Select the option from the given choices that corresponds to the relationship in the equation (function) given below:

Appeared in: ESIC Nursing Officer - 2019 (Shift-2)

Explanation

The problem defines two new mathematical operators: ⊗ for the sum of squares (M2 + N2) and □ for the square of the sum ((P + Q)2).
The solution relies on the fundamental algebraic identity for the expansion of a binomial square: (P + Q)2 = P2 + Q2 + 2PQ.
By substituting the definitions of the operators into this identity, we can establish the relationship between them.
The term (P + Q)2 is replaced by P □ Q.
The term P2 + Q2 is replaced by P ⊗ Q.
This substitution directly yields the equation: P □ Q = (P ⊗ Q) + 2PQ, which matches the correct option.

Option A: This option incorrectly subtracts 2PQ. The expansion of (P+Q)2 results in a positive 2PQ term. This expression would be related to the expansion of (P-Q)2, which is P2 - 2PQ + Q2.
Option B: This option incorrectly rearranges the terms. It states P2 + Q2 = (P + Q)2 + 2PQ. Substituting the expansion of (P+Q)2 gives P2 + Q2 = (P2 + 2PQ + Q2) + 2PQ, which simplifies to 0 = 4PQ. This is not a valid general identity.
Option D: This option, P ⊗ Q = (P □ Q) - 2PQ, is an algebraically correct rearrangement of the identity (P2 + Q2 = (P+Q)2 - 2PQ). However, the most direct derivation comes from expanding the P □ Q term, which leads directly to Option C.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving mathematical reasoning problems involving custom-defined operators and algebraic identities as background academic context rather than a clinical decision trigger.
While this is a mathematical reasoning question, the underlying skills are essential in nursing.
Nurses must use strong logical and analytical skills for critical tasks like calculating medication dosages, interpreting vital signs and lab results, and making sound clinical judgments.
Problem-solving by breaking down a complex issue into smaller, manageable parts is a core competency in providing safe and effective patient care.

How to Approach the Question

First, carefully analyze the image to understand the definition of each custom symbol or operator. Write them down clearly: ⊗ means sum of squares, and □ means square of the sum.
Identify the underlying mathematical concept being tested. The question asks for a relationship between (P+Q)2 and P2+Q2.
Recall the relevant algebraic identity: (a+b)2 = a2 + 2ab + b2.
Translate the algebraic identity into the language of the custom operators. Replace (P+Q)2 with P □ Q and P2+Q2 with P ⊗ Q.
This substitution gives the equation: P □ Q = (P ⊗ Q) + 2PQ.
Compare this derived equation with the given options to find the one that matches.

Concept Tested & Keywords

Concept Tested: Solving mathematical reasoning problems involving custom-defined operators and algebraic identities.
Stem keywords: mathematical relationship, function, equation, operator
Lead-in keywords: Select the option

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