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Question

If $fraction numerator x to the power of 2 end exponent minus 10 x plus 13 over denominator left parenthesis x minus 1 right parenthesis open parentheses x to the power of 2 end exponent minus 5 x plus 6 close parentheses end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator x minus 2 end fraction plus fraction numerator K over denominator x minus 3 end fraction$ then K=

-1
-2
-3
-4

Hint:

Take LCM on RHS and simplify

Then take a value of x which satisfies the requirement.

The correct answer is: -4

Given :
$fraction numerator x to the power of 2 end exponent minus 10 x plus 13 over denominator left parenthesis x minus 1 right parenthesis open parentheses x to the power of 2 end exponent minus 5 x plus 6 close parentheses end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator x minus 2 end fraction plus fraction numerator K over denominator x minus 3 end fraction$

Taking LCM on RHS and factorizing on LHS
$rightwards double arrow fraction numerator x squared minus 10 x plus 13 over denominator left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis end fraction equals fraction numerator A left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis space plus space B left parenthesis x minus 1 right parenthesis left parenthesis x minus 3 right parenthesis space plus space K left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis end fraction rightwards double arrow x squared minus 10 x plus 13 space equals space A left parenthesis x squared space minus 5 x space plus 6 right parenthesis plus space B left parenthesis x squared minus 4 x space plus 3 right parenthesis space plus space K left parenthesis x squared space minus 3 x space plus 2 right parenthesis rightwards double arrow x squared minus 10 x plus 13 space equals space A left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis space plus space B left parenthesis x minus 1 right parenthesis left parenthesis x minus 3 right parenthesis space plus space K left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis P u t space x equals space 3 rightwards double arrow 9 minus 30 plus 13 space equals space 0 plus 0 right parenthesis space plus space 2 K rightwards double arrow 2 K space equals space minus 8 rightwards double arrow space K space equals space minus 4$

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