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माना $\alpha=\max _{x \in R }\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}$ तथा $\beta=\min _{ n \in R }\left\{8^{2 \sin 3 n } \cdot 4^{4 \cos 3 x }\right\}$ हैं। यदि द्विघातीय समीकरण $8 x ^{2}+ bx + c =0$ के मूल $\alpha^{1 / 5}$ तथा $\beta^{1 / 5}$ है, तो $c - b$ का मान बराबर है
$43$
$42$
$50$
$47$
Solution
$\alpha=\max \left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}$
$=\max \left\{2^{6 \sin 3 x} \cdot 2^{8 \cos 3 x}\right\}$
$=\max \left\{2^{6 \sin 3 x+8 \cos 3 x}\right\}$
and $\beta=\min \left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}=\min \left\{2^{6 \sin 3 x+8 \cos 3 x}\right\}$
Now range of $6 \sin 3 x+8 \cos 3 x$
$=\left[-\sqrt{6^{2}+8^{2}},+\sqrt{6^{2}+8^{2}}\right]=[-10,10]$
$\alpha=2^{10} \,\&\, \beta=2^{-10}$
$\text { So, } \alpha^{1 / 5}=2^{2}=4$
$\Rightarrow \beta^{1 / 5}=2^{-2}=1 / 4$
$\text { quadratic } 8 x^{2}+b x+c=0, c-b=$
$8 \times[\text { (product of roots })+(\text { sum of roots })]$
$=8 \times\left[4 \times \frac{1}{4}+4+\frac{1}{4}\right]=8 \times\left[\frac{21}{4}\right]=42$