8. Sequences and Series
hard

यदि एक वास्तविक संख्या $x$ के लिए $1$ , $\log _{10}(4 x-2)$ तथा $\log _{10}\left(4^{x}+\frac{18}{5}\right)$ एक समान्तर श्रेढ़ी में है, तो सारणिक $\left|\begin{array}{ccc}2\left( x -\frac{1}{2}\right) & x -1 & x ^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|$ का मान बराबर है......।

A

$5$

B

$4$

C

$1$

D

$2$

(JEE MAIN-2021)

Solution

$2 \log _{10}\left(4^{ x }-2\right)=1+\log _{10}\left(4^{ x }+\frac{18}{5}\right)$

$\left(4^{ x }-2\right)^{2}=10\left(4^{ x }+\frac{18}{5}\right)$

$\left(4^{ x }\right)^{2}+4-4\left(4^{ x }\right)-32=0$

$\left(4^{ x }-16\right)\left(4^{ x }+2\right)=0$

$4^{ x }=16$

$x =2$

$\left|\begin{array}{lll}3 & 1 & 4 \\ 1 & 0 & 2 \\ 2 & 1 & 0\end{array}\right|=3(-2)-1(0-4)+4(1)$

$=-6+4+4=2$

Standard 11
Mathematics

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