The modulus is calculated as √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

The discriminant D = 2² - 4*1*5 = 4 - 20 = -16, which indicates complex roots.

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

To find inflection points, set f''(x) = 0. f''(x) = 6x - 12 = 0, x = 2. f(2) = 4.

Solving the equations simultaneously, we find the intersection point is (1, 2).

Solving the equations simultaneously, we find x = 2 and y = 3. Thus, the point of intersection is (2, 3).

Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting back, y = 2(1) + 1 = 3. The intersection is (1, 3).

Setting 3x + 1 = -x + 5 gives 4x = 4, thus x = 1. Substituting x back gives y = 4. The intersection point is (1, 4).

Number of kings = 4. Probability = 4/52 = 1/13.

Number of queens = 4. Probability = 4/52 = 1/13.

There are 2 outcomes: heads or tails. Probability of heads = 1/2.

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

9 × 6 = 54.

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

(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i.

The product of the roots is given by c/a = 6/2 = 3.

The product of the roots is given by c/a = 9/3 = 3.

The product of the roots is given by c/a = 2/3.

The product of the roots is given by c/a = 6/1 = 6.

The product of the roots is given by c/a = 24/1 = 24.

The product of the roots is given by c/a = 15/1 = 15.

proj_B(A) = (A · B / |B|^2)B = ((3*1 + 4*1) / (1^2 + 1^2))(i + j) = (7/2)(i + j) = 3.5.

Circumference = 2πr; 62.8 = 2πr; r = 10 cm.

Area = πr²; 154 = 22/7 * r²; r² = 154 * 7/22 = 49; r = 7 units.

Circumference = 2πr; 31.4 = 2πr; r = 31.4/(2π) ≈ 5 m.

Circumference = 2πr; 62.8 = 2πr; r = 62.8/(2π) ≈ 10 cm.

Radius is half of the diameter. Therefore, radius = 14 cm / 2 = 7 cm.

The real part of the complex number 4 + 5i is 4.

The real part of the complex number 7 - 4i is 7.

Slope = (6-2)/(3-1) = 4/2 = 2.