Real Analysis – Mathematics Optional PYQs (2008–2026)

Here are all 89 Real Analysis questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 6 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

2.1] Real number system and sequences

13 questions

1. Prove that the sequence $\left(u_n\right)$ defined by the recursion formula $u_{n+1}=\sqrt{7+u_n}$, $u_1=\sqrt{7}$, converges to the positive root of $x^2-x-7=0$. [2026/15m/2b]

2. Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy condition? [2025/15m/2a]

3. Using Cauchy’s general principle of convergence, examine the convergence of the sequence $⟨f_n⟩$, where $f_n=1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}$. [2024/15m/2a]

4. Prove that the oscillation of a real-valued bounded function $f$ defined on $[a,b]$ is the supremum of the set $\{|f\left(x_1\right)-f\left(x_2\right)|:x_1,x_2\in [a,b]\}$. [2023/15m/4a]

5. Prove that the sequence $\left(a_n\right)$ satisfying the condition $|a_{n+1}-a_n|\le \alpha |a_n-a_{n-1}|$, $0<\alpha <1$ for all natural numbers $n\ge 2$, is a Cauchy sequence. [2020/10m/1c]

6. Let $x_1=2$ and $x_{n+1}=\sqrt{x_n+20}$, $n=1,2,3,\dots$. Show that the sequence $x_1,x_2,x_3,\dots$ is convergent. [2017/10m/1a]

7. Two sequences $\{x_n\}$ and $\{y_n\}$ are defined inductively by the following : $x_1=\frac{1}{2},y_1=1\mathrm{\text{and}}{x}_n=\sqrt{x_{n-1}{y}_{n-1}},n=2,3,4,\dots$ $\frac{1}{y_n}=\frac{1}{2}(\frac{1}{x_n}+\frac{1}{y_{n-1}}),n=2,3,4,\dots$ Prove that $x_{n-1}<x_n<y_n<y_{n-1},n=2,3,4,\dots$ and deduce that both the sequences converge to the same limit $l$, where $\frac{1}{2}<l<1$. [2016/10m/1c]

8. Show that every open subset of $ℝ$ is a countable union of disjoint open intervals. [2013/14m/2d]

9. Discuss the convergence of the sequence $\{x_n\}$ where $x_n=\frac{sin\left(\frac{n\pi }{2}\right)}{8}.$ [2010/12m/1c]

10. Define $\{x_n\}$ by $x_1=5$ and $x_{n+1}=\sqrt{4+x_n}$ for $n>1$. Show that the sequence converges to $\frac{(1+\sqrt{17})}{2}$. [2010/12m/1d]

11. Show that a bounded infinite subset of $ℝ$ must have a limit point. [2009/15m/3d]

12. Let $T=\{\frac{1}{n},n\in N\}\cup \{1+\frac{3}{2n},n\in N\}\cup \{6-\frac{1}{3n},n\in N\}.$ Find derived set $T\prime$ of $T$ and find Supremum of $T$ and greatest number of $T$. [2008/6m/1c(ii)]

13. Prove that the sets $A=[0,1]$, $B=]0,1[$ are equivalent sets. [2008/6m/3d(i)]

2.2] Series

11 questions

1. Examine whether the series ${\sum }_{n=1}^{\infty }\frac{{(-1)}^{n-1}}{n}$ is absolutely or conditionally convergent. [2025/10m/1c]

2. Test the convergence of the series ${\sum }_{n=1}^{\infty }\frac{1.3.5\dots (2n-1)}{2.4.6\dots \left(2n\right)}\cdot \frac{x^{2n+1}}{(2n+1)},x>0$ [2023/10m/1c]

3. Test for convergence or divergence of the series $x+\frac{2^2{x}^2}{2!}+\frac{3^3{x}^3}{3!}+\frac{4^4{x}^4}{4!}+\frac{5^5{x}^5}{5!}+\cdots (x>0)$ [2022/15m/4b]

4. Find the range of $p(>0)$ for which the series : $\frac{1}{{(1+a)}^p}-\frac{1}{{(2+a)}^p}+\frac{1}{{(3+a)}^p}-\dots ,a>0,$ is (i) absolutely convergent and (ii) conditionally convergent. [2018/10m/1d]

