before adding or subtracting
• Check that a quadratic can be divided by x to produce the reciprocal form before using it

Who this beginner set suits

Aspirants for SSC, banking, CAT and other aptitude exams who are starting algebra and want a firm base in reciprocal identities.