Question 86

For what value of k, the expression $$x^6 - 18x^3 + k$$ will be a perfect square?

Solution

Expression : $$x^6 - 18x^3 + k$$

Let $$x^3=y$$

= $$y^2-18y+k$$

= $$y^2-2(y)(9)+k$$

For above expression to be a perfect square, $$k=(9)^2=81$$

= $$y^2-2(y)(9)+81$$

= $$(y-9)^2$$

= $$(x^3-9)^2$$, which is a perfect square

=> Ans - (D)

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