The positive value of $x$ satisfying the equation $\int_x^1(1-t) d t=\frac{1}{2}$ is
The positive value of $x$ satisfying the equation $\int_x^1(1-t) d t=\frac{1}{2}$ is
- $1$
- $\sqrt{2}$
- $3$
- $2$
Solution
$\begin{aligned}
& \text {} \int_x^1(1-t) d t=\frac{1}{2} \Rightarrow\left[t-\frac{t^2}{2}\right]_x^1=\frac{1}{2} \\
& \Rightarrow\left[1-\frac{1}{2}-x+\frac{x^2}{2}\right]=\frac{1}{2} \\
& \Rightarrow \frac{1-2 x+x^2}{2}=\frac{1}{2} \Rightarrow(1-x)^2=1 \\
& \Rightarrow 1-x= \pm 1 \Rightarrow x=0,2
\end{aligned}$
$\therefore$ There are 2 positive values of $x$ satisfying the equation.
Asked in: AP EAMCET 2023 (17 May Shift 1)
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