1. If \(a \propto b\), \(b \propto \frac{1}{c}\), and \(c \propto d\), then \(a \propto \frac{1}{d}\).

2. If \(a ∝ \frac{1}{c}\) and \(c ∝ \frac{1}{b}\), then determine the relationship between \(a\) and \(b\).

3. If \(x \propto \sqrt{y}\) and \(z \propto \sqrt[3]{x}\), then what is the relationship between \(y\) and \(z\)?

(a) \(z^6 \propto y\) (b) \(y^6 \propto z\) (c) \(z^2 \propto y\) (d) \(y^2 \propto z\)

4. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

5. If \(a\), \(b\), and \(c\) are in continuous geometric progression and both \(a\) and \(c\) have negative signs, then \(b\) will also be positive in sign. Here’s the full English translation of your sentence: If \(a\), \(b\), and \(c\) are in continuous geometric proportion and both \(a\) and \(c\) have negative signs, then \(b\) will have a ——— sign.