# GATE Real Analysis 2021 Solutions

GATE Real Analysis 2021 Solutions, you must be looking for if you are a GATE aspirant. You are in the right place and you’ll have here the solutions of GATE 2021 on the topic “Real Analysis.” 2021 GATE question paper consists of questions from Metric Space, Function of Several Variable, Directional Derivative, and Uniform Convergence. In the article below you’ll get the latest previous year GATE question paper solution. ## GATE Real Analysis 2021 Solutions

GATE Real Analysis 2021 Problem 1:
Let f_{n}:[0,10]\rightarrow \mathbb{R} be given by f_{n}(x)=n x^{3} e^{-n x} for n=1,2,3, \ldots .
Consider the following statements:
P: \left(f_{n}\right) is equicontinuous on [0,10].
Q: \sum_{n=1}^{\infty} f_{n} does NOT converge uniformly on [0,10].
Then
(A) both \mathrm{P} and \mathrm{Q} are TRUE
(B) \mathrm{P} is TRUE and \mathrm{Q} is FALSE
(C) P is FALSE and Q is TRUE
(D) both \mathrm{P} and \mathrm{Q} are FALSE  GATE Real Analysis 2021 Problem 2:
Let f: \mathbb{R}^{2} \rightarrow \mathbb{R} be given by
\begin{array}{cl} f(x, y)= & \left\{\begin{array}{cl} \sqrt{x^{2}+y^{2}} \sin \left(y^{2} / x\right) & \text { ir } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. \end{array}
Consider the following statements:
P: f is continuous at (0,0) but f is NOT differentiable at (0,0).
Q: The directional derivative D_{u} f(0,0) of f at (0,0) exists in the direction of every unit vector u \in \mathbb{R}^{2}.
Then
(A) both \mathrm{P} and \mathrm{Q} are TRUE
(B) \mathrm{P} is TRUE and \mathrm{Q} is FALSE
(C) P is FALSE and Q is TRUE
(D) both \mathrm{P} and \mathrm{Q} are FALSE  GATE Real Analysis 2021 Problem 3:
Let f:\left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R} be given by f(x)=\frac{\pi}{2}+x-\tan ^{-1} x. Consider the following statements:

P: |f(x)-f(y)|<|x-y| for all x, y \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)
Q: f has a fixed point.
(A) both \mathrm{P} and \mathrm{Q} are TRUE
(B) \mathrm{P} is TRUE and \mathrm{Q} is FALSE
(C) P is FALSE and Q is TRUE
(D) both \mathrm{P} and \mathrm{Q} are FALSE GATE Real Analysis 2021 Problem 4:
Let f: \mathbb{R}^{2} \rightarrow \mathbb{E} be given by f(x, y)=4 x y-2 x^{2}-y^{4} . Then f has
(A) a point of local maximum and a saddle point
(B) a point of local minimum and a saddle point
(C) a point of local maximum and a point of local minimum GATE RA 2021 Problem 5:
Consider the following statements:
P: d_{1}(x, y)=\left|\log \left(\frac{x}{y}\right)\right| is a metric on (0,1).
Q: d_{2}(x, y)=\left\{\begin{array}{cl}|x|+|y| . & \text { if } x \neq y \\ 0, & \text { if } x=y\end{array} \quad\right. is a mefric on (0,1).
Then
(A) both \mathrm{P} and \mathrm{Q} are TRUE
(B) \mathrm{P} is TRUE and \mathrm{Q} is FALSE
(C) P is FALSE and Q is TRUE
(D) both \mathrm{P} and \mathrm{Q} are FALSE  I Hope, these GATE solutions have helped you, please comment on which previous year gate question paper solutions. mathdart.com brings you many helpful notes for GATE preparation.