HPAS 2018 Maths Optional Paper-1 Question 1(a)
Find the order and degree of the differential equation whose general solution is \(y^2 = 2c(x+\sqrt{c})\), where c is a positive parameter.
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HPAS 2018 Maths Optional Paper-1 Question 1(b)
How many solutions does the following system of linear equations have?
\[ \begin{cases} -x+5y = -1 \\ x-y = 2 \\ x+3y = 3 \end{cases} \]
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HPAS 2018 Maths Optional Paper-1 Question 1(c)
Find the intervals in which the function \(f(x) = 10 – 6x – 2x^2\) is strictly increasing or strictly decreasing.
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HPAS 2018 Maths Optional Paper-1 Question 1(d)
Find the unit outward normal vector at the point \((\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)\) for the surface \(x^2+y^2+z^2=1\).
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HPAS 2018 Maths Optional Paper-1 Question 2(a)
Two solutions of the ordinary differential equation \(y” – 2y’ + y = 0\) are \(e^x\) and \(5e^x\). Is \(y = Ae^x + B(5e^x)\) the general solution of the differential equation?
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HPAS 2018 Maths Optional Paper-1 Question 2(b)
If the integrating factor of the differential equation \((x^7y^2+3y)dx + (3x^8y-x)dy=0\) is \(x^m y^n\), then find the values of m and n.
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HPAS 2018 Maths Optional Paper-1 Question 3(a)
The direction of a vector A is radially outward from the origin, and its magnitude is \(|\vec{A}| = kr^n\), where \(r^2 = x^2+y^2+z^2\) and k is a constant. Find the value of n for which \(\nabla \cdot \vec{A} = 0\).
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HPAS 2018 Maths Optional Paper-1 Question 3(b)
If P, Q, and R are three points having Cartesian coordinates (3, -2, -1), (1, 3, 4), and (2, 1, -2) respectively in the XYZ Cartesian plane, then find the distance from point P to the plane OQR, where O is the origin.
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HPAS 2018 Maths Optional Paper-1 Question 4(a)
Suppose a function f(x) satisfies the conditions: (i) \(f(0)=2\), \(f(1)=1\); (ii) f has a minimum value at \(x=5/2\); and (iii) \(f'(x) = 2ax+b\) for all x. Determine the constants a, b and the function f(x).
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HPAS 2018 Maths Optional Paper-1 Question 4(b)
Change the order of integration in the integral \(\iint f(x,y)dxdy\). The area of integration is enclosed by the curves \(y = x \tan\alpha\), \(y = \sqrt{a^2-x^2}\), \(x=0\) and \(x = a\cos\alpha\).
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HPAS 2018 Maths Optional Paper-1 Question 5(a)
Let U and V be vector spaces and \(T:U \to V\) be a surjective linear mapping. If \(\dim U=6\) and \(\dim V=3\), find \(\dim(\text{Ker } T)\).
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HPAS 2018 Maths Optional Paper-1 Question 5(b)
Let \(\mathbb{R}^3(\mathbb{R})\) be a vector space with respect to ordinary addition and scalar multiplication. Find the rank of the linear transformation \(T: \mathbb{R}^3 \to \mathbb{R}^3\) defined by \(T(x,y,z)=(y,0,z)\).
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HPAS 2018 Maths Optional Paper-1 Question 6(a)
Let A be a \(3\times3\) square matrix with eigenvalues 1, -1, and 0. Find the value of \(\det(I+A^{100})\).
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HPAS 2018 Maths Optional Paper-1 Question 6(b)
Let \(\mathbb{R}^4(\mathbb{R})\) be a vector space and let S be its subspace spanned by the vectors (1,2,3,0), (2,3,0,1), and (3,0,1,2). Find the dimension of the quotient space \(\mathbb{R}^4/S\).
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HPAS 2018 Maths Optional Paper-1 Question 7(a)
In a finite-dimensional inner product space V, let \(\{w_1, w_2, \dots, w_n\}\) be an orthonormal subset of V such that
\[ \sum_{i=1}^{n} |\langle w_i, v \rangle|^2 = ||v||^2 \]
for all \(v \in V\). Find a basis of V.
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HPAS 2018 Maths Optional Paper-1 Question 7(c)
Let \(V=\mathbb{R}^2\) be a finite-dimensional standard inner product space. Prove that \{(-1, 0), (0, -1)\} forms an orthonormal basis of V.
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HPAS 2018 Maths Optional Paper-1 Question 8(a)
With usual notations, prove that the angular acceleration in the direction of motion of a point moving in a plane is
\[ \frac{v}{\rho}\frac{dv}{ds} – \frac{v^2}{\rho^2}\frac{d\rho}{ds} \]
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HPAS 2018 Maths Optional Paper-1 Question 8(b)
Three forces P, Q, and R act along the sides of the triangle taken in order, formed by the lines \(x+y=1\), \(y-x=1\), and \(y=2\). Find the equation of the line of action of their resultant.
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