मराठी

If a and B Denote the Sum of the Coefficients in the Expansions of ( 1 − 3 X + 10 X 2 ) N and ( 1 + X 2 ) N Respectively, Then Write the Relation Between a and B.

Advertisements
Advertisements

प्रश्न

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.

 
 
 
Advertisements

उत्तर

\[Here, \]
\[a = 1 - 3 + 10 = 8 = 2^3 \]
\[b = 1 + 1 = 2\]
\[ \Rightarrow a = b^3\]

shaalaa.com
Introduction of Binomial Theorem
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 18: Binomial Theorem - Exercise 18.3 [पृष्ठ ४५]

APPEARS IN

आर.डी. शर्मा Mathematics [English] Class 11
पाठ 18 Binomial Theorem
Exercise 18.3 | Q 9 | पृष्ठ ४५

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  :

(vi) \[\left( \frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}} \right)^6\]

 


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)  \[\left( 1 + 2x - 3 x^2 \right)^5\]

 


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

(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

(vi)  \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]


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, 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: 

(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: 

(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?

 

 


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.

 
 

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 x8 y10 in the expansion of (x + y)18 is


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×