If $$(x^2 + \frac{1}{x}) = 119$$ and $$x > 1$$, then the value of $$(x^2 - \frac{1}{x})$$ is?

Appeared in: UPPSC-2021

Explanation

The solution is found by systematically reducing the power of the variable 'x' using algebraic identities.
First, the expression x⁴ + 1/x⁴ = 119 is used to find x² + 1/x² by applying the identity (a+b)², which results in x² + 1/x² = 11.
Next, using the value of x² + 1/x² and the identity (a-b)², the value of x - 1/x is calculated as 3.
Finally, the target expression x³ - 1/x³ is calculated using the identity a³ - b³ = (a-b)³ + 3ab(a-b) and the previously found value of x - 1/x.
Substituting x - 1/x = 3 into the identity gives (3)³ + 3(1)(3) = 27 + 9 = 36.

Option A: This value is incorrect. It may arise from a calculation error, such as incorrectly calculating the cube of 3 or misapplying the final identity.
Option B: This value is incorrect. It might be obtained if one incorrectly assumes the answer is (x - 1/x)², which is 9, and then doubles it, or some other miscalculation.
Option C: This value is incorrect. There is no clear mathematical path that leads to 76 from the given information and standard algebraic identities.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving algebraic expressions using standard identities as background academic context rather than a clinical decision trigger.
This question tests general intelligence and numerical ability, which are components of many nursing entrance examinations.
Strong problem-solving skills are essential for nurses for critical thinking and making quick decisions in clinical settings, although this specific type of math is not directly used.
What if the question asked for x³ + 1/x³? Then, you would first need to calculate x + 1/x. From x² + 1/x² = 11, you get (x + 1/x)² = 11 + 2 = 13, so x + 1/x = √13. The answer would be (√13)³ - 3(√13) = 13√13 - 3√13 = 10√13.

How to Approach the Question

First, carefully read the question and identify the given expression and the expression to be found. Note any conditions, like x > 1.
Recognize that this type of problem requires using standard algebraic identities to relate the given expression to the target expression.
Work backward from the given expression's power (e.g., power of 4) to a lower power (power of 2), and then to an even lower power (power of 1).
In this case, start with x⁴ + 1/x⁴ = 119 and use the identity for (x² + 1/x²)² to find the value of x² + 1/x².
Then, use the value of x² + 1/x² and the identity for (x - 1/x)² to find the value of x - 1/x.
Finally, use the value of x - 1/x and the identity for x³ - 1/x³ to calculate the final answer.

Concept Tested & Keywords

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Practise the full UPPSC-2021

Attempt every question from this paper in a timed mock, then review the full solution for each one.