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Maths-

General

Easy

Question

The number of solution of the equation $l o g subscript vertical line x minus 1 vertical line end subscript invisible function application open vertical bar x to the power of 2 end exponent minus 1 close vertical bar equals left square bracket vertical line s i n invisible function application x vertical line plus vertical line c o s invisible function application x vertical line right square bracket text (where end text left square bracket. right square bracket$ denotes the greatest integer function) is

3
2
1
None of these

The correct answer is: 1

$log subscript x minus 1 end subscript invisible function application open vertical bar x to the power of 2 end exponent minus 1 close vertical bar equals open square brackets open vertical bar sin invisible function application x close vertical bar plus open vertical bar cos invisible function application x close vertical bar close square brackets$
$text Period of end text vertical line s i n invisible function application x vertical line plus vertical line c o s invisible function application x vertical line text is end text fraction numerator pi over denominator 2 end fraction text and the range of the function end text$
$vertical line s i n invisible function application x vertical line plus vertical line c o s invisible function application x vertical line element of left square bracket 1 comma square root of 2 right square bracket text then the value end text left square bracket vertical line s i n invisible function application x vertical line plus vertical line c o s invisible function application x vertical line right square bracket equals 1$
$l o g subscript 1 x minus 1 end subscript invisible function application open vertical bar x to the power of 2 end exponent minus 1 close vertical bar equals 1$
According to the definition of log,
$open vertical bar x to the power of 2 end exponent minus 1 close vertical bar not equal to 0 rightwards double arrow x not equal to plus-or-minus 1$
$vertical line x minus 1 vertical line not equal to 01 rightwards double arrow x minus 1 not equal to negative 10 comma 1$
$rightwards double arrow x not equal to 01 comma 2$
$text So, end text x not equal to negative 10 comma 12$
By property of log,
$open vertical bar x to the power of 2 end exponent minus 1 close vertical bar equals vertical line x minus 1 vertical line$
$vertical line left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis vertical line equals vertical line x minus 1 vertical line$
$vertical line x plus 1 vertical line equals 1 left parenthesis vertical line x minus 1 vertical line not equal to 0 right parenthesis$
$x plus 1 equals negative 11$
$x equals negative 20$
Value n f $x minus 7$
$text Number of solutions is end text 1 text . end text$

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