HPAS 2023 Maths Optional Paper-1 Question 1(a)
Show that the roots of the equation \(P_n(x)=0\) are real and lie between -1 and 1, where \(P_n(x)\) is the Legendre’s polynomial of degree \(n\), and \(n\) is a positive integer.
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HPAS 2023 Maths Optional Paper-1 Question 1(b)
Let the function \(f\) be continuous on the real line \(\mathbb{R}\). Then show that the set \(A = \{x : f(x) = 0\}\) is closed.
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HPAS 2023 Maths Optional Paper-1 Question 1(c)
For what values of \(a\) and \(b\) is the vector field \(\vec{F} = (x+z)\mathbf{i} + a(y+z)\mathbf{j} + b(x+y)\mathbf{k}\) a conservative field?
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HPAS 2023 Maths Optional Paper-1 Question 1(d)
Give an example of a diagonalizable matrix that does not have distinct eigen values.
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HPAS 2023 Maths Optional Paper-1 Question 1(e)
A particle executes Simple Harmonic Motion with a period of 10 seconds and an amplitude of 5 cm. Calculate the maximum velocity.
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HPAS 2023 Maths Optional Paper-1 Question 2(a)
Using the Cauchy-Schwarz inequality, show that the cosine of an angle has an absolute value of at most 1.
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HPAS 2023 Maths Optional Paper-1 Question 2(b)
Let V be a finite-dimensional vector space and W be a subspace of V. Show that \(\dim A(W) = \dim V – \dim W\), where \(A(W)\) is the annihilator of W.
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HPAS 2023 Maths Optional Paper-1 Question 2(c)
Find the equations of the lines in which the plane \(2x+y-z=0\) cuts the cone \(4x^2 – y^2 + 3z^2 = 0\).
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HPAS 2023 Maths Optional Paper-1 Question 3(a)
Show that the radius of curvature of the lemniscate \((x^2+y^2)^2 = a^2(x^2-y^2)\) at any point where the tangent is parallel to the x-axis, is \(\frac{\sqrt{2}a}{3}\).
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HPAS 2023 Maths Optional Paper-1 Question 3(b)
Evaluate
\[ \lim_{x\to0} x^m (\log_e x)^n \]
where \(m\) and \(n\) are positive integers.
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HPAS 2023 Maths Optional Paper-1 Question 3(c)
Let \(T: V \to W\) be a linear transformation. Then show that \(\text{Rank}(T) + \text{Nullity}(T) = \dim(V)\).
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HPAS 2023 Maths Optional Paper-1 Question 4(a)
If
\[ u(x,y) = \sin^{-1}\left(\left(\frac{x^{1/3}+y^{1/3}}{x^{1/2}+y^{1/2}}\right)^{1/2}\right) \]
then show that
\[ x^2\frac{\partial^2 u}{\partial x^2} + 2xy\frac{\partial^2 u}{\partial x \partial y} + y^2\frac{\partial^2 u}{\partial y^2} = \frac{1}{144}\tan u(13 + \tan^2 u) \]
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HPAS 2023 Maths Optional Paper-1 Question 4(b)
Find the magnitude and the equations of the shortest distance between the lines:
\[ \frac{x}{2} = \frac{-y}{3} = \frac{z}{1} \quad \text{and} \quad \frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2} \]
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HPAS 2023 Maths Optional Paper-1 Question 5(a)
Determine the general and singular solutions of the differential equation \(9p^2(2-y)^2 = 4(3-y)\), where \(p = \frac{dy}{dx}\).
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HPAS 2023 Maths Optional Paper-1 Question 5(b)
Solve the differential equation
\[ x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + \frac{xy}{2} = 0 \]
in terms of Bessel functions.
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HPAS 2023 Maths Optional Paper-1 Question 5(c)
Using the method of variation of parameters, solve the differential equation \((D^2 – 2D + 2)y = e^x \tan x\), where \(D = \frac{d}{dx}\).
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HPAS 2023 Maths Optional Paper-1 Question 6(a)
Given \(\vec{F} = y\mathbf{i} – z^3\mathbf{j} + x^2\mathbf{k}\), use Stokes’s theorem to evaluate \(\int_C \vec{F} \cdot d\vec{r}\), where C is the boundary of the area S formed by the part of the plane \(x+4y+z=4\) that lies in the first octant. The integration around the boundary C is in the clockwise direction.
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HPAS 2023 Maths Optional Paper-1 Question 6(b)
Find the directional derivative of \(f(x,y,z) = x^2 + 3y^2 + 2z^2\) in the direction of the vector \(2\mathbf{i} – \mathbf{j} – 2\mathbf{k}\) and determine its value at the point (1, -3, 2).
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HPAS 2023 Maths Optional Paper-1 Question 7(a)
Let V be the vector space of real-valued functions \(y=f(x)\) satisfying
\[ \frac{d^3y}{dx^3} – 6\frac{d^2y}{dx^2} + 11\frac{dy}{dx} – 6y = 0 \]
Then show that V is a 3-dimensional vector space over \(\mathbb{R}\).
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HPAS 2023 Maths Optional Paper-1 Question 7(b)
Three forces P, Q, and R act on a particle and keep it in equilibrium. If the angle between P and Q, and between Q and R, is \(120^\circ\) each, then show that \(P=Q=R\).
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HPAS 2023 Maths Optional Paper-1 Question 8(a)
A particle performing Simple Harmonic Motion has a mass of 2.5 gm and a frequency of vibration of 10 Hz. It is oscillating with an amplitude of 2 cm. Calculate the total energy of the particle.
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HPAS 2023 Maths Optional Paper-1 Question 8(b)
The motion of a particle under the influence of a central force is described by \(r = a \sin\theta\). Find an expression for the force.
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