The coefficient of x12 in the expansion of \(\left(3x^2 + \frac 1 x\ri

1.	The coefficient of x12 in the expansion of \(\left(3x^2 + \frac 1 x\right)^{30}\) is:1. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {15} \end{array}} \right){3^{15}}\)2. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {10} \end{array}} \right){3^{12}}\)3. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {12} \end{array}} \right){3^{12}}\)4. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {16} \end{array}} \right){3^{14}}\)
Answer» Correct Answer - Option 4 : \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {16} \end{array}} \right){3^{14}}\) Concept:* The expansion of (x + y)ⁿ is given by the binomial expansion. The expansion will be (x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 y + ⁿC₂ x^n-2 y² + ... + ⁿC_r x^n-r y^r + ... + ⁿC_n yⁿ; The general term is given by T_r = nCr xn-r yr ; Calculation: Given expansion is \(\left(3x^2 + \frac 1 x\right)^{30}\) Let the r^th term has x¹² and the coefficient be A; ⇒ T_r = A x¹²; The general r^th term of the given expansion will be T_r = \(^{30}C_r (3x^2)^{30-r}(\frac {1}{x})^{r} \) ⇒ T_r = ³⁰C_r 3^30-r × x^60-2r × x^-r = 30Cr 330-r × x60-3r Equating the power of x with 12, ⇒ 60 - 3r = 12 ⇒ r = 16 Now the coefficient will be A = 30Cr 330-r = ³⁰C₁₆ 3¹⁴;

The coefficient of x12 in the expansion of \(\left(3x^2 + \frac 1 x\right)^{30}\) is:1. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {15} \end{array}} \right){3^{15}}\)2. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {10} \end{array}} \right){3^{12}}\)3. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {12} \end{array}} \right){3^{12}}\)4. \(\left( {\begin{array}{{20}{c}} {30}\\ C\\ {16} \end{array}} \right){3^{14}}\)

Answer» Correct Answer - Option 4 : \(\left( {\begin{array}{*{20}{c}} {30}\\ C\\ {16} \end{array}} \right){3^{14}}\)

Concept:

The expansion of (x + y)ⁿ is given by the binomial expansion. The expansion will be

(x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^n-1 y + ⁿC₂ x^n-2 y² + ... + ⁿC_r x^n-r y^r + ... + ⁿC_n yⁿ;

The general term is given by

T_r = nCr xn-r yr ;

Calculation:

Given expansion is \(\left(3x^2 + \frac 1 x\right)^{30}\)

Let the r^th term has x¹² and the coefficient be A;

⇒ T_r = A x¹²;

The general r^th term of the given expansion will be

T_r = \(^{30}C_r (3x^2)^{30-r}(\frac {1}{x})^{r} \)

⇒ T_r = ³⁰C_r 3^30-r × x^60-2r × x^-r = 30Cr 330-r × x60-3r

Equating the power of x with 12,

⇒ 60 - 3r = 12 ⇒ r = 16

Now the coefficient will be

A = 30Cr 330-r = ³⁰C₁₆ 3¹⁴;