GATE 2026 IN GA Q.8: solution and answer

Official answer: BSource: Official answer key, GATE 2026 IN (IIT Guwahati)

1. Write the statements as set relations. Let $C$, $B$ and $D$ be the sets of patients with high cholesterol, high BP and diabetes. Everyone with high cholesterol also has high BP, so $C\subseteq B$. No one has both high cholesterol and diabetes, so $C\cap D=\varnothing$. The data give $|C\cup B\cup D|=75$, $|C|=10$, $|B|=45$, and only high BP $=20$.

2. Split the high-BP group into parts that do not overlap. Because $C\subseteq B$ and $C\cap D=\varnothing$, no patient has all three conditions, so the high-BP group has exactly three parts: (i) high BP with high cholesterol and no diabetes $=|C|=10$; (ii) only high BP $=20$; (iii) high BP with diabetes and no high cholesterol $=x$.

3. The three parts make up the whole high-BP group: $$10+20+x=|B|=45\Rightarrow x=15$$

4. Check against the total. Since $C\subseteq B$, the union is $C\cup B\cup D=B\cup D$, so the number with only diabetes is $75-45=30$. That is a valid non-negative count, and $x=15>0$ agrees with "Some patients with high BP also have diabetes". The data are consistent.

5. Because $C\cap D=\varnothing$, every patient with both diabetes and high BP is in part (iii). So $|B\cap D|=15$, which is option (B).