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(C) The given quadratic equation is $2x^{2} + (a - 10)x + (\frac{33}{2} - 2a) = 0$.For the equation to have real roots,the discriminant $D$ must be gr

Vedclass · Accepted Answer

$8$

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(C) The given quadratic equation is $2x^{2} + (a - 10)x + (\frac{33}{2} - 2a) = 0$.For the equation to have real roots,the discriminant $D$ must be greater than or equal to zero $(D \geq 0)$.$D = b^{2} - 4ac = (a - 10)^{2} - 4(2)(\frac{33}{2} - 2a) \geq 0$.Expanding the expression: $(a^{2} - 20a + 100) - 8(\frac{33}{2} - 2a) \geq 0$.$a^{2} - 20a + 100 - 132 + 16a \geq 0$.$a^{2} - 4a - 32 \geq 0$.Factoring the quadratic inequality: $(a - 8)(a + 4) \geq 0$.The solution set for this inequality is $a \in (-\infty, -4] \cup [8, \infty)$.Since we are looking for the least positive value of $a$,we consider the interval $[8, \infty)$.The least positive value is $8$.

The least positive value of $a$ for which the equation $2x^{2} + (a - 10)x + \frac{33}{2} = 2a$ has real roots is

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