Advertisements
Advertisements
प्रश्न
Find the middle terms(s) in the expansion of:
(ix) \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]
Advertisements
उत्तर
\[\left( \frac{p}{x} + \frac{x}{p} \right)^9 \]
\[\text{ Here, n is an odd number } . \]
\[\text{ Therefore, the middle terms are } \left( \frac{9 + 1}{2} \right)^{th} \text{ and } \left( \frac{9 + 1}{2} + 1 \right)^{th} , i . e . , 5^{th} \text{ and } 6^{th}\text{ terms } . \]
\[\text{ Now, we have } \]
\[ T_5 = T_{4 + 1} \]
\[ = ^{9}{}{C}_4 \left( \frac{p}{x} \right)^{9 - 4} \left( \frac{x}{p} \right)^4 \]
\[ = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \times \left( \frac{p}{x} \right)\]
\[ = \frac{126 p}{x}\]
\[\text{ And,} \]
\[ T_6 = T_{5 + 1} \]
\[ =^{9}{}{C}_5 \left( \frac{p}{x} \right)^{9 - 5} \left( \frac{x}{p} \right)^5 \]
\[ = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \times \left( \frac{x}{p} \right)\]
\[ = \frac{126 x}{p}\]
APPEARS IN
संबंधित प्रश्न
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 the middle term in the expansion of:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
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:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
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:
(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) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
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 coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
Find the sum of the coefficients of two middle terms in the binomial expansion of \[\left( 1 + x \right)^{2n - 1}\]
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] 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
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find the middle term in the expansion of `(2ax - b/x^2)^12`.
Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^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.
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`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
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.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
The sum of the co-efficients of all even degree terms in x in the expansion of `(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6, (x > 1)` is equal to ______.
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 coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) is ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 is ______.
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 ______.
