PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].

If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].

For any two complex numbers z₁ and z₂ and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].

If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .

Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.

If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.

Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].

Disclaimer: There is a misprinting in the question. It should be  \[\left( 1 + i\sqrt{3} \right)\]  instead of \[\left( 1 + \sqrt{3} \right)\].

The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.

If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =

If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then

The polar form of (i²⁵)³ is

If i² = −1, then the sum i + i² + i³ +... upto 1000 terms is equal to

If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is

If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]

The principal value of the amplitude of (1 + i) is

The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.

If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to

If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]

\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to

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