Question

Find the mode of data $1,2,5,3,2,3,0,5,2$.

• A. $3$
• B. $1$
• C. $5$
• D. None of these

Answer 1

The correct option is A $3$

$\begin{array}{c}\Rightarrow \frac{dy}{dx}=\frac{y^2-2y}{x-1}\\ \Rightarrow \frac{dy}{y^2-2y}=\frac{dx}{\left(x-1\right)}\\ \Rightarrow \int \frac{dy}{y\left(y-2\right)}=\frac{dx}{x-1}\\ \Rightarrow \int \frac{dy}{y\left(y-2\right)}=\log \left(x-1\right)+c\\ Let\\ \Rightarrow \frac{1}{y\left(y-2\right)}=\frac{A}{y}+\frac{B}{\left(y-2\right)}\\ \Rightarrow 1=Ay-2A+By\\ \Rightarrow 1+2A\\ \Rightarrow A=\frac{-1}{2}\\ and\left(A+B\right)y=0,\\ A+B=0\to \left(i\right)\\ PutA=\frac{-1}{2}inequation\left(i\right)\\ \frac{-1}{2}+B=0,B=\frac{1}{2}\\ \Rightarrow \int \frac{A}{y}dy+\int \frac{B}{\left(y-2\right)}dy=\log \left(x-1\right)+c\\ \Rightarrow \frac{-1}{2}\int \frac{dy}{y}+\frac{1}{2}\int \frac{1}{\left(y-2\right)}=\log (x-1+c\\ \Rightarrow \frac{-1}{2}\log y+\frac{1}{2}\log \left(y-2\right)=\log \left(x-2\right)+\log c\\ \Rightarrow \frac{-1}{2}\left[\log \left(\frac{y-2}{y}\right)\right]=\log \left(x-1\right)c\\ \Rightarrow \log \sqrt{\frac{y-2}{y}}=\log \left(x-1\right)c\\ \therefore \sqrt{\frac{\left(y-2\right)}{y}}=\left(x-1\right)c\\ \Rightarrow \frac{y}{y}\frac{-2}{y}={\left(x-1\right)}^2{c}^2\\ \Rightarrow 1-{\left(x-1\right)}^2{c}^2=\frac{2}{y}\\ \therefore y=\frac{2}{-1{\left(x-1\right)}^2{c}^2}Ans.\end{array}$