If 1, _ 10 (4^ x -2 ) and _ 10 (4^ x +18 5 ) are in arithmetic pr… - Vidyadip

MCQ

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
$1$
✓
$2$

✓

Answer

Correct option: D.

$2$

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

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

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

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

$4^{ x }=16$

$x =2$

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

$=-6+4+4=2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $y\sec x + \tan x + {x^2}y = 0$, then ${{dy} \over {dx}} =$

Consider the given data with frequency distribution

$\mathrm{x}_{\mathrm{i}}$ $\ \ 3\ \ 8\ \ 11\ \ 10\ \ 5\ \ 4$

$\mathrm{f}_{\mathrm{i}}$ $\ \ 5 \ \ 2 \ \ 3 \ \ 2 \ \ 4 \ \ 4$

Match each entry in List-$I$ to the correct entries in List-$II$.

List-$I$	List-$II$
($P$) The mean of the above data is	$(1) 2.5$
($Q$) The median of the above data is	$(2) 5$
($R$) The mean deviation about the mean of the above data is	$(3) 6$
($S$) The mean deviation about the median of the above data is	$(4) 2.7$
	$(5) 2.4$

The correct option is :

If $0 < x < \frac{1}{\sqrt{2}}$ and $\frac{\sin ^{-1} x}{\alpha}=\frac{\cos ^{-1} x}{\beta}$, then a value of $\sin \left(\frac{2 \pi \alpha}{\alpha+\beta}\right)$ is$......$

If $2\sin \theta + \tan \theta = 0$, then the general values of $\theta $ are

Let $a, b \in \mathbb{R}$ and $a^2+b^2 \neq 0$. Suppose $S=\left\{z \in \mathbb{C}: z=\frac{1}{a+i b t}, t \in \mathbb{R}, t \neq 0\right\}$, where $i=\sqrt{-1}$, If $z=x+i y$ and $z \in S$, then $(x, y)$ lies on

($A$) the circle with radius $\frac{1}{2 a}$ and centre $\left(\frac{1}{2 a}, 0\right)$ for $a>0, b \neq 0$

($B$) the circle with radius $-\frac{1}{2 a}$ and centre $\left(-\frac{1}{2 a}, 0\right)$ for $a<0, b ; 0$

($C$) the $x$-axis for $a \neq 0, b=0$

($D$) the $y$-axis for $a=0, b \neq 0$

If $f:R \to R$, then the range of the function $f(x) = \frac{{{x^2}}}{{{x^2} + 1}}$ is

$\int\limits_1^e {\left( {(x + 1} \right).{e^x}\ln x} )dx\, = $

If $y = {\sec ^{ - 1}}{{2x} \over {1 + {x^2}}} + {\sin ^{ - 1}}{{x - 1} \over {x + 1}}$, then ${{dy} \over {dx}}$ is equal to

$\frac{{3 + 2i\sin \theta }}{{1 - 2i\sin \theta }}$ will be purely imaginary, if $\theta = $

[Where $n$ is an integer]

All the pairs $(x, y)$ that satisfy the inequality ${2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1$ also Satisfy the equation