HPAS 2021 Maths Optional Paper-2 Question 1(a)
Show that every closed subspace of a complete metric space \( (X, d) \) is complete.
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HPAS 2021 Maths Optional Paper-2 Question 1(b)
Find a transformation \(w = f(z)\) which maps the real axis of the \(z\)-plane onto the real axis in the \(w\)-plane.
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HPAS 2021 Maths Optional Paper-2 Question 1(c)
Give an example of a finite abelian group which is not cyclic.
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HPAS 2021 Maths Optional Paper-2 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.
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HPAS 2021 Maths Optional Paper-2 Question 1(e)
Determine the inverse Laplace transform of:
\[ \log\frac{s+c}{s+d} \]
where \(c\) and \(d\) are constants.
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HPAS 2021 Maths Optional Paper-2 Question 2(a)
Show that a finite group having more than two elements has a non-trivial automorphism.
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HPAS 2021 Maths Optional Paper-2 Question 2(b)
Prove that every quotient group of a cyclic group is cyclic. Does the converse of this statement hold? Justify your answer with an example.
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HPAS 2021 Maths Optional Paper-2 Question 2(c)
If \(f: [a,b] \to \mathbb{R}\) is a step function, then show that \(f\) is Riemann integrable.
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HPAS 2021 Maths Optional Paper-2 Question 3(a)
Let \(X\) be the set of all continuous real-valued functions on \([0, 1]\), and let:
\[ d(x,y) = \int_{0}^{1} |x(t) – y(t)| dt \]
Show that the metric space \( (X, d) \) is not complete.
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HPAS 2021 Maths Optional Paper-2 Question 3(b)
Using the concept of residue, determine the value of the integral:
\[ \int_{0}^{\infty} \frac{\sin(mx)}{x} dx \]
when \(m > 0\).
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HPAS 2021 Maths Optional Paper-2 Question 4(a)
Test the series for convergence:
\[ \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\log n}{n} \]
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HPAS 2021 Maths Optional Paper-2 Question 4(b)
Show that the sequence \((s_n)\) defined by:
\[ s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} \]
is divergent.
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HPAS 2021 Maths Optional Paper-2 Question 4(c)
If the partial sums of the series \(\sum a_n\) are bounded, show that the series
\[ \sum_{n=1}^{\infty} a_n e^{-nt} \]
converges for \(t > 0\).
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HPAS 2021 Maths Optional Paper-2 Question 5(a)
Solve the following partial differential equation using the Lagrange method:
\[ p(z+e^x) + q(z+e^y) = z^2 – e^{x+y} \]
where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2021 Maths Optional Paper-2 Question 5(b)
Find the root of the equation \(x \sin x + \cos x = 0\) by using the Newton-Raphson method.
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HPAS 2021 Maths Optional Paper-2 Question 5(c)
Find the complete integral of the partial differential equation:
\[ 2\sqrt{p} + 3\sqrt{q} = 6x + 2y \]
where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2021 Maths Optional Paper-2 Question 6(a)
Solve the partial differential equation:
\[ (D^2 + DD’ – 6D’^2)z = y \cos x \]
where \(D = \frac{\partial}{\partial x}\) and \(D’ = \frac{\partial}{\partial y}\).
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HPAS 2021 Maths Optional Paper-2 Question 6(b)
Solve the partial differential equation using Monge’s method:
\[ r – t \sin^2 x – p \cot x = 0 \]
where \(p = \frac{\partial z}{\partial x}\), \(r = \frac{\partial^2 z}{\partial x^2}\), and \(t = \frac{\partial^2 z}{\partial y^2}\).
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HPAS 2021 Maths Optional Paper-2 Question 7(a)
Determine the inverse Laplace transform of:
\[ \frac{1}{s^2 – e^{-as}} \]
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HPAS 2021 Maths Optional Paper-2 Question 7(b)
Using the concept of Laplace Transform, find the solution of the initial value problem:
\[ t\frac{d^2y}{dt^2} + 2t\frac{dy}{dt} + 2y = 2 \]
with \(y(0) = 1\), and \(y'(0)\) is arbitrary.
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HPAS 2021 Maths Optional Paper-2 Question 7(c)
Test for an extremum of the functional:
\[ I[y(x)] = \int_{0}^{1} (xy + y^2 – 2y^2 y’) dx \]
with boundary conditions \(y(0) = 1\), \(y(1) = 2\), where \(y’ = \frac{dy}{dx}\).
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HPAS 2021 Maths Optional Paper-2 Question 8(a)
Determine the maximum error in evaluating the integral:
\[ \int_{0}^{\pi/2} \cos x \,dx \]
by both the Trapezoidal and Simpson’s rules using four subintervals.
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HPAS 2021 Maths Optional Paper-2 Question 8(b)
Apply Euler’s modified method to find the value of \(y\) at \(x = 0.1\) correct to five decimal places, given:
\[ \frac{dy}{dx} = x^2 + y \]
with the initial condition \(y(0) = 0.94\).
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