GMAT Data Insights Quiz: Algebraic Constraint Testing

What this quiz covers

This quiz focuses on Algebraic Constraint Testing, giving you a quick way to practice the rules, question types, and explanations that matter most for GMAT Data Insights.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient to answer the question asked.

Explanation: First, factor the expression: x³ + y³ - x²y - xy² = x³ - x²y + y³ - xy² = x²(x - y) + y²(y - x) = x²(x - y) - y²(x - y) = (x - y)(x² - y²) = (x - y)²(x + y). So we need (x - y)²(x + y) > 0. Since (x - y)² ≥ 0 always, this inequality holds if and only if (x - y)² > 0 and x + y > 0, OR (x - y)² = 0 and x + y > 0. The first condition means x ≠ y and x + y > 0. Statement (1): x > y > 0 implies x ≠ y and x + y > 0, so (x - y)²(x + y) > 0. This is sufficient. Statement (2): x + y > 0 and xy > 0 means both x and y have the same sign and their sum is positive, so both are positive. However, this doesn't tell us whether x ≠ y. If x = y, then (x - y)²(x + y) = 0, making the inequality false. If x ≠ y, the inequality is true. Statement (2) is insufficient.

Question 2

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (correct answer)
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient to answer the question asked.

Explanation: For m/n to be in simplest form, we need gcd(m,n) = 1. Statement (1): gcd(m+1, n+1) = 1 doesn't guarantee gcd(m,n) = 1. For example, if m = 4 and n = 6, then gcd(5,7) = 1, but gcd(4,6) = 2 ≠ 1. Alternatively, if m = 3 and n = 5, then gcd(4,6) = 2 ≠ 1, so the condition isn't even satisfied. Let me try m = 2, n = 4: gcd(3,5) = 1, but gcd(2,4) = 2. Or m = 1, n = 3: gcd(2,4) = 2. The statement is insufficient. Statement (2): If m and n are consecutive Fibonacci numbers, then gcd(m,n) = 1. This is a well-known property: consecutive Fibonacci numbers are always coprime. This can be proven by the Euclidean algorithm: if F_k and F_{k+1} are consecutive Fibonacci numbers, then gcd($F_{k+1}$, F_k) = gcd(F_k, $F_{k+1}$ - F_k) = gcd(F_k, $F_{k-1}$) = ... = gcd($F_2$, $F_1$) = gcd(1,1) = 1. Therefore, statement (2) is sufficient.

Question 3

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient to answer the question asked. (correct answer)

Explanation: A system is inconsistent when the coefficient ratios are equal but the constant ratio is different. For 2x + 3y = 7 and ax + by = c, inconsistency occurs when a/2 = b/3 but c/7 ≠ a/2 (equivalently c/7 ≠ b/3). Statement (1): This directly states the condition for inconsistency, so the system is inconsistent. Sufficient. Statement (2): a = 4, b = 6, c = 15. Check ratios: a/2 = 2, b/3 = 2, c/7 = 15/7 ≈ 2.14. Since a/2 = b/3 = 2 but c/7 ≠ 2, the system is inconsistent. Sufficient.

Question 4

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient to answer the question asked. (correct answer)

Explanation: For f(x) = x² + px + q to have two distinct real roots, the discriminant must be positive: Δ = p² - 4q > 0. Statement (1): p² - 4q > 0 directly gives us the discriminant condition. This is sufficient for two distinct real roots. Statement (2): f(0) = q and f(1) = 1 + p + q. If f(0)·f(1) < 0, then q(1 + p + q) < 0, meaning q and (1 + p + q) have opposite signs. By the Intermediate Value Theorem, since f is continuous and f(0) and f(1) have opposite signs, there exists at least one root between 0 and 1. However, we need to verify this guarantees two distinct roots. If f(0) and f(1) have opposite signs, then the parabola crosses the x-axis at least once between x = 0 and x = 1. For a upward-opening parabola (coefficient of x² is 1 > 0), this crossing between 0 and 1, combined with the continuous nature and the fact that the parabola goes to +∞ as x → ±∞, guarantees exactly two distinct real roots. Statement (2) is sufficient.