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Maths-

General

Easy

Question

Number of solutions of the equation |3x 2| = 3[x] + 2{x}, where [.] and {.} denote the integral and fractional parts of x respectively, is

0
1
2
infinite

The correct answer is: 1

case-I when $x less than fraction numerator 2 over denominator 3 end fraction$ , we have
$2 minus 3 x equals 3 left square bracket x right square bracket plus 2 left curly bracket x right curly bracket$
$2 minus 3 left square bracket x right square bracket minus 3 left curly bracket x right curly bracket equals 3 left square bracket x right square bracket plus 2 left curly bracket x right curly bracket$
$5 left curly bracket x right curly bracket equals 2 minus 6 left square bracket x right square bracket$
$0 less or equal than 2 minus 6 left square bracket x right square bracket less than 5$ i.e. $negative 2 less or equal than negative 6 left square bracket x right square bracket less than 3$
$negative fraction numerator 1 over denominator 2 end fraction less than left square bracket x right square bracket less or equal than fraction numerator 1 over denominator 3 end fraction$
$x equals fraction numerator 2 over denominator 5 end fraction$
Case-II when $x greater or equal than fraction numerator 2 over denominator 3 end fraction$ we have
$3 x minus 2 equals 3 left square bracket x right square bracket plus 2 left curly bracket x right curly bracket$
{x} = 2 (not possible)
Number of solutions is 1

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