Question

Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$.

Consider the following statements:
$1$. The term containing $x^2$ does not exist in the given expansion.
$2$. The sum of the coefficient of all the terms in the given expansion is $2^{15}$.
Which of the statements is$/$ are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

The correct option is B $2$ only

1.The genral term of it can be written as $T_{r+1}{=}^{15}{C}_r.x^{2.(15-r)}.\frac{1}{x^r}{=}^{15}{C}_r.x^{30-3r}$

the term containing power of x=2

hence $\Rightarrow 30-3r=2$ $\Rightarrow r=\frac{28}{3}$ but term can't be fractional so,it does not exsist.

2.To find the sum of coefficent of all the terms we just put x=1

${(1+1)}^{15}$=$2^{15}$

So statement 2 is only correct