Let \( {F}_3 \) be the field with exactly 3 elements. The number of elements in \( GL_2({F}_3) \) is equal to (answer in integer):
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- Let \( T, S : {R}^4 \to {R}^4 \) be two non-zero, non-identity \( {R} \)-linear transformations. Assume \( T^2 = T \). Which of the following is/are true?
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Let \( p_1<p_2 \) be the two fixed points of the function \( g(x) = e^x - 2 \), where \( x \in {R} \). For \( x_0 \in {R} \), let the sequence \( (x_n)_{n \geq 1} \) be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?
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- The problem: Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function such that \[ f(x, y) = \begin{cases} \left( 1 - \cos\left(\frac{x^2}{y^2}\right) \right) \sqrt{x^2 + y^2}, & \text{if } y \neq 0, x \in \mathbb{R}, \\ 0, & \text{otherwise.} \end{cases} \] Which of the following is/are correct?
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For an integer \( n \), let \( f_n(x) = xe^{-nx }\), where \( x \in [0, 1] \). Let \( S := \{f_n : n \geq 1\} \). Consider the metric space \( (C([0, 1]), d) \), where \[ d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|, \quad f, g \in C([0, 1]). \] Which of the following statement(s) is/are true?}
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- Let \( T : {R}^4 \to {R}^4 \) be an \( {R} \)-linear transformation such that 1 and 2 are the only eigenvalues of \( T \). Suppose the dimensions of \({Kernel}(T - I_4)\) and \({Range}(T - 2I_4)\) are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of \( T \)?
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Let \( \{(a, b) : a, b \in {R, a<b \} }\) be a basis for a topology \( \tau \) on \( {R} \). Which of the following is/are correct?
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- mathematical reasoning
- Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem \[ y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \] with the boundary conditions \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]
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- Core Mathematics
Let \( H \) be the subset of \( S_3 \) consisting of all \( \sigma \in S_3 \) such that \[ {Trace}(A_1 A_2 A_3) = {Trace}((A_1 \sigma(A_2) A_3)), \] for all \( A_1, A_2, A_3 \in M_2(\mathbb{C}) \). The number of elements in \( H \) is equal to ……… (answer in integer).
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- Core Mathematics
Let \( k \in \mathbb{R} \) and \( D = \{(r, \theta) : 0<r<2, 0<\theta<\pi\ \). Let \( u(r, \theta) \) be the solution of the following boundary value problem \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad (r, \theta) \in D, \] \[ u(r, 0) = u(r, \pi) = 0, \quad 0 \leq r \leq 2, \] \[ u(2, \theta) = k \sin(2\theta), \quad 0<\theta<\pi. \] If \( u\left(\frac{1}{4}, \frac{\pi}{4}\right) = 2 \), then the value of \( k \) is equal to ………. (round off to TWO decimal places).
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- Let \( A \) be a \( 2 \times 2 \) non-diagonalizable real matrix with a real eigenvalue \( \lambda \) and \( v \) be an eigenvector of \( A \) corresponding to \( \lambda \). Which one of the following is the general solution of the system \( y' = Ay \) of first-order linear differential equations?
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- Time Speed and Distance