bySayantani Barman Experta en el extranjero
Question: Is x^3 - 6x^2 + 11x - 6 < 0
(1) 1 < x <= 2
(2) 2 <= x < 3
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- 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 Statement 2: For 2 Approach Solution 3: Statement 1: 1 < x <= 2 Statement 2: 2 <= x< 3 “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. Suggested GMAT Data Sufficiency Samples:
Therefore, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
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.
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.
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.
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