Complex Numbers and Quadratic Equations — Practice

Question 1

Evaluate: i⁹⁷ + i⁹⁸ + i⁹⁹ + i¹⁰⁰.

Accepted Answer

97 = 4×24+1, so i⁹⁷ = i¹ = i. 98 = 4×24+2, so i⁹⁸ = i² = −1. 99 = 4×24+3, so i⁹⁹ = i³ = −i. 100 = 4×25, so i¹⁰⁰ = i⁰ = 1. Sum = i+(−1)+(−i)+1 = 0.

Question 2

Express (3+4i)/(1−2i) in the form a+ib and find its modulus.

Accepted Answer

Multiply by conjugate: (3+4i)(1+2i)/[(1−2i)(1+2i)] = (3+6i+4i+8i²)/(1+4) = (3+10i−8)/5 = (−5+10i)/5 = −1+2i. Modulus = √((−1)²+2²) = √(1+4) = √5.

Question 3

Solve: x² − 2x + 5 = 0 and verify that the roots are complex conjugates.

Accepted Answer

Discriminant = (−2)²−4(1)(5) = 4−20 = −16 < 0. So x = (2 ± √(−16))/2 = (2 ± 4i)/2 = 1 ± 2i. Roots: α = 1+2i and β = 1−2i. Verification: α and β are complex conjugates (same real part, opposite imaginary parts). Sum = (1+2i)+(1−2i) = 2 = −(−2)/1 ✓. Product = (1+2i)(1−2i) = 1+4 = 5 = 5/1 ✓.

Practice — Complex Numbers and Quadratic Equations

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