Advertisements
Advertisements
Question
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Advertisements
Solution
(iii) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( 2 x^2 - \frac{3}{x^3} \right)^{25} \]
\[ T_{r + 1} = ^{25}{}{C}_r (2 x^2 )^{25 - r} \left( \frac{- 3}{x^3} \right)^r \]
\[ = ( - 1 )^r {25}{}{C}_r \times 2^{25 - r} \times 3^r x^{50 - 2r - 3r} \]
\[\text{ For this term to be independent of x, we must have} : \]
\[50 - 5r = 0\]
\[ \Rightarrow r = 10\]
\[\text{ Therefore, the required term is the 11th term .} \]
\[\text{ Now, we have } \]
`( - 1 )^{10} "^25C_{10} \times 2^{25 - 10} \times 3^{10} `
` = "^25C_{10 ( 2^{15} \times 3^{10} )`
APPEARS IN
RELATED QUESTIONS
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find the middle terms in the expansions of `(x/3 + 9y)^10`
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1 .
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
Find the middle terms in the expansion of:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the middle terms(s) in the expansion of:
(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]
Find the term independent of x in the expansion of the expression:
(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
If in the expansion of (1 + x)n, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where \[p \neq q\]
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Find a, b and n in the expansion of (a + b)n, if the first three terms in the expansion are 729, 7290 and 30375 respectively.
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
Find n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7th term from the beginning to the 7th term from the end is `1/6`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` is ______.
Let the coefficients of the middle terms in the expansion of `(1/sqrt(6) + βx)^4, (1 - 3βx)^2` and `(1 - β/2x)^6, β > 0`, common difference of this A.P., then `50 - (2d)/β^2` is equal to ______.
The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to ______.
The sum of the real values of x for which the middle term in the binomial expansion of `(x^3/3 + 3/x)^8` equals 5670 is ______.
