WAEC 2014 · Paper 2 · Q4

(a)
Without using tables or a calculator, evaluate $\log_{10}\left(\frac{75}{10}\right) - 2\log_{10}\left(\frac59\right) + \log_{10}\left(\frac{100}{243}\right)$ .
(b)
Given that $X = \{x : 10 \le x < 15\}$ and $Y = \{\text{even numbers} < 18\}$ are subsets of $U = \{10, 11, 12, \ldots, 20\}$ , find: (i) $X \cap Y$ ; (ii) $n(X' \cap Y)$ .

Worked solution (try it first)

(a)

Move the 2 inside as a power: $2\log_{10}\frac59 = \log_{10}\frac{25}{81}$ .
Then combine: $\log_{10}\left(\frac{75}{10} \times \frac{100}{243} \div \frac{25}{81}\right)$ .
Simplify: $\frac{75}{10} \times \frac{100}{243} \times \frac{81}{25} = \frac{75 \times 100 \times 81}{10 \times 243 \times 25}$
$= 10$ .
So the answer is $\log_{10} 10 = 1$ .

(b)

$X = \{10, 11, 12, 13, 14\}$ and $Y$ (even numbers in $U$ less than 18) $= \{10, 12, 14, 16\}$ .

(i)

$X \cap Y = \{10, 12, 14\}$ .

(ii)

$X' = \{15, 16, 17, 18, 19, 20\}$ , so $X' \cap Y = \{16\}$ and $n(X' \cap Y) = 1$ .

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