Question

Statement-I : If roots of the equation $x^2-bx+c=0$ are two consecutive integers, then $b^2-4c=1$.
Statement-II : If $a,b,c$ are odd integers, then the roots of the equation $4abc{x}^2+(b^2-4ac)x-b=0$ are real and distinct.

• A. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I
• B. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I
• C. Statement-I is true, Statement-II is false
• D. Statement-I is false, Statement-II is true

Answer 1

The correct option is B Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I

If roots of the equation $x^2-bx+c=0$ are two consecutive integers.
$\Rightarrow$ Difference between roots is $1$.
$\Rightarrow \frac{\sqrt{\Delta }}{a}=1$
$\therefore b^2-4c=1$
$4abc{x}^2+(b^2-4ac)x-b=0$
$\Delta =(b^2-4ac{)}^2+164a{b}^{2}c=(b^2+4ac{)}^2>0$ ($\because a,b,c$ are odd integers.)
$\therefore$ Roots are real and distinct.
Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I
Hence, option B.