Wave speed = frequency × wavelength = 50 Hz × 2 m = 100 m/s.

Period (T) = 1/Frequency = 1/50 Hz = 0.02 s.

Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.

cos(90°) = 0, so θ = 90°

cos(60°) = 0.5, so θ = 60°

cos(60°) = 0.5, so θ = 60°.

sin²(θ) + cos²(θ) = 1; sin²(θ) = 1 - 0.6² = 0.64; sin(θ) = √0.64 = 0.8.

cos(45°) = √2/2 ≈ 0.707, so θ = 45°.

cos(60°) = 1/2, so θ = 60°

The determinant of E can be calculated using the rule of Sarrus or cofactor expansion, resulting in 0.

The determinant of E is calculated as (a*d) - (b*c) = ad - bc.

Det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

The determinant of F is calculated as (2*3) - (0*0) = 6.

Det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*0) + 3(1*1 - 0*0) = 2(0 - 2) - 1(1) + 3(1) = -4 - 1 + 3 = -2.

Using the determinant formula, det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*3) + 3(1*1 - 0*3) = 2(0 - 2) - 1(1 - 6) + 3(1 - 0) = -4 + 5 + 3 = 4.

Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.

The determinant of F is (2*3) - (1*1) = 6 - 1 = 5.

f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.

Determinant of G = (1*-1) - (1*1) = -1 - 1 = -2.

Det(H) = (1*-1) - (1*1) = -1 - 1 = -2.

The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

The determinant of H is (1*4) - (2*2) = 4 - 4 = 0.

Det(H) = (2*5) - (3*4) = 10 - 12 = -2.

Det(H) = (2*5) - (3*4) = 10 - 12 = -2.

Using cofactor expansion, det(I) = 1(1*4 - 3*0) - 0 + 2(0*0 - 1*1) = 4 - 2 = 2.

Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 3*1) - 0(0 - 3*1) + 2(0 - 1*1) = 0 - 0 - 2 = -2.

The determinant of I is 0 because the rows are linearly dependent.

Det(J) = (1*1) - (1*1) = 1 - 1 = 0.

Determinant of J = 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.