Each of the questions given below consists of a statement and/or a question…

2023

Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the given question.

Read both the statements and Give answer

If x, y are integers, then (x2 + y2)1/2 is an integer?

Statement I – x2 + y2 is an integer

Statement II – x2 – 3y2 = 0

  1. A.

    if the data in Statement I alone is sufficient to answer the question, while the data in Statement II alone is not sufficient to answer the question.

  2. B.

    if the data in Statement II alone is sufficient to answer the question, while the data in Statement I alone is not sufficient to answer the question.

  3. C.

    if the data in each Statement I and Statement II alone is sufficient to answer the question.

  4. D.

    if the data even in both Statements I and II together are not sufficient to answer the question.

Show answer & explanation

Correct answer: B

Concept: For integers x and y, √(x2 + y2) is an integer only when x2 + y2 is a perfect square.

A statement is sufficient in a data-sufficiency question only if it forces the same answer for every pair of integers x, y consistent with it — a relation that is automatically true for all integers adds no new restriction, while an equation tying x to y can be substituted into the target expression to test whether it always reduces to a perfect square.

Applying it here:

  1. Statement I alone: “x2 + y2 is an integer” is true for every pair of integers x, y, so it adds no new restriction. With x = 1, y = 1, x2 + y2 = 2 and √2 is not an integer; with x = 3, y = 4, x2 + y2 = 25 and √25 = 5 is an integer. Two pairs consistent with Statement I give different answers, so Statement I alone is not sufficient.

  2. Statement II alone: x2 − 3y2 = 0 gives x2 = 3y2. Substituting this into the expression: x2 + y2 = 3y2 + y2 = 4y2, so √(x2 + y2) = √(4y2) = 2|y|. Since y is an integer, 2|y| is an integer for every y satisfying this equation, so Statement II alone is sufficient.

Cross-check: Squaring the result back, (2y)2 = 4y2, and 4y2 = 3y2 + y2 = x2 + y2 whenever x2 = 3y2 — confirming the substitution is consistent.

Answer: Statement II alone is sufficient to answer the question, while Statement I alone is not.

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