Question

JEE MAINS MATHS QUESTION SEPTEMBER 2020 ..

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

The (r+1)th term of an Binomial Expansion of

${(a + x)}^{n} \: \: is \:$

$\large{ {}^{n} C_r} \: \times {a}^{n - r} \times {x}^{r}$

EVALUATION

Let coefficient of x occurs in (r+1)th term of an Binomial Expansion of

$\displaystyle \: { \bigg( {x}^{m} + \frac{1}{ {x}^{2} } \: \bigg ) }^{22}$

Now the (r+1)th term of

$\displaystyle \: { \bigg( {x}^{m} + \frac{1}{ {x}^{2} } \: \bigg ) }^{22} \: \: is \:$

$= \displaystyle \: \large{ {}^{22} C_r} \: \times { ({x}^{m} )}^{22 - r} \times { \bigg( \frac{1}{ {x}^{2} } \bigg)}^{r}$

$= \displaystyle \: \large{ {}^{22} C_r} \: \times { {x}^{} }^{22 m- mr - 2r}$

By the given condition

$\large{ {}^{22} C_r} = 1540 \: \: \: and \: \: \: {22 m- mr - 2r} = 1$

Now

${22 m- mr - 2r} = 1 \: \: gives \:$

$\displaystyle \: m = \frac{2r + 1}{ 22 - r} \: \: ....(1)$

Again

$\displaystyle \: \large{ {}^{22} C_r} = 1540$

$\implies \: \displaystyle \: \large{ {}^{22} C_r} = 22 \times 7 \times 5 \times 2$

$\implies \: \displaystyle \: \large{ {}^{22} C_r} = \frac{22 \times 21 \times 20 }{3 \times 2}$

$\displaystyle \: \implies \: \large{ {}^{22} C_r} = \large{ {}^{22} C_{19}}$

$\implies \: r = 19$

From Equation (1)

$\displaystyle \: m = \frac{(2 \times 19) + 1}{ 22 - 19} \: \:$

$\implies \: \displaystyle \: m = \frac{39}{3} \: \:$

$\implies \: \displaystyle \: m = 13$

RESULT

Hence the required value of m is 13