According to Raoult's Law, the vapor pressure of the solvent in the solution is 0.75 * 100 mmHg = 75 mmHg.

The mole fraction of the solute is 1 - X_solvent = 1 - 0.8 = 0.2.

The mole fraction of the solvent is 1 - X_solute = 1 - 0.2 = 0.8.

Angular momentum L = Iω = 10 kg m² * 5 rad/s = 50 kg m²/s.

Rotational kinetic energy is given by KE = 1/2 I ω² = 1/2 * 10 * 5² = 125 J.

Using τ = Iα, we have α = τ/I = 20 N·m / 10 kg·m² = 2 rad/s².

Angular momentum L = Iω, so if I is doubled and ω remains constant, L also doubles.

Rotational kinetic energy is given by KE = 1/2 I ω^2. If I is doubled and ω remains constant, KE also doubles.

Angular acceleration α = τ/I. If I is doubled and τ remains constant, α is halved.

Angular momentum L = Iω; if I is doubled and ω remains constant, L also doubles.

According to Newton's second law for rotation, τ = Iα, if I increases and τ is constant, α must decrease.

Expected deaths = 5% of 2000 = 0.05 * 2000 = 100 deaths.

Damped frequency (ω_d) = ω_n√(1-ζ^2) = 5√(1-0.2^2) = 5√(0.96) ≈ 4.8 rad/s.

'a' represents the first term of the harmonic progression in the formula for the nth term.

The nth term of a harmonic progression can be expressed as the harmonic mean of n and m, which is 1/(1/n + 1/m).

The first term corresponds to n=1, which gives 1/1 = 1.

Substituting n = 7 into the formula gives a_7 = 5*7 - 3 = 35 - 3 = 32.

a_4 = 4^2 + 4 = 16 + 4 = 20.

The 4th term is 4^2 + 4 = 16 + 4 = 20.

In base-6, 'A' = 3*6^1 + 0*6^0 = 18 + 0 = 18. Therefore, 'A' is 42 in decimal.

In base-5, 'XYZ' = X*5^2 + Y*5^1 + Z*5^0 = 100. Solving gives X = 3.

The binary number 1011 converts to decimal as follows: 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = 8 + 0 + 2 + 1 = 11.

To convert from binary to decimal, calculate: 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = 8 + 0 + 2 + 1 = 11.

In hexadecimal, A represents 10. Therefore, 1A = 1*16^1 + 10*16^0 = 16 + 10 = 26.

The prime factorization of 36 is 2^2 * 3^2, as 36 = 2 * 2 * 3 * 3.

Arranging the numbers: 3, 5, 7, 8, 12. The median is the middle number, which is 7.

The mode is 5, as it appears 3 times, more than any other number.

When the object distance equals the focal length, the image is formed at infinity.

When the object is at the focal length, the image is formed at infinity.

Probability = Odds in favor / (Total odds) = 3 / (3 + 2) = 3/5 = 0.6.