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4-2.Quadratic Equations and Inequations
hard
If $x$ is real, the expression $\frac{{x + 2}}{{2{x^2} + 3x + 6}}$ takes all value in the interval
A
$\left( {\frac{1}{{13}},\frac{1}{3}} \right)$
B
$\left[ { - \frac{1}{{13}},\frac{1}{3}} \right]$
C
$\left( { - \frac{1}{3},\frac{1}{{13}}} \right)$
D
None of these
(IIT-1969)
Solution
(b) If the given expression be $y$, then $y = 2{x^2}y + (3y – 1)x + (6y – 2) = 0$
If $y \ne 0$then $\Delta \ge 0$ for real $x$ i.e. ${B^2} – 4AC \ge 0$
or -$39{y^2} + 10y + 1 \ge 0$ or $(13y + 1)(3y – 1) \le 0$
==> $ – 1/13 \le y \le 1/3$
If $y = 0$ then $x = – 2$ which is real and this value of $y$ is included in the above range.
Standard 11
Mathematics
