QJEEM16Mafc24: JEE Main 2016 Mathematics - Class 11 JEEM16Mafc24 Solution: If the coefficients of x −2 and x −4 in the expansion of are m…

Question

Verified Solution • 2025–26 Session moderate

If the coefficients of x⁻² and x⁻⁴ in the expansion of

{\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x &gt; 0} \right),

are m and n respectively, then

{m \over n}

is equal to :

Accepted Answer

T_r+1 = ¹⁸C_r

{\left( {{x^{{1 \over 3}}}} \right)^{18 - r}}

.

{\left( {{1 \over {2{x^{{1 \over 3}}}}}} \right)^r}

= ¹⁸C_r

{\left( {{1 \over 2}} \right)^r}\,\,.\,\,{x^{{{18 - 2r} \over 3}}}

For coefficient of x^{$-$ 2},

{{18 - 2r} \over 3}

=

-

2

\Rightarrow

r = 12

\therefore

Coefficient of x^{$-$ 2} is (m) = ¹⁸C₁₂

{\left( {{1 \over 2}} \right)^{12}}

For coefficient of x^{$-$ 4},

{{18 - 2r} \over 3}

=

-

4

\Rightarrow

r = 15

\therefore

Coefficient of x^{$-$ 4} is (n) = ¹⁸C₁₅

\left( {{1 \over {2}}} \right)

¹⁵

\therefore

{m \over n} = {{^{18}{C_{12}}{{\left( {{1 \over 2}} \right)}^{12}}} \over {^{18}{C_{15}}{{\left( {{1 \over 2}} \right)}^{15}}}}

=

{{{}^{18}{C_6} \times {{\left( 2 \right)}^3}} \over {{}^{18}{C_3}}}

= 182

Step-by-Step Verified Solution