PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

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 

(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

(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

(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]

Evaluate the

(viii)  \[\left( 0 . 99 \right)^5 + \left( 1 . 01 \right)^5\]

Evaluate the

(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]

Evaluate the

(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]

Find  \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .

Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .

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 .

Using binomial theorem, prove that \[2^{3n} - 7n - 1\] is divisible by 49, where \[n \in N\] .