If P = 7 + 4√3 and PQ = 1, then what is the value of 1/P^2 + 1/Q^2?

Introduction / Context:
This question uses properties of surds and reciprocal pairs. When the product PQ is 1, P and Q are reciprocals of each other. For expressions involving 1/P^2 and 1/Q^2, recognising that Q = 1/P allows us to reduce the problem to finding P^2 + 1/P^2, which can be computed using simple algebra.

Given Data / Assumptions:

• P = 7 + 4√3.
• PQ = 1, so Q = 1/P.
• We must find 1/P^2 + 1/Q^2.

Concept / Approach:
Since Q = 1/P, we have: 1/P^2 + 1/Q^2 = 1/P^2 + Q^2. But Q^2 = (1/P)^2 = 1/P^2, so it is simpler to note that 1/P^2 + 1/Q^2 = P^2 + 1/P^2 because Q = 1/P. A more straightforward route is to first compute Q explicitly using the conjugate of P, then compute P^2 + Q^2 and use that as the required value.

Step-by-Step Solution:
We have P = 7 + 4√3. If PQ = 1, then Q must be 1/P. For surds of the form a + b√3, the reciprocal is a - b√3 divided by a^2 - 3b^2. Compute P * (7 - 4√3) = (7 + 4√3)(7 - 4√3) = 7^2 - (4√3)^2 = 49 - 16*3 = 49 - 48 = 1. So Q = 7 - 4√3. Now compute P^2: (7 + 4√3)^2 = 7^2 + 2*7*4√3 + (4√3)^2 = 49 + 56√3 + 48 = 97 + 56√3. Similarly, Q^2 = (7 - 4√3)^2 = 49 - 56√3 + 48 = 97 - 56√3. Then P^2 + Q^2 = (97 + 56√3) + (97 - 56√3) = 194. Because PQ = 1, 1/P^2 = Q^2 and 1/Q^2 = P^2, so 1/P^2 + 1/Q^2 = P^2 + Q^2 = 194.

Verification / Alternative check:
Note that P * Q = 1 implies P^2 * Q^2 = 1, so P^2 and Q^2 are also reciprocals. The sum of a number and its reciprocal is symmetric, so the result being a rational integer 194 makes sense and matches our explicit computation.

Why Other Options Are Wrong:
196, 206 and 182 could appear if one computes only P^2 or only Q^2 rather than their sum, or if 7^2 + 48 is miscalculated. 200 is an attractive round number but does not match the exact algebraic simplification.

Common Pitfalls:
Some learners forget to use the conjugate to find Q, or they incorrectly compute (4√3)^2 as 16√3 instead of 48. Another common issue is mixing up 1/P^2 + 1/Q^2 with P^2 + Q^2; for reciprocal pairs arising from surds, these two sums are actually equal, which is a powerful shortcut.

Final Answer:
The value of 1/P^2 + 1/Q^2 is 194.