HPAS Maths Optional PYQs: Real Analysis
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Based on your query and the provided excerpts from the HPAS Maths Optional Question Papers (2014–2024), here is the compilation of questions categorized under the topic of Real Analysis.
I. The Riemann Integral (Definition, Existence, Properties)
This section includes questions related to the existence, definition, properties, and theorems (Darboux and Mean Value) associated with the Riemann integral.
HPAS 2024 Real Analysis Questions
Question 2(b): If \(f:[a,b] \to R\) is a monotone function on \([a,b]\), then show that \(f\) is Riemann integrable (\(f \in R[a,b]\)).
Question 8(b): Let \(f\) and \(g\) be integrable functions on \([a,b]\). Then, show that: (i) \(f \cdot g\) is integrable on \([a,b]\); (ii) \(\max(f,g)\) and \(\min(f,g)\) are integrable on \([a,b]\).
HPAS 2023 Real Analysis Questions
Question 1(b): If \(f\) is a Riemann integrable function on the interval \([a,b]\), then show that \(f^2\) is also a Riemann integrable function.
Question 3(a): Show that the function \(f(x)=x\) (when x is rational) and \(f(x)=-x\) (when x is not rational) is not Riemann integrable in the interval \([a,b]\), but \(|f|\) is Riemann integrable.
HPAS 2021 Real Analysis Questions
Question 1(d): Show by an example that if functions \(f\) and \(g\) are not Riemann integrable, then their product \(f \cdot g\) can be Riemann integrable.
Question 2(c): If \(f:[a,b] \to R\) is a step function, then show that \(f\) is Riemann integrable.
HPAS 2020 Real Analysis Questions
Question 3(b): State the second mean value theorem of integral calculus. Let \(f: \to R\) be a continuous function such that \(\int_{0}^{1} |f(x)| dx = 0\). Show that \(f\) is identically zero on \(\mathbf{R}\).
HPAS 2019 Real Analysis Questions
Question 8(c): Let \(f\) be a continuous function. Then show that \(f\) is Riemann integrable.
HPAS 2018 Real Analysis Questions
Question 5(b): For any partition P of \([a,b]\), let \(L(P,f)\) and \(U(P,f)\) denote the lower and upper Darboux sums. Let P and Q be two partitions of \([a,b]\). Then prove that \(L(P,f) \le L(Q,f) \le U(Q,f) \le U(P,f)\).
HPAS 2017 Real Analysis Questions
Question 1(b): Let \(p_1\) and \(p_2\) be two partitions of \([a,b]\). Show that \(L(f,p_1) \le U(f,p_2)\) and \(L(f,p_2) \le U(f,p_1)\).
Question 3(a): Let \(f(x)=x\) for all \(x \in [0,1]\). Show that \(f\) is Riemann-integrable over \([0,1]\) and that \(\int_{0}^{1} f(x) dx = \frac{1}{2}\).
HPAS 2016 Real Analysis Questions
Question 5(b): If \(f:[a,b] \to R\) is continuous on \([a,b]\) then show that the function is Riemann-integrable on \([a,b]\).
Question 6(a): Show that the identity: \(\int_{a}^{b} f'(x) dx = f(b) – f(a)\) is not always valid, with the help of an example.
HPAS 2015 Real Analysis Questions
Question 3(a): State and prove Darboux theorem.
Question 3(b): If function \(f(x)=\sin x, x \in [0, \frac{\pi}{2}]\) and P is a given partition of \([0, \frac{\pi}{2}]\), then prove that \(f \in R[0, \frac{\pi}{2}]\). Also find \(L(f,P)\), \(U(f,P)\), \(\sup\{L(f,P)\}\) and \(\inf\{U(f,P)\}\).
HPAS 2014 Real Analysis Questions
Question 3(a): Let \(f\) be bounded on \([a,b]\). Then prove that \(f\) is R-integrable over \([a,b]\) if and only if given \(\epsilon > 0\) there exists a partition P of \([a,b]\) such that: \(0 \le U(f,P) – L(f,P) < \epsilon\).
Question 3(b): For the functions: \(f(x)=x\), \(g(x)=e^x\), then verify the second mean value theorem in the interval \([-1,1]\).
II. Improper Integrals and their Convergence
This section focuses on testing the convergence of improper integrals and related criteria.
HPAS 2023 Real Analysis Questions
Question 3(b): For what values of \(m\) and \(n\) is the integral \(\int_{0}^{1} x^{m-1} (1-x)^{n-1} \log x dx\) convergent?.
HPAS 2020 Real Analysis Questions
Question 2(c): State Abel’s test for convergence of improper integrals. Test the convergence of \(\int_{0}^{\infty} \frac{\sin x}{\log(x+2)} dx\).
HPAS 2019 Real Analysis Questions
Question 4(b): Show that the integral \(\int_{a}^{\infty} (1+x^4) \frac{\sin^2 x}{x} dx\) is divergent.
HPAS 2018 Real Analysis Questions
Question 3(a): Determine the values of \(s\) for which the improper integral \(\int_{0}^{\infty} e^{-sx} dx\) converges.
HPAS 2017 Real Analysis Questions
Question 3(b): Test the convergence of the following integral: \(\int_{0}^{\infty} e^{-ax} \frac{\sin x}{x} dx\) (\(a \ge 0\)).
HPAS 2014 Real Analysis Questions
Question 1(b): Test the convergence of the integral: \(\int_{0}^{\infty} e^{-x^2} dx\).