HPAS 2020 Maths Optional Paper-2 Question 1(a)
Show that
\[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n-\log n} \]
is a conditionally convergent series.
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HPAS 2020 Maths Optional Paper-2 Question 1(b)
Determine all group homomorphisms from \(S_3\) to \(\mathbb{Z}_3\).
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HPAS 2020 Maths Optional Paper-2 Question 1(c)
Solve:
\(x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -xy\)
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HPAS 2020 Maths Optional Paper-2 Question 1(d)
Find the value of \(y(0.5)\) for the initial value problem \(\frac{dy}{dx}=y\), \(y(0)=1\), using Euler’s method with a step size of \(h=0.1\).
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HPAS 2020 Maths Optional Paper-2 Question 2(a)
Prove that any subgroup of a cyclic group is cyclic.
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HPAS 2020 Maths Optional Paper-2 Question 2(b)
Define the order of an element in a group. Let G be a finite group of even order. Show that G has an element of order 2 and that the number of elements of order 2 in G is odd.
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HPAS 2020 Maths Optional Paper-2 Question 2(c)
State Abel’s test for convergence of improper integrals. Test the convergence of
\[ \int_{0}^{\infty} \frac{\sin x}{\log(x+2)} dx \]
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HPAS 2020 Maths Optional Paper-2 Question 3(a)
State the logarithmic test for convergence of a series. If \(\{a_n\}\) is a sequence of real numbers such that \(\{n^2a_n\}\) is a convergent sequence, show that \(\sum_{n=1}^{\infty} a_n\) is an absolutely convergent series.
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HPAS 2020 Maths Optional Paper-2 Question 3(b)
State the second mean value theorem of integral calculus. Let \(f:[0,1] \to \mathbb{R}\) be a continuous function such that \(\int_{0}^{1} |f(x)| dx = 0\). Show that f is identically zero on \([0,1]\).
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HPAS 2020 Maths Optional Paper-2 Question 3(c)
Let \(f(z) = \frac{z-1}{z+1}\) and let L be the line in the z-plane through \(z=0\) and \(z=1+i\). Find the image of L under f.
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HPAS 2020 Maths Optional Paper-2 Question 4(a)
Find the bilinear transformation which maps the points 0, i, \(1+i\) onto 2i, -1, and 0, respectively.
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HPAS 2020 Maths Optional Paper-2 Question 4(b)
Prove that every bounded entire function is constant. Is \(f(z) = \sin z\) bounded on \(\mathbb{C}\)? Support your answer.
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HPAS 2020 Maths Optional Paper-2 Question 4(c)
Define a connected metric space. Prove that every closed interval of the real line \(\mathbb{R}\) is connected.
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HPAS 2020 Maths Optional Paper-2 Question 5(a)
Define a continuous function between metric spaces. Prove that any real-valued continuous function on a compact metric space is bounded.
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HPAS 2020 Maths Optional Paper-2 Question 5(b)
Obtain the interpolating polynomial by Newton’s divided difference formula for the following data. Also, find the value of \(f(3.5)\).
x: -3, -1, 0, 3, 5
f(x): -30, -22, -12, 330, 3458
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HPAS 2020 Maths Optional Paper-2 Question 6(a)
Find the shortest distance between the parabola \(y=x^2\) and the straight line \(x-y=5\).
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HPAS 2020 Maths Optional Paper-2 Question 6(b)
Solve the following initial value problem using the Laplace transform:
\[ \frac{d^2y}{dx^2} – 3\frac{dy}{dx} – 4y = x^2 \]
with \(y(0)=2\) and \(y'(0)=1\).
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HPAS 2020 Maths Optional Paper-2 Question 6(c)
Find the Laplace transform of the following periodic function \(f(x)\) with period \(2\pi\):
\[ f(x) = \begin{cases} x & , \ 0 \le x \le \pi \\ 2\pi – x & , \ \pi \le x \le 2\pi \end{cases} \]
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HPAS 2020 Maths Optional Paper-2 Question 7(a)
Find the general solution of the following partial differential equation:
\[ D(D^2 – D’ + 1)(D + 2D’ + 1)^3 z = 0 \]
where \(D \equiv \frac{\partial}{\partial x}\) and \(D’ \equiv \frac{\partial}{\partial y}\).
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HPAS 2020 Maths Optional Paper-2 Question 7(b)
Apply Charpit’s method to find the complete integral of the partial differential equation \(xp^2 + yq^2 = z\), where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2020 Maths Optional Paper-2 Question 7(c)
Find a real root of the equation \(xe^x – \cos x = 0\) using the Regula-Falsi method.
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HPAS 2020 Maths Optional Paper-2 Question 8(a)
Evaluate the integral
\[ I = \int_{0}^{1} \frac{dx}{1+x} \]
correct to three decimal places by using the trapezoidal rule and Simpson’s one-third rule, taking \(h=0.25\).
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HPAS 2020 Maths Optional Paper-2 Question 8(b)
Find the extremal of the following functional
\[ v[y(x)] = \int_{0}^{1} (2y + (y”)^2) dx \]
that satisfies the conditions \(y(0)=0\), \(y'(0)=1\), \(y(1)=1\), and \(y'(1)=1\).
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HPAS 2020 Maths Optional Paper-2 Question 8(c)
Write a flow chart in C language for the Runge-Kutta method of the fourth order.
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