Advertisements
Advertisements
Question
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
Advertisements
Solution
General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^15"C"_r ((3x^2)/2)^(15 - r) (- 1/(3x))^r`
= `""^15"C"_r (3/2)^(15 - r) * (x)^(30 - 2r) * (- 1/3)^r * 1/x^r`
= `""^15"C"_r (3/2)^(15 - r) * (x)^(30 - 2r - r) (-1)^r * 1/(3)^r`
= `""^15"C"_r (3/2)^(15 - r) * x^(30 - 3r) (-1)^r * 1/(3)^r`
For getting the term independent of x
30 – 3r = 0
⇒ r = 10
On putting the value of r in the above expression, we get
= `""^15"C"_10 (3/2)^(15 - 10) (-1)^10 * 1/(3)^10`
= `""^15"C"_10 (3)^5/(2)^5 * 1/(3)^10`
= `""^15"C"_10 * 1/((2)^5 * (3)^5)`
= `""^15"C"_10 (1/6)^5`
Hence, the required term = `""^15"C"_10 (1/16)^5`
APPEARS IN
RELATED QUESTIONS
Find the coefficient of a5b7 in (a – 2b)12
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the 4th term in the expansion of (x – 2y)12 .
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
Find the middle term in the expansion of:
(i) \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]
Find the middle term in the expansion of:
(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]
Find the middle terms in the expansion of:
(iv) \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]
Find the middle terms(s) in the expansion of:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
Find the middle terms(s) in the expansion of:
(vii) \[\left( 3 - \frac{x^3}{6} \right)^7\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the middle terms(s) in the expansion of:
(x) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Find the term independent of x in the expansion of the expression:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
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.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in A.P., show that \[2 n^2 - 9n + 7 = 0\]
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.
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.
If the term free from x in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] is 405, find the value of k.
Write the middle term in the expansion of `((2x^2)/3 + 3/(2x)^2)^10`.
Find the sum of the coefficients of two middle terms in the binomial expansion of \[\left( 1 + x \right)^{2n - 1}\]
If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
Let for the 9th term in the binomial expansion of (3 + 6x)n, in the increasing powers of 6x, to be the greatest for x = `3/2`, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k + n0 is equal to ______.
If the coefficient of x10 in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.
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 ______.
