Advertisements
Advertisements
प्रश्न
Find \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .
Advertisements
उत्तर
The expression \[(a + b )^4 - (a - b )^4\] can be written as
\[(a + b )^4 - (a - b )^4 = 2[ ^{4}{}{C}_1 a^3 b^1 + ^{4}{}{C}_3 a^1 b^3 ] \]
\[ = 2[4 a^3 b + 4a b^3 ]\]
\[ = 8( a^3 b + a b^3 )\]
\[\text{ Putting a } = \sqrt{3} \text{ and } b = \sqrt{2}, \text{ we get } : \]
\[ (\sqrt{3} + \sqrt{2} )^4 - (\sqrt{3} - \sqrt{2} )^4 = 8[(\sqrt{3} )^3 \times \sqrt{2} + \sqrt{3} \times (\sqrt{2} )^3 ]\]
\[ = 8(3\sqrt{6} + 2\sqrt{6})\]
\[ = 40\sqrt{6}\]
\[\therefore (\sqrt{3} + \sqrt{2} )^4 - (\sqrt{3} - \sqrt{2} )^4 = 40\sqrt{6}\]
APPEARS IN
संबंधित प्रश्न
Using binomial theorem, write down the expansions .
(i) \[\left( 2x + 3y \right)^5\]
Using binomial theorem, write down the expansions .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
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) \[\left( x + 1 - \frac{1}{x} \right)\]
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
(v) \[\left( 3 + \sqrt{2} \right)^5 - \left( 3 - \sqrt{2} \right)^5\]
Evaluate the
(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
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
Find the coefficient of:
(vi) x in the expansion of \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
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?
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 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
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 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
