List of top Real Analysis Questions asked in GATE MA

Let \( D = \{(x, y) \in \mathbb{R}^2 : 0 \leq x \leq 2,\ 0 \leq y \leq 2\} \) and let \( f(t) \) denote the smallest integer greater than or equal to \( t \). Then the value of the integral
\[ \iint_D f(x+y)\, dx\, dy \]
is ______.

Let \( \alpha = \iint_S \vec{F} \cdot \hat{n} \, dS \), where \( \vec{F} = (2x + 3z)\hat{i} + (xz - y)\hat{j} + (y^2 + 2z)\hat{k} \) and \( S \) is the sphere with centre at \( (3, -1, 2) \) and radius \( 9 \). Here \( \hat{n} \) is the unit normal drawn outward and \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors.

Then the value of \( \dfrac{1}{36\pi}\alpha \) is equal to ______. (Answer in integer)

Let \[ \alpha = \lim_{n \to \infty} \int_0^{1} \frac{n^2 + (\sin e^x)^n}{7n^2 + x^8} \, dx. \] The value of \(14\alpha\) is equal to ______. (Answer in integer)

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be a function defined as follows:
\[ f(x,y) = \begin{cases} \dfrac{x\sqrt{x^2+y^2}}{|x|} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{cases} \]
Which of the following statements is/are TRUE?

Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined as
\[ f(x, y) = (e^x \cos y,\ e^x \sin y) \]
Which one of the following is TRUE?

Let \( I \) be the integral defined as follows:
\[ I = \int_0^1 \int_0^{\sqrt{y}} dx\, dy \;+\; \int_1^2 \int_{\sqrt{y-1}}^{1} dx\, dy \]
If the order of integration is changed, then which one of the following is the correct expression for \( I \)?

Let \( C \) denote the Cantor set and \( f: [0,1] \to \mathbb{R} \) be defined as follows:
\[ f(x) = \begin{cases} x^{2026} & \text{for } x \in C \\ \cos(\pi x) & \text{for } x \in \left[0, \dfrac{1}{2}\right] \setminus C \\ \sin(\pi x) & \text{for } x \in \left[\dfrac{1}{2}, 1\right] \setminus C \end{cases} \]
The value of the Lebesgue integral of \( f(x) \) over the interval \( [0,1] \) is equal to