Advertisements
Advertisements
प्रश्न
Find the coefficient of:
(iii) \[x^{- 15}\] in the expansion of \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]
उत्तर
(iii) Suppose x−15 occurs at the (r + 1)th term in the given expression.
Then, we have:
\[\Rightarrow T_{r + 1} = ( - 1 )^r {10}{}{C}_r \left( 3^{10 - r - r} \right)\left( x^{20 - 2r - 3r} \right)\left( a^r \right)\]
\[\text{ For this term to contain } x^{- 15} ,\text{ we must have} : \]
\[20 - 5r = - 15\]
\[ \Rightarrow 5r = 20 + 15\]
\[ \Rightarrow r = 7\]
\[ \therefore \text{ Coefficient of } x^{- 15} = ( - 1 )^7 {10}{}{C}_7 3^{10 - 14} \left( a^7 \right) = - \frac{10 \times 9 \times 8}{3 \times 2 \times 9 \times 9} a^7 = - \frac{40}{27} a^7\]
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 :
(iv) \[\left( 1 - 3x \right)^7\]
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) \[\left( x + 1 - \frac{1}{x} \right)\]
Evaluate the
(i)\[\left( \sqrt{x + 1} + \sqrt{x - 1} \right)^6 + \left( \sqrt{x + 1} - \sqrt{x - 1} \right)^6\]
Evaluate the
(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]
Evaluate the
(iv) \[\left( \sqrt{2} + 1 \right)^6 + \left( \sqrt{2} - 1 \right)^6\]
Evaluate the
(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]
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 \[2^{3n} - 7n - 1\] is divisible by 49, where \[n \in N\] .
Using binomial theorem, prove that \[3^{2n + 2} - 8n - 9\] is divisible by 64, \[n \in N\] .
Find the coefficient of:
(i) x10 in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]
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 \[\left( x^2 - \frac{1}{3x} \right)^9\]
Find the coefficient of:
(vii) \[a^5 b^7\] in the expansion of \[\left( a - 2b \right)^{12}\]
Find the coefficient of:
(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]
Which term in the expansion of \[\left\{ \left( \frac{x}{\sqrt{y}} \right)^{1/3} + \left( \frac{y}{x^{1/3}} \right)^{1/2} \right\}^{21}\] contains x and y to one and the same power?
Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]
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 are coefficients of xn in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, 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
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 \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
If the coefficients of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is
