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 13th term in the expansion of `(9x - 1/(3sqrtx))^18 , x != 0`
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of `(root4 2 + 1/ root4 3)^n " is " sqrt6 : 1`
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:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
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 middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\] is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
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 A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] is
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\] , the term independent of x is
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] after simplification is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
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 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 ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` is ______.
If the 4th term in the expansion of `(ax + 1/x)^n` is `5/2` then the values of a and n respectively are ______.
