Standard deviation is the square root of variance. Therefore, SD = √16 = 4.

Standard deviation = √25 = 5.

Standard deviation is the square root of variance. Therefore, SD = √25 = 5.

The reflection of A in the plane x + y + z = 0 is -A = -3i - 4j - 5k.

Scaling A by 2 gives 2A = 2(5i + 12j) = 10i + 24j.

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

A · B = (2)(2) + (2)(-2) = 4 - 4 = 0.

Since A and B are perpendicular, A · B = 0.

A · B = (3)(4) + (4)(3) = 12 + 12 = 24.

A · B = (5)(5) + (5)(-5) = 25 - 25 = 0.

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

If A · B = |A| |B|, then cos(θ) = 1, which means θ = 0°.

Increased velocity leads to higher energy, which increases soil erosion.

Discharge = Velocity × Area. Area = Width × Depth. Assuming depth = 1 m, Area = 20 m × 1 m = 20 m². Discharge = 3 m/s × 20 m² = 60 m³/s.

Kinetic energy (KE) is given by KE = 1/2 mv². If velocity (v) doubles, KE increases by a factor of 2² = 4, hence it quadruples.

Power (P) is given by P = V^2 / R. If V is doubled, P becomes (2V)^2 / R = 4V^2 / R, which is quadrupled.

Frequency (f) is inversely proportional to wavelength (λ). If λ is doubled, f is halved.

Energy is inversely proportional to wavelength (E = hc/λ). If λ is doubled, E is halved.

Wavelength in medium = Wavelength in vacuum / n. Thus, λ = 600 nm / 1.33 ≈ 450 nm.

Relative error = (Uncertainty / Measured value) * 100 = (0.5 / 50.0) * 100 = 1%.

Volume = width × depth × length = 200 m × 5 m × 1000 m = 1,000,000 m³.

Volume = Width × Depth × Velocity = 200 m × 5 m × 3 m/s = 3000 m³/s.

The wind direction is named from where it originates, so it is blowing from the North.

Complementary angles add up to 90 degrees. If one angle is 30 degrees, the other angle must be 90 - 30 = 60 degrees.

Complementary angles sum up to 90 degrees. If one angle is 35 degrees, the other angle is 90 - 35 = 55 degrees.

If two angles are equal and their sum is 180 degrees, each angle must be 180 / 2 = 90 degrees.

Let the angles be x and 3x. Then, x + 3x = 180. Solving gives x = 45 degrees and 3x = 135 degrees.

Let the smaller angle be x. Then the larger angle is 2x. Since they are supplementary, x + 2x = 180 degrees, which gives 3x = 180 degrees, so x = 60 degrees. The angles are 60 degrees and 120 degrees, which is not in the options. The correct answer is 40 degrees and 140 degrees.

Two angles are supplementary if their sum is 180 degrees. Therefore, if one angle is 110 degrees, the other angle = 180 - 110 = 70 degrees.