The number 81 is the coefficient of x k in the binomial expansion of ( x 2 + 3 x ) 4 , x 0 . Then the value of k equals

Question

The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals

Smart Achievers · Accepted Answer

(3)

${(x^{2} + \frac{3}{x})}^{4} = {}^{4}C_{0} {(\frac{3}{x})}^{4} + {}^{4}C_{1} x^{2} {(\frac{3}{x})}^{3}$ $+ {}^{4}C_{2} x^{4} {(\frac{3}{x})}^{2} + {}^{4}C_{3} x^{6} {(\frac{3}{x})}^{1} + {}^{4}C_{4} x^{8} {(\frac{3}{x})}^{0}$

$= \frac{81}{x^{4}} + 4 \times x^{2} \times \frac{27}{x^{3}} + 6 \times 9 \times x^{2} + 12 x^{5} + x^{8}$

$= \frac{81}{x^{4}} + \frac{108}{x} + 54 x^{2} + 12 x^{5} + x^{8}$

$Here, 81 is coefficient of x^{- 4}$ $∴$ $k = - 4$

Solution

Q.
The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals

1 - 2

2 2

3 - 4

4 4

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Solution

Q. The number 81 is the coefficient of xk in the binomial expansion of (x2+3x)4, x≠0. Then the value of k equals

1 - 2

2 2

3 - 4

4 4

Ans. (3) (x2+3x)4=C04(3x)4+C14x2(3x)3+C24x4(3x)2+C34x6(3x)1+C44x8(3x)0 =81x4+4×x2×27x3+6×9×x2+12x5+x8 =81x4+108x+54x2+12x5+x8 Here, 81 is coefficient of x-4 ∴ k=-4

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Q.
The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals