JEEAdvanced
For a complex number z with |z| = 2, find the maximum value of the…
Question
For a complex number z with |z| = 2, find the maximum value of the product |z^2 - z + 1| * |z^2 + z + 1|. Report the exact integer value.
✓ Verified answer: 21checked by our engine — not a guess
Step-by-step solution
Since (z^2 - z + 1)(z^2 + z + 1) = z^4 + z^2 + 1, we must maximize |z^4 + z^2 + 1| on |z| = 2.
Write u = z^2, so |u| = 4, and we maximize |u^2 + u + 1| with |u| = 4.
By the triangle inequality |u^2 + u + 1| <= |u|^2 + |u| + 1 = 16 + 4 + 1 = 21, with equality when u^2, u, 1 are positive real multiples of each other, i.e.
when u is a positive real number, u = 4 (z = +/-2).
Then z^4 + z^2 + 1 = 16 + 4 + 1 = 21.
Hence the maximum is 21.
Final answer21
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Complex Numbers solutions
Let w_1, w_2, ..., w_6 denote the six seventh roots of unity other…JEE · MathA complex number z satisfies |z| = 1. Determine the maximum possible…JEE · MathCompute the product (2 - 2cos(2*pi*1/13)) * (2 - 2cos(2*pi*2/13)) *…JEE · MathLet w = e^{2*pi*i/10} be a primitive 10th root of unity. Evaluate the…JEE · MathConsider the thirty 30th roots of unity z = e^{2*pi*i*k/30} for k =…JEE · MathThree complex numbers z_1, z_2, z_3 are the vertices of an…JEE · MathLet w = e^{i*pi/3} = cos(60 degrees) + i sin(60 degrees). Evaluate…JEE · MathLet w_0, w_1, ..., w_6 be the seven 7th roots of unity (the roots of…JEE · Math