Question

Which of the following is/are correct for the quadratic equation $(k+4)x^2+(k+1)x+1=0$

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

Answer 1

Answer: (C), (D)

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.

$(k+4)x^2+(k+1)x+1=0$
Discriminant of the equation is,
$\Delta =(k+1{)}^2-4(k+4)=k^2+2k+1-4k-16=k^2-2k-15=(k-5\left)\right(k+3)$

For $k=-5$,
$\Delta =20>0$
Hence, for $k=-5$, roots are real and distinct.

For imaginary roots,
$\Delta <0$
$\Rightarrow (k-5)(k+3)<0$
$\Rightarrow k\in (-3,5)$
Possible integral values of $k$ are $-2,-1,0,1,2,3,4$

For $k=5,-3$,
$\Delta =0$ and hence, roots are real and equal.