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15 Nov 2022, 02:30
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Difficulty:
15%
(low)
Question Stats:
83% (01:29) correct
17%
(01:27)
wrong
based on 78
sessions
History
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Is x < 0 ?
(1) x^5 < x^6
(2) 5^x > 6^x
(1) x^5 < x^6
(2) 5^x > 6^x
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15 Nov 2022, 03:02
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Is x < 0 ?
(1) x^5 < x^6
(2) 5^x > 6^x
From A
x^5<x^6
x^5-x^6<0
x^5(1-X)<0
here x^5(x-1)>0
implies X>0 or X-1>0
implies x>0 or x>1
leads to X>1
another case
X<0 or X-1<0
X<0 or X<1
X<0
So from we can say that
x>1 or x<0
hence we cannot say exactly about this from A
so answer is false
From B
5^x>6^x
put x=0
we find that it doesnot work as
we check for -1,-2,-3
we can say that it will work
and we put the value between 0 to 1
we find that it will not satisfy.
but we put the value less than 0. even we put the value between 0 to -1.
we find that it will is true and satisfy.
only B
(1) x^5 < x^6
(2) 5^x > 6^x
From A
x^5<x^6
x^5-x^6<0
x^5(1-X)<0
here x^5(x-1)>0
implies X>0 or X-1>0
implies x>0 or x>1
leads to X>1
another case
X<0 or X-1<0
X<0 or X<1
X<0
So from we can say that
x>1 or x<0
hence we cannot say exactly about this from A
so answer is false
From B
5^x>6^x
put x=0
we find that it doesnot work as
we check for -1,-2,-3
we can say that it will work
and we put the value between 0 to 1
we find that it will not satisfy.
but we put the value less than 0. even we put the value between 0 to -1.
we find that it will is true and satisfy.
only B
15 Nov 2022, 03:22
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Bookmarks
We need to find if x < 0.
Statement 1 \(x^5 < x^6\)
The number line has 4 important regions where the relations between the exponent vary.
Region 1: x > 1
In this region \(x^6 > x^5\), you simply take an example of x = 2.
This is a valid region.
Region 2: 1 > x > 0
In this region \(x^6 < x^5\), you simply take an example of x = \(\frac{1}{2}\)
This is not a valid region.
Region 1: -1 > x > 0
In this region \(x^6 > x^5\), even power must be positive.
This is a valid region.
Region 1: x > 1
In this region \(x^6 > x^5\), even power must be positive.
This is a valid region.
Hence we can say x < 0 or x > 1.
This statement is insufficient and we eliminate options A and D.
Statement 2: \(5^x > 6^x\)
Region 1: x > 1
It does not satisfy when we have x = 2.
This is not a valid region.
Region 2: 1 > x > 0
It does not satisfy for x = \(\frac{1}{2}\) as we can have \(\sqrt{5} \)< \(\sqrt{6}\)
Region 1: -1 > x > 0
It satisfies for x = -\(\frac{1}{2}\) as\(\frac{1}{\sqrt{6}}\) < \(\frac{1}{\sqrt{5}}\)
Region 1: x > 1
It satisfies for x = -2 as \(\frac{1}{6^2}\) < \(\frac{1}{5^2}\)
So, the answer is B.
Statement 1 \(x^5 < x^6\)
The number line has 4 important regions where the relations between the exponent vary.
Region 1: x > 1
In this region \(x^6 > x^5\), you simply take an example of x = 2.
This is a valid region.
Region 2: 1 > x > 0
In this region \(x^6 < x^5\), you simply take an example of x = \(\frac{1}{2}\)
This is not a valid region.
Region 1: -1 > x > 0
In this region \(x^6 > x^5\), even power must be positive.
This is a valid region.
Region 1: x > 1
In this region \(x^6 > x^5\), even power must be positive.
This is a valid region.
Hence we can say x < 0 or x > 1.
This statement is insufficient and we eliminate options A and D.
Statement 2: \(5^x > 6^x\)
Region 1: x > 1
It does not satisfy when we have x = 2.
This is not a valid region.
Region 2: 1 > x > 0
It does not satisfy for x = \(\frac{1}{2}\) as we can have \(\sqrt{5} \)< \(\sqrt{6}\)
Region 1: -1 > x > 0
It satisfies for x = -\(\frac{1}{2}\) as\(\frac{1}{\sqrt{6}}\) < \(\frac{1}{\sqrt{5}}\)
Region 1: x > 1
It satisfies for x = -2 as \(\frac{1}{6^2}\) < \(\frac{1}{5^2}\)
So, the answer is B.
