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.

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.

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.

z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.

z1 + z2 = (2 + 2i) + (3 - i) = 5 + i.

Arranging the ages: 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 20. The median is the 8th number, which is 16.

Probability = Number of girls / Total students = 12/30 = 2/5.

The number 60 appears most frequently (3 times), so the mode is 60.

Arrange the ages: 16, 17, 18, 19, 20. The median is the middle value, which is 18.

The mode is 55, as it appears three times in the data set.

The mode is 1, which appears 3 times in the data set.

The mode is B, as it appears most frequently (3 times) in the data set.

For an even number of observations, the median is the average of the two middle numbers, which are the 5th and 6th numbers.

Replacing 5 with 7 removes the most frequent number, so the new mode is 7.

The mode is the number that appears most frequently. Here, 5 appears 3 times, which is more than any other number.

The number 5 appears most frequently (3 times), so the mode is 5.

Arrange the numbers: 2, 3, 4, 5, 6, 7, 8. The median is the middle number, which is 5.

The mode is the number that appears most frequently. Here, 7 appears 3 times, which is more than any other number.

The mode is the number that appears most frequently. Here, 7 appears three times, which is more than any other number.

The mode is the number that appears most frequently. Here, 7 appears three times, which is more than any other number.

The only scenario where there are no girls is if all three children are boys, which has a probability of (1/2)^3 = 1/8. Therefore, the probability of having at least one girl is 1 - 1/8 = 7/8.

Both 2 and 3 appear 3 times, but since 2 is the smallest, the mode is 2.