The number of values of $k$ for which $[x^{2} - (k - 2) x + k^{2}] [x^{2} + k x + (2 k - 1)]$ is a perfect square is

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none of these

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Solution

The correct option is B 1
$[x^{2} - (k - 2) x + k^{2}] [x^{2} + k x + (2 k - 1)]$ for this expression to be a perfect square there can two possible ways:
1) When both the quadratic expression are perfect square for a particular value,
For that to happen $k - 2 = 2 k \Rightarrow k = - 2$
Now for $k = - 2$ in the second expression we get $x^{2} - 2 x - 5$ which is not a perfect square.
2) The other is when both the quadratic equation are same,
$∴ - (k - 2) = k \Rightarrow k = 1 k^{2} = 2 k - 1 \Rightarrow k = 1$
Therefore, at $k = 1$ the expression is perfect square.

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