TS ICET 23 May 2018 Shift 2 Paper

Let the remainder, when p(x) is divided by ${\mathit{x}}^2-5\mathit{x}+6$ be $\mathit{a}\mathit{x}+\mathit{b}$ b and quotient be q(x) by division algorithm.
$\mathit{p}\left(\mathit{x}\right)=\left({\mathit{x}}^2-5\mathit{x}+6\right)\mathit{q}\left(\mathit{x}\right)+\mathit{a}\mathit{x}+\mathit{b}$
$=\left(\mathit{x}-2\right)\left(\mathit{x}-3\right)\mathit{q}\left(\mathit{x}\right)+\mathit{a}\mathit{x}+\mathit{b}$
Given $\mathit{p}\left(2\right)=6;\mathit{p}\left(3\right)=4$
$\mathit{p}\left(2\right)=2\mathit{a}+\mathit{b}\Rightarrow 2\mathit{a}+\mathit{b}=6$→ (1)
$\mathit{p}\left(3\right)=3\mathit{a}+\mathit{b}\Rightarrow 3\mathit{a}+\mathit{b}=4$ → (2)
Solving (1) and (2), $\mathit{a}=-2;\mathit{b}=10$
∴ Required remainder $=-2\mathit{x}+10$