Question

Let $1+i$ and $1+2i$ are two roots of the equation
$x^5+k{x}^4+l{x}^3+m{x}^2+nx-5=0,k,l,m,n\in R,$ then the value of $k$ is

• A. $\frac{9}{2}$
• B. $-\frac{9}{2}$
• C. $\frac{5}{2}$
• D. $-\frac{5}{2}$

Answer 1

The correct option is B $-\frac{9}{2}$

The roots given are:
$1+i,1+2i$
We know that complex roots occur in conjugate pair, so all the roots of the equation are,
$a,1+i,1-i,1+2i,1-2i$
So, the sum of roots is,
$a+2+2=-k\Rightarrow a+4=-k$
Product of the roots will be,
$a\times 2\times 5=5\Rightarrow a=\frac{1}{2}$
Now,
$k=-(4+\frac{1}{2})=-\frac{9}{2}$