GATE 2024 IN GA Q.3: solution and answer

Official answer: DSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

1. The correct answer is $pq$. The student's answer is $\frac{p}{q}$.

2. Percentage error is measured against the correct value: $$\text{Error}=\frac{\left|pq-\frac{p}{q}\right|}{pq}\times 100=\left|1-\frac{1}{q^2}\right|\times 100$$

3. Set this equal to 80%: $\left|1-\frac{1}{q^2}\right|=0.8$.

4. Case 1: $1-\frac{1}{q^2}=0.8\Rightarrow \frac{1}{q^2}=0.2\Rightarrow q^2=5\Rightarrow q=\sqrt{5}$, since $q>0$.

5. Case 2: $1-\frac{1}{q^2}=-0.8\Rightarrow \frac{1}{q^2}=1.8\Rightarrow q^2=\frac{5}{9}\Rightarrow q=\frac{\sqrt{5}}{3}$. This is not one of the options.

6. Check: with $q=\sqrt{5}$, the correct value is $\sqrt{5}p$ and the student's value is $\frac{p}{\sqrt{5}}=\frac{\sqrt{5}p}{5}$, which is 20% of the correct value. So the error is 80%, and the answer is (D).