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Q10.    Find the expansion of $(3x^2 - 2ax +3a^2)^3$ using binomial theorem.

Given  By Binomial Theorem It can also be written as  Now, Again By Binomial Theorem, From (1) and (2) we get,

Q9.    Expand using Binomial Theorem $\left(1 + \frac{x}{2} - \frac{2}{x} \right )^4, \ x\neq 0$

Given the expression, Binomial expansion of this expression is  Now Applying Binomial Theorem again, And  Now, From (1), (2) and (3) we get,

Q8.    Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}} \right )^n$is $\sqrt6 :1$

Given, the expression  Fifth term from the beginning  is  And Fifth term from the end is, Now, As given in the question, So, From Here , From here, Hence the value of n is 10.

Q7. Find an approximation of (0.99)5 using the first three terms of its expansion.

As we can write 0.99 as 1-0.01, Hence the value of     is 0.951 approximately.

Q6.     Find the value of $\left(a^2 + \sqrt{a^2 -1} \right )^4 + \left(a^2 - \sqrt{a^2 -1} \right )^4$

First, lets simplify the expression  using binomial expansion, And Now, Now, Putting  we get,

Q5.    Evaluate $\left(\sqrt3 + \sqrt2 \right )^6 - \left(\sqrt{3} - \sqrt2 \right )^6$.

First let's simplify the expression  using binomial theorem, So, And  Now, Now, Putting  we get

Q4.    If a and b are distinct integers, prove that$a - b$ is a factor of $a^n - b^n$ , whenever n is a positive integer.
[Hint: write $a^n = (a - b + b)^n$ and expand]

we need to prove,    where k is some natural number. Now let's add and subtract b from a so that we can prove the above result, Hence, is a factor of .

Q3.    Find the coefficient of $x^5$ in the product $(1 + 2x)^6 (1 - x)^7$ using binomial theorem.

First, lets expand both expressions individually, So, And  Now, Now, for the coefficient of , we multiply and add those terms whose product gives .So, The term which contain are, Hence the coefficient of  is 171.

Q1.    Find a, b and n in the expansion of $(a + b)^n$ if the first three terms of the expansion are 729, 7290 and 30375, respectively.

As we know the Binomial expansion of  is given by  Given in the question, Now, dividing (1) by (2) we get, Now, Dividing (2) by (3) we get,  Now, From (4) and (5), we get,

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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