Test: Algebra - GMAT MCQ

To determine the value of |x + 7|, let's evaluate each statement:

Statement (1): |x + 3| = 14

Considering the absolute value, we can rewrite the equation as two separate cases:

x + 3 = 14:
Solving this equation gives us x = 11. Therefore, |x + 7| = |11 + 7| = |18| = 18.

-(x + 3) = 14:
Simplifying, we have -x - 3 = 14.
Rearranging, we get -x = 17, and dividing by -1, we have x = -17. Therefore, |x + 7| = |-17 + 7| = |-10| = 10.

Statement (1) alone gives us two possible values for |x + 7|: 18 or 10.

Statement (2): (x + 2)² = 169

Taking the square root of both sides, we have x + 2 = ±13.

Case 1: x + 2 = 13
Solving this equation gives us x = 11. Therefore, |x + 7| = |11 + 7| = |18| = 18.

Case 2: x + 2 = -13
Solving this equation gives us x = -15. Therefore, |x + 7| = |-15 + 7| = |-8| = 8.

Statement (2) alone gives us two possible values for |x + 7|: 18 or 8.

Combining both statements, we see that the only common value for |x + 7| is 18.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient. The answer is (C).

To determine the value of a + b, let's evaluate each statement:

Statement (1): a² - b² = 0

This equation can be factored as (a - b)(a + b) = 0. Since a ≠ b, the only possibility for the product to be zero is if (a + b) = 0. Therefore, statement (1) tells us that a + b = 0.

Statement (2): ab = 12

This equation gives us the product of a and b, but it doesn't provide any direct information about their sum, a + b.

Considering both statements together, we know that a + b = 0 from statement (1), but we have no additional information about the value of a + b from statement (2).

Therefore, statements (1) and (2) together are not sufficient to answer the question. Additional data is needed. The answer is (E).

To determine the value of abcd + a + b + c + d, let's evaluate each statement:

Statement (1): a² + b² + c² + d = 249

This equation provides information about the sum of the squares of a, b, c, and d, but it doesn't directly give us the value of abcd + a + b + c + d.

Statement (2): d ≥ 249

This statement provides information about the value of d, but it doesn't provide any information about the values of a, b, or c.

Considering both statements together, we can see that statement (1) gives us some information about a, b, c, and d, while statement (2) only provides a condition on the value of d. However, this information is not sufficient to determine the value of abcd + a + b + c + d because we still don't know the specific values of a, b, and c.

Therefore, both statements together are not sufficient to answer the question. Additional data is needed. The answer is (C).

To determine the value of 1 Ω 1, let's evaluate each statement:

Statement (1): 2 Ω 2 = 4

From this statement, we know that 2 Ω 2 equals 4. However, we don't have any specific information about the operation Ω. We cannot determine the value of 1 Ω 1 based on this statement alone.

Statement (2): 0 Ω 1 = 0

This statement tells us that the result of 0 Ω 1 is 0. Again, we don't have any specific information about the operation Ω, and we cannot determine the value of 1 Ω 1 based on this statement alone.

Considering both statements together, we still don't have any specific information about the operation Ω or how it behaves with different values. Both statements are insufficient to determine the value of 1 Ω 1.

Therefore, statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked. The answer is (B).

To determine if a equals b, let's evaluate each statement:

Statement (1): a² = b²

From this statement, we know that the squares of a and b are equal. However, this does not necessarily mean that a and b themselves are equal. For example, if a = 2 and b = -2, then a² = b² = 4, but a ≠ b. Therefore, statement (1) alone is not sufficient to determine if a equals b.

Statement (2): a = 1

This statement tells us that a is equal to 1. However, we don't have any information about the value of b. Without knowing the value of b, we cannot determine if a equals b. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have information about a and b being squares of each other (statement 1) and a being equal to 1 (statement 2). However, this still does not give us enough information to determine if a equals b. We could have scenarios where a = 1 and b = -1, or a = 1 and b = 1, which would lead to a ≠ b or a = b, respectively.

Therefore, statements (1) and (2) together are not sufficient to answer the question. Additional data is needed. The answer is (E).

