Advertisements
Advertisements
प्रश्न
Using binomial theorem, write down the expansions .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
Advertisements
उत्तर
(iii) \[\left( x - \frac{1}{x} \right)^6 \]
\[ = ^{6}{}{C}_0 x^6 \left( \frac{1}{x} \right)^0 - ^{6}{}{C}_1 x^5 \left( \frac{1}{x} \right)^1 +^{6}{}{C}_2 x^4 \left( \frac{1}{x} \right)^2 - ^{6}{}{C}_3 x^3 \left( \frac{1}{x} \right)^3 + ^{6}{}{C}_4 x^2 \left( \frac{1}{x} \right)^4 -^6 C_5 x^1 \left( \frac{1}{x} \right)^5 + ^{6}{}{C}_6 x^0 \left( \frac{1}{x} \right)^6 \]
\[ = x^6 - 6 x^5 \times \frac{1}{x} + 15 x^4 \times \frac{1}{x^2} - 20 x^3 \times \frac{1}{x^3} + 15 x^2 \times \frac{1}{x^4} - 6 x \times \frac{1}{x^5} + \frac{1}{x^6}\]
\[ = x^6 - 6 x^4 + 15 x^2 - 20 + \frac{15}{x^2} - \frac{6}{x^4} + \frac{1}{x^6}\]
APPEARS IN
संबंधित प्रश्न
Using binomial theorem, write down the expansions .
(i) \[\left( 2x + 3y \right)^5\]
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 :
(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
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Evaluate the
(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]
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
(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 .
(iv) (98)5
Using binomial theorem, prove that \[3^{2n + 2} - 8n - 9\] is divisible by 64, \[n \in N\] .
Find the coefficient of:
(v) \[x^m\] in the expansion of \[\left( x + \frac{1}{x} \right)^n\]
Find the coefficient of:
(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]
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.
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 x4 in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] 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
