Advertisements
Advertisements
Question
Find the coefficient of:
(ii) x7 in the expansion of \[\left( x - \frac{1}{x^2} \right)^{40}\]
Solution
(ii) Suppose x7 occurs at the (r + 1) th term in the given expression.
Then, we have:
\[T_{r + 1} = ^{40}{}{C}_r x^{40 - r} \left( \frac{- 1}{x^2} \right)^r \]
`= (-1)^r "^40C_r x^(40-r-2r)`
\[\text{ For this term to contain } x^7 , \text{ we must have: } \]
\[40 - 3r = 7\]
\[ \Rightarrow 3r = 40 - 7 = 33\]
\[ \Rightarrow r = 11\]
\[ \therefore \text{ Coefficient of } x^7 = ( - 1 )^{11} \]` "^40C_11 = "^-40 C_11`
APPEARS IN
RELATED QUESTIONS
Using binomial theorem, write down the expansions :
(ii) \[\left( 2x - 3y \right)^4\]
Using binomial theorem, write down the expansions :
(iv) \[\left( 1 - 3x \right)^7\]
Using binomial theorem, write down the expansions :
(v) \[\left( ax - \frac{b}{x} \right)^6\]
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 :
(x) \[\left( 1 - 2x + 3 x^2 \right)^3\]
Evaluate the
(i)\[\left( \sqrt{x + 1} + \sqrt{x - 1} \right)^6 + \left( \sqrt{x + 1} - \sqrt{x - 1} \right)^6\]
Evaluate the
(iv) \[\left( \sqrt{2} + 1 \right)^6 + \left( \sqrt{2} - 1 \right)^6\]
Evaluate the
(vi) \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]
Evaluate the
(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]
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 \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .
Using binomial theorem evaluate :
(i) (96)3
Using binomial theorem evaluate .
(ii) (102)5
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:
(iii) \[x^{- 15}\] in the expansion of \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]
Find the coefficient of:
(v) \[x^m\] in the expansion of \[\left( x + \frac{1}{x} \right)^n\]
Find the coefficient of:
(vi) x in the expansion of \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
Does the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)\] contain any term involving x9?
Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]
If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
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 =
The coefficient of x5 in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
