Factoring the polynomial P(x) gives (x - 2)(x - 3), so the roots are 2 and 3.

Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

P(2) = 2^3 - 6(2^2) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Population after one year = Current Population * (1 + Growth Rate) = 1 million * (1 + 0.02) = 1 million * 1.02 = 1.02 million.

Population after 2 years = 100,000 * (1 + 0.05)^2 = 100,000 * 1.1025 = 110,250.

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13

Velocity vector = dr/dt = (2, 3, 4)

Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.

Projection of P onto a = ((P · a) / |a|^2) * a.

Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).

Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).

Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).

Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.

The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

The x-component of r is 2.

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Distance (d) = Potential (V) / Electric field (E) = 5 V / 2 V/m = 2.5 m.

The energy stored in a capacitor is given by U = 1/2 CV^2. If V is halved, the energy becomes U' = 1/2 C(V/2)^2 = 1/8 CV^2, which is halved.

The charge (Q) stored in a capacitor is directly proportional to the potential difference (V) across it, given by Q = CV.

Work done (W) = Charge (Q) × Potential difference (V) = 2 C × 10 V = 20 J.

The angle θ can be found using cos(θ) = power factor. Thus, θ = cos⁻¹(0.8) ≈ 36.87°.

According to Boyle's Law, P * V = constant at constant temperature, so if P increases, V must decrease.

Let the original price be x. The new price after a 10% increase is x + 0.1x = 1.1x. Setting this equal to $22 gives us 1.1x = 22, so x = 22/1.1 = 20. Therefore, the original price was $20.

Let the original price be $100. After a 10% increase, the price becomes $110. After a 10% decrease, the price becomes $110 - $11 = $99. The net change is -1%, so the answer is 0%.

Let the original price be x. Then, x + 0.2x = 24, or 1.2x = 24. Thus, x = 24 / 1.2 = 20.

Let the original price be Rs. 100. After a 20% increase, price = 120. After a 20% decrease, price = 120 - 24 = 96. Net change = (96 - 100)/100 * 100 = -4%.

Let the original price be $100. After a 20% increase, the price becomes $120. After a 20% decrease, the price becomes $120 - 0.20 × 120 = $96. The net change is (96 - 100)/100 × 100 = -4%.

Percentage increase = ((25 - 20) / 20) * 100 = 25%