A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.

x² + y² = (x + y)² - 2xy = 10² - 2*21 = 100 - 42 = 58.

|-3| = 3, so |x| + x = 3 - 3 = 0.

The absolute value |x| of -5 is 5.

1/x = 1/(1/2) = 2.

x² + y² = 3² + 4² = 9 + 16 = 25.

x² - y² = 4² - 5² = 16 - 25 = -9.

2x - 3 = 2(4) - 3 = 8 - 3 = 5.

3x - 2 = 3(4) - 2 = 12 - 2 = 10.

(x - 1)² = (5 - 1)² = 4² = 16.

Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.

The modulus of z = -1 + i is |z| = √((-1)² + 1²) = √(1 + 1) = √2. The square of the modulus is |z|² = 2.

The real part of a complex number z = a + bi is a. Here, the real part is -3.

|z|² = (1)² + (1)² = 1 + 1 = 2.

The argument of a complex number z = a + bi is given by θ = tan⁻¹(b/a). Here, θ = tan⁻¹(1/1) = π/4.

z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.

The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 4.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = 8.

Using Euler's formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

The modulus of z is given by |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

The conjugate of z = 4 - 3i is z* = 4 + 3i.

z = 4(cos(π/3) + i sin(π/3)) = 4(1/2 + i√3/2) = 2 + 2√3i.

The argument of z = 5 + 12i is θ = arctan(12/5).

z = 5(cos(π/3) + i sin(π/3)) = 5(1/2 + i√3/2) = 2.5 + 5i√3/2.