List of top Mathematics Questions on Real Analysis asked in IIT JAM MA

Let \( f: \mathbb{R} \to \mathbb{R} \) be the function given by \[ f(x) = \begin{cases} \cos x, & \text{if } x \in \mathbb{Q} \\ 0, & \text{if } x \notin \mathbb{Q} \end{cases} \]
Which of the following statements is/are TRUE?

Find the radius of convergence of the series
\[ \sum_{n=0}^{\infty} \frac{\binom{n}{6}^2}{(2n)!} x^n \]

Find the number of automorphisms of the cyclic group $\mathbb{Z}_n$ for $n = 30$.

Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \] is convergent or divergent.

Let $G = P(N)$, where the operation is
\[ A \Delta B = A \cup B - A \cap B \] Which of the following is true?

Find the radius of convergence of the series \[ \sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}\, x^n. \]

Consider the four functions from \(\mathbb{R}\) to \(\mathbb{R}\): \[ f_1(x) = x^4 + 3x^3 + 7x + 1, \quad f_2(x) = x^3 + 3x^2 + 4x, \quad f_3(x) = \arctan(x), \] and \[ f_4(x) = \begin{cases} x, & \text{if } x \notin \mathbb{Z}, \\ 0, & \text{if } x \in \mathbb{Z}. \end{cases} \] Which of the following subsets of \(\mathbb{R}\) are open?

Consider the function \[ f(x) = \begin{cases} 1, & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cup \{0\}, \\ 1 - \frac{1}{p}, & \text{if } x = \frac{n}{p},\ n \in \mathbb{Z}\setminus\{0\},\ p \in \mathbb{N},\ \gcd(n,p)=1 . \end{cases} \] Then

The value of \[ \lim_{n \to \infty} \int_0^1 e^{x^2} \sin(nx)\,dx \] is _________.

Consider those continuous functions \( f : \mathbb{R} \to \mathbb{R} \) that have the property that for every \(x \in \mathbb{R},\) \[ f(x) \in \mathbb{Q} \text{ if and only if } f(x + 1) \notin \mathbb{Q}. \] The number of such functions is _________.

Consider the set \[ A = \{ a \in \mathbb{R} : x^2 = a(a+1)(a+2) \text{ has a real root} \}. \] The number of connected components of \(A\) is _________.

Which of the following subsets of \(\mathbb{R}\) are connected?

Let \( f : [0,1] \to [0,1] \) be a non-constant continuous function such that \( f \circ f = f. \) Define \[ E_f = \{x \in [0,1] : f(x) = x\}. \] Then

For every \( n \in \mathbb{N} \), let \( f_n : \mathbb{R} \to \mathbb{R} \) be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]

Which one of the following subsets of \(\mathbb{R}\) has a non-empty interior?

Let \( P \) and \( Q \) be two non-empty disjoint subsets of \( \mathbb{R} \). Which of the following is (are) FALSE?

Which of the following subsets of \( \mathbb{R} \) is (are) connected?

Let \( S \) be a subset of \( \mathbb{R} \) such that 2018 is an interior point of \( S \). Which of the following is (are) TRUE?