QJEEM23Ma3756: JEE Main 2023 Mathematics - Class 11 JEEM23Ma3756 Solution: If the value of real number for which and have a common real…

${x^2} - 5\alpha x + 1 = 0$ ..... (1)

${x^2} - \alpha x - 5 = 0$ ...... (2)

have a common root.

Subtracting (1) with (2) we'll get $x = {6 \over {4\alpha }}$

Substituting in (1)

${{36} \over {16{\alpha ^2}}} - {{30} \over 4} + 1 = 0$

$\Rightarrow {\alpha ^2} = {9 \over {26}}$

$\alpha = {3 \over {\sqrt {2 \times 13} }}$

$\therefore$ $\beta = 13$