HPAS 2020 Maths Optional Paper-1 Question 1(a)
Obtain the general solution of the following differential equation:
\(x\frac{dy}{dx}+(2-x)y=e^{3x}\), for \(x>0\).
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HPAS 2020 Maths Optional Paper-1 Question 1(b)
Find a linear transformation \(T:\mathbb{R}^{3}\to\mathbb{R}^{3}\) such that its image space is the plane \(x+y+z=0\).
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HPAS 2020 Maths Optional Paper-1 Question 1(c)
If \(f:[0,6] \to \mathbb{R}\) is a continuous function such that \(f(0)=f(6)\), then show that \(f(x)=f(x+3)\) for some \(x\) in \([0,3]\).
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HPAS 2020 Maths Optional Paper-1 Question 1(d)
Find the velocity of a particle moving on the surface of a right circular cylinder of radius \(b\).
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HPAS 2020 Maths Optional Paper-1 Question 2(a)
Define a basis of a vector space. Find a basis of the subspace of the vector space \(\mathbb{R}^4(\mathbb{R})\) generated by the subset:
\[ \{(1,1,0,-1), (2,4,6,0), (-2,-3,-3,1), (-1,-2,-2,2), (4,6,4,-6)\} \]
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HPAS 2020 Maths Optional Paper-1 Question 2(b)
Let \(A = \begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}\). Find an invertible \(2 \times 2\) matrix P such that \(PAP^{-1}\) is a diagonal matrix.
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HPAS 2020 Maths Optional Paper-1 Question 3(a)
Find the locus of the point of intersection of three mutually perpendicular tangent planes to \(ax^2+by^2+cz^2=1\).
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HPAS 2020 Maths Optional Paper-1 Question 3(b)
Show that \(\lim_{x\to0} \cos\frac{1}{x}\) does not exist.
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HPAS 2020 Maths Optional Paper-1 Question 3(c)
Evaluate \(\lim_{x\to\infty}(\sqrt{x^2+3x}-x)\) and \(\lim_{x\to0^{+}}(1-\sin x)^{1/x}\).
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HPAS 2020 Maths Optional Paper-1 Question 4(a)
State a set of sufficient conditions for a local maximum or minimum at a point for a twice continuously differentiable function \(f(x,y)\). Test the function \(f(x,y)=x^3+y^3-9xy+1\) for local maximum or minimum.
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HPAS 2020 Maths Optional Paper-1 Question 4(b)
Let \(f:\mathbb{R}^2 \to \mathbb{R}^2\) be defined by \(f(x,y)=(x^2+y^2, xy)\). Compute the total derivative of f at the point (1,2).
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HPAS 2020 Maths Optional Paper-1 Question 4(c)
Reduce the following equation to the standard form:
\[ 3x^2+5y^2+3z^2+2yz+2zx+2xy-4x-8z+5=0 \]
Find the nature of the conicoid, its center, and the equations of its axes.
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HPAS 2020 Maths Optional Paper-1 Question 5(a)
Find the volume of the solid region that is interior to both the sphere \(x^2+y^2+z^2=4\) and the cylinder \((x-1)^2+y^2=1\).
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HPAS 2020 Maths Optional Paper-1 Question 5(b)
A particle of mass m moves with a central attractive force \(\mu(r^5-c^4r)\) towards the origin. It is projected from an apse at distance c with velocity \(\sqrt{\frac{2\mu}{3}}c^3\). Show that the equation of the central orbit is \(x^4+y^4=c^4\).
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HPAS 2020 Maths Optional Paper-1 Question 6(a)
State the statement of Stokes’ theorem and verify it for the line integral
\[ \oint_C [(x+y)dx + (2x-z)dy + (y+z)dz] \]
where C is the boundary of the triangle with vertices (2,0,0), (0,3,0), and (0,0,6).
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HPAS 2020 Maths Optional Paper-1 Question 6(b)
Define a wrench of a system of forces. Three forces P, Q, and R act along the three straight lines \(x=0, y-z=a\); \(y=0, z-x=a\); and \(z=0, x-y=a\) respectively. Find the vector symmetrical equation of the central axis and the pitch of the equivalent wrench.
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HPAS 2020 Maths Optional Paper-1 Question 7(a)
The tangential acceleration of a particle moving along a circle of radius a is \(\lambda\) times the normal acceleration. If its speed at a certain time is u, then prove that it will return to the same point after a time \(\frac{a}{\lambda u}(1-e^{-2\pi\lambda})\).
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HPAS 2020 Maths Optional Paper-1 Question 7(b)
Solve the following differential equation:
\(x\frac{d^2y}{dx^2} – \frac{dy}{dx} = x^2e^x\), for \(x>0\).
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HPAS 2020 Maths Optional Paper-1 Question 8(a)
A heavy uniform rod rests with one end against a smooth vertical wall and with a point in its length resting on a smooth peg. Find the position of equilibrium, and show that it is unstable.
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HPAS 2020 Maths Optional Paper-1 Question 8(b)
Apply the method of power series to solve the following differential equation:
\[ (1-x^2)\frac{d^2y}{dx^2} – 2x\frac{dy}{dx} + 12y = 0 \]
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