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JEE Advance | PYQ

Consider the function \( f : (0,\infty) \to (-\infty,\infty) \) given by
\( f(x) = \sqrt{x}\,\log_e(x) - x + 1 \).
Hence which one of the following statements is TRUE?

Let \( S \) be the set of all twice differentiable functions \( f \) from \( \mathbb{R} \) to \( \mathbb{R} \) such that \( \frac{d^{2}f}{dx^{2}}(x)>0 \) for all \( x\in(-1,1) \). For \( f\in S \), let \( X_{f} \) be the number of points \( x\in(-1,1) \) for which \( f(x)=x \). Then which of the following statements is(are) true?

Let the function \( f : \mathbb{R} \to \mathbb{R} \) be defined by
\( f(x) = \frac{\sin x}{e^{\pi x}} \left( \frac{x^{2023} + 2024x + 2025}{x^{2} - x + 3} \right) + \frac{2}{e^{\pi x}} \left( \frac{x^{2023} + 2024x + 2025}{x^{2} - x + 3} \right) \).
Then the number of solutions of \( f(x) = 0 \) in \( \mathbb{R} \) is ________.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^{2} - 3x - 6}{x^{2} + 2x + 4} \). Then which of the following statements is (are) TRUE?

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by

Then which of the following statements is (are) TRUE?

Let \( \alpha=\displaystyle\sum_{k=1}^{\infty}\sin^{2k}\!\left(\frac{\pi}{6}\right) \). Let \( g:[0,1]\to\mathbb{R} \) be the function defined by \( g(x)=2^{\alpha x}+2^{\alpha(1-x)} \). Then, which of the following statements is/are TRUE?

Let \( f_1 : (0,\infty) \to \mathbb{R} \) and \( f_2 : (0,\infty) \to \mathbb{R} \) be defined by
\( f_1(x) = \displaystyle\int\limits_{0}^{x} \prod_{j=1}^{21} (t - j)^{j} \, dt, \quad x > 0 \) and \(f_2(x) = 98 (x-1)^{50} - 600 (x-1)^{49} + 2450, \quad x > 0, \)
Where, for any positive integer \( n \) and real numbers \( a_1, a_2, \ldots, a_n \), \( \prod_{i=1}^{n} a_i \) denotes the product of \( a_1, a_2, \ldots, a_n \). Let \( m_i \) and \( n_i \) respectively, denote the number of points of local minima and the number of points of local maximum of function \( f_i, \, i=1,2, \) in the interval \( (0,\infty) \).
The value of \( \displaystyle 6m_2 + 4n_2 + 8m_2n_2 \) is ________.

Let \( f_1 : (0,\infty) \to \mathbb{R} \) and \( f_2 : (0,\infty) \to \mathbb{R} \) be defined by
\( f_1(x) = \displaystyle\int\limits_{0}^{x} \prod_{j=1}^{21} (t - j)^{j} \, dt, \quad x > 0 \) and \(f_2(x) = 98 (x-1)^{50} - 600 (x-1)^{49} + 2450, \quad x > 0, \)
Where, for any positive integer \( n \) and real numbers \( a_1, a_2, \ldots, a_n \), \( \prod_{i=1}^{n} a_i \) denotes the product of \( a_1, a_2, \ldots, a_n \). Let \( m_i \) and \( n_i \) respectively, denote the number of points of local minima and the number of points of local maximum of function \( f_i, \, i=1,2, \) in the interval \( (0,\infty) \).
The value of \( 2m_1 + 3n_1 + m_1 n_1 \) is ________.