1. On the Argand diagram, z = 2 + 3i and w = (1 + i)z. Which Cartesian form gives w?
Expand w = (1 + i)(2 + 3i) = 2 + 3i + 2i + 3i2 = -1 + 5i.
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Practice functions, complex numbers, matrices, calculus, proof, and advanced problem solving.
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Expand w = (1 + i)(2 + 3i) = 2 + 3i + 2i + 3i2 = -1 + 5i.
Since b . a = 14 and a . a = 20, proj_a b = (14/20)a. Hence p = b - proj_a b = (-9/5, 18/5), with magnitude √(405)/5 = 9√5/5.
Use the A column: 120 x 0.70 + 80 x 0.40 = 84 + 32 = 116.
Factor the equation as (2cos x - 1)(cos x + 1) = 0. Thus cos x = 1/2 or cos x = -1. On 0 ≤ x < 2π, these occur at π/3, 5π/3, and π, giving three solutions.
Velocity remains non-negative on 0 ≤ t ≤ 6. The distance is the triangular area under the graph: 1/2 x 6 x 12 = 36 m.
The next sum adds the next cube to the assumed k-case. Therefore Sₖ₊₁ starts as [k(k + 1)/2]2 + (k + 1)3 before simplification.
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