Live practice

Take |a| as the base. The perpendicular height is |b|sinθ.

Define the sample space and count each favourable outcome once.

The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.

Use the mean and squared deviations.

Use rectangular-coordinate definitions and keep corresponding coordinates aligned.

The final relation satisfies the original coordinate or distance condition, completing the verification.

After fixing 6 and 7, three positions remain and eight digits are available.

Expand and replace i² by −1.

For an arc, l=rθ where θ is measured in radians.

1. False. \(P\times Q=\{(m,n),(m,m),(n,n),(n,m)\}\).
2. True by the definition of Cartesian product; choosing one member from each set produces a pair.
3. True, because \(B\cap\varnothing=\varnothing\) and \(A\times\varnothing=\varnothing\).

1. \(\{3n:n\in\mathbb N,1\le n\le4\}\).
2. \(\{2^n:n\in\mathbb N,1\le n\le5\}\).
3. \(\{5^n:n\in\mathbb N,1\le n\le4\}\).
4. \(\{2n:n\in\mathbb N\}\).
5. \(\{n^2:n\in\mathbb N,1\le n\le10\}\).

1. When present in manageable amounts, they are decomposed into humus and mineral nutrients and thus recycle matter and improve soil fertility.
2. If accumulated in large quantities, decomposition can release foul-smelling gases and provide breeding places for flies and disease-causing organisms. In water, microbial decomposition can consume dissolved oxygen and harm aquatic life.