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Maths-

General

Easy

Question

The number of positive integral solutions of the inequation $fraction numerator x squared left parenthesis 3 x minus 4 right parenthesis cubed left parenthesis x minus 2 right parenthesis to the power of 4 over denominator left parenthesis x minus 5 right parenthesis to the power of 5 left parenthesis 2 x minus 7 right parenthesis to the power of 6 end fraction less or equal than 0$ is

2
0
3
4

The correct answer is: 3

Step by step solution:
$fraction numerator x squared left parenthesis 3 x minus 4 right parenthesis cubed left parenthesis x minus 2 right parenthesis to the power of 4 over denominator left parenthesis x minus 5 right parenthesis to the power of 5 left parenthesis 2 x minus 7 right parenthesis to the power of 6 end fraction less or equal than 0$
Here, $x squared greater than 0$ , $open parentheses x minus 2 close parentheses to the power of 4 greater than 0$ , $open parentheses 2 x minus 7 close parentheses to the power of 6 greater than 0$ $rightwards double arrow space x space equals space 0 comma space 2$
CASE 1 : $3 x minus 4 less than 0 comma x minus 5 greater than 0$
$rightwards double arrow 3 x less than 4 space comma space x greater than 5$
$rightwards double arrow x less than 4 over 3 comma space space x space greater than 5$
CASE 2: $3 x minus 4 greater than 0 comma x minus 5 less than 0$
$rightwards double arrow 3 x greater than 4 space comma space x less than 5$
$rightwards double arrow x greater than 4 over 3 comma space x less than 5$
Therefore, $x space equals space 0 comma space 2 comma space a n d space 4$ [ since, $4 over 3 less than x less than 5$ and integral value in the range is 4]
Thus, there are 3 positive integral values.
Hence, option(c) is the correct option.

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