5. Let ${\sum }_{n=1}^{\infty }{x}_n$ be a conditionally convergent series of real numbers. Show that there is a rearrangement ${\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}$ of the series ${\sum }_{n=1}^{\infty }{x}_n$ that converges to 100. [2017/20m/4c]

6. Show that the series ${\sum }_{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n+1}$ is conditionally convergent. (If you use any theorem(s) to show it, then you must give a proof of that theorem(s).) [2016/15m/2a]

7. Test the convergence and absolute convergence of the series ${\sum }_{n=1}^{\infty }{(-1)}^{n+1}\frac{n}{n^2+1}$. [2015/10m/1c]

8. Show that the series ${\sum }_{n=1}^{\infty }{\left(\frac{\pi }{\pi +1}\right)}^n{n}^6$ is convergent. [2012/12m/1e]

9. Show that the series : ${\left(\frac{1}{3}\right)}^2+{\left(\frac{1\cdot 4}{3\cdot 6}\right)}^2+\cdots \cdots +{\left(\frac{1\cdot 4\cdot 7\cdots \cdots (3n-2)}{3\cdot 6\cdot 9\cdots \cdots 3n}\right)}^2+\cdots \cdots$ converges. [2009/15m/2c]

10. Discuss the convergence of the series $\frac{x}{2}+\frac{1\cdot 3}{2\cdot 4}{x}^2+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}{x}^3+\cdots \cdots ,x>0.$ [2008/15m/2c]

11. Show that the series $\sum \frac{1}{n(n+1)}$ is equivalent to $\frac{1}{2}{\prod }_2^{\infty }(1+\frac{1}{n^2-1}).$ [2008/15m/2d]

2.3] Continuity and uniform continuity

15 questions

1. Find the points of discontinuity of the function $f:(0,1)\to ℝ$ defined by $f\left(x\right)={lim}_{n\to \infty }\frac{{(1+sin\frac{\pi }{x})}^n-1}{{(1+sin\frac{\pi }{x})}^n+1},x\in (0,1).$ [2026/10m/1c]

2. Find the maximum and minimum values of $f\left(x\right)=x^3-9x^2+26x-24$ for $0\le x\le 1$. [2021/15m/2a]

3. Prove that the function $f\left(x\right)=sin{x}^2$ is not uniformly continuous on the interval $[0,\infty [$. [2020/15m/2b]

4. Show that if a function $f$ defined on an open interval $(a,b)$ of $ℝ$ is convex, then $f$ is continuous. Show, by example, if the condition of open interval is dropped, then the convex function need not be continuous. [2018/15m/2c]

5. Find the supremum and the infimum of $\frac{x}{sinx}$ on the interval $(0,\frac{\pi }{2}]$. [2017/10m/1c]

6. For the function $f:(0,\infty )\to ℝ$ given by $f\left(x\right)=x^{2}sin\frac{1}{x},0<x<\infty ,$ show that there is a differentiable function $g:ℝ\to ℝ$ that extends $f$. [2016/10m/1b]

7. Let $f:ℝ\to ℝ$ be a continuous function such that ${lim}_{x\to +\infty }f\left(x\right)$ and ${lim}_{x\to -\infty }f\left(x\right)$ exist and are finite. Prove that $f$ is uniformly continuous on $ℝ$. [2016/15m/4b]

8. Let $S=(0,1]$ and $f$ be defined by $f\left(x\right)=\frac{1}{x}$ where $0<x\le 1$ (in $ℝ$). Is $f$ uniformly continuous on $S$ ? Justify your answer. [2011/12m/1b]

9. Define the function $f\left(x\right)=\{\begin{array}{cc}\hfill x^{2}sin\frac{1}{x},\hfill & \hfill \mathrm{\text{if}}x\ne 0\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{if}}x=0\hfill \end{array}$ Find $f\prime \left(x\right)$. Is $f\prime \left(x\right)$ continuous at $x=0$ ? Justify your answer. [2010/15m/2c]

10. State Rolle’s theorem. Use it to prove that between two roots of $e^{x}cosx=1$ there will be a root of $e^{x}sinx=1$. [2009/12m/1c]

