WAEC 2025 · Paper 2 · Q1✱✱

(a)
Factorize $px - 2qx - 4qy + 2py$ .
(b)
Given $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ , $P = \{1, 2, 4, 6, 10\}$ and $Q = \{2, 3, 6, 9\}$ , show that $(P \cup Q)' = P' \cap Q'$ .
Show the answer
Both sides equal $\{5, 7, 8\}$ .

Worked solution (try it first)

(a)

Group the terms in $x$ and the terms in $y$ : $(px - 2qx) + (2py - 4qy) = x(p - 2q) + 2y(p - 2q)$ .
$(p - 2q)$ is a common factor: $px - 2qx - 4qy + 2py = (p - 2q)(x + 2y)$ .

(b)

Left side: $P \cup Q = \{1, 2, 3, 4, 6, 9, 10\}$ , so $(P \cup Q)' = \{5, 7, 8\}$ .
Right side: $P' = \{3, 5, 7, 8, 9\}$ and $Q' = \{1, 4, 5, 7, 8, 10\}$ .
The elements in both: $P' \cap Q' = \{5, 7, 8\}$ .
Both sides equal $\{5, 7, 8\}$ , so $(P \cup Q)' = P' \cap Q'$ .

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