In an examination, there are 100 questions divided into 3 parts A, B, C, and each part should contain at least one question. Each question in parts A, B, and C carry 1, 2 and 3 marks respectively. Part A is for at least 60% of the total marks and part B should contain 23 questions. How many questions must be set in part C?

Given: Total questions = 100, Part B = 23 questions (2 marks each).

Let Part A = \(a\) questions (1 mark), Part C = \(c\) questions (3 marks).

So, \(a + 23 + c = 100 \Rightarrow a + c = 77\).
Total marks = \(a + 46 + 3c\).
Part A marks ≥ 60% total marks:
\(\Rightarrow a \geq 0.6(a + 46 + 3c)\)
\(\Rightarrow a \geq 0.6a + 27.6 + 1.8c\)
\(\Rightarrow 0.4a \geq 27.6 + 1.8c\)
\(\Rightarrow a \geq 69 + 4.5c\).

Using \(a = 77 - c\):
\(77 - c \geq 69 + 4.5c \Rightarrow 8 \geq 5.5c \Rightarrow c \leq 1.45\).
So, \(c = 1\).

Answer: Part C has 1 question.