List of top Real Analysis Questions asked in IIT JAM MA

Suppose \[ a_n = \frac{3^n + 3}{5^n - 5} \quad \text{and} \quad b_n = \frac{1}{(1 + n^2)^{1/4}} \quad \text{for} \ n = 2, 3, 4, \ldots \] Then which one of the following is true?

Consider the series \[\sum_{n=1}^{\infty} n^m \left(\frac{1}{1 + \frac{1}{n^p}}\right)\] where \( m \) and \( p \) are real numbers. Under which of the following conditions does the above series converge?

Let \( T \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots + (-1)^{n+1} \frac{1}{n} + \ldots\]
and let \( S \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \sum_{n=1}^{\infty} a_n\]
where
\[ a_{3m-2} = \frac{1}{2m-1} , a_{3m-1} = 0, \text{ and } a_{3m} = \frac{-1}{4m} \text{ for } m \in \mathbb{N}.\]
Then which one of the following is true?

The value of
\[ \lim_{n \to \infty} \sum_{k=2}^n \frac{\sqrt{n} + 1 - \sqrt{n}}{k(\ln k)^2} \]
is equal to

The sum of the series
\(∑^∞_{ n=1} \frac{1}{(4n-3)(4n+1) }\) 
is equal to_____.(Round off to two decimal places)

Let r be the radius of convergence of the power series
\(\frac{1}{3}+\frac{x}{5}+\frac{x^2}{3^2}+\frac{x^3}{5^2}+\frac{x^4}{3^3}+\frac{x^5}{5^3}+\frac{x^6}{3^4}+\frac{x^7}{5^4}+...\)
Then the value of r² is equal to _________. (Rounded off to two decimal places)

Consider the system of linear equations\[ \begin{cases}x + y + t = 4, \\2x - 4t = 7, \\x + y + z = 5, \\x - 3y - z - 10t = \lambda,\end{cases}\]where \( x, y, z, t \) are variables and \( \lambda \) is a constant. Then which one of the following is true?

For t ∈ \(\R\), let [𝑡] denote the greatest integer less than or equal to t.
Let D = {(x, y) ∈ \(\R^2\) ∶ x2 + y2 < 4}. Let f : D → \(\R\) and g : D → \(\R\) be defined by f(0, 0) = g(0, 0) = 0 and
\(f(x,y)=[x^2+y^2]\frac{x^2y^2}{x^4+y^4},\ \ \ g(x,y)=[y^2]\frac{xy}{x^2+y^2}\)
for (x, y) ≠ (0, 0). Let E be the set of points of D at which both f and g are discontinuous. The number of elements in the set E is _________.

Let (x_n) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

A real-valued function y(x) defined on \(\R\) is said to be periodic if there exists a real number \(T\gt0\) such that y(x + T) = y(x) for all \(x\isin\R\). Consider the differential equation
\(\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)\)
where \(a\isin\R\) is a constant. 
Then which of the following is/are true?