Is k^2 + k 2 > 0? (1) k is an integer greater than zero.

GMATinsight, given equation is k^2+k-1 and not k^2-k+1. The red part above is incorrect.

Made correction, Thank you ...

Our Question is
\(k^2+k-2>0 ?\)

Restate the given as \(k^2+2k-1k-2>0\) ------> \((k-1)(k+2)>0\) ---------> k > 1 or k > -2

So our question now is Is k>1 or k> -2 ?

Statement 1 --- k > 0

Take k = 1 we get No
k = 2 we get Yes
So Not Sufficient

Statement 2 --- \(\frac{k}{2}\) = Integer our in other words k can be even

Take k = 2 we get Yes
Take k = -2 we get No
So Not Sufficient

Together 1 + 2 we get
K is even and k > 0

So It is sufficient and answer is C

Statement 1: k is an integer>0
Given: (k-1)(k+2)>0 ?
Put k=1, equation=0
Put k=2, equation>0. Insufficient

Statement 2: k divided by 2 is an integer. k could be 0,2,4,6 etc
Insufficient

Using both, k =2,4,6 etc.
Answer C

800score Official Solution:

The first thing we need to do is factor k² + k – 2 into (k + 2)(k – 1). From these factors, we can see that the value of k² + k – 2 = 0 when k = -2 or 1.

When k > 1, both k + 2 and k – 1 will be positive, so the product of the two (the original expression) will be positive.

Similarly, when k is between -2 and 1, the product of the two expressions will be negative.

Finally, when k is less than -2, the product will be positive.

So we've learned that k² + k – 2 > 0 as long as k is greater than 1 or less than -2.

Statement (1) tells us that k is an integer greater than zero. This is not sufficient because k² + k – 2 > 0 when k is greater than 1, but it will equal zero when k is 1.

Statement (2) tells us k is an even integer. This is also insufficient, because all even integer values of k will make k² + k – 2 greater than zero except -2 and 0. k = -2 will make the expression equal to zero, and k = 0 will make the expression negative. (Remember: 0 is an even integer!)

Combined, the two statements are sufficient, because the only possible values for k are positive even numbers (2 or greater), all of which make k² + k – 2 > 0.

Since the statements are insufficient individually, but sufficient when combined, the correct answer is choice (C).

Target question: Is k² + k – 2 > 0?
This is a good candidate for rephrasing the target question.

What needs to happen in order for k² + k – 2 to be positive?
Factor to get: k² + k – 2 = (k + 2)(k - 1)
For (k + 2)(k - 1) to be positive, we need one of two scenarios:

Scenario A: (k + 2) and (k - 1) are both POSITIVE
For this to occur, k must be greater than 1

Scenario B: (k + 2) and (k - 1) are both NEGATIVE
For this to occur, k must be less than -2

In other words, for k² + k – 2 to be positive, it must be the case that EITHER k is greater than 1 OR k is less than -2
REPHRASED target question: Is it true that EITHER k is greater than 1, OR k is less than -2?

Once we've rephrased the target question, we can head to the two statements....

Statement 1: k is an integer greater than zero.
There are several values of k that satisfy statement 1. Here are two:
Case a: k = 2, in which case the answer to the REPHRASED target question is YES
Case b: k = 1, in which case the answer to the REPHRASED target question is NO
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: k divided by 2 is an integer
There are several values of k that satisfy statement 2. Here are two:
Case a: k = 2, in which case the answer to the REPHRASED target question is YES
Case b: k = 0, in which case the answer to the REPHRASED target question is NO
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that k is positive
Statement 2 tells us that k divided by 2 is an integer.
In other words, k/2 must be a positive integer
Some possible values of k: 2, 4, 6, 8, . . .
In all of these cases, the answer to the REPHRASED target question is YES
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer: C

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