COMPLEX NUMBERS AND QUADRATIC EQUATIONS
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a + ib and their representation in a plane, Argand diagram, algebra of complex number, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions Relations between roots and coefficient, nature of roots, the formation of quadratic equations with given roots.
- The least value of |z| where z is complex number which satisfies the inequality Answer: (1) 3 Read the inequality as exp((|z|+3)(|z|-1)/|z|+1log_e 2) ≥ log_(sqrt2)|5sqrt7 + 9i| . Right side: |5sqrt7 + 9i| = √(175 + 81) =... 1
- The area of the triangle with vertices A(z), B(iz) and C (z + iz) is : Answer: (2) 1/2|z|² Look at the sides meeting at C: overrightarrowCA = z - (z + iz) = -iz , length |z| . overrightarrowCB = iz - (z + iz) =... 1
- Let S1, S2 and S3 be three sets defined as S1 = {z ∈ C : |z - 1| ≤ √2} Answer: (3) Has infinitely many elements Let z = x + iy . S₁ : |z-1| ≤ sqrt2 is the closed disc with centre (1, 0) and radius sqrt2 . S₂ : (... 1
- If the equation a|z|^2 + bar (bar α z + α bar z) + d = 0 represents a circle Have a look in image for answer 2
- Let a complex number be w = 1 – √3i . Let another complex number z be such that |zw| = 1 and arg(z) – arg(w) Correct Answer is (2) 1/2 Explaination:: Image:: 1