HPAS 2017 Maths Optional Paper-2 Question 1(a)
If \(a \ne e\) is the only element of order 2 in a group G, then prove that \(ax = xa\) for all \(x \in G\).
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HPAS 2017 Maths Optional Paper-2 Question 1(b)
Let \(p_1\) and \(p_2\) be two partitions of \([a, b]\). Show that \(L(f, p_1) \le U(f, p_2)\) and \(L(f, p_2) \le U(f, p_1)\).
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HPAS 2017 Maths Optional Paper-2 Question 1(c)
Evaluate the Laplace transform of the function \(f(x) = (x+2)^2 e^x\).
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HPAS 2017 Maths Optional Paper-2 Question 1(d)
On which curves can the functional
\[ H[y(x)] = \int_{0}^{1} [(y’)^2 + 12xy] dx \]
with boundary conditions \(y(0)=0\) and \(y(1)=1\), be extremized?
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HPAS 2017 Maths Optional Paper-2 Question 2(a)
Prove that a finite group of prime order does not have any proper subgroup.
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HPAS 2017 Maths Optional Paper-2 Question 2(b)
Prove that the kernel of a homomorphism f from a group G to a group G’ is a normal subgroup of G.
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HPAS 2017 Maths Optional Paper-2 Question 3(a)
Let a real-valued function f be defined on \([0, 1]\) by \(f(x)=x\) for all \(x \in [0, 1]\). Show that f is Riemann-integrable over \([0, 1]\) and that \(\int_{0}^{1} f(x) dx = \frac{1}{2}\).
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HPAS 2017 Maths Optional Paper-2 Question 3(b)
Test the convergence of the following integral:
\[ \int_{0}^{\infty} e^{-ax} \frac{\sin x}{x} dx \quad (a \ge 0) \]
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HPAS 2017 Maths Optional Paper-2 Question 4(a)
If (A, d) is a metric space, then show that:
\[ |d(x_1, y_1) – d(x_2, y_2)| \le d(x_1, x_2) + d(y_1, y_2) \]
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HPAS 2017 Maths Optional Paper-2 Question 4(b)
Test the convergence of the series:
\[ x^2(\log 2)^q + x^3(\log 3)^q + x^4(\log 4)^q + \dots \]
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HPAS 2017 Maths Optional Paper-2 Question 5(a)
Prove that the function \(u(x,y) = x^3 – 3xy^2\) is harmonic and obtain its conjugate.
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HPAS 2017 Maths Optional Paper-2 Question 5(b)
Evaluate:
\[ \int_C \frac{e^{3z}}{z-\pi i} dz \]
where C is the circle \(|z-1|=4\).
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HPAS 2017 Maths Optional Paper-2 Question 6(a)
Solve the following equation by Charpit’s method:
\((p^2+q^2)y = qz\)
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HPAS 2017 Maths Optional Paper-2 Question 6(b)
Solve:
\(x^2r + 2xys + y^2t = 0\)
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HPAS 2017 Maths Optional Paper-2 Question 7(a)
Show that:
\( L\left\{ \int_{0}^{x} \frac{1-e^{-u}}{u} du \right\}(p) = \frac{1}{p}\log\left(1+\frac{1}{p}\right) \)
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HPAS 2017 Maths Optional Paper-2 Question 7(b)
Apply the convolution theorem to find:
\( L^{-1}\left\{ \frac{p^2}{(p^2+a^2)^2} \right\}(x) \)
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HPAS 2017 Maths Optional Paper-2 Question 8(a)
Find the path on which a particle, in the absence of friction, will slide from one fixed point to another point not in the same vertical line in the shortest time under the action of gravity.
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HPAS 2017 Maths Optional Paper-2 Question 8(b)
Find a root of the equation \(x \log_{10}x – 1.2 = 0\) correct to four places of decimals.
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