If $x + y = 12$, then the value of $(x - 5)^3 + (y - 7)^3$ is

Appeared in: UPPSC 2017

Explanation

The problem can be simplified by substituting a = x - 5 and b = y - 7.
The sum of these new variables is a + b = (x - 5) + (y - 7) = x + y - 12.
Given that x + y = 12, the sum a + b becomes 12 - 12 = 0.
The expression to be evaluated is a³ + b³.
Since a + b = 0, we have a = -b. Substituting this gives (-b)³ + b³ = -b³ + b³ = 0.
Alternatively, using the identity a³ + b³ = (a + b)(a² - ab + b²), and knowing a + b = 0, the entire expression becomes 0.

Option A: 144 is the square of 12 (12²). This might be chosen by mistakenly squaring the given sum instead of performing the correct algebraic manipulation on the cubic expression.
Option B: 125 is the cube of 5 (5³). This might be chosen by incorrectly focusing on the number 5 in the expression (x - 5)³ and ignoring the other terms and relationships.
Option C: 81 is 9². This value does not arise from any straightforward, albeit incorrect, calculation based on the given numbers. It represents a random calculation error.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Algebraic Identities and Substitution as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical and logical reasoning skills, which are essential for nursing.
Accurate calculation is critical in clinical settings for tasks like medication dosage calculation, IV drip rate monitoring, and interpreting lab results.
What if the equation was x + y = 14? Then a + b = x + y - 12 = 14 - 12 = 2. The expression would be a³ + b³ = (a+b)(a²-ab+b²) = 2((a+b)² - 3ab) = 2(4 - 3ab). The value would no longer be zero and would depend on the specific values of x and y.

How to Approach the Question

First, carefully read the given equations and the expression you need to evaluate.
Look for a way to simplify the target expression. Notice that the terms inside the cubes, x - 5 and y - 7, are related to the given equation x + y = 12.
Use substitution to simplify the problem. Let a = x - 5 and b = y - 7.
Find the relationship between a and b by adding them: a + b = (x - 5) + (y - 7) = x + y - 12.
Substitute the given x + y = 12 into this relationship to find that a + b = 0.
Finally, evaluate the simplified expression a³ + b³ knowing that a = -b. This leads directly to the answer 0.

Concept Tested & Keywords

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Practise the full UPPSC 2017

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