Complex Numbers and Quadratic Equations — Practice

Question 1

Evaluate i^33.

Accepted Answer

33 = 4(8) + 1, so i^33 = i^1 = i. Answer: i^33 = i.

Question 2

Express (2+3i) + (4-5i) in the form a+ib.

Accepted Answer

Adding real and imaginary parts separately: (2+4) + i(3-5) = 6 - 2i. Answer: The result is 6 - 2i.

Question 3

Evaluate sqrt(-16) . sqrt(-25), showing why the naive sqrt(a).sqrt(b)=sqrt(ab) rule cannot be used directly.

Accepted Answer

sqrt(-16) = 4i and sqrt(-25) = 5i, so the product is (4i)(5i) = 20i^2 = -20. Treating it as sqrt((-16)(-25)) = sqrt(400) = 20 gives the wrong sign, since both numbers under the root are negative. Answer: sqrt(-16).sqrt(-25) = -20.

Question 4

Express (2+i)/(3-4i) in the form a+ib.

Accepted Answer

Multiply numerator and denominator by the conjugate (3+4i): (2+i)(3+4i) / [(3-4i)(3+4i)] = (6+8i+3i+4i^2) / (9+16) = (6+11i-4)/25 = (2+11i)/25. Answer: The result is 2/25 + 11i/25.

Question 5

Find the modulus and conjugate of z = -5 + 12i.

Accepted Answer

Modulus: |z| = sqrt((-5)^2+12^2) = sqrt(25+144) = sqrt(169) = 13. Conjugate: z-bar = -5 - 12i. Answer: |z| = 13 and z-bar = -5-12i.

Question 6

Find the multiplicative inverse of 3 + 4i.

Accepted Answer

z-bar = 3-4i, |z|^2 = 9+16 = 25. z^(-1) = (3-4i)/25 = 3/25 - 4i/25. Answer: The multiplicative inverse is 3/25 - 4i/25.

Question 7

If z1 = 1+2i and z2 = 3-i, verify that the conjugate of z1 z2 equals (conjugate of z1)(conjugate of z2).

Accepted Answer

z1 z2 = (1+2i)(3-i) = 3-i+6i-2i^2 = 3+5i+2 = 5+5i, so conjugate(z1 z2) = 5-5i. Separately: conjugate(z1)=1-2i, conjugate(z2)=3+i, and their product is (1-2i)(3+i) = 3+i-6i-2i^2 = 3-5i+2 = 5-5i. Both routes give 5-5i. Answer: conjugate(z1 z2) = conjugate(z1).conjugate(z2) = 5 - 5i, confirmed by both methods.

Question 8

Plot z = 3-4i on the Argand plane and find its distance from the point representing z-bar.

Accepted Answer

z=3-4i is the point (3,-4); z-bar=3+4i is the point (3,4), its mirror image across the real axis. The distance between (3,-4) and (3,4) is the vertical difference: |4-(-4)| = 8. Answer: The distance between z and its conjugate's point is 8.

Question 9

Find the value of i + i^2 + i^3 + ... + i^40.

Accepted Answer

Every consecutive block of 4 powers sums to zero: i+i^2+i^3+i^4 = i-1-i+1 = 0. Since 40 is a multiple of 4, the sum splits into exactly 10 such blocks, each contributing 0. Answer: i + i^2 + ... + i^40 = 0.

Question 10

If (x+iy)(2-3i) = 4+i, find the real values of x and y.

Accepted Answer

Expanding: (x+iy)(2-3i) = 2x-3xi+2yi-3yi^2 = (2x+3y) + i(2y-3x). Equating with 4+i: 2x+3y=4 and 2y-3x=1. Solving simultaneously (multiply the first by 2 and the second by 3, then subtract) gives x=5/13, y=14/13. Answer: x = 5/13 and y = 14/13.

Practice — Complex Numbers and Quadratic Equations

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities