Advertisements
Advertisements
Question
Evaluate the
(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]
Advertisements
Solution
(iii) \[(1 + 2\sqrt{x} )^5 + (1 - 2\sqrt{x} )^5 \]
\[ = 2[ ^{5}{}{C}_0 (2\sqrt{x} )^0 +^{5}{}{C}_2 (2\sqrt{x} )^2 + ^{5}{}{C}_4 (2\sqrt{x} )^4 ]\]
\[ = 2[1 + 10 \times 4x + 5 \times 16 x^2 ]\]
\[ = 2[1 + 40x + 80 x^2 ]\]
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 :
(vi) \[\left( \frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}} \right)^6\]
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
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Evaluate the
(viii) \[\left( 0 . 99 \right)^5 + \left( 1 . 01 \right)^5\]
Evaluate the
(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]
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:
(iii) \[x^{- 15}\] in the expansion of \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]
Find the coefficient of:
(iv) \[x^9\] in the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\]
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 \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
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?
Does the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)\] contain any term involving x9?
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 coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
The coefficient of \[\frac{1}{x}\] in the expansion of \[\left( 1 + x \right)^n \left( 1 + \frac{1}{x} \right)^n\] 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 \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
The coefficient of x8 y10 in the expansion of (x + y)18 is
If the coefficients of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is
