QQ.1. State whether the following statements are True or False. Give a reason in support of your answer.
- a)) Between any two different irrational numbers there is another irrational number. (90 words)
- b)) ∫(x^11 + x^21 + x^31)dx = 0 (90 words)
- c)) The determinant |p x q+r| |q x r+p| |r x p+q| = 0 (90 words)
- d)) The range of the data shown in the following frequency distribution is 350. Classes: 20-25, 25-30, 30-35, 35-40, 40-45, 45-50, 50-55; Frequencies: 0, 7, 3, 8, 4, 0, 0 (90 words)
Key Points:
- Irrational numbers are dense: between any two distinct irrationals, another irrational always exists.
- Indefinite integrals result in functions + C, not a constant value of zero.
- The determinant of vectors (p x q + r, q x r + p, r x p + q) is not generally zero; counterexamples exist.
- Range of grouped data is max class boundary minus min class boundary, or max data value minus min data value.
Answer: This response evaluates four mathematical statements from the MST-001 course content, determining their truth value and providing comprehensive reasoning based on fundamental principles of number systems, calculus, vector algebra, and statistics. Each sub-question adheres to specific word limits and formatting requirements, ensuring clarity and academic rigor. The answers explain why each statement is true or false, drawing upon relevant definitions and calculation methods taught in the course.