Question

If $f\left(x\right)=x^3-12x^2+34x-k$ and the roots of $f\left(x\right)=0$ are in A.P., then which of the following is/are correct?

• A. $f\left(x\right)=0$ has one rational root.
• B. $f\left(x\right)=0$ has two irrational roots.
• C. $k=-8$
• D. $k=8$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $f\left(x\right)=0$ has one rational root.
B $f\left(x\right)=0$ has two irrational roots.
D $k=8$

$f\left(x\right)=x^3-12x^2+34x-k$
Let the roots of $f\left(x\right)=0$ be $a-d,a,a+d,$ where $d$ is the common
difference of the A.P.

Now, using sum of roots
$(a-d)+a+(a+d)=12\Rightarrow a=4$

$a(a-d)+a(a+d)+(a-d)(a+d)=34\Rightarrow 4(4-d)+4(4+d)+(4-d)(4+d)=34\Rightarrow 32+16-d^2=34\Rightarrow d=\pm \sqrt{14}$

So, the roots are $4-\sqrt{14},4,4+\sqrt{14}$
​
Now, product of roots is,
$k=4(4-\sqrt{14})(4+\sqrt{14})\therefore k=8$