Find the value of 'm', if $m x^{3} + 2 x^{2} - 3$ and $x^{2} - m x + 4$ leave the same remainder when each is divided by x - 2.

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Solution

Let f(x) = $m x^{3} + 2 x^{2} - 3$
g(x) = $x^{2} - m x + 4$
It is given that f(x) and g(x) leave the same remainder when divided by (x – 2). Therefore, we have:
f (2) = g (2)
m ${(2)}^{3}$ + 2 ${(2)}^{2}$ – 3 = ${(2)}^{2}$ – m(2) + 4
8m + 8 – 3 = 4 – 2m + 4
10m = 3
m = $\frac{3}{10}$

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