Let \( f(x) = \sqrt{x} + \alpha x, \; x > 0 \) and \[ g(x) = a_0 + a_1(x - 1) + a_2(x - 1)^2 \] be the sum of the first three terms of the Taylor series of \( f(x) \) around \( x = 1 \). If \( g(3) = 3 \), then \( \alpha \) is .............
Consider the expansion of the function \( f(x) = \dfrac{3}{(1 - x)(1 + 2x)} \) in powers of \( x \), valid in \( |x| < \dfrac{1}{2}. \) Then the coefficient of \( x^4 \) is ................
Let \( S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\} \) and \( f : S \to \mathbb{R} \) be defined by \( f(x) = \frac{1}{x}. \) Then \[ \max \left\{ \delta : |x - \tfrac{1}{3}| < \delta \Rightarrow |f(x) - f(\tfrac{1}{3})| < 1 \right\} \] is ............. (rounded off to two decimal places).
Let \( f : [0, 1] \to \mathbb{R} \) be a continuous function such that \( f\left(\dfrac{1}{2}\right) = -\dfrac{1}{2} \) and \[ |f(x) - f(y) - (x - y)| \le \sin(|x - y|^2) \] for all \( x, y \in [0, 1]. \) Then \( \int_0^1 f(x) \, dx \) is
Let \( a \in \mathbb{R} \). If \( f(x) = \begin{cases} (x + a)^2, & x \leq 0 \\ (x + a)^3, & x > 0 \end{cases} \), then
The radius of convergence of the power series \( \displaystyle \sum_{n=1}^{\infty} \left( \dfrac{n+2}{n} \right)^{n^2} x^n \) is
Let \( s_n = 1 + \dfrac{(-1)^n}{n}, \, n \in \mathbb{N}. \) Then the sequence \(\{s_n\}\) is
Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( f, f', f'' \) are continuous with \( f > 0, f' > 0, f'' > 0. \) Then \( \displaystyle \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} \) is ............
Let \( f \) be a real-valued function of a real variable, such that \( |f^{(n)}(0)| \leq K \) for all \( n \in \mathbb{N} \), where \( K > 0. \) Which of the following is/are true?
If \( s_n = \dfrac{(-1)^n}{2^n + 3} \) and \( t_n = \dfrac{(-1)^n}{4n - 1}, \, n = 0, 1, 2, \dots, \) then
Let \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \) be defined by \( f(x) = x + \dfrac{1}{x^3} \). On which of the following interval(s) is \( f \) one-to-one?
For \( x > -\dfrac{1}{2} \), let \( f_1(x) = \dfrac{2x}{1+2x} \), \( f_2(x) = \log_e(1 + 2x) \) and \( f_3(x) = 2x \). Then which one of the following is TRUE?
Let \( f : \mathbb{R} \to \mathbb{R} \) be a function and let \( J \) be a bounded open interval in \( \mathbb{R} \). Define \[ W(f, J) = \sup \{ f(x) | x \in J \} - \inf \{ f(x) | x \in J \} \] Which one of the following is FALSE?
For \( x \in \mathbb{R} \), let \( f(x) = \begin{cases} x^3 \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases} \) . Then which one of the following is FALSE?
Let \( a_n = \dfrac{(-1)^{n}}{\sqrt{1+n}} \) and let \( c_n = \sum_{k=0}^{n} a_{n-k} a_k \), where \( n \in \mathbb{N} \cup \{0\} \). Then which one of the following is TRUE?
Let \( a_n = n + \frac{1}{n} \), \( n \in \mathbb{N} \). Then the sum of the series \( \sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{a_{n+1}}{n!} \) is
Let \( a, b, c \in \mathbb{R} \). Which of the following values of \( a, b, c \) do NOT result in the convergence of the series \[ \sum_{n=3}^{\infty} \frac{a^n}{n^b (\log_e n)^c} ? \]