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Maths-

General

Easy

Question

$text If end text theta element of left square bracket 05 pi right square bracket text and end text r element of R text such that end text 2 s i n invisible function application theta equals r to the power of 4 end exponent minus 2 r to the power of 2 end exponent plus 3$ then the maximum number of values of the pair $left parenthesis r comma theta right parenthesis$ is

8
10
6
None of these

The correct answer is: 6

Hint: For real values of x, the discriminant of a quadratic equation is greater than or equal to zero.
$text Given equation is end text r to the power of 4 end exponent minus 2 r to the power of 2 end exponent plus 3 minus 2 s i n invisible function application theta equals 0 text . end text$
$text The above equation is quadratic in end text r to the power of 2 end exponent text . For the real values of end text r to the power of 2 end exponent text , end text text discriminant > 0. Therefore, end text$
$table row cell 4 minus 4 left parenthesis 3 minus 2 s i n invisible function application theta right parenthesis greater or equal than 0 end cell row cell rightwards double arrow 1 minus 3 plus 2 s i n invisible function application theta greater or equal than 0 end cell row cell rightwards double arrow negative 2 plus 2 s i n invisible function application theta greater or equal than 0 rightwards double arrow s i n invisible function application theta greater or equal than 1 end cell end table$
$text That is, the above inequality holds only at end text sin invisible function application theta equals 1 text . end text$
$text Possible value of end text theta equals fraction numerator pi over denominator 2 end fraction comma 2 pi plus fraction numerator pi over denominator 2 end fraction comma 4 pi plus fraction numerator pi over denominator 2 end fraction$
$text And end text r to the power of 4 end exponent minus 2 r to the power of 2 end exponent plus 3 minus 2 equals 0 rightwards double arrow open parentheses r to the power of 2 end exponent minus 1 close parentheses to the power of 2 end exponent equals 0 rightwards double arrow r equals negative 11$
$text Hence, number of ordered pair of end text left parenthesis r comma theta right parenthesis text is end text 6.$

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