HPAS 2024 Maths Optional Paper-1 Question 1(a)
Show that similar matrices have the same minimal polynomial.
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HPAS 2024 Maths Optional Paper-1 Question 1(b)
Find the asymptotes of the curve:
\[ x^{3}+x^{2}y+xy^{2}+y^{3}+2x^{2}+3xy-4y^{2}+7x+2y=0 \]
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HPAS 2024 Maths Optional Paper-1 Question 1(c)
Compute the general solution of the non-linear differential equation \(y = xy’ + (y’)^2\) where \(y’ = \frac{dy}{dx}\).
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HPAS 2024 Maths Optional Paper-1 Question 1(d)
Show that the vector field defined by the vector function \(\vec{V} = xyz(yz\mathbf{i} + xz\mathbf{j} + xy\mathbf{k})\) is conservative.
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HPAS 2024 Maths Optional Paper-1 Question 2(a)
Show that the inner product space is a normed vector space but the converse is not true.
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HPAS 2024 Maths Optional Paper-1 Question 2(b)
If the plane \(x+y+z=1\) cuts the cylinder \(x^2+y^2=1\) in an ellipse, then determine the points on the ellipse that lie closest to and farthest from the origin.
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HPAS 2024 Maths Optional Paper-1 Question 3(a)
Find the volume of the solid enclosed between the surfaces \(x^2+y^2=a^2\) and \(x^2+z^2=a^2\).
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HPAS 2024 Maths Optional Paper-1 Question 3(b)
Find the equation of the plane passing through the line of intersection of the planes \(a_1x+b_1y+c_1z+d_1=0\) and \(a_2x+b_2y+c_2z+d_2=0\) and perpendicular to the xy-plane.
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HPAS 2024 Maths Optional Paper-1 Question 4(a)
Show that every non-zero finite-dimensional inner product space has an orthonormal basis.
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HPAS 2024 Maths Optional Paper-1 Question 4(b)
Using the concept of diagonalizability, determine \(A^5\) where
\[ A = \begin{pmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{pmatrix} \]
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HPAS 2024 Maths Optional Paper-1 Question 5(a)
Solve the differential equation:
\[ x^2\frac{d^2y}{dx^2} – 2x\frac{dy}{dx} + 2y = x + x^2\log x + x^3 \]
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HPAS 2024 Maths Optional Paper-1 Question 5(b)
Determine the power series solution about the origin of the differential equation:
\[ (1-x^2)\frac{d^2y}{dx^2} – 4x\frac{dy}{dx} + 2y = 0 \]
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HPAS 2024 Maths Optional Paper-1 Question 6(a)
Find the work done by the force \(\vec{F}=(x^2-y^2)\mathbf{i}+(x+y)\mathbf{j}\) in moving a particle along the closed path C containing the curves \(x+y=0\), \(x^2+y^2=16\), and \(y=x\) in the first and the fourth quadrants.
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HPAS 2024 Maths Optional Paper-1 Question 6(b)
Evaluate the surface integral \(\iint_{S} \vec{F} \cdot \mathbf{n} \,dA\) where \(\vec{F} = z^2\mathbf{i} + xy\mathbf{j} – y^2\mathbf{k}\) and S is the portion of the surface of the cylinder \(x^2+y^2=36\) for \(0 \le z \le 4\) included in the first quadrant.
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HPAS 2024 Maths Optional Paper-1 Question 7(a)
A particle is projected in a plane with velocity \(\sqrt{\frac{\mu}{3a^6}}\) at a distance \(a\) from the center of force, attracting according to the law \(\frac{\mu}{r^7}\), in a direction inclined at \(30^\circ\) to the radius vector. Show that the orbit is \(r^2 = 2a^2 \cos(2\theta)\).
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HPAS 2024 Maths Optional Paper-1 Question 7(b)
R is the resultant of forces P and Q acting on a particle. If P is reversed, with Q remaining the same, the resultant becomes \(R’\). If R and \(R’\) are perpendicular to each other, show that \(P=Q\).
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HPAS 2024 Maths Optional Paper-1 Question 8(a)
Show that every real n-dimensional vector space is isomorphic to \(\mathbb{R}^n\).
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HPAS 2024 Maths Optional Paper-1 Question 8(b)
The amplitude of a simple harmonic oscillator is doubled. How does this affect the time period, total energy, and maximum velocity of the oscillator?
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