​ 1. Find the term independent of x in the expansion of ( x^2+2/x)^6 ​

Question

1. Find the term independent of x in the expansion of (\(x^2\) + \(\frac{2}{x}\))⁶ 

2. If the middle term in the expansion of \(\big(x^m + \frac{2}{x}\big)^6\) is independent of x, find the value of m

Answer 1

1. t_r+1 = ⁿC_r a^n-r b^r = ⁶C_r (x²)^6-r \((\frac{2}{x})^r\) 

= ⁶C_r x^12-2r x^-r (2)^r = ⁶C_r x^12-3r (2)^r 

For term independent of x; 

12 – 3r = 0 r = 4 

t₅ = ⁶C₄ (2)⁴ = ⁶C₂ × 16 = \(\frac{6 \times 5}{2 \times 2}\) × 16 = 240 

2. m = 1