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

The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is 

The value of (i⁵ + i⁶ + i⁷ + i⁸ + i⁹) / (1 + i) is

\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals

The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is 

The value of \[(1 + i )^4 + (1 - i )^4\] is

If \[z = a + ib\]  lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if

If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is

A real value of x satisfies the equation  \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]

The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on

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)  \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]