Test: Algebra - GMAT MCQ

Statement (1): x/a < y/a
Dividing both sides of the inequality by a, we get x < y.
However, we don't have any information about the values of x or y, or the value of a.
Therefore,
we cannot determine whether x is greater than y based on this statement alone.

Statement (2): x > 2y
This statement provides a direct comparison between x and 2y.
However, it doesn't give us any information about the relationship between x and y.
For example, if y = 1 and x = 3, then x is greater than 2y.
However, if y = 2 and x = 5, then x is not greater than 2y.
Therefore, we cannot determine whether x is greater than y based on this statement alone.

When we consider both statements together, we still don't have enough information to determine the relationship between x and y.
Statement (1) tells us that x < y, while Statement (2) tells us that x > 2y.
These statements do not provide a clear relationship between x and y, as they can be satisfied by different values of x and y.

Therefore, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): One of the integers is 2 greater than the other integer.

This statement tells us that the two integers have a difference of 2.
Let's assume the smaller integer is x, so the larger integer would be x + 2.

The absolute difference between the cubes of the two integers can be expressed as (x + 2)³ - x³.

Expanding this expression, we get (x³ + 6x² + 12x + 8) - x³, which simplifies to 6x² + 12x + 8.

We can see that the expression does not yield a unique value for the absolute difference between the cubes. For example, if x = 1, the expression becomes 26, but if x = 2, the expression becomes 56. Therefore, statement (1) alone is not sufficient to determine the absolute difference between the cubes of the two integers.

Statement (2): The square of the sum of the integers is 49 greater than the product of the integers.

This statement can be expressed as (x + (x + 2))² = x(x + 2) + 49.

Simplifying this equation, we get (2x + 2)² = x² + 2x + 49.

Expanding and simplifying further, we have 4x² + 8x + 4 = x² + 2x + 49.

Rearranging terms, we get 3x² + 6x - 45 = 0.

Factoring this quadratic equation, we have (x + 5)(3x - 9) = 0.

From this, we find two possible solutions: x = -5 and x = 3.
However, we are given that the integers are non-negative, so x = -5 is not a valid solution.
Thus, x = 3 is the valid solution.

Now that we know x = 3, we can find the two integers: 3 and 5.
The absolute difference between the cubes of these integers is (5³ - 3³) = 122.

Therefore, statement (2) alone is sufficient to determine the absolute difference between the cubes of the two integers.

Combined sufficiency:

By combining both statements, we know that one integer is 2 greater than the other (statement 1) and the integers are 3 and 5 (statement 2).
With this information, we can calculate the absolute difference between the cubes, which is 122.

Thus, both statements together are sufficient to determine the absolute difference between the cubes of the two integers.

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

Statement 1

x² − 10x = 24
x² − 10x − 24 = 0
x² − 12x + 2x - 24 = 0
(x + 2)(x − 12) = 0
Therefore x can be either -2 or x can be 12.
As we are getting two possible values of x, the statement is not sufficient.

Statement 2

20x = 2(x² − 2x)
Dividing by 2 on both sides of the equation
10x = x² − 2x
x² − 12x = 0
x(x−12) = 0
x = 0 or x = 12
Hence x = 0 is not a valid value.
x = 12
This statement is sufficient.

Viete's theorem states that for the roots x1 and x² of a quadratic equation ax2 + bx + c = 0:
x₁ + x₂ = -b/a and x₁∗x₂ = c/a.

(1) Both roots are prime number.
According to the theorem:  x₁ + x₂ = -b/a = 7.7 can be expressed as the sum of two primes only in one way 2 + 5 = 7. Knowing the roots we can find the value of k. Sufficient.

(2) 2 is one of root of the equation. Plug x = 2: 2² − 7*2 + k = 0. We can find k. Sufficient.

Statement 1 Alone:

9∗200² + 200b + c = 54000. However, we have two variables so there is no unique solution for b or c. Then statement 1 alone is insufficient.