To determine the value of r, let's evaluate each statement:

Statement (1): r³ - r² = 0

From this statement, we have an equation involving r. By factoring out r from the terms, we get r²(r - 1) = 0. This equation tells us that either r² = 0 or r - 1 = 0.

For the case of r² = 0, the only solution is r = 0.

For the case of r - 1 = 0, the solution is r = 1.

Therefore, statement (1) alone is sufficient to determine the value of r. It tells us that r can be either 0 or 1.

Statement (2): r = -r

This equation implies that r is equal to its own negative. The only value that satisfies this condition is r = 0.

Therefore, statement (2) alone is also sufficient to determine the value of r. It tells us that r is 0.

Both statements independently provide the same information, so either statement alone is sufficient to answer the question. The answer is (B).

To determine the value of x - y, let's evaluate each statement:

Statement (1): (x + y)² = 4xy

Expanding the left side of the equation, we have x² + 2xy + y² = 4xy. Rearranging terms, we get x² - 2xy + y² = 0. This equation can be factored as (x - y)² = 0. Taking the square root of both sides, we have x - y = 0.

Therefore, statement (1) alone tells us that x - y is equal to 0.

Statement (2): x² - y² = 0

This equation can be factored as (x - y)(x + y) = 0. From this equation, we can determine that either x - y = 0 or x + y = 0. However, we are interested in the value of x - y, not x + y. So, statement (2) alone does not provide sufficient information to determine the value of x - y.

Combining both statements, we know from statement (1) that x - y = 0. This information is sufficient to determine the value of x - y.

Therefore, statement (1) alone is sufficient to answer the question, while statement (2) alone is not. The answer is (A).

To determine if integer x < -20, let's evaluate each statement:

Statement (1): x² + 40x + 391 = 0

We have a quadratic equation here, but it doesn't provide any direct information about the value of x. To determine if x < -20, we need to find the roots of the quadratic equation. However, without solving the equation or factoring it, we can't determine if x is less than -20 based on this statement alone.

Statement (2): x² = 529

This equation can be rewritten as (x - 23)(x + 23) = 0. From this equation, we can determine that either x - 23 = 0 or x + 23 = 0. Solving each equation, we find two possible values for x: x = 23 and x = -23. However, neither of these values satisfies the condition x < -20. Therefore, statement (2) alone does not provide sufficient information to answer the question.

Considering both statements together, we have no overlapping information that helps determine if x < -20. Statement (1) does not provide specific values for x, and statement (2) does not provide values of x that satisfy the condition.

Therefore, both statements together are not sufficient to answer the question. Additional data is needed. The answer is (C).

To determine the value of y, let's evaluate each statement:

Statement (1): x² - y² = 5

This equation provides a relationship between the squares of x and y, but it doesn't give us specific values for either variable. Without additional information or a way to isolate y, we can't determine the value of y based on this statement alone.

Statement (2): x and y are each positive integers

This statement tells us that both x and y are positive integers. While this provides some information about the nature of the variables, it doesn't give us any specific values for y. Without a specific value or range for y, we cannot determine its value based on this statement alone.

Considering both statements together, statement (1) gives us a relationship between the squares of x and y, and statement (2) tells us that both x and y are positive integers. However, this still doesn't provide us with enough information to determine the value of y.

Therefore, both statements together are not sufficient to answer the question. Additional data is needed. The answer is (C).

To determine the value of ab, let's analyze the given statements:

(1) a = b + 1

Statement (1) provides a relationship between a and b, but it does not give us any specific values. Without additional information, we cannot determine the value of ab based solely on this statement.

(2) a² = b + 1

Statement (2) provides a quadratic equation relating a and b. However, it still doesn't give us a unique solution for the values of a and b. We can determine the value of a in terms of b by taking the square root of both sides of the equation, but we won't have a unique solution for ab.

When we consider both statements together, we still don't have enough information to determine the value of ab. Although we have two equations, they are not sufficient to uniquely determine the values of a and b. We need additional data or constraints to solve for ab.

Therefore, the correct answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.