Question

I. lf one root of the equation $5x^2+13x+k=0$ is the reciprocal of the other, then $k=5$
II. lf the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal, then $a,b,c$ are in H.P.
Which of the above statement is true?

• A. only I
• B. only II
• C. both I and II
• D. neither I nor II

Answer 1

Answer: (C)

both I and II

If one root is reciprocal of other, then product of
roots i one.
$\Rightarrow \frac{k}{5}>1\Rightarrow k=5$
If roots of $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal, then.
$\left(b\right(c-a){)}^2=4ac(b-c\left)\right(a-b)$
$b^2(c^2+a^2-2ac)=4ac(ab-ac+bc-b^2)$
$b^2(c+a{)}^2=4ac(ab-ac+bc)$
$=4ac\left(\right(a+c)b-ac)$
For $a+c=\frac{2ac}{b}$
$b^2(\frac{2ac}{b}{)}^2=4ac(\frac{ac}{b})$
$\Rightarrow 4a^2{c}^2=4a^2{c}^2$
$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}\Rightarrow$ $a,b,c$ are in H.P