from fpdf import FPDF # Creating a clean version of the content to avoid encoding issues content_final = """ Solutions: 1. If f(a+h) = f(a) + hf(a + theta*h), then: Answer: A. 0 <= theta <= 1. 2. In Cauchy’s Mean Value Theorem, the Lagrange’s form of reminder Rn is given by: Answer: A. h^n/n! f^(n)(a + theta*h). 3. For f(x) = x(x^2 - 1) on [0, 1], if Rolle's theorem is applied, c = ?: Answer: C. 1/sqrt(3). 4. If Cauchy’s Mean Value Theorem is applied to f(x) = x and g(x) = x^2, the value of c is: Answer: C. 1/2. 5. If f(x) satisfies Rolle's theorem, the tangent is: Answer: A. Parallel to x-axis. 6. lim (x -> 0) log(x)/cot(x) = ?: Answer: D. None of these. 7. lim (x -> 0) (1 + x)^(1/x) = ?: Answer: A. e. 8. lim (x -> 0) tan(x)log(x) = ?: Answer: A. 0. 9. lim (x -> pi/2) log(sin(x))/(pi/2 - x) = ?: Answer: C. -1. 10. A function f(x) has a maximum value at x = a if: Answer: A. f(a) > 0. 11. The extreme value of x^(1/x) is: Answer: B. (1/e)^e. """ # Generate PDF pdf = FPDF() pdf.set_auto_page_break(auto=True, margin=15) pdf.add_page() pdf.set_font("Arial", size=12) pdf.multi_cell(0, 10, content_final) # Save PDF file_path_clean = "/mnt/data/Solutions_clean.pdf" pdf.output(file_path_clean) file_path_clean