Statement 2 Alone:
Using quadratic equation knowledge, we can find b/a = 20 + 9 = 29 and c/a 20*9, however that is not enough to find b or c. Then statement 2 alone is insufficient.

(There may be a typo in this statement since a was not declared, so the intended equation might have been 9x² - bx + c, and this statement would be sufficient in that case).

Statement 2 Alone:

Combined we have three equations with three variables. Plug b = 29a and c = 180a into the first equation to find a, then b and c. Thus by combining statements, it is sufficient.

Given:
y = 9 - x

We need to determine the value of y, given that x and y are positive integers.

Statement 1: x < 8
This statement alone tells us about the range of values that x can take.
Since x is a positive integer and x < 8, the possible values for x are 1, 2, 3, 4, 5, 6, and 7.
Using the equation y = 9 - x, we can calculate the corresponding values of y for each value of x.
For example, if x = 1, then y = 9 - 1 = 8. Similarly, for x = 2, y = 9 - 2 = 7, and so on.
Since we have a unique value of y for each value of x within the given range, we can determine the value of y.
Therefore, statement 1 alone is sufficient.

Statement 2: y > 1
This statement gives us a condition for the value of y.
Since y = 9 - x, if y > 1, then 9 - x > 1.
Solving this inequality, we have x < 8.
This condition is the same as statement 1.
As we've already established that statement 1 alone is sufficient, statement 2 alone is also sufficient.

Since both statement 1 and statement 2 individually provide enough information to determine the value of y, the correct answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement 1

(1) n² + 3n = 4
n² + 3n − 4 = 0
n² + 4n − n − 4 = 0
(n − 1)(n + 4) = 0
n = 1 or n = -4

As we have two possible values of n, the statement is not sufficient. We can eliminate A and D.

Statement 2

(2) (n + 3)ⁿ = 1
n = 0; (0 + 3) 0 = 1
n = −2;(−2 + 3)⁻² = 1⁻² = 1
The statement is not sufficient. Hence eliminate B.

Combined
The statements combined give n = -4
Hence sufficient.

Elevation = −t² + 12t − 35

(1) t < 8
Let's notice what happens at exactly t = 8
Elevation = -(8)² + 12*8 - 35 = -3
It means the whale is under water at this time.

Put t = 7
Elevation = -(7)² + 12*7 - 35 = 0
So the whale is exactly at the surface at this time.

Put t = 6
Elevation = -(6)² + 12*6 - 35 = 1
The whale has a positive elevation at this time.

Let's find out what happens at t = 5
Elevation = -(5)² + 12*5 - 35 = 0
So the whale is exactly at the surface at this time.

We see what is happening here. The whale is exactly at the surface at t = 5, is out of water and then back in the water at t = 7.

Given that t < 8, the whale is underwater if t > 7 or t < 5. Not sufficient.

(2) t > 5
If t = 6, the whale is out of water but if t > 7, it is underwater. Not sufficient.

Using both, If t = 6, the whale is out of water but if t > 7, the whale is underwater. Not sufficient.

Statement (1): (x + y)(x - y) = 5

The product of (x + y) and (x - y) is equal to 5. We need to find the value of x - y, so let's analyze the possibilities for the product:

• If (x + y) = 5 and (x - y) = 1, then x = 3 and y = 2. However, this violates the given condition that x < y < 0, so this case is not valid.
• If (x + y) = 1 and (x - y) = 5, then x = 3 and y = -2. This satisfies the given condition, and in this case, x - y = 3 - (-2) = 3 + 2 = 5.

Therefore, we can determine the value of x - y based on statement (1) alone.

Statement (2): xy = 6

The product of x and y is equal to 6. This statement provides information about the product of x and y, but it does not provide any information about the difference between x and y. Therefore, we cannot determine the value of x - y based on statement (2) alone.
Considering both statements together:
By combining both statements, we have additional information about the relationship between x and y. However, statement (2) alone does not provide enough information to determine the value of x - y. Therefore, the correct answer is (A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.
In summary, we can determine the value of x - y based on statement (1) alone, but we cannot determine it based on statement (2) alone.