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?

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 \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]

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?

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?}

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 \( T \)?

Let \( L^2([-1, 1]) \) denote the space of all real-valued Lebesgue square-integrable functions on \( [-1, 1] \), with the usual norm \( \|\cdot\| \). Let \( P_1 \) be the subspace of \( L^2([-1, 1]) \) consisting of all the polynomials of degree at most 1. Let \( f \in L^2([-1, 1]) \) be such that \[ \|f\|^2 = \frac{18}{5}, \quad \int_{-1}^1 f(x) dx = 2, \quad {and} \quad \int_{-1}^1 xf(x) dx = 0. \] Then \[ \inf_{g \in P_1} \|f - g\|^2 = ........... \]

The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).

Let \( K \subseteq \mathbb{C} \) be the field extension of \( \mathbb{Q} \) obtained by adjoining all the roots of the polynomial equation \( (x^2 - 2)(x^2 - 3) = 0 \). The number of distinct fields \( F \) such that \( \mathbb{Q} \subseteq F \subseteq K \) is equal to ……….. (answer in integer).

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).