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Are the following statements 'True' or 'False'? Justify your answers.
The only value of \( k \) for which the quadratic polynomial \( k x^{2}+x+k \) has equal zeros is \( \frac{1}{2} \)
Given:
The only value of \( k \) for which the quadratic polynomial \( k x^{2}+x+k \) has equal zeros is \( \frac{1}{2} \)
To do:
We have to find whether the given statement is true or false.
Solution:
Let $f(x) = kx^2 + x + k$
For equal roots, discriminant of $f(x)$ should be zero.
$D = b^2 - 4ac = 0$
Therefore,
$D=1^2-4(k)(k) = 0$
$1=4k^2$
$k^2=\frac{1}{4}$
$k =\sqrt{\frac{1}{4}}$
$k=\pm \frac{1}{2}$
So, for two values of $k$, the given quadratic polynomial has equal zeroes.
Hence, the given statement is false.
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