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29 Mar 2024, 01:30
Kudos
Bookmarks
D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
59% (01:30) correct
41%
(01:30)
wrong
based on 280
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Not Attempted Yet
If \((x^3+y^6)\) is positive, which of the following must be true?
A. \(x\) is positive
B. y is positive
C. \(x^3\) is negative
D. \((x+y^2)\) and \((x^2+y^4−xy^2)\) are either both positive or both negative
E. \(∣x^3∣ < y^6\)
A. \(x\) is positive
B. y is positive
C. \(x^3\) is negative
D. \((x+y^2)\) and \((x^2+y^4−xy^2)\) are either both positive or both negative
E. \(∣x^3∣ < y^6\)
Updated:
A. \(x\) is positive:
C. \(x^3\) is negative:
D. \((x+y^2) \text{and} (x^2+y^4-xy^2)\) are either both positive or both negative:
E. \( |x^3| < y^6\)
A. \(x\) is positive:
- If \(x\) is posiive, then \(x^3+y^6\) is positive. However, if \(x < 0\), then \(x^3 + y^6\) can still be positive. For example, \(x = -1, y = 2\), then \(x^3 + y^6 = -1 + 8 = 7 > 0\). Therefore, \(x\) need not to be positive.
- The exponent of \(y\) is an even, \(y^6\) must be positive regardless of sign. Hence, \(y\) begin positive or negative not a necessity for \(x^3 + y^6\) to be positive.
C. \(x^3\) is negative:
- If \(x^3 = 0\) or \(x^3\) is positive, \(x^3 + y^6\) can still be positive.
D. \((x+y^2) \text{and} (x^2+y^4-xy^2)\) are either both positive or both negative:
- \((x+y^2) \times (x^2+y^4-xy^2) = x^3 + xy^4 - x^2y^2 + x^y^2 + y^6 - xy^4 = x^3 + y^6\) and we get is positive, which follows that the \((x+y^2) \text{and} (x^2+y^4-xy^2)\) must have the same sign.
- Hence, Option D represents a must be true condition.
E. \( |x^3| < y^6\)
- Not necessarily condition. For example, if \(x = 2\) and \(y = 1\), then \(|2^3| = 8\) which is not less than \(1^6 = 1\), yet \((x^3 + y^6 > 0)\).
Quote:
29 Mar 2024, 02:56
Kudos
Bookmarks
Factorisation of the expression gives \((x+y^2)*(x^2+y^4-x*y^2) \\
using. formula \\
{a^3+b^3=(a+b)*(a^2+b^2-ab)}\)
The expression is positive if both these terms are either simultaneously positive or simultaneously negative.
So IMO
Ans D
Posted from my mobile device
using. formula \\
{a^3+b^3=(a+b)*(a^2+b^2-ab)}\)
The expression is positive if both these terms are either simultaneously positive or simultaneously negative.
So IMO
Ans D
Posted from my mobile device
