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

Write −1 + i \[\sqrt{3}\] in polar form .

Write the argument of −i.

Write the least positive integral value of n for which  \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.

Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .

Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]

If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.

If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of  \[x^2 + y^2\].

Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].

Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.

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