How many of the following inequalities are possible for at least one

Answer should be C. iii is incorrect because when x > x2, which is when 0
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Answer is D.

For (i) - Consider x=1/3. => We can have X= 2/27 which is between 1/9 and 1/27

For (ii)- Consider x= 1/2= > We can have X=0 which is less than 1/4 and 1/8.

For (iii) - Consider x= 2 => We can have X = 5 which is between 8 and 4.

Hence, we can have each of the above statements valid for different vales of x and X.

NOTE : I have solved the above question based on the assumption that x and X are two different variables.

if the highlighted part is correct then there is no need of any option there always exist a number in between

Karishma

Please give your 2 cents also because source mentioned is veritas...

Correct bro. I agree to what you said.

Can anyone please help here?

Responding to a pm:

Note that square of a real number is never negative.

If

\(x = -1/2,\)

\(x^2 = (-1/2)^2 = (-1/2)*(-1/2) = 1/4\)

\(x^3 = (-1/2)*(-1/2)*(-1/2) = -1/8\)

So
\(x^2 > x^3 > x\)

Bunuel: Can you please check this question? I did not find any values for which (iii) holds. Shouldn't OA be C?

Yes. The correct answer is C, not D. Edited. Thank you.

For a problem that hinges on a conceptual understanding of properties of numbers with exponents, plugging in is a natural first approach. So try values in meaningfully different ranges. Say x is 2: that makes
\(x^2\)=4
and \(x^3\)=8
\(So, x^3 > x^2 > x\)
which doesn't work for any of the inequalities.
Now say x is -2: that makes \(x^2\)
\(x^3\)=-8

\(So, x^2 > x > x^3\)

Having tried both negative and positive integers, consider what other ranges are left: fractions between 0 and 1 and fractions between 0 and 1. Inequality (ii) will be true for any fraction between 0 and -1, so it's possible, but inequality (iii) never works: the only numbers for which x > \(x^2\) are fractions between 0 and 1, but for all such numbers \(x^2\) is also greater than \(x^3\)

Answer C

i think for these kind of questions knowing how the graph looks and intersects helps a lot
x,x^2 and x^3 should be something that must be remembered

even graphs for 1/x and 1/(x^2) should be remembered
and we can easily see all of them intersect at 1 or -1 depending on the graph and thats where the > order changes.

once we know the above graphs. graphical transformation helps a lot
x-1 is nothing but x shifted by 1 unit down. so instead of cutting at x=0 it cuts at x=1
|x+5| is x shifted by 5 unit up and then taking a mirror image along x axis

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