If sum of the coefficients of first- second and third terms in the expansion of x2 - 1xm is 46- then the coefficient of the term that is independent of x- is

We are given nbsp; ^mC_0 + ^mC_1 + ^mC_2 = 46 ⇒ 1 +m+ m(m 1)2 = 46 ⇒ 2m + m(m 1) = 90 ⇒ m^2 + m 90 = 0 ⇒ m = 9 or m= 10 ⇒ m = 9as m gt; 0 Now, (r + 1)th ...

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Standard XI

Mathematics

Term Independent of x

If sum of the...

Question

If sum of the coefficients of first, second and third terms in the expansion of ${(x^{2} + \frac{1}{x})}^{m}$ is $46,$ then the coefficient of the term that is independent of $x,$ is

96

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84

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78

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88

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Solution

The correct option is B $84$
We are given
$^{m} C_{0} +^{m} C_{1} +^{m} C_{2} = 46$
$\Rightarrow 1 + m + \frac{m (m - 1)}{2} = 46$
$\Rightarrow 2 m + m (m - 1) = 90$
$\Rightarrow m^{2} + m - 90 = 0$
$\Rightarrow m = 9$ or $m = - 10$
$\Rightarrow m = 9$ as $m > 0$
Now, $(r + 1) t h$ term of ${(x^{2} + \frac{1}{x})}^{m}$ is $^{m} C_{r} (x^{2})^{m - r} {(\frac{1}{x})}^{r}$
$=^{m} C_{r} x^{2 m - 3 r}$
For this to be independent of $x$ , $2 m - 3 r = 0 \Rightarrow r = 6$
$∴$ Coefficient of the term independent of $x$ is $^{9} C_{6} = 84.$

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If sum of the coefficients of first, second and third terms in the expansion of (x2+1x)m is 46, then the coefficient of the term that is independent of x, is

If sum of the coefficients of first, second and third terms in the expansion of ${(x^{2} + \frac{1}{x})}^{m}$ is $46,$ then the coefficient of the term that is independent of $x,$ is