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4-2.Quadratic Equations and Inequations
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समीकरण $\log _{(3 x-1)}(x-2)=\log _{\left(9 x^2-6 x+1\right)}\left(2 x^2-10 x-2\right)$ के हल $x$ का मान निम्न है :
A
$9-\sqrt{15}$
B
$3+\sqrt{15}$
C
$2+\sqrt{5}$
D
$6-\sqrt{5}$
(KVPY-2015)
Solution
(b)
We have,
$\log _{3 x-1}(x-2)$
$=\log _{\left(9 x^2-6 x+1\right)}\left(2 x^2-10 x-2\right)$
$\log _{(3 x-1)}(x-2)$
$=\log _{(3 x-1)^2}\left(2 x^2-10 x-2\right)$
$\log _{3 x-1}(x-2)$
$=\frac{1}{2} \log _{3 x-1}\left(2 x^2-10 x-2\right)(x-2)^2$
$(x-2)^2=2 x^2-10 x-2$
$x^2-4 x+4=2 x^2-10 x-2$
$x^2-6 x-6=0$
$x=3 \pm \sqrt{15}$
$x=3-\sqrt{15}, 3+\sqrt{15}$
$x=3-\sqrt{15} \Rightarrow x-2 < 0$
$x=3+\sqrt{15}$
Standard 11
Mathematics
