Advertisements
Advertisements
प्रश्न
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
Advertisements
उत्तर
\[(1 + 2a )^4 (2 - a )^5 \]
\[ = [ ^{4}{}{C}_0 (2a )^0 + ^{4}{}{C}_1 (2a )^1 +^{4}{}{C}_2 (2a )^2 + ^{4}{}{C}_3 (2a )^3 +^{4}{}{C}_4 (2a )^4 ] \times \]
\[ [ ^{5}{}{C}_0 (2 )^5 ( - a )^0 +^{5}{}{C}_1 (2 )^4 ( - a )^1 + ^{5}{}{C}_2 (2 )^3 ( - a )^2 + ^{5}{}{C}_3 (2 )^2 ( - a )^3 + ^{5}{}{C}_4 (2 )^1 ( - a )^4 + ^{5}{}{C}_5 (2 )^0 ( - a )^5 ]\]
\[ = [1 + 8a + 24 a^2 + 32 a^3 + 16 a^4 ] \times [32 - 80a + 80 a^2 - 40 a^3 + 10 a^4 - a^5 ]\]
\[\text{ Coefficient of } a^4 = 10 - 320 + 1920 - 2560 + 512 = - 438\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – y)6
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 the middle terms in the expansions of `(x/3 + 9y)^10`
In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal.
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(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 middle terms(s) in the expansion of:
(ix) \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]
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:
(ii) \[\left( 2x + \frac{1}{3 x^2} \right)^9\]
Find the term independent of x in the expansion of the expression:
(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
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 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
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 \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find 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 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
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
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] 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
Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
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.
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
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)!)`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
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 ______.
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 ______.
