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

2025

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

What is the probability that x3 – 8 = 0 when x is selected from a set of 8 integers?

Statement I. The smallest number in the set is -11

Statement II. The arithmetic mean of the set is 1/8.

  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: D

In a Data Sufficiency question, a statement — or a combination of statements — is sufficient only when every scenario consistent with it produces the SAME unique answer; if two scenarios that both satisfy the given statement(s) disagree on the answer, the data is insufficient. Also, for a real number, an equation of the form x³ = k has exactly one real solution (its other two cube roots are complex).

Applying this here: x3 = 8 gives x = 2 as the only real solution, so the event “x3 – 8 = 0” is identical to the event “x = 2”. The required probability is (number of times the value 2 occurs among the 8 integers) ÷ 8, so a statement is sufficient here only if it fixes this count uniquely — not merely some summary property of the set as a whole.

Testing each statement against this requirement:

  1. Statement I alone: only the minimum value (-11) is known. The sets {-11, -9, 1, 2, 3, 4, 5, 6} (contains 2) and {-11, -10, -9, -8, -7, -6, -5, -4} (does not contain 2) are both 8 distinct integers with minimum -11, so Statement I alone cannot fix whether — or how often — 2 appears.

  2. Statement II alone: the arithmetic mean is 1/8, so the sum of the 8 integers is 1. The sets {-6, -2, -1, 0, 1, 2, 3, 4} (contains 2, sum = 1) and {-5, -4, -3, -1, 0, 1, 3, 10} (no 2, sum = 1) are both 8 distinct integers summing to 1, so Statement II alone cannot fix the count either.

  3. Statements I and II together: even fixing both the minimum (-11) and the sum (1), the remaining numbers are still not uniquely determined — see the two combined-consistent sets below — so the count of 2's in the set remains undetermined.

This is confirmed by two sets of 8 distinct integers that both satisfy Statement I and Statement II but disagree on the answer:

  • {-11, -9, 1, 2, 3, 4, 5, 6} — minimum is -11, sum is 1, and 2 occurs once, giving probability 1/8.

  • {-11, -7, 0, 1, 3, 4, 5, 6} — minimum is -11, sum is 1, but 2 never occurs, giving probability 0.

Since both statements together still allow two different probabilities, the data — even combined — is not sufficient to answer the question. The correct choice is that the data in both Statements I and II together is not sufficient.

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