if a\= 1+ √2 and b \= 1 - √2 , so value of a² + b² ?& x20; - ESIC Nursing Officer -2019 (Shift -1)

ESIC Nursing Officer -2019 (Shift -1)

Edition 3

Medium

Appeared in: ESIC Nursing Officer -2019 (Shift -1)

Explanation

The problem requires calculating the sum of the squares of two given numbers, a = 1 + √2 and b = 1 - √2.
One method is to square each number individually. a² = (1 + √2)² = 1 + 2√2 + 2 = 3 + 2√2.
Similarly, b² = (1 - √2)² = 1 - 2√2 + 2 = 3 - 2√2.
Adding these two results: a² + b² = (3 + 2√2) + (3 - 2√2) = 6.
Alternatively, using the identity a² + b² = (a + b)² - 2ab, we find a+b = 2 and ab = -1. Substituting these gives (2)² - 2(-1) = 4 + 2 = 6.

Option A: This value (8) is the result of calculating (a - b)². Here, a - b = (1 + √2) - (1 - √2) = 2√2, and (2√2)² = 4 × 2 = 8. This is a common error of calculating the wrong expression.
Option C: This value (2) is the result of calculating (a + b). Here, a + b = (1 + √2) + (1 - √2) = 2. The student may have forgotten to perform the squaring operation required by the question.
Option D: This value (5) is likely the result of a calculation error, such as incorrectly squaring the terms or making a mistake during addition. For example, incorrectly calculating (1²+1²) + (√2) = 2+√2, or 1²+2²=5.

Visual 1: Infographic - A visual breakdown of the algebraic identities (x + y)², (x - y)², and (x+y)(x-y) with geometric proofs (e.g., showing squares and rectangles). This helps in understanding how these formulas are derived.
Visual 2: Flowchart - A step-by-step flowchart showing the two different methods (Direct Squaring vs. Using Identities) to solve the problem, highlighting the calculations at each stage.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Algebraic identities and simplification of expressions involving square roots as background academic context rather than a clinical decision trigger.
This question tests basic mathematical aptitude, which is a non-nursing subject but essential for the nursing profession.
Nurses frequently use mathematical skills for critical tasks such as calculating drug dosages, determining IV drip rates, and interpreting patient data and lab results.
A strong foundation in mathematics ensures patient safety by preventing medication errors and enabling accurate monitoring.

How to Approach the Question

First, carefully read the question to identify the given values (a and b) and what needs to be calculated (a² + b²).
Recognize that the numbers 'a' and 'b' are conjugates (1 ± √2). This is a clue that using algebraic identities might simplify the calculation.
Decide on a method. You can either square 'a' and 'b' directly or use an algebraic identity like a² + b² = (a + b)² - 2ab.
If using the identity method, first calculate the sum (a + b) and the product (ab).
Substitute these intermediate results back into the identity to find the final answer.
Double-check your arithmetic, especially when dealing with square roots and negative signs, to avoid common errors.

Concept Tested & Keywords

Concept Tested: Algebraic identities and simplification of expressions involving square roots.
Stem keywords: algebra, square root, algebraic expression
Lead-in keywords: value of

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