If α and β are the roots of the equation ax2 + bx + c = 0, is α equal to β? (1) α + β, α2 + β2, α3 + β3 are in G.P. (2) α ≠ 0, β ≠ 0.
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Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
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Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
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BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
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EACH statement ALONE is sufficient.
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Statements (1) and (2) TOGETHER are NOT sufficient.
Statement 1 implies a relationship between roots that can lead to equality, but doesn't guarantee it without knowing the coefficients. Statement 2 ensures roots are non-zero. Together, they allow solving for the condition where the roots are equal.
Statement 1 establishes that the terms alpha + beta, alpha squared + beta squared, and alpha cubed + beta cubed form a geometric progression. Using the condition for a geometric progression, we find that alpha and beta must satisfy an equation that implies alpha equals beta, but if alpha and beta are both zero, the common ratio is undefined, creating an ambiguity. Statement 2 clarifies this by stating alpha and beta are non-zero, which guarantees the common ratio is well-defined. Because statement 1 requires statement 2 to resolve this edge case, both statements together are sufficient, but neither statement alone is sufficient.