QJEEM16Ma50c6: JEE Main 2016 Mathematics - Class 11 JEEM16Ma50c6 Solution: If the equations x 2 + bx−1 = 0 and x 2 + x + b = 0 have a…

Given,

x² + bx $-$ 1 = 0 . . . . .(1)

and x² + x + b = 0 . . . . . (2)

Performing (1) $-$ (2) we get,

bx $-$ 1 $-$ x $-$ b = 0

$\Rightarrow$     x(b $-$ 1) = b + 1

$\Rightarrow$    x = ${{b + 1} \over {b - 1}}$

putting value of x in equation (2),

${\left( {{{b + 1} \over {b - 1}}} \right)^2} + \left( {{{b + 1} \over {b - 1}}} \right) + b = 0$

$\Rightarrow$    (b + 1)² + (b + 1) (b $-$ 1) + b (b $-$ 1)² = 0

$\Rightarrow$     b² + 2b + 1 + b² $-$ 1 + b (b² $-$ 2b + 1) = 0

$\Rightarrow$     2b³ + 2b + b³ $-$ 2b² + b = 0

$\Rightarrow$     b³ + 3b = 0

$\Rightarrow$     b(b² + 3) = 0

b² = $-$ 3, b = 0

$\therefore$    b = $\pm \sqrt 3 i$

$\Rightarrow$     $\left| b \right|$ = $\sqrt 3$