Advertisements
Advertisements
प्रश्न
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
Advertisements
उत्तर
Given expression is (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
Let us consider `(3/2 x^2 - 1/(3x))^9`
General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
`"T"_(r + 1) = ""^9"C"_r (3/2 x^2)^(9 - r) (- 1/(3x))^r`
= `""^9"C"_r (3/2)^(9 - r) (x)^(18 - 2r) * (- 1/3)^r * 1/(x)^r`
= `""^9"C"_r (3/2)^(9 - r) (x)^(18 - 2r - r) * (- 1/3)^r`
= `""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * x^(18 - 3r)`
So, the general term in the expansion of
`(1 + x + 2x^3) (3/2 x^2 - 1/(3x))^9`
= `""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(18 - 3r) + ""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(19 - 3r) + 2 * ""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(21 - 3r)`
For getting the term independent of x,
Put 18 – 3r = 0, 19 – 3r = 0 and 21 – 3r = 0, we get
r = 6
r = `19/3` and r = 7
The possible value of r are 6 and 7 ```.....(because r ≠ 19/3)`
∴ The term independent of x is
= `""^9"C"_6 (3/2)^(9 - 6) (- 1/3)^6 + 2 * ""^9"C"_7 (3/2)^(9 - 7) (- 1/3)^7`
= `(9 xx 8 xx 7 xx 6!)/(3 xx 2 xx 1 xx 6!) * 3^3/2^3 * 1/3^6 - 2 * (9 xx 8 xx 7!)/(7!2 xx 1) * 3^2/2^2 * 1/3^7`
= `84/8 * 1/3^3 - 36/4 * 2/3^5`
= `7/18 - 2/27`
= `(21 - 4)/54`
= `17/54`
Hence, the required term = `17/54`
APPEARS IN
संबंधित प्रश्न
Find the coefficient of x5 in (x + 3)8
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:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
Find the middle term in the expansion of:
(iv) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
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:
(iii) \[\left( 1 + 3x + 3 x^2 + x^3 \right)^{2n}\]
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:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
Find the term independent of x in the expansion of the expression:
(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]
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 the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find n.
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, 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.
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.
If p is a real number and if the middle term in the expansion of \[\left( \frac{p}{2} + 2 \right)^8\] is 1120, find p.
Write the middle term in the expansion of `((2x^2)/3 + 3/(2x)^2)^10`.
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
If an the expansion of \[\left( 1 + x \right)^{15}\] , the coefficients of \[\left( 2r + 3 \right)^{th}\text{ and } \left( r - 1 \right)^{th}\] terms are equal, then the value of r is
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
If rth term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\] then \[\left( r + 3 \right)^{th}\] term is
Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
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`
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.
If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by ______.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` is ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 is ______.
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 ______.
