If a^3 + 3a^2 + 9a = 1, then what is the value of a^3 + 3/a?

Introduction / Context:
This problem tests algebraic manipulation using a given cubic equation in a. Instead of solving explicitly for a, we are expected to transform the given relation to evaluate another expression involving a. This technique appears frequently in algebra and aptitude questions to save time and avoid unnecessary root calculations.

Given Data / Assumptions:

• a is a real number satisfying a^3 + 3a^2 + 9a = 1.
• We need the value of a^3 + 3/a.

Concept / Approach:
The idea is to manipulate the given equation to express 1/a in terms of a, then substitute into a^3 + 3/a. Dividing the original equation by a is helpful because it creates 1/a on one side. Once we express 1/a as a polynomial in a, we can combine terms to reach a constant value.

Step-by-Step Solution:
Given a^3 + 3a^2 + 9a = 1. Divide both sides by a (a is non zero, otherwise the left side would be 0, not 1): a^2 + 3a + 9 = 1/a. We want the value of a^3 + 3/a. Using 1/a = a^2 + 3a + 9, we can rewrite: a^3 + 3/a = a^3 + 3(a^2 + 3a + 9). Expand the right side: a^3 + 3a^2 + 9a + 27. From the original equation, a^3 + 3a^2 + 9a = 1. So a^3 + 3a^2 + 9a + 27 = 1 + 27 = 28. Hence a^3 + 3/a = 28.

Verification / Alternative check:
We could, in principle, solve the cubic equation for a and then compute the expression numerically, but that is unnecessary. The direct manipulation approach already yields a unique constant, and any real root will satisfy this derived relation. Numerical software also confirms that the real root of the equation indeed gives approximately 28 when substituted into a^3 + 3/a.

Why Other Options Are Wrong:
26, 24, 30 and 31 could arise from misplacing the constant term 27 or incorrectly using the relation for 1/a. For example, forgetting to add 27 or miscomputing 3*9 as 18 instead of 27 would give an incorrect answer.

Common Pitfalls:
A frequent mistake is dividing the original equation incorrectly or assuming that a is 0, which contradicts the equation. Another issue is trying to solve the cubic exactly, which is time consuming and unnecessary here. Recognising that the combination a^3 + 3a^2 + 9a already appears in the target expression is the main insight.

Final Answer:
The value of a^3 + 3/a is 28.