HPAS 2023 Maths Optional Paper-2 Question 1(a)
Suppose G is a finite group of order \(pq\), where \(p\) and \(q\) are prime numbers such that \(p>q\). Show that G has at most one subgroup of order \(p\).
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HPAS 2023 Maths Optional Paper-2 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.
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HPAS 2023 Maths Optional Paper-2 Question 1(c)
Find the equation whose roots are \(2\cos\frac{\pi}{7}\), \(2\cos\frac{3\pi}{7}\), and \(2\cos\frac{5\pi}{7}\).
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HPAS 2023 Maths Optional Paper-2 Question 1(d)
Determine the inverse Laplace Transform of the function:
\[ \tan^{-1}\left(\frac{2}{s^2}\right) \]
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HPAS 2023 Maths Optional Paper-2 Question 1(e)
Show that every closed sphere is a closed set.
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HPAS 2023 Maths Optional Paper-2 Question 2(a)
Show that every finitely generated subgroup of \(\langle \mathbb{Q}, +\rangle\) is cyclic, where \(\mathbb{Q}\) is the set of rational numbers.
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HPAS 2023 Maths Optional Paper-2 Question 2(b)
Show that a subgroup of an infinite cyclic group is infinite.
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HPAS 2023 Maths Optional Paper-2 Question 2(c)
Give an example of an infinite group in which every element is of finite order. Justify your answer.
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HPAS 2023 Maths Optional Paper-2 Question 3(a)
Show that the function
\[ f(x) = \begin{cases} x & \text{when x is rational} \\ -x & \text{when x is not rational} \end{cases} \]
is not Riemann integrable in the interval \([a, b]\), but \(|f|\) is Riemann integrable.
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HPAS 2023 Maths Optional Paper-2 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?
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HPAS 2023 Maths Optional Paper-2 Question 4(a)
Let \(\langle a_n \rangle\) be a sequence such that \(\lim_{n\to\infty} a_n = l\). Then show that
\[ \lim_{n\to\infty} \frac{a_1+a_2+a_3+\dots+a_n}{n} = l \]
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HPAS 2023 Maths Optional Paper-2 Question 4(b)
Show that every compact subset F of a metric space \((X, d)\) is closed.
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HPAS 2023 Maths Optional Paper-2 Question 4(c)
Let \((X, d)\) be a metric space. Then show that any disjoint pair of closed sets in \(X\) can be separated by disjoint open sets in \(X\).
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HPAS 2023 Maths Optional Paper-2 Question 5(a)
Determine the analytic function \(f(z) = u+iv\) if
\[ u-v = \frac{\cos x + \sin x – e^{-y}}{2(\cos x – \cosh y)} \]
and \(f\left(\frac{\pi}{2}\right) = 0\).
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HPAS 2023 Maths Optional Paper-2 Question 5(b)
Show that the transformation \(w = z + \frac{1}{z}\) converts the straight line \(\text{arg}(z) = \alpha\) (where \(|\alpha| < \frac{\pi}{2}\)) into a branch of a hyperbola with eccentricity \(\sec\alpha\).
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HPAS 2023 Maths Optional Paper-2 Question 6(a)
Solve the partial differential equation
\[ \left(\frac{y-z}{yz}\right)p + \left(\frac{z-x}{zx}\right)q = \frac{x-y}{xy} \]
where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2023 Maths Optional Paper-2 Question 6(b)
Using Charpit’s method, find the solution of the partial differential equation \(p^2x + q^2y = z\), where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2023 Maths Optional Paper-2 Question 6(c)
Solve the partial differential equation \((D^2 – DD’ + D’ – 1)z = \cos(x+2y) + e^y\), where \(D = \frac{\partial}{\partial x}\) and \(D’ = \frac{\partial}{\partial y}\).
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HPAS 2023 Maths Optional Paper-2 Question 7(a)
On which curve can the functional
\[ \int_{0}^{\pi/2} (y’^2 – y^2 + 2xy) dx \]
with boundary conditions \(y(0)=0\) and \(y(\frac{\pi}{2})=0\), be extremized? (where \(y’ = \frac{dy}{dx}\))
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HPAS 2023 Maths Optional Paper-2 Question 7(b)
By applying Gauss’s quadrature formula, compute the integral
\[ \int_{5}^{12} \frac{1}{x} dx \]
Also find the error.
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HPAS 2023 Maths Optional Paper-2 Question 7(c)
Show that the rate of convergence of the Newton-Raphson method is quadratic, and determine a root of the equation \(x^{10}-1=0\) with the initial point \(x_0 = 0.5\).
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HPAS 2023 Maths Optional Paper-2 Question 8(a)
Let the polynomial \(\phi(x)\) be of the form \(\phi(x) = \sum_{i=0}^{n} L_i(x)y_i\), where each Lagrangian function \(L_i(x)\) is a polynomial in x of degree less than or equal to n. Then show that \(\sum_{i=0}^{n} L_i(x) = 1\).
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HPAS 2023 Maths Optional Paper-2 Question 8(b)
Using the 4th order Runge-Kutta method, solve the differential equation \(\frac{dy}{dx} = -xy^2\) with \(y(0)=1\). Taking a step size of \(h=0.2\), determine \(y(0.4)\).
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