Question: Is x^3 - 6x^2 + 11x - 6 < 0

(1) 1 < x <= 2
(2) 2 <= x < 3

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.

Correct Answer: A
Solution and Explanation:
Approach Solution 1:
The problem statement asked to find whether x^3 - 6x^2 + 11x - 6 < 0
We know the method of dealing with quadratic equations but rarely do we come across third-degree equations.
If by any chance we get a third-degree equation, it will have a very easy root which is 0 or 1 or -1 or 2 or -2.
We can try some of these values to get the first root.
Here we see that x = 1 works. This is easy to figure out as you have two 6s and an 11 as the coefficients.
Now you know that (x-1) is a factor.
We need to calculate the quadratics which if multiplied by (x-1) gives a third-degree expression
(x−1) (x^2−5x+6) = x^3 − 6x^2 + 11x − 6
The way to derive this quadratic is explained below.
(x-1) (ax^2+bx+c) = x^3 - 6x^2 + 11x – 6
The coefficient of x^3 on the right-hand side is 1.
So now it is known that all we need is x^2 so that it can multiply with x to give x^3 on the left-hand side as well. So, a must be 1.
(x-1) (x^2 +bx + c) = x^3 – 6x^2 +11x - 6
The constant term is also very easy to figure out. It should multiply with -1 to give -6 on the right-hand side. So, c must be 6.
(x-1) (x^2+bx+6) = x^3 – 6x^2 +11x – 6
The middle term is a bit more complex. bx multiplies with x to give x^2 term on the right-hand side i.e -6x ^2, however you also get x^2 term by multiplying -1 with x^2.
We need -6x^2 and we have -x^2 hence we need another -5x^2 from bx^2.
Hence b must be -5.
(x-1) (x^2-5x+6) = x^3 – 6x^2 + 11x – 6
We can now factorize the quadratic in a usual way and use the wavy curve to solve inequality.
(x-1) (x-2) (x-3) < 0
2 < x< 3 or x< 1
Now, let's analyse each statement:
Statement 1: 1< x <= 2
Between 1 to 2 (inclusive), the expression is not negative. It is either 0 or positive.
Sufficient.
Statement 2: 2 <= x < 3
Between 2 and 3, the expression becomes negative but it becomes 0 when x = 2.
Hence it is insufficient.
Therefore, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Approach Solution 2:
The problem statement asked to find whether x^3 - 6x^2 + 11x - 6 < 0
First, we need to factorize the given expression.
x^3 - 6x^2 + 11x - 6
= x^3 – 6x^2 + 5x+ 6x – 6
= x (x^2- 6x +5) + 6(x-1)
= x(x-5) (x-1) + 6(x-1)
= (x-1) (x^2-5x+6)
= (x-1) (x-2) (x-3)
The question now comes down to this: Is x^3 - 6x^2 + 11x - 6 < 0?
So, the given expression when put in negative and positive ranges results in:
---(-ve) ----1-----(+ve)----2----(-ve)------3----(+ve)-----

Statement 1: For 1<x<2, the given expression is always positive. For the boundary condition of x=2, the given expression is zero. In both cases, the expression is not negative. Hence, statement 1 alone is sufficient.

Statement 2: For 2<x<3, the expression is negative, but for the boundary condition i.e x=2, the expression is zero and not negative. Hence statement 2 is insufficient.
Hence the correct option is A.
Therefore, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Approach Solution 3:
The problem statement asked to find whether x^3 - 6x^2 + 11x - 6 < 0
x^3 - 6x^2 + 11x - 6
= x^3 – 6x^2 + 5x+ 6x – 6
= x (x^2- 6x +5) + 6(x-1)
= x(x-5) (x-1) + 6(x-1)
= (x-1) (x^2-5x+6)
= (x-1) (x-2) (x-3)
Hence, the roots of x^3 - 6x^2 + 11x - 6 = 0 are 1,2 and 3 so these are the boundary points.

Statement 1: 1 < x <= 2
The only root present in this region is 2.
Hence for 1 < x < 2, the expression x^3 - 6x^2 + 11x – 6 is positive that is not <0 or zero.
Hence, the answer is no.
Also, for x=2, x^3 - 6x^2 +11x -6 is not less than 0. Thus, the statement is sufficient.

Statement 2: 2 <= x< 3
The only root present in this region is 2.
For x = 2, x^3 – 6x^2+ 11x-6 = 0.
Hence, the answer is No.
For 2< x< 3, x^3 -6x^2 + 11x- 6 is negative.
Thus, the answer is Yes.
This is a double case and the statement is not sufficient.
Therefore, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

“Is x^3 - 6x^2 + 11x - 6 < 0”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Data Sufficiency Prep Course”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include 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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