HPAS 2019 Maths Optional Paper-2 Question 1(a)
Discuss the convergence of the series:
\[ 1 + \frac{x}{2} + \frac{2!}{3}x^2 + \frac{3!}{4}x^3 + \dots \]
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HPAS 2019 Maths Optional Paper-2 Question 1(b)
Examine if the function \(u = e^{2xy}\sin(x^2 – y^2)\) is harmonic. Find the complex function \(f(z)\) in terms of z, where \(f(z) = u+iv\).
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HPAS 2019 Maths Optional Paper-2 Question 1(c)
Obtain the Laplace transform of the function \(f(t) = e^{-3t}u(t-2)\).
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HPAS 2019 Maths Optional Paper-2 Question 1(d)
Obtain the smallest positive root of the equation \(x \log_{10}x – 1.2 = 0\), correct to 2 decimal places, using the Newton-Raphson method.
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HPAS 2019 Maths Optional Paper-2 Question 2(a)
Find the transformation which maps the semi-infinite strip of width \(\pi\), bounded by the lines \(v=0\), \(v=\pi\), and \(u=0\), into the upper half of the z-plane.
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HPAS 2019 Maths Optional Paper-2 Question 2(b)
Evaluate
\[ \int_C \frac{3z^2+z}{z^2-1} dz \]
where C is the circle \(|z-1|=1\).
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HPAS 2019 Maths Optional Paper-2 Question 3(a)
Show that a cyclic group is necessarily abelian. Show by an example that the converse may not be true.
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HPAS 2019 Maths Optional Paper-2 Question 3(b)
Examine if there exists a one-to-one correspondence between the right and left cosets of H in G, if H is any subgroup of G.
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HPAS 2019 Maths Optional Paper-2 Question 3(c)
If \(f:G \to G’\) is a homomorphism, then prove that Im(f) is a subgroup of \(G’\).
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HPAS 2019 Maths Optional Paper-2 Question 4(a)
Define a compact metric space. Prove that a closed subset of a compact space is compact.
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HPAS 2019 Maths Optional Paper-2 Question 4(b)
Show that the integral
\[ \int_{a}^{\infty} \frac{x}{1+x^4\sin^2x} dx \]
is divergent.
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HPAS 2019 Maths Optional Paper-2 Question 5(a)
Solve the following system of equations by the Gauss-Seidel method, correct to 2 decimal places:
\[ \begin{cases} x_1 + 6x_2 + 2x_3 = 6 \\ 5x_1 + x_2 – x_3 = 12 \\ 3x_1 – 2x_2 + 8x_3 = -4 \end{cases} \]
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HPAS 2019 Maths Optional Paper-2 Question 5(b)
From the following table, obtain \(f(2.07)\) using the best formula:
x: 2.00, 2.05, 2.10, 2.15, 2.20, 2.25
f(x): 0.69315, 0.71784, 0.74194, 0.76547, 0.78846, 0.81093
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HPAS 2019 Maths Optional Paper-2 Question 6(a)
Obtain the inverse Laplace Transform of \(f(s) = \log\frac{s+1}{s-1}\).
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HPAS 2019 Maths Optional Paper-2 Question 6(b)
Find the Laplace transform of the following periodic function \(f(t)\) with period 2c:
\[ f(t) = \begin{cases} t & , \ 0 < t < c \\ 2c - t & , \ c < t < 2c \end{cases} \]
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HPAS 2019 Maths Optional Paper-2 Question 6(c)
Solve the following Initial Value Problem using the Laplace transform:
\[ x” + 2x’ + 5x = e^{-t}\sin t \]
with initial conditions \(x(0)=0\) and \(x'(0)=1\).
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HPAS 2019 Maths Optional Paper-2 Question 7(a)
Find the series solution of the following differential equation:
\[ (1+x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} – y = 0 \]
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HPAS 2019 Maths Optional Paper-2 Question 7(b)
Solve the partial differential equation
\[ (x^2 – y^2 – z^2)p + 2xyq = 2xz \]
where \(p = \frac{\partial z}{\partial x}\) and \(q = \frac{\partial z}{\partial y}\).
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HPAS 2019 Maths Optional Paper-2 Question 8(a)
What do the following explain in the C Language:
(i) ++nc
(ii) else if (condition)
(iii) tolower(c)
(iv) Pointer
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HPAS 2019 Maths Optional Paper-2 Question 8(b)
Using the Euler equation, find the extremal of the following functional:
\[ \int_{a}^{b} (12xy(x) + (y’)^2) dx \]
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HPAS 2019 Maths Optional Paper-2 Question 8(c)
Let \(f\) be a continuous function. Then show that \(f\) is Riemann integrable.
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