JEE Mains · Maths · STD 11 - 8. sequence and series
If \(1, \log _{10}\left(4^{x}-2\right)\) and \(\log _{10}\left(4^{x}+\frac{18}{5}\right)\) are in
arithmetic progression for a real number \(x\) then the value of the determinant \(\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|\) is equal to ...... .
- A \(5\)
- B \(4\)
- C \(1\)
- D \(2\)
Answer & Solution
Correct Answer
(D) \(2\)
Step-by-step Solution
Detailed explanation
\(2 \log _{10}\left(4^{ x }-2\right)=1+\log _{10}\left(4^{ x }+\frac{18}{5}\right)\) \(\left(4^{ x }-2\right)^{2}=10\left(4^{ x }+\frac{18}{5}\right)\) \(\left(4^{ x }\right)^{2}+4-4\left(4^{ x }\right)-32=0\) \(\left(4^{ x }-16\right)\left(4^{ x }+2\right)=0\) \(4^{ x }=16\)…
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