First slide

Theory of equations

Question

$If e^{λ} and e^{- λ} are the roots of 3 x^{2} - (a + b) x + 2 a = 0, a, b, λ \in R, λ \neq 0 then least integral value of b is$

Moderate

A

4
B

5
C

9
D

10

Solution

$f (x) = 3 x^{2} - (a + b) x + 2 a = 0 has roots e^{λ} and e^{- λ} Product of the roots = 1$
$\begin{array}{l} \Rightarrow \frac{2 a}{3} = 1 \Rightarrow a = \frac{3}{2} \\ Now, sum of the roots = e^{λ} + e^{- λ} = \frac{a + b}{3} > 2 \\ \Rightarrow b > 6 - a \\ \Rightarrow b > 6 - \frac{3}{2} \Rightarrow b > \frac{9}{2} \end{array}$
$Therefore, least integral value of b is 5 .$

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