First slide

Theory of expressions

Question

The set of all values of ‘a’ for which the roots of the equation $(a + 1) x^{2} - 3 a x + 4 a = 0 (a \neq - 1) are real and greater than 1 is$

Moderate

A

$[\frac{- 10}{7}, 1]$
B

$[\frac{- 12}{7}, 0]$
C

$[\frac{- 16}{7}, - 1)$
D

$[\frac{- 16}{7}, 0)$

Solution

$\begin{array}{l} (a + 1) x^{2} - 3 a x + 4 a = 0 \\ D = 9 a^{2} - 16 a (a + 1) \geq 0 \\ \Rightarrow - 7 a^{2} - 16 a \geq 0 \Rightarrow a (7 a + 16) \leq 0 \\ \Rightarrow a \in [\frac{- 16}{7}, 0] . . . . . . . (1) \\ x_{1} > 1, x_{2} > 1 \\ ∴ (a + 1) f (1) > 0 and \frac{3 a}{2 (a + 1)} > 1 \\ \Rightarrow (a + 1) (2 a + 1) > 0 \end{array}$
$\begin{array}{l} a<-1 or a > \frac{-1}{2} and \frac{a - 2}{a + 1} > 0 \dots \dots . . . (2) \\ from (1), (2) \\ ∴ a \in [- \frac{16}{7}, - 1) \end{array}$

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