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Maths-

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Easy

Question

lf the function $f left parenthesis x right parenthesis equals left curly bracket fraction numerator square root of 2 plus blank c o s blank x end root minus 1 over denominator k comma left parenthesis pi minus x right parenthesis squared end fraction comma x not equal to pi right curly bracket$ is continuous at $x equals pi$ , then $k$ equals

$0$
$\frac{1}{2}$
2
$\frac{1}{4}$

The correct answer is: $fraction numerator 1 over denominator 4 end fraction$

$chi rightwards arrow J tau I i m fraction numerator square root of 2 plus blank c o s blank x end root minus 1 over denominator left parenthesis pi minus x right parenthesis to the power of 2 end exponent end fraction equals vertical line i m fraction numerator square root of 2 plus blank c o s blank left parenthesis pi plus h right parenthesis end root minus 1 over denominator left parenthesis pi minus pi minus h right parenthesis to the power of 2 end exponent end fraction h rightwards arrow 0 equals vertical line i m fraction numerator square root of 2 minus blank c o s blank h end root minus 1 over denominator h to the power of 2 end exponent end fraction h rightwards arrow 0$
$left parenthesis 2 minus blank c o s blank h minus 1 right parenthesis$
$equals I i m minus equals I i m stack 1 minus blank c o s blank h with _ below equals I i m stack 2 s i n to the power of 2 end exponent fraction numerator h over denominator 2 end fraction with _ below$ $h rightwards arrow 0 square root of 2 minus blank c o s blank h end root$ $h rightwards arrow 0 square root of 2 minus blank c o s blank h end root$ $h rightwards arrow 0 square root of 2 minus blank c o s blank h end root$ $equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator 2 s i n to the power of 2 end exponent fraction numerator h over denominator 2 end fraction over denominator h to the power of 2 end exponent end fraction equals I i m 2 left square bracket fraction numerator h over denominator 2 end fraction fraction numerator blank s i n blank fraction numerator h over denominator 2 end fraction over denominator blank end fraction right parenthesis cross times subscript h rightwards arrow 0 end subscript vertical line i m fraction numerator 1 over denominator 4 left parenthesis square root of 2 minus blank c o s blank h end root plus 1 right parenthesis end fraction$
$negative cross times 4 left parenthesis square root of 2 minus blank c o s blank h end root plus 1 right parenthesis 4 blank h rightwards arrow 0$
$equals Z cross times fraction numerator 1 over denominator 4 left parenthesis X right parenthesis end fraction equals fraction numerator 1 over denominator 4 end fraction$
Therefore, $k equals fraction numerator 1 over denominator 4 end fraction$
Hence, the correct answer is option (D).

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