HPAS 2024 Maths Optional Paper-2 Question 1(a)
If the sequences \(\langle a_n \rangle\) and \(\langle b_n \rangle\) converge to finite limits \(a\) and \(b\), respectively, then show that
\[ \lim_{n\to\infty} \frac{a_1b_n + a_2b_{n-1} + \dots + a_nb_1}{n} = ab \]
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HPAS 2024 Maths Optional Paper-2 Question 1(b)
Evaluate the contour integral
\[ \oint_{C} \frac{e^{3z}}{(z-\log 2)^4} dz \]
Where C is the square with vertices at \(\pm 1, \pm i\).
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HPAS 2024 Maths Optional Paper-2 Question 1(c)
Find the Laplace transform of:
\[ \frac{1}{t} \int_{0}^{t} e^{u} \sin u \,du \]
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HPAS 2024 Maths Optional Paper-2 Question 1(d)
Show that the Newton-Raphson process has a quadratic convergence.
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HPAS 2024 Maths Optional Paper-2 Question 2(a)
Let G be a group. Let Aut(G) denote the set of all automorphisms of G and let \(A(G)\) be the group of all permutations of G. Show that Aut(G) is a subgroup of \(A(G)\).
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HPAS 2024 Maths Optional Paper-2 Question 2(b)
Let \(\mathcal{R}[a, b]\) be the set of all Riemann integrable functions on the interval \([a, b]\). If \(f:[a,b] \to \mathbb{R}\) is a monotone function on \([a, b]\), then show that \(f \in \mathcal{R}[a, b]\).
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HPAS 2024 Maths Optional Paper-2 Question 3(a)
If \(\langle S_n \rangle\) is a sequence of positive real numbers such that \(S_n = \frac{1}{2}(S_{n-1} + S_{n-2})\) for all \(n>2\), then show that \(\langle S_n \rangle\) converges and find \(\lim_{n\to\infty} S_n\).
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HPAS 2024 Maths Optional Paper-2 Question 3(b)
Let \(R_\infty\) be the extended set of real numbers. The function \(d\) is defined by \(d(x,y) = |f(x) – f(y)|\) for all \(x, y \in R_\infty\), where \(f(x) = \frac{x}{1+|x|}\) when \(-\infty < x < \infty\), \(f(x)=1\) when \(x=\infty\), and \(f(x)=-1\) when \(x=-\infty\). Show that \((R_\infty, d)\) is a bounded metric space.
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HPAS 2024 Maths Optional Paper-2 Question 4(a)
Let \((X, d)\) be a complete metric space and \(Y\) be a subspace of \(X\). Show that \(Y\) is complete if and only if it is closed in \((X, d)\).
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HPAS 2024 Maths Optional Paper-2 Question 4(b)
Let G be a region and suppose that \(f:G \to \mathbb{C}\) is analytic such that \(f(G)\) is a subset of a circle. Then show that \(f\) is constant.
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HPAS 2024 Maths Optional Paper-2 Question 5(a)
Solve the partial differential equation:
\[ (x^2 – yz)p + (y^2 – zx)q = z^2 – xy \]
where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2024 Maths Optional Paper-2 Question 5(b)
Using Monge’s method, solve the wave equation \(r = a^2t\) (for \(a>0\)), where \(r = \frac{\partial^2 z}{\partial x^2}\) and \(t = \frac{\partial^2 z}{\partial y^2}\).
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HPAS 2024 Maths Optional Paper-2 Question 6(a)
Is the Jacobi condition fulfilled for the extremal of the functional
\[ \int_{0}^{a} (y’^2 – 4y^2 – e^{-x^2}) dx, \quad a \ne \frac{n\pi}{2} \]
with fixed boundaries A(0,0) and B(a,0)? (where \(y’ = dy/dx\))
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HPAS 2024 Maths Optional Paper-2 Question 6(b)
Using Euler’s method, solve the initial value problem \(\frac{dy}{dt} = 1 – t + 4y\) with \(y(0)=1\), in the interval \(0 \le t \le 0.5\) with \(h=0.1\). If the exact solution is \(y = -\frac{9}{16} + \frac{1}{4}t + \frac{19}{16}e^{4t}\), then compute the error and the percentage error.
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HPAS 2024 Maths Optional Paper-2 Question 7(a)
If \( C(t) = \int_{t}^{\infty} \frac{\cos x}{x} dx \), show that the Laplace transform of \(C(t)\) is \( \log\frac{\sqrt{s^2+1}}{s} \).
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HPAS 2024 Maths Optional Paper-2 Question 7(b)
Write the algorithm for the Bisection method for finding a real root of the equation \(f(x)=0\) which lies in the interval \([a, b]\). Further, develop a simple program in C language for finding a real root of the equation \(x^3 – 2x – 1 = 0\) using the Bisection method.
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HPAS 2024 Maths Optional Paper-2 Question 8(a)
Let C be the unit circle \(z = e^{i\theta}\), for \(-\pi \le \theta \le \pi\). First, show that for any real constant \(a\),
\[ \int_{C} \frac{e^{az}}{z} dz = 2\pi i \]
Then, write this integral in terms of \(\theta\) to derive the integration formula:
\[ \int_{0}^{\pi} e^{a \cos\theta} \cos(a \sin\theta) d\theta = \pi \]
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HPAS 2024 Maths Optional Paper-2 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 \([a, b]\).
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