To pull a box up a rough slope, the force required will be least when it is applied : a) horizontaly
b) vertically
c) perpendicular to the plane
d) parallel to the plane

The correct option is;

d) Parallel to the plane

Explanation:

The forces acting on the box of mass, m are;

Weight of the box acting an angle, θ, equal to the inclination of the plane to the perpendicular of the plane

Weight of the box acting along the plane = m×g×sin(θ)

The force of friction along the plane = μ×m×g×cos(θ)

The total force acting downward along the plane , = m×g×sin(θ) + μ×m×g×cos(θ)

The Force needed to pull the box up along the plane F = The total force acting downward along the plane

F = m×g×sin(θ) + μ×m×g×cos(θ) = m×g×(sin(θ) + μ×cos(θ))

When the force, Fₐ is applied vertically, the force acting along the plane = Fₐ×cos(θ)

When the force is applied perpendicular, the force acting along the plane = Fₐ×sin(θ)

When the force is applied horizontally, the force acting along the plane = Fₐ×cos(θ)

When the force is applied parallel to the plane, the force acting along the plane = Fₐ

Therefore, since Fₐ > Fₐ×cos(θ) and Fₐ > Fₐ×sin(θ), for acute angles, we have that the least force is required when the force is acting parallel to the plane.

d) physical means can be used to separate the components of air.

scattering of light by tiny particles in air to produce the colors we see on sky, is dependent on the following factors:

(i) wavelength of light

(ii) size of the scattering particle  

(iii) distance traveled by light through the atmosphere.

according to rayleigh's law of scattering, the amount of scattering is inversely proportional to the fourth power of its wavelength. shorter the wavelength, greater is its scattering. hence violet/blue light is scattered more than red.

sun's light is a composite light ( visible light) with wavelengths in the range 400 nm (violet) to 700 nm (red). as sunlight passes through our atmosphere, the molecules of air scatter light according to their wavelengths. during sunrise and sunset, the sun's position is below the horizon. hence light from the sun travels a greater distance in the atmosphere when compared to its position a noon.

as sunlight enters the earth's atmosphere, blue light is scattered away and at the end of its journey in the sky, only red and orange wavelengths remain in the beam.

hence at sunrise and sunset, the sky appears reddish orange.