Inequalities, modulus and exponents go together very well with data sufficiency. When will the comparison asked in this question hold good - what happens if one of the numbers is negative? Unless stated explicity, do not assume numbers to take only positive values or only integer values.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 3: **If m and n are non-zero integers, is m ^{n} > n^{n}?**

**Statement 1**: |m| = n

**Statement 2**: m < n

INR

**Q1. What kind of an answer will the question fetch?**

An "Is" question will fetch an "Yes" or a "No" as an answer.

The data provided in the statements will be considered sufficient if the question is answered with a conclusive Yes or a conclusive No.

**Q2. When is the answer an "Yes"?**

If m^{n} > n^{n}, the answer to the question is a conclusive Yes.

**Q3. When is the answer a "No"?**

If m^{n} ≤ n^{n}, the answer to the question is a conclusive No.

Note: When m^{n} = n^{n}, the answer is No.

**Q4. What values can m and n take?**

From the information available from the question stem, both m and n can take only integer values.

So, we need not worry about values such as 0.5 or 1.2.

However, both m and n can be either positive or negative. Neither can be 0.

**Statement 1**: |m| = n

We can infer the following information about m and n from the statement.

- The modulus of a number is always positive. n = |m|. Hence, n is positive
- m can take either positive or negative values.

**Example**: Let m = -3 and n = 3. |m| = n holds good.

(-3)^{3} < 3^{3}. So, the answer to the question is **NO.**

**Counter Example**: Let m = -2 and n = 2. |m| = n still holds good.

(-2)^{2} = 2^{2}. So, the answer to the question is **NO.**

Notice that we are not able to come up with a counter example. Both examples returned NO as answer.

Not finding a counter example might be our limitation. Let us reason why we seem to be getting NO as answer and will it hold good for all values satisfying statement 1.

When **a negative number is raised to an odd power**, the result is negative. So, LHS < RHS. **Answer is NO.**

When **a negative number is raised to an even power**, the result is positive. So, LHS = RHS. It is still not greater. **So, the answer will still be NO.**

The values that m and n takes based on __ statement 1 gives a conclusive answer__ to the question.

Hence, statement 1 is sufficient.

**Statement 2**: m < n

We need to check whether we get a conclusive Yes or No using this statement to determine whether statement 2 alone is sufficient.

Let us look for counter examples

**Example**: Let m = 2 and n = 3

2^{3} < 3^{3}. So, the answer to the question is **NO.**

**Counter Example**: Let m = -3 and n = 2

(-3)^{2} > 2^{2}. So, the answer to the question is **YES.**

The values that m and n takes based on statement 2 __ do not give a conclusive answer__ to the question.

Hence, statement 2 is not sufficient.

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