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 square box symbol P □ Q is defined as the square of the sum of P and Q, which algebraically is (P + Q)2.
The circle-cross symbol P ⊗ Q is defined as the sum of the squares of P and Q, which is P2 + Q2.
The standard algebraic identity for the square of a sum is (P + Q)2 = P2 + Q2 + 2PQ.
By substituting the given symbolic definitions into this standard identity, we find the relationship: P □ Q = P ⊗ Q + 2PQ.

Option A: This option incorrectly equates the square of the sum (P + Q)2 with the sum of the squares P2 + Q2. It omits the + 2PQ term from the algebraic expansion.
Option B: This option suggests (P + Q)2 = 2(P2 + Q2). Expanding both sides gives P2 + Q2 + 2PQ = 2P2 + 2Q2, which simplifies to P2 + Q2 - 2PQ = 0 or (P - Q)2 = 0. This is not a general identity.
Option D: This option incorrectly subtracts the 2PQ term. The expression P ⊗ Q - 2PQ is equivalent to P2 + Q2 - 2PQ, which is the expansion of (P - Q)2, not (P + Q)2.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Algebraic identities and symbolic representation in mathematical reasoning as background academic context rather than a clinical decision trigger.
This question tests analytical and logical reasoning skills, which are fundamental for nurses.
These skills are applied daily in clinical practice for tasks such as calculating medication dosages, interpreting vital signs and lab results, and making sound clinical judgments based on patient data.
What if the square box symbol represented (P - Q)2? Then the correct relationship would be P □ Q = P ⊗ Q - 2PQ, making Option D the correct answer.

How to Approach the Question

First, carefully analyze the image to understand the definition of each new mathematical symbol or operation.
Identify that ⊗ means 'sum of squares' (M2 + N2) and □ means 'square of the sum' ((P + Q)2).
Next, recall relevant standard mathematical formulas. Here, the algebraic identity (a + b)2 = a2 + b2 + 2ab is the key.
Substitute the definitions of the symbols into the standard formula, using P and Q as the variables for consistency.
Replace (P + Q)2 with P □ Q and P2 + Q2 with P ⊗ Q.
This results in the equation P □ Q = P ⊗ Q + 2PQ, which you can then match with the correct option.

Concept Tested & Keywords

Concept Tested: Algebraic identities and symbolic representation in mathematical reasoning.
Stem keywords: mathematical function, equation, symbolic relationship
Lead-in keywords: choose one, corresponds to

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