A particle P starts from the point $z_0=1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from the origin by 5 units and then vertically away from the origin by 3 units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\frac{\pi }{2}$ in the anticlockwise direction on a circle with centre at the origin, to reach a point $z_2$. The point $z_2$ is given by

Complex Numbers — Preparation

This question tests your understanding of Complex Numbers, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A complex number has the form z = a + ib, where a and b are real numbers and i is the imaginary unit with i^2 = -1. This topic covers powers of i (which repeat in a cycle of four: i, -1, -i, 1), algebra of complex numbers, the conjugate, the modulus |z| = sqrt(a^2 + b^2), and the argument (amplitude). It also covers the polar and exponential forms of a complex number, De Moivre's theorem, the cube roots of unity (1, omega and omega^2, where 1 + omega + omega^2 = 0 and omega^3 = 1), square roots of complex numbers, and representing complex numbers in the Argand plane. Questions involving powers of omega are very common, and the two identities above solve most of them in a line or two. For the argument, always check the quadrant of the point before deciding the angle. Practise the complex numbers questions below, each with a step-by-step solution.