11. Let $f\left(x\right)=\{\begin{array}{cc}\hfill \frac{\left|x\right|}{2}+1\hfill & \hfill \mathrm{\text{if}}x<1\hfill \\ \hfill \frac{x}{2}+1\hfill & \hfill \mathrm{\text{if}}1\le x<2\hfill \\ \hfill -\frac{\left|x\right|}{2}+1\hfill & \hfill \mathrm{\text{if}}2\le x\hfill \end{array}$ What are the points of discontinuity of $f$, if any ? What are the points where $f$ is not differentiable, if any ? Justify yours answers. [2009/12m/1d]

12. Show that if $f:[a,b]\to ℝ$ is a continuous function then $f([a,b])=[c,d]$ for some real numbers $c$ and $d$, $c\le d$. [2009/15m/2d]

13. For $x>0$, show $\frac{x}{1+x}<log(1+x)<x$. [2008/6m/1c(i)]

14. If $f:ℝ\to ℝ$ is continuous and $f(x+y)=f\left(x\right)+f\left(y\right),$ for all $x,y\in ℝ$, then show that $f\left(x\right)=xf\left(1\right)$ for all $x\in ℝ$. [2008/12m/1d]

15. Prove that $\frac{tanx}{x}>\frac{x}{sinx},x\in ]0,\frac{\pi }{2}[.$ [2008/9m/3d(ii)]

2.4] Riemann integral

21 questions

1. A function $f$ is defined on $[0,1]$ by $f\left(x\right)=\{\begin{array}{cc}\hfill x^2,\hfill & \hfill \mathrm{\text{when}}x\mathrm{\text{is rational}}\hfill \\ \hfill x^3,\hfill & \hfill \mathrm{\text{when}}x\mathrm{\text{is irrational}}\hfill \end{array}$ Evaluate lower integral ${\underset{\_}{\int }}_0^{1}f$ and upper integral ${\overline{\int }}_0^{1}f$. Does the integral ${\int }_0^{1}f$ exist? [2026/20m/3b]

2. Prove that the improper integral ${\int }_0^{\infty }\frac{1}{1+x^2{sin}^{2}x}dx$ is divergent. [2026/15m/4b]

3. Prove that every continuous function is Riemann integrable. [2025/15m/4b]

4. Test the convergence of ${\int }_0^2\frac{logx}{\sqrt{(2-x)}}dx$. [2024/10m/1c]

5. Find the upper and lower Riemann integrals for the function $f$ defined on $[0,1]$ as follows : $f\left(x\right)=\{\begin{array}{cc}\hfill {(1-x^2)}^{1/2},\hfill & \hfill \mathrm{\text{if}}x\mathrm{\text{is rational.}}\hfill \\ \hfill (1-x),\hfill & \hfill \mathrm{\text{if}}x\mathrm{\text{is irrational.}}\hfill \end{array}$ Hence, show that $f$ is not Riemann integrable on $[0,1]$. [2024/15m/4b]

6. Test the convergence of ${\int }_0^{\infty }\frac{cosx}{1+x^2}dx$. [2022/10m/1c]

7. Let $f\left(x\right)=x^2$ on $[0,k]$, $k>0$. Show that $f$ is Riemann integrable on the closed interval $[0,k]$ and ${\int }_0^{k}fdx=\frac{k^3}{3}$. [2022/15m/2a]

8. If a function $f$ is monotonic in the interval $[a,b]$, then prove that $f$ is Riemann integrable in $[a,b]$. [2021/10m/1c]

9. Show that ${\int }_0^{\pi /2}\frac{{sin}^{2}x}{sinx+cosx}dx=\frac{1}{\sqrt{2}}{log}_e(1+\sqrt{2})$ [2020/15m/4b]

10. Evaluate ${\int }_0^{\infty }\frac{{tan}^{-1}\left(ax\right)}{x(1+x^2)}dx,a>0,a\ne 1.$ [2019/10m/1c]

11. Discuss the convergence of ${\int }_1^2\frac{\sqrt{x}}{l_{n}x}dx$. [2019/15m/4c]

12. Prove the inequality : $\frac{{\pi }^2}{9}<{\int }_{\frac{\pi }{6}}^{\frac{\pi }{2}}\frac{x}{sinx}dx<\frac{2{\pi }^2}{9}$. [2018/10m/1b]

