PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

If z is a complex number, then

Which of the following is correct for any two complex numbers z₁ and z₂?

If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on

Using binomial theorem, write down the expansions  :

(iii)  \[\left( x - \frac{1}{x} \right)^6\]

\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]

\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
\[ = 32 x^5 + 240 x^4 y + 720 x^3 y^2 + 1080 x^2 y^3 + 810x y^4 + 243 y^5 \]

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  .

(iii)  \[\left( x - \frac{1}{x} \right)^6\]

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  :

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

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