A quadratic with roots m and n satisfies the equation mm−1+nn−1=a2−7a2−4. Given that m,n∈(1,4) , mn=4(a∈R), which of the following is/are INCORRECT?

The number of integral values of a is 2 , The maximum integral value of a is 4 , The minimum integral value of a is -3

A quadratic with roots m and n satisfies the equation mm−1+nn−1=a2−7a2−4. Given that m,n∈(1,4) , mn=4(a∈R), which of the following is/are INCORRECT?

Simplifying m n = 4 ⇒ m + n = a 2 + 1 ∴ m, n are the roots of x 2 − ( a 2 + 1 ) x + 4 = 0 m , n ∈ ( 1 , 4 ) ⇒ a ∈ ( − 2 , − 3 ] ∪ [ 3 , 2 )

The least positive value of a equal 3

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A quadratic with roots m and n satisfies the equation $\frac{m}{m - 1} + \frac{n}{n - 1} = \frac{a^{2} - 7}{a^{2} - 4}$ . Given that $m, n \in (1, 4)$ , $m n = 4 (a \in R)$ , which of the following is/are INCORRECT?

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The number of integral values of a is 2

The least positive value of a equal $\sqrt{3}$

The maximum integral value of a is 4

The minimum integral value of a is -3

answer is A, C, D.

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Detailed Solution

Simplifying $m n = 4$
$\Rightarrow m + n = a^{2} + 1$
$∴$ m, n are the roots of $x^{2} - (a^{2} + 1) x + 4 = 0$
$m, n \in (1, 4) \Rightarrow a \in (- 2, - \sqrt{3}] \cup [\sqrt{3}, 2)$

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