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Question

Which of the following is/are correct for the quadratic equation (k+4)x2+(k+1)x+1=0

A
For k=5, the roots are real and equal.
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B
There are 9 integral values of k for which the roots are imaginary.
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C
There are 7 integral values of k for which the roots are imaginary.
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D
For k=5, the roots are real and equal.
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Solution

The correct options are
C There are 7 integral values of k for which the roots are imaginary.
D For k=5, the roots are real and equal.
(k+4)x2+(k+1)x+1=0
Discriminant of the equation is,
Δ=(k+1)24(k+4) =k2+2k+14k16 =k22k15 =(k5)(k+3)

For k=5,
Δ=20>0
Hence, for k=5, roots are real and distinct.

For imaginary roots,
Δ<0
(k5)(k+3)<0
k(3,5)
Possible integral values of k are 2,1,0,1,2,3,4

For k=5,3,
Δ=0 and hence, roots are real and equal.

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