Question

(i) If $\alpha ,\beta$ are the roots of $x^2+Kx+2=0$, and $\alpha -\beta =1$ then $K=\pm 3$
(ii) The difference of the roots of $x^2-bx+c=0$ is same as the difference of the roots of $x^2-cx+b=0$. If $b\ne c$, then $b+c=-3$.
Which of the above statements is correct.

• A. only (i)
• B. only (ii)
• C. Both (i) and (ii)
• D. Neither (i) nor (ii)

Answer 1

The correct option is A only (i)

Given equation is $x^2+Kx+2=0$
$\alpha +\beta =-K;\alpha \beta =2$
$\alpha -\beta =\sqrt{(\alpha +{\beta }^2)-4\alpha \beta }$
$=\sqrt{k^2-8}$
If $\alpha -\beta =1\Rightarrow K=\pm 3$
So, (i) is true.
Also given is $x^2-bx+c=0$
$\Rightarrow \alpha +\beta =b;\alpha \beta =c$
$\Rightarrow \alpha -\beta =\sqrt{(\alpha +\beta {)}^2-4\alpha \beta }$
$=\sqrt{b^2-4c}$
For $x^2-cx+b;{\alpha }_1-{\beta }_1=\sqrt{c^2-4b}$
Given both are equal
$\Rightarrow c^2-4b=b^2-4c$
$\Rightarrow c^2-b^2=4(b-c)$
$\Rightarrow c+b=4$
So, (ii) is false.