If the term independent of x in the expansion of is 105 then a2 is equal to :

Engineering

Mathematics

Introduction to Determinants

Question

If the term independent of x in the expansion of ${(\sqrt{a} x^{2} + \frac{1}{2 x^{3}})}^{10}$ is 105 then a² is equal to :

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$T_{r + 1} = {}^{10}C_{r} {(\sqrt{a} x^{2})}^{10 - r} {(\frac{1}{2 x^{3}})}^{r}$

2(10 – r) – 3r = 0
r = 4

$T_{5} = {}^{10}C_{4} (\sqrt{a})^{6} {(\frac{1}{2})}^{4} = 105$

$= \frac{10 \times 9.8.7}{4.3.2.1} \times \frac{9^{3}}{2 \cdot 2.2.2} = 5 \times 7 \times 3$

a³ = 8, a = 2 ⇒ a² = 4

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