x + 1/x Type Questions Practice Questions with Detailed Solutions – Level 3

x Plus 1 Over x Identities Level 3 Practice Quiz: Higher Powers and Surds

Built for quant learners who have mastered the basic reciprocal identities, this Level 3 practice quiz on Aptitude360 covers harder x plus 1 over x type questions. It takes sums and differences of x and its reciprocal to higher powers. It also uses surd values of x, quadratic conditions, cube roots of unity and the special case of a sum equal to minus two.

Topics covered in this x plus 1 over x Level 3 set

• Higher powers of x plus its reciprocal found from a given sum
• Differences such as x minus its reciprocal and their higher odd and even powers
• Turning a quadratic condition into a reciprocal sum by dividing through by x
• Surd values of x whose reciprocal is the conjugate surd
• Working backwards from a fourth power to recover the original sum or difference
• Special cases such as cube roots of unity and sums equal to two or minus two

Key formulas and rules for reciprocal identity questions

• Let \( t=x+\dfrac{1}{x} \) and \( d=x-\dfrac{1}{x} \)
• Square: \( x^{2}+\dfrac{1}{x^{2}}=t^{2}-2 \)
• Cube: \( x^{3}+\dfrac{1}{x^{3}}=t^{3}-3t \)
• Link: \( t^{2}-d^{2}=4 \)
• Difference of squares: \( x^{2}-\dfrac{1}{x^{2}}=t\times d \)
• With \( a_{n}=x^{n}+\dfrac{1}{x^{n}} \): \( a_{5}=a_{2}a_{3}-a_{1} \)

Revision tips for harder reciprocal problems

• When x is a surd, rationalise to find its reciprocal; it is often the conjugate, which makes the sum a whole number.
• For a quadratic condition, divide every term by x to read off the reciprocal sum directly.
• Use the given sign condition on x before taking a square root while working backwards.
• If the sum equals two, x is one; if it equals minus two, x is minus one, so very large powers become simple.

Who should practise this level

Students preparing the algebra part of quantitative aptitude who have finished the earlier levels of this topic and want advanced reciprocal identity practice.