Question: What is the value of k?

1. \(k^4 = \frac{1}{625}\)
2. \(k^3 < k^2\)

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.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

(1) \(k^4 = \frac{1}{625}\)
\(k^4 < 5^{-4}\)
\(\Rightarrow k=5^{-1} = \frac{1}{5} or k=\frac{-1}{5}\)
Not sufficient

(2) \(k^3 < k^2\)

This statement is also not sufficient
Let’s combine (1) and (2)

\(k^3 < k^2 \Rightarrow (\frac{-1}{5})^3 < (\frac{-1}{5})^2 \Rightarrow \frac{-1}{125} < \frac{1}{25} \)--True
\(k^3 < k^2 \Rightarrow (\frac{1}{5})^3 < (\frac{-1}{5})^2 \Rightarrow \frac{1}{125} < \frac{1}{25}\)--True

Correct Option: E

Approach Solution (2):

In order to find the value, we need:
Exact value of k
Any equation to solve for k
Any characteristics of k
Statement Analysis:

S1:

\(k^4 = \frac{1}{625}\)
\(k^4 - \frac{1}{625} = 0\)
\(k^4 - (\frac{1}{5})^4 = 0\)
\((k^2+\frac{1}{25})( k+\frac{1}{5})( k-\frac{1}{5}) = 0\)
\(\therefore k = \frac{1}{5},\frac{-1}{5} \)

Insufficient
Hence option a and d is eliminated

S2: 
\(k^3 < k^2\)

This holds for infinite values of k.
Hence insufficient
Hence option b is eliminated
S1 and S2: From these both statements, we got k = 1/5, -1/5. Putting these values in the equation from St2, we get k = -1/5 as well as 1/5 holds for the equation.
Hence insufficient
Option c is eliminated

Correct Option: E

Approach Solution (3):

S1:
\(k^4 = \frac{1}{625}\)
\(\rightarrow k^2 = \frac{1}{25}\)
k= \( \pm\frac{1}{5}\)

Not sufficient
S2:
\(k^3 < k^2\)

This could be true in two cases:

1. k is negative. So k is cube is negative while k square is positive
2. k is positive and between 0 and 1. Increasing powers of k become smaller

Not sufficient

(1) + (2) together are also insufficient

Correct Option: E

“What is the value of k?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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