Live practice

Only the component of force along the displacement contributes to the scalar product.

Principle. Average speed uses path length, whereas average velocity uses endpoint displacement.

Use I=MR²/2 for a solid cylinder.

An event and its complement exhaust the sample space.

Adding a constant shifts the centre but does not change deviations from the centre.

Here a=2 and r=3. Use Sₙ=a(rⁿ−1)/(r−1).

Use the sequence definition or geometric-progression formula, keeping the first-term offset explicit.

The displayed substitutions cover every requested part and verify the result against the original data.

On subtraction, the even-power terms cancel.

Multiply numerator and denominator by the conjugate 4+3i.

The condition is \(y=3x\). Both values must lie in \(A\), so \(x=1,2,3,4\), giving \(R=\{(1,3),(2,6),(3,9),(4,12)\}\).

Thus domain \(=\{1,2,3,4\}\), codomain \(=A\), and range \(=\{3,6,9,12\}\).