HPAS 2017 Maths Optional Paper-1 Question 1(a)
If A is a nilpotent matrix of index 2, show that \(A(I \pm nA) = A\) for any positive integer n.
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HPAS 2017 Maths Optional Paper-1 Question 1(b)
Examine the curve \(x=6t^2\), \(y=4t^3-3t\) for concavity and convexity.
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HPAS 2017 Maths Optional Paper-1 Question 1(c)
Find the degree and order of the following differential equation:
\[ \left|1+\left(\frac{dy}{dx}\right)^2\right|^{2/3} = \rho\frac{d^2y}{dx^2} \]
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HPAS 2017 Maths Optional Paper-1 Question 1(d)
If \(\vec{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), then show that the vector \(\vec{r}\) is an irrotational vector.
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HPAS 2017 Maths Optional Paper-1 Question 2(a)
Find the matrix representation of a linear transformation t on \(V_3(\mathbb{R})\) defined as \(t(x,y,z)=(2y+z, x-4y, 3x)\) corresponding to the basis \(B = \{(1,0,0), (0,1,0), (0,0,1)\}\).
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HPAS 2017 Maths Optional Paper-1 Question 2(b)
State and prove the Cauchy-Schwarz inequality.
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HPAS 2017 Maths Optional Paper-1 Question 3(a)
Find the curve on which the three points of intersection of the curve \(x^2y – xy^2 + xy + y^2 + x – y = 0\) with its asymptotes lie.
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HPAS 2017 Maths Optional Paper-1 Question 3(b)
Prove that a bounded function is not necessarily Riemann integrable.
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HPAS 2017 Maths Optional Paper-1 Question 4(a)
Find the condition that the straight line \(\frac{l}{r} = A\cos\theta + B\sin\theta\) may touch the circle \(r = 2a\cos\theta\).
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HPAS 2017 Maths Optional Paper-1 Question 4(b)
Find the equation of a sphere which passes through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) and has the smallest possible radius.
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HPAS 2017 Maths Optional Paper-1 Question 5(a)
Solve the following differential equation by the method of variation of parameters:
\[ (1-x)\frac{d^2y}{dx^2} + x\frac{dy}{dx} – y = (1-x)^2 \]
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HPAS 2017 Maths Optional Paper-1 Question 5(b)
For Bessel’s function \(J_n(x)\), show that:
\(2nJ_n(x) = x[J_{n-1}(x) + J_{n+1}(x)]\)
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HPAS 2017 Maths Optional Paper-1 Question 6(a)
Obtain the Serret-Frenet formulas.
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HPAS 2017 Maths Optional Paper-1 Question 6(b)
Verify Stokes’ theorem for the function \(\vec{F} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k}\), where C is the unit circle in the xy-plane bounding the hemisphere \(z = \sqrt{1-x^2-y^2}\).
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HPAS 2017 Maths Optional Paper-1 Question 7(a)
Five weightless rods of equal length are jointed together to form a rhombus ABCD with one diagonal BD. If a weight W is attached to C and the system is suspended from A, show that there is a thrust in BD equal to \(\frac{W}{\sqrt{3}}\).
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HPAS 2017 Maths Optional Paper-1 Question 7(b)
Two forces act, one along the line \(y=0, z=0\) and the other along the line \(x=0, z=c\). As the forces vary, show that the surface generated by the central axis of their equivalent wrench is \((x^2+y^2)z = cy^2\).
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HPAS 2017 Maths Optional Paper-1 Question 8(a)
A particle moves in a curve such that its tangential and normal accelerations are equal and the angular velocity of the tangent is constant. Find the path.
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HPAS 2017 Maths Optional Paper-1 Question 8(b)
A particle describes an ellipse under a force \(\frac{\mu}{(\text{distance})^2}\) towards a focus. If it was projected with velocity V from a point at a distance r from the center of force, show that its periodic time is
\[ \frac{2\pi}{\sqrt{\mu}} \left[ \frac{2}{r} – \frac{V^2}{\mu} \right]^{-3/2} \]
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