13. Let $f\left(t\right)={\int }_0^t[x]dx,$ where $[x]$ denotes the largest integer less than or equal to $x$. (i) Determine all the real numbers $t$ at which $f$ is differentiable. (ii) Determine all the real numbers $t$ at which $f$ is continuous but not differentiable. [2017/15m/2a]

14. Is the function $f\left(x\right)=\{\begin{array}{cc}\hfill \frac{1}{n},\hfill & \hfill \frac{1}{n+1}<x\le \frac{1}{n}\hfill \\ \hfill 0,\hfill & \hfill x=0\hfill \end{array}$ Riemann integrable? If yes, obtain the value of ${\int }_0^{1}f\left(x\right)dx$. [2015/15m/2b]

15. Test the convergence of the improper integral ${\int }_1^{\infty }\frac{dx}{x^2(1+e^{-x})}$. [2014/10m/1b]

16. Integrate ${\int }_0^{1}f\left(x\right)dx$, where $f\left(x\right)=\{\begin{array}{cc}\hfill 2xsin\frac{1}{x}-cos\frac{1}{x},\hfill & \hfill x\in ]0,1]\hfill \\ \hfill 0,\hfill & \hfill x=0\hfill \end{array}$ [2014/15m/2b]

17. Let $f\left(x\right)=\{\begin{array}{cc}\hfill \frac{x^2}{2}+4\hfill & \hfill \mathrm{\text{if}}x\ge 0\hfill \\ \hfill \frac{-x^2}{2}+2\hfill & \hfill \mathrm{\text{if}}x<0\hfill \end{array}$ Is $f$ Riemann integrable in the interval $[-1,2]$ ? Why ? Does there exist a function $g$ such that $g\prime \left(x\right)=f\left(x\right)$ ? Justify your answer. [2013/10m/1c]

18. Let $[x]$ denote the integer part of the real number $x$, i.e., if $n\le x<n+1$ where $n$ is an integer, then $[x]=n$. Is the function $f\left(x\right)=[x{]}^2+3$ Riemann integrable in $[-1,2]$ ? If not, explain why. If it is integrable, compute ${\int }_{-1}^2([x{]}^2+3)dx$. [2013/10m/3d]

19. Let $f\left(x\right)$ be differentiable on $[0,1]$ such that $f\left(1\right)=f\left(0\right)=0$ and ${\int }_0^1{f}^2\left(x\right)dx=1$. Prove that ${\int }_0^{1}xf\left(x\right)f\prime \left(x\right)dx=-\frac{1}{2}$. [2012/15m/3b]

20. Give an example of a function $f\left(x\right)$, that is not Riemann integrable but $\left|f\left(x\right)\right|$ is Riemann integrable. Justify. [2012/20m/4b]

21. Let $f$ be a continuous function on $[0,1]$. Using first mean value theorem on Integration, prove that ${lim}_{n\to \infty }{\int }_0^1\frac{nf\left(x\right)}{1+n^2{x}^2}dx=\frac{\pi }{2}f\left(0\right).$ [2008/15m/3c]

2.5] Sequences and series of functions

12 questions

1. Consider the series ${\sum }_{n=1}^{\infty }{U}_n\left(x\right)$, $0\le x\le 1$, the sum of whose first $n$ terms is given by $S_n\left(x\right)=\frac{1}{2n^2}log(1+n^4{x}^2)$, $x\in [0,1]$. Show that the given series can be differentiated term-by-term, though ${\sum }_{n=1}^{\infty }{U}_n\prime \left(x\right)$, does not converge uniformly on $[0,1]$. [2024/20m/3b]

2. Test the uniform convergence of the series $x^4+\frac{x^4}{1+x^4}+\frac{x^4}{{(1+x^4)}^2}+\frac{x^4}{{(1+x^4)}^3}+\cdots$ on $[0,1]$. [2021/10m/1b]

3. Discuss the uniform convergence of $f_n\left(x\right)=\frac{nx}{1+n^2{x}^2},\forall x\in ℝ(-\infty ,\infty )$ $n=1,2,3,\dots$ [2019/15m/3a]

4. Test the series of functions ${\sum }_{n=1}^{\infty }\frac{nx}{(1+n^2{x}^2)}$ for uniform convergence. [2015/15m/3b]

5. Show that the series ${\sum }_1^{\infty }\frac{{(-1)}^{n-1}}{n+x^2}$, is uniformly convergent but not absolutely for all real values of $x$. [2013/13m/2c]

