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Question
The values of k for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are
Options
−11, −3
5, 7
5, −7
none of these
Solution
5, −7
The given equation is \[k x^2 + 1 = kx + 3x - 11 x^2\] which can be written as.
\[k x^2 + 11 x^2 - kx - 3x + 1 = \]
\[ \Rightarrow \left( k + 11 \right) x^2 - \left( k + 3 \right)x + 1 = 0\]
For equal and real roots, the discriminant of
\[\therefore \left( k + 3 \right)^2 - 4\left( k + 11 \right) = 0\]
\[ \Rightarrow k^2 + 2k - 35 = 0\]
\[ \Rightarrow \left( k - 5 \right)\left( k + 7 \right) = 0\]
\[ \Rightarrow k = 5, - 7\]
Hence, the equation has real and equal roots when \[k = 5 , - 7 .\]
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