In this aptitude (simplification and trigonometry) question, evaluate the expression: sin(60 degrees) + 2 / sqrt(3), using exact standard trigonometric values and rational simplification, then choose the correct simplified result.

Introduction / Context:
This question checks familiarity with standard trigonometric values and basic algebraic simplification. Instead of using decimal approximations, we are expected to use exact values for special angles and combine them into a single simplified expression. Such problems are common in quantitative aptitude and engineering entrance examinations.

Given Data / Assumptions:
We must evaluate sin(60 degrees) + 2 / sqrt(3).
Use the exact value sin(60 degrees) = sqrt(3) / 2.
We treat sqrt(3) as an exact irrational quantity and combine terms algebraically.

Concept / Approach:
First substitute the exact trigonometric value for sin(60 degrees). Then we need to add two fractions that have different denominators. A common strategy is to express both terms with a common denominator, then sum numerators. Finally, we simplify the resulting fraction as much as possible without rationalising the denominator unless required. Here the main aim is to match the given exact form among the options.

Step-by-Step Solution:
Start from sin(60 degrees) + 2 / sqrt(3). Replace sin(60 degrees) with sqrt(3) / 2. So the expression becomes sqrt(3) / 2 + 2 / sqrt(3). To add these, use common denominator 2 * sqrt(3). Rewrite sqrt(3) / 2 as (sqrt(3) * sqrt(3)) / (2 * sqrt(3)) = 3 / (2 * sqrt(3)). Rewrite 2 / sqrt(3) as (2 * 2) / (2 * sqrt(3)) = 4 / (2 * sqrt(3)). Now add: 3 / (2 * sqrt(3)) + 4 / (2 * sqrt(3)) = 7 / (2 * sqrt(3)).

Verification / Alternative check:
We can check numerically to gain confidence. Approximate sqrt(3) as about 1.732. Then sin(60 degrees) is approximately 0.866. The second term 2 / sqrt(3) is about 1.155. Their sum is roughly 2.021. Now compute 7 / (2 * sqrt(3)) using the same approximation: 2 * sqrt(3) is about 3.464, and 7 divided by 3.464 is again close to 2.021. This confirms that 7 / (2 * sqrt(3)) is the correct exact representation of the sum.

Why Other Options Are Wrong:
Option 3 is much larger than the approximate value near 2. Option (2*sqrt(3)+1)/sqrt(3) simplifies to 2 + 1 / sqrt(3), which does not match. Option 2 + sqrt(3) is clearly larger than 2 plus about 1.732, so it is far above the required value. Option 5 / (2 * sqrt(3)) is smaller than 7 / (2 * sqrt(3)) and does not match the computed sum.

Common Pitfalls:
Typical mistakes include using the wrong standard value for sin(60 degrees), confusing degrees with radians, or using decimal approximations too early and then trying to match them with complicated exact options. Another error is mishandling fraction addition with radical denominators. Working systematically with a common denominator avoids these issues.

Final Answer:
The exact simplified value of the expression is 7/(2*sqrt(3)).