HPAS 2018 Maths Optional Paper-2 Question 1(a)
Let X be any non-empty set and \(S(X)\) be the set of all bijections of X onto itself. Then prove that \((S(X), \circ)\) is an abelian group if and only if X is a set with one or two elements, where \(\circ\) is the operation of composition of functions.
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HPAS 2018 Maths Optional Paper-2 Question 1(b)
Let \((X, d)\) be a metric space and let \(A \subseteq X\). Then prove that \(\overline{A} = \{x \in X : d(x,A) = 0\}\).
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HPAS 2018 Maths Optional Paper-2 Question 1(c)
If \(X = \sqrt{-1}\), then find the value of \(X^X\).
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HPAS 2018 Maths Optional Paper-2 Question 1(d)
Find the \(\limsup\) and \(\liminf\) of the sequence \((-1)^n + \frac{1}{n}\).
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HPAS 2018 Maths Optional Paper-2 Question 2(a)
Let G be a group that has two subgroups of orders 45 and 75. If \(|G| < 400\), find \(|G|\).
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HPAS 2018 Maths Optional Paper-2 Question 2(b)
Let \(f:G \to H\) be a group homomorphism with kernel K. If the orders of G, H, and K are 75, 45, and 15 respectively, find the order of the image \(f(G)\).
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HPAS 2018 Maths Optional Paper-2 Question 3(a)
Determine the values of s for which the improper integral \(\int_{0}^{\infty} e^{-sx} dx\) converges.
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HPAS 2018 Maths Optional Paper-2 Question 3(b)
Find the value of \(L\{x^{7/2}\}\).
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HPAS 2018 Maths Optional Paper-2 Question 4(a)
Obtain the partial differential equation for the set of all right circular cones whose axes coincide with the z-axis.
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HPAS 2018 Maths Optional Paper-2 Question 4(b)
Prove that any sufficiently differentiable function of the form \(F(x+kt)\) satisfies the wave equation \(F_{xx} = (1/k^2)F_{tt}\).
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HPAS 2018 Maths Optional Paper-2 Question 5(a)
Let
\[ f(x) = \begin{cases} c, & 0 \le x \le c \\ 2c, & c < x \le 1 \end{cases} \]
If \(\int_{0}^{1} f(x) dx = \frac{7}{16}\), find the value of c.
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HPAS 2018 Maths Optional Paper-2 Question 5(b)
Let \(f:[0,1] \to \mathbb{R}\) be a continuous function. For any partition P of \([0, 1]\), let \(L(P,f)\) and \(U(P,f)\) denote the lower and upper Darboux sums respectively. Let \(P = \{0, 0.01, 0.02, \dots, 1\}\) and \(Q = \{0, 0.001, 0.002, \dots, 1\}\) be two partitions of \([0, 1]\). Then prove that \(L(P,f) \le L(Q,f) \le U(Q,f) \le U(P,f)\).
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HPAS 2018 Maths Optional Paper-2 Question 6(a)
Define an open sphere in a metric space. Describe with a figure the open sphere of unit radius centered at (0,0) for the metric \(d(z_1, z_2) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\) defined on \(\mathbb{R}^2\), where \(z_1 = (x_1, y_1)\) and \(z_2 = (x_2, y_2)\) are any two points of \(\mathbb{R}^2\).
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HPAS 2018 Maths Optional Paper-2 Question 6(b)
Define a closed sphere in a metric space. Describe with a figure the closed sphere of unit radius centered at (0,0) for the metric \(d(z_1, z_2) = |x_1-x_2| + |y_1-y_2|\) defined on \(\mathbb{R}^2\), where \(z_1 = (x_1, y_1)\) and \(z_2 = (x_2, y_2)\) are any two points of \(\mathbb{R}^2\).
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HPAS 2018 Maths Optional Paper-2 Question 7(a)
Test the convergence of the series
\[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}} \]
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HPAS 2018 Maths Optional Paper-2 Question 7(b)
Test the convergence of the series
\[ \sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{-n^2} \]
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HPAS 2018 Maths Optional Paper-2 Question 8(a)
Write an algorithm and draw a flow chart for integrating \(\int_{a}^{b} f(x) dx\) by the Trapezoidal rule, taking a step size h.
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HPAS 2018 Maths Optional Paper-2 Question 8(b)
Write an algorithm and draw a flow chart for finding the value of y at \(x=x_n\) for the differential equation \(\frac{dy}{dx}=f(x,y)\), taking a step size h, when the initial values of x and y are given, by Euler’s method.
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