Advertisements
Advertisements
प्रश्न
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
Advertisements
उत्तर
(i) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9 \]
\[ T_{r + 1} =^{9}{}{C}_r \left( \frac{3}{2} x^2 \right)^{9 - r} \left( \frac{- 1}{3x} \right)^r \]
\[ = ( - 1 )^r {9}{}{C}_r . \frac{3^{9 - 2r}}{2^{9 - r}} \times x^{18 - 2r - r} \]
\[\text{ For this term to be independent of x, we must have} \]
\[18 - 3r = 0\]
\[ \Rightarrow 3r = 18\]
\[ \Rightarrow r = 6\]
\[\text{ Hence, the required term is the 7th term } . \]
\[\text{ Now, we have } \]
\[ ^{9}{}{C}_6 \times \frac{3^{9 - 12}}{2^{9 - 6}}\]
\[ = \frac{9 \times 8 \times 7}{3 \times 2} \times 3^{- 3} \times 2^{- 3} \]
\[ = \frac{7}{18}\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – y)6
In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal.
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:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
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:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms(s) in the expansion of:
(i) \[\left( x - \frac{1}{x} \right)^{10}\]
Find the middle terms(s) in the expansion of:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
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:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find 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.
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find n.
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 middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
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
If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] , \[x^{- 17}\] occurs in rth term, then
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
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 term independent of x in the expansion of `(3x - 2/x^2)^15`
Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
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`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The last two digits of the numbers 3400 are 01.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 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 ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 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 ______.
