Advertisements
Advertisements
प्रश्न
Find the coefficient of:
(vi) x in the expansion of \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
Advertisements
उत्तर
Suppose x occurs at the (r + 1)th term in the given expression.
Then, we have:
\[(1 - 2 x^3 + 3 x^5 ) \left( 1 + \frac{1}{x} \right)^8 \]
\[ = \left( 1 - 2 x^3 + 3 x^5 \right)\left( ^{8}{}{C}_0 + ^{8}{}{C}_1 \left( \frac{1}{x} \right) + ^{8}{}{C}_2 \left( \frac{1}{x} \right)^2 + ^{8}{}{C}_3 \left( \frac{1}{x} \right)^3 +^{8}{}{C}_4 \left( \frac{1}{x} \right)^4 + ^{8}{}{C}_5 \left( \frac{1}{x} \right)^5 + ^{8}{}{C}_6 \left( \frac{1}{x} \right)^6 +^{8}{}{C}_7 \left( \frac{1}{x} \right)^7 +^{8}{}{C}_8 \left( \frac{1}{x} \right)^8 \right)\]
\[ \text{ x occurs in the above expresssion at } - 2 x^3 . ^{8}{}{C}_2 \left( \frac{1}{x^2} \right) + 3 x^5 . ^{8}{}{C}_4 \left( \frac{1}{x} \right)^4 . \]
\[ \therefore \text{ Coefficient of x } = - 2\left( \frac{8!}{2! 6!} \right) + 3\left( \frac{8!}{4! 4!} \right) = - 56 + 210 = 154\]
APPEARS IN
संबंधित प्रश्न
Using binomial theorem, write down the expansions :
(iii) \[\left( x - \frac{1}{x} \right)^6\]
\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]
\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
\[ = 32 x^5 + 240 x^4 y + 720 x^3 y^2 + 1080 x^2 y^3 + 810x y^4 + 243 y^5 \]
Using binomial theorem, write down the expansions .
(i) \[\left( 2x + 3y \right)^5\]
Using binomial theorem, write down the expansions .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
Using binomial theorem, write down the expansions :
(iv) \[\left( 1 - 3x \right)^7\]
Using binomial theorem, write down the expansions :
(vi) \[\left( \frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}} \right)^6\]
Using binomial theorem, write down the expansions :
(vii) \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]
Using binomial theorem, write down the expansions :
(viii) \[\left( 1 + 2x - 3 x^2 \right)^5\]
Using binomial theorem, write down the expansions :
(ix) \[\left( x + 1 - \frac{1}{x} \right)\]
Using binomial theorem, write down the expansions :
(x) \[\left( 1 - 2x + 3 x^2 \right)^3\]
Evaluate the
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Evaluate the
(v) \[\left( 3 + \sqrt{2} \right)^5 - \left( 3 - \sqrt{2} \right)^5\]
Evaluate the
(vi) \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]
Evaluate the
(viii) \[\left( 0 . 99 \right)^5 + \left( 1 . 01 \right)^5\]
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
Find \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .
Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .
Using binomial theorem evaluate :
(i) (96)3
Using binomial theorem evaluate .
(iii) (101)4
Using binomial theorem evaluate .
(iv) (98)5
Using binomial theorem, prove that \[3^{2n + 2} - 8n - 9\] is divisible by 64, \[n \in N\] .
Find the coefficient of:
(ii) x7 in the expansion of \[\left( x - \frac{1}{x^2} \right)^{40}\]
Find the coefficient of:
(iv) \[x^9\] in the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\]
If a and b denote respectively the coefficients of xm and xn in the expansion of \[\left( 1 + x \right)^{m + n}\], then write the relation between a and b.
If a and b denote the sum of the coefficients in the expansions of \[\left( 1 - 3x + 10 x^2 \right)^n\] and \[\left( 1 + x^2 \right)^n\] respectively, then write the relation between a and b.
The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is
The coefficient of x4 in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] is
If \[T_2 / T_3\] in the expansion of \[\left( a + b \right)^n \text{ and } T_3 / T_4\] in the expansion of \[\left( a + b \right)^{n + 3}\] are equal, then n =
If the sum of the binomial coefficients of the expansion \[\left( 2x + \frac{1}{x} \right)^n\] is equal to 256, then the term independent of x is
The coefficient of x5 in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
The coefficient of x8 y10 in the expansion of (x + y)18 is
