Physics

The vectors î, ĵ, and k̂ are the standard unit vectors in Cartesian coordinates. The cross product ĵ × k̂ equals î, and the dot product î · î is 1, so the magnitude of î · (ĵ × k̂) is 1.

Magnetic flux (Φ) is defined as the product of the magnetic flux density (B) and the area (A) through which the field lines pass, so Φ = B × A. This relationship directly connects the two quantities.

The permeability of free space (also called the magnetic constant) is exactly 4π × 10⁻⁷ henry per meter (H/m), which is equivalent to Wb·A⁻¹·m⁻¹ in SI units. This value is fundamental in electromagnetism.

The root mean square (r.m.s) value of an alternating current is defined as the square root of the mean (average) of the square of the instantaneous current values over one complete cycle. This matches option B.

The signs of the rectangular components (x and y) of a vector depend on the quadrant in which the vector lies. In the 1st quadrant, both x and y components are positive, so they have the same sign. In the 3rd quadrant, both x and y components are negative, so they also have the same sign. Therefore, the two rectangular components of a vector have the same sign in both the 1st and 3rd quadrants. Since option C states 'Both 1st and 3rd', it is the correct answer. The question asks for the quadrant(s) where the components have the same sign, and both these quadrants satisfy that condition.

The cutoff frequency (also called the -3dB frequency) of an RC circuit is given by the formula f = 1/(2πRC). Among the given options, the closest correct formula is 1/(2RC), which is option D. The exact formula includes 2π in the denominator, so none of the options perfectly match the exact formula, but option D is the closest and most correct approximation compared to the others.

Centripetal force acts perpendicular to the direction of the particle's instantaneous velocity, so it does no work on the particle as work requires a component of force along the displacement.

Geothermal energy originates from multiple sources within the Earth. The primary contributors are: (A) the decay of radioactive elements such as uranium, thorium, and potassium isotopes, which produce significant heat; (B) the compression of material in the Earth, including frictional heating and phase changes as the core solidifies; and (C) the residual heat from the Earth's original formation and accretion. Together, these processes contribute to the Earth's internal heat that drives geothermal energy. Therefore, the most accurate answer is (D) All of the above (A, B, and C).

When unpolarized light passes through a Polaroid filter, it becomes plane polarized because the Polaroid only allows light waves oscillating in one plane to pass through.

The standard value for solar irradiance at the Earth's surface on a clear day at noon is approximately 1.4 kW/m², which is the closest to the correct scientific value among the options provided.