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.

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.

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.

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.

Det(J) = (1*4) - (2*2) = 4 - 4 = 0.

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 10.

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 9.

Using the root, substitute x = 3: 3² - 7*3 + p = 0, which gives p = 6.

sin(30°) = 0.5, so θ = 30°.

Using sin²(θ) + cos²(θ) = 1, cos²(θ) = 1 - 0.64 = 0.36, cos(θ) = 0.6

Using sin²(θ) + cos²(θ) = 1, cos(θ) = √(1 - 0.8²) = √(0.36) = 0.6.

sin(60°) = √3/2 ≈ 0.866, so θ = 60°.

sin(90°) = 1, so θ = 90°.

sin(30°) = 1/2, so θ = 30°.

sin(45°) = 1/√2, so θ = 45°

tan(45°) = 1, so θ = 45°.

Energy of a wave is proportional to the square of the amplitude. If amplitude is doubled, energy increases by (2^2) = 4 times.

The energy carried by a wave is proportional to the square of the amplitude. Therefore, if the amplitude is doubled, the energy increases by a factor of four.

Capacitive reactance XC = 1/(ωC). If capacitance C increases, XC decreases.

The energy stored in a capacitor is proportional to the square of the charge (U = Q^2/(2C)). Doubling the charge increases the energy by a factor of 4.

Circumference = 2πr; 31.4 = 2 * π * r; r = 5 cm.

Circumference = 2πr, so r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm.

Circumference = 2πr. Therefore, r = Circumference / (2π) = 31.4 cm / (2 * 3.14) = 5 cm.

For a second-order reaction, if concentration is tripled, the rate increases by 3² = 9 times.