HPAS 2021 Maths Optional Paper-1 Question 1(a)
Show that two finite-dimensional vector spaces over a field \(F\) are isomorphic if and only if they have the same dimension.
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HPAS 2021 Maths Optional Paper-1 Question 1(b)
Determine the directional derivative of the function \(f(x,y,z) = 4e^{2x-y+z}\) at the point (1, 1, 1) in the direction towards the point (-3, 5, 6).
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HPAS 2021 Maths Optional Paper-1 Question 1(c)
Show that the tangent planes at the extremities of any diameter of an ellipsoid are parallel.
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HPAS 2021 Maths Optional Paper-1 Question 1(d)
Solve the ordinary differential equation \(xp^2 – yp – y = 0\), where \(p = dy/dx\).
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HPAS 2021 Maths Optional Paper-1 Question 1(e)
Determine the value of the integral:
\[ \int_{0}^{2a} \int_{0}^{\sqrt{2ay-y^2}} dx \,dy \]
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HPAS 2021 Maths Optional Paper-1 Question 2(a)
Let V be a non-zero inner product space of dimension \(n\). Then show that V has an orthonormal basis.
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HPAS 2021 Maths Optional Paper-1 Question 2(b)
Let V be the space of all real-valued continuous functions. Define \(T: V \to V\) by:
\[ (Tf)(x) = \int_{0}^{x} f(t) dt \]
Show that T has no eigenvalues.
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HPAS 2021 Maths Optional Paper-1 Question 3(a)
Show that the equation \(ax^2+by^2+cz^2+2ux+2vy+2wz+d=0\) represents a cone if:
\[ \frac{u^2}{a} + \frac{v^2}{b} + \frac{w^2}{c} = d \]
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HPAS 2021 Maths Optional Paper-1 Question 3(b)
Find the maximum and minimum values of \(u^2+v^2+w^2\) subject to the conditions \(\frac{u^2}{4} + \frac{v^2}{5} + \frac{w^2}{25} = 1\) and \(w=u+v\).
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HPAS 2021 Maths Optional Paper-1 Question 3(c)
Using the concept of Gamma and Beta functions, show that:
\[ \int_{0}^{\pi/2} \sqrt{\tan x} \,dx = \frac{\pi}{\sqrt{2}} \]
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HPAS 2021 Maths Optional Paper-1 Question 4(a)
Find the directional derivative of \(f(x,y) = x^2y^3 + xy\) at the point (2,1) in the direction of a unit vector which makes an angle of \(\pi/3\) with the x-axis.
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HPAS 2021 Maths Optional Paper-1 Question 4(b)
Show that a function \(f\) defined on the real line \(\mathbb{R}\) is continuous if and only if for each open set \(G\) in \(\mathbb{R}\), \(f^{-1}(G)\) is an open set in \(\mathbb{R}\).
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HPAS 2021 Maths Optional Paper-1 Question 4(c)
Test the convergence of the integral:
\[ \int_{0}^{4} \frac{\sin^2 x}{\sqrt{x}(x-1)} dx \]
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HPAS 2021 Maths Optional Paper-1 Question 5(a)
Show that the smallest root of the equation \(J_0(x)=0\) lies in the interval \((2, \sqrt{8})\), where \(J_0(x)\) is the Bessel’s function of order zero.
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HPAS 2021 Maths Optional Paper-1 Question 5(b)
Solve the ordinary differential equation:
\[ \{x^2D^2 – (2m-1)xD + (m^2+n^2)\}y = n^2x^m \log x \]
where \(D=d/dx\).
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HPAS 2021 Maths Optional Paper-1 Question 5(c)
Find the series solution near \(x=0\) of the differential equation:
\[ x(1-x)\frac{d^2y}{dx^2} + (1-x)\frac{dy}{dx} – y = 0 \]
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HPAS 2021 Maths Optional Paper-1 Question 6(a)
Show that the radius of curvature R at any point \((r, \theta)\) on the curve \(r^2 = a^2 \sec(2\theta)\) is proportional to \(r^3\).
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HPAS 2021 Maths Optional Paper-1 Question 6(b)
Evaluate \(\int_C (x+y)dx – x^2dy + (y+z)dz\), where C is the curve defined by \(x^2=4y\), \(z=x\), and \(0 \le x \le 2\).
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HPAS 2021 Maths Optional Paper-1 Question 6(c)
Verify Stokes’s theorem for the vector field \(\vec{v} = (3x-y)\mathbf{i} – 2yz^2\mathbf{j} – 2y^2z\mathbf{k}\), where S is the surface of the sphere \(x^2+y^2+z^2=16\) and \(z>0\).
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HPAS 2021 Maths Optional Paper-1 Question 7(a)
Determine the center and radius of the circle in which the sphere \(x^2+y^2+z^2+2x-2y-4z-19=0\) is cut by the plane \(x+2y+2z+7=0\).
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HPAS 2021 Maths Optional Paper-1 Question 7(b)
Three forces act perpendicular to the sides of a triangle at their middle points and are proportional to the sides. Show that they are in equilibrium.
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HPAS 2021 Maths Optional Paper-1 Question 7(c)
Show that the only law for a central attraction, for which the velocity in a circle at any distance is equal to the velocity acquired in falling from infinity to that distance, is that of the inverse cube.
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HPAS 2021 Maths Optional Paper-1 Question 8(a)
The resultant of two forces acts along a line perpendicular to one force and is in magnitude half the other. Compute the angle between the forces.
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HPAS 2021 Maths Optional Paper-1 Question 8(b)
A particle of mass \(m\) is attached to a light wire which is stretched tightly between two fixed points with a tension \(T\). If \(a\) and \(b\) are the distances of the particle from the two ends, then show that the period of the small transverse oscillation of \(m\) is \(2\pi\sqrt{\frac{mab}{T(a+b)}}\).
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