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Q12.    Find a positive value of m for which the coefficient of $x^2$ in the expansion $(1 + x)^ m$ is 6.

As we know that the general   term   in the binomial expansion of    is given by  So, the general   term   in the binomial expansion of    is  will come when . So, The coeficient of   in the binomial expansion of    = 6  Hence the positive value of m for which the coefficient of  in the expansion is 6, is 4.

Q11.    Prove that the coefficient of $x^n$ in the expansion of $(1+x)^{2n}$ is twice the coefficient of $x^n$ in the expansion of $(1+x)^{2n-1}$.

As we know that the general   term   in the binomial expansion of    is given by  So, general   term   in the binomial expansion of   is,  will come when , So, Coefficient of  in the binomial expansion of   is, Now, the general   term   in the binomial expansion of   is, Here also  will come when , So, Coefficient of  in the binomial expansion of   is, Now, As we can see Hence, the...

Q9.    In the expansion of $(1 + a)^{m+n}$ , prove that coefficients of $a^m$ and $a^n$ are equal

As we know that the general   term   in the binomial expansion of    is given by  So, the general  term   in the binomial expansion of    is given by  Now, as we can see  will come when  and  will come when  So,  Coefficient of  : CoeficientCoefficient of  : As we can see . Hence it is proved that the coefficients of  and  are equal.

Find the middle terms in the expansion of

Q8.    $\left(\frac{x}{3} + 9y \right )^{10}$

As we know that the middle term in the expansion of   when n is even is, , Hence the middle term of the expansion        is, Which is  Now,  As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of    is Hence the middle term of the expansion of   is  .

Find the middle terms in the expansion of

Q7.    $\left(3 - \frac{x^3}{6} \right )^7$

As we know that the middle  terms in the expansion of   when n is odd are, Hence the middle term of the expansion       are  Which are  Now,  As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of    is And the  Term of the expansion of     is, Hence the middle terms of the expansion of given expression are

Q6.    Find the 13th term in the expansion of    $\left(9x - \frac{1}{3\sqrt x} \right )^{18},\ x\neq 0$

As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of        is

Find the 4th term in the expansion of  $(x-2y)^{12}$.

As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of  is .

Write the general term in the expansion of

Q4.    $(x^2 - xy)^{12}, \ x\neq 0$

As we know that the general   term   in the binomial expansion of    is given by  So the general term of the expansion of  is .

Write the general term in the expansion of

Q3.    $(x^2 - y)^6$

As we know that the general   term   in the binomial expansion of    is given by  So the general term of the expansion of   : .

Find the coefficient of

Q2.    $a^5b^7$  in $(a- 2b)^{12}$

As we know that the  term   in the binomial expansion of    is given by  Now let's assume  happens in the  term of the binomial expansion of  So, On comparing the indices of x we get, Hence the coefficient of the    in  is

Find the coefficient of

Q1.    $x^5$ in $(x + 3)^8$

As we know that the  term   in the binomial expansion of    is given by  Now let's assume  happens in the  term of the binomial expansion of  So, On comparing the indices of x we get, Hence the coefficient of the   in  is
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