If $K$ is the coefficient of $x^{4}$ in the expansion of $(1 + x + a x^{2})^{10}$ , then the absolute value of $a$ for which $K$ is minimum, is

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Solution

We know that,
$(1 + x + a x^{2})^{10} = 10 \sum r = 0^{10} C_{r} (1 + x)^{10 - r} (a x^{2})^{r}$
The coefficient of $x^{4}$ is,
$K =^{10} C_{0}^{10} C_{4} +^{10} C_{1}^{9} C_{2} a +^{10} C_{2}^{8} C_{0} a^{2} \Rightarrow K = 45 a^{2} + 45 \times 8 a + \frac{45 \times 14}{3} \Rightarrow K = 45 [a^{2} + 8 a + \frac{14}{3}]$
Minimum value of $K$ occurs at $- \frac{B}{2 A} = - \frac{8}{2} = - 4$

Hence, the required value of $a$ is $4.$

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