Question

16. Solve for x:
$\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}=\frac{2}{3},\times \ne 1,2,3$

17. A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. $\left(\mathrm{Use}\mathrm{\pi }=\frac{22}{7}\right)$.

Answer 1

Dear student
Que16
$$\begin{gathered} Wehave, \\ \frac{1}{(x-1)(x-2)}+\frac{1}{\left(x-2\right)\left(x-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{x-3+x-1}{\left(x-1\right)\left(x-2\right)\left(x-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{2x-4}{\left(x-1\right)\left(x-2\right)\left(x-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{2\left(x-2\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)}=\frac{2}{3} \\ \Rightarrow \frac{1}{\left(x-1\right)\left(x-3\right)}=\frac{1}{3} \\ \Rightarrow 3=x^2-4x+3 \\ \Rightarrow x^2-4x=0 \\ \Rightarrow x\left(x-4\right)=0 \\ \Rightarrow x=0orx=4 \end{gathered}$$
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