Circumference = 2πr. If radius is halved, new circumference = 2π(r/2) = πr, which is halved.

Area increases by a factor of (3r)²/r² = 9.

Max revenue occurs at x = -b/(2a) = 100/(2*2) = 25.

Using the sum and product of roots: 1 + (-3) = -2 and 1 * (-3) = -3, thus k = 3.

The discriminant must be greater than zero: 2² - 4*1*k > 0, which simplifies to k < 1.

Using the product of roots: k = (-2)(-3) = 6.

Using the sum and product of roots: k = 1*4 = 4, and sum = 1 + 4 = 5, thus k = 7.

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find p = 12.

Using the sum of the roots: p = -(3 + 4) = -7.

The possible outcomes are HH, HT, TH, TT. The outcomes with at least one tail are HT, TH, TT (3 outcomes). So, the probability is 3/4.

Possible combinations for sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 combinations. Total outcomes = 36. Probability = 6/36 = 1/6.

Substituting x = 5, we get 2(5) + 3 = 10 + 3 = 13.

|z| = √((-2)^2 + (2)^2) = √(4 + 4) = √8 = 2√2.

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

z² = (1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.

|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.

The conjugate of z = 2 - 3i is 2 + 3i.

The imaginary part of the complex number z = 3 + 4i is 4.

z + z* = (3 + 4i) + (3 - 4i) = 6.

z + z̅ = (3 + 4i) + (3 - 4i) = 6.

The real part of the complex number z = 4 + 3i is 4.

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

The determinant of \( B \) is \( 2*4 - 3*1 = 8 - 3 = 5 \).

Number of boys = 30 - 18 = 12. Probability = 12/30 = 2/5.

Number of non-smokers = 50 - 20 = 30. Probability = 30/50 = 3/5.

There are 50 - 20 = 30 non-smokers. The probability is 30/50 = 3/5.

Probability = Number of winning tickets / Total tickets = 2/10 = 1/5.

Probability = Number of winning tickets / Total tickets = 3/10.

The number of ways to award 3 prizes to 5 students is 5P3 = 60.

The number of arrangements of 4 distinct flags is 4! = 24.