6. Let $f_n\left(x\right)=\{\begin{array}{cc}\hfill 0,\hfill & \hfill \mathrm{\text{if}}x<\frac{1}{n+1},\hfill \\ \hfill sin\frac{\pi }{x},\hfill & \hfill \mathrm{\text{if}}\frac{1}{n+1}\le x\le \frac{1}{n},\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{if}}x>\frac{1}{n}.\hfill \end{array}$ Show that $f_n\left(x\right)$ converges to a continuous function but not uniformly. [2012/12m/1b]

7. Let $f_n\left(x\right)=nx{(1-x)}^n$, $x\in [0,1]$ Examine the uniform convergence of $\{f_n\left(x\right)\}$ on $[0,1]$. [2011/15m/2b]

8. Show that the series for which the sum of first $n$ terms $f_n\left(x\right)=\frac{nx}{1+n^2{x}^2},0\le x\le 1$ cannot be differentiated term-by-term at $x=0$. What happens at $x\ne 0$ ? [2011/15m/3b]

9. Show that if $S\left(x\right)={\sum }_{n=1}^{\infty }\frac{1}{n^3+n^4{x}^2}$, then its derivative $S\prime \left(x\right)=-2x{\sum }_{n=1}^{\infty }\frac{1}{n^2{(1+n{x}^2)}^2},$ for all $x$. [2011/20m/4b]

10. Consider the series ${\sum }_{n=0}^{\infty }\frac{x^2}{{(1+x^2)}^n}$. Find the values of $x$ for which it is convergent and also the sum function. Is the convergence uniform ? Justify your answer. [2010/15m/2d]

11. Let $f_n\left(x\right)=x^n$ on $-1<x\le 1$ for $n=1,2,\dots$. Find the limit function. Is the convergence uniform ? Justify your answer. [2010/15m/3c]

12. Show that : ${lim}_{x\to 1}{\sum }_{n=1}^{\infty }\frac{n^2{x}^2}{n^4+x^4}={\sum }_{n=1}^{\infty }\frac{n^2}{n^4+1}.$ Justify all steps of your answer by quoting the theorems you are using. [2009/15m/3c]

2.6] Functions of several variables

17 questions

1. Show that the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ is $\frac{8abc}{3\sqrt{3}}$. [2025/20m/3b]

2. Using the method of Lagrange’s multipliers, find the minimum and maximum distances of the point $P(2,6,3)$ from the sphere $x^2+y^2+z^2=4$. [2023/15m/2b]

3. Find the maximum and minimum values of $\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$, when $lx+my+nz=0$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. Interpret the result geometrically. [2022/20m/3b]

4. Find the stationary values of $x^2+y^2+z^2$ subject to the conditions $a{x}^2+b{y}^2+c{z}^2=1$ and $lx+my+nz=0$. Interpret the result geometrically. [2021/20m/3b]

5. If $u={tan}^{-1}\frac{x^3+y^3}{x-y}$, $x\ne y$ then show that $x^2\frac{{\partial }^{2}u}{\partial x^2}+2xy\frac{{\partial }^{2}u}{\partial x\partial y}+y^2\frac{{\partial }^{2}u}{\partial y^2}=(1-4{sin}^{2}u)sin2u$ [2020/20m/3c]

6. Show that the function $f(x,y)=\{\begin{array}{cc}\hfill \frac{x^2-y^2}{x-y},\hfill & \hfill (x,y)\ne (1,-1),(1,1)\hfill \\ \hfill 0,\hfill & \hfill (x,y)=(1,1),(1,-1)\hfill \end{array}$ is continuous and differentiable at $(1,-1)$. [2019/10m/1b]

7. Using differentials, find an approximate value of $f(4.1,4.9)$ where $f(x,y)={(x^3+x^{2}y)}^{\frac{1}{2}}$. [2019/15m/2c]

8. Find the maximum value of $f(x,y,z)=x^2{y}^2{z}^2$ subject to the subsidiary condition $x^2+y^2+z^2=c^2$, $(x,y,z>0)$. [2019/15m/4a]

9. Find the relative maximum and minimum values of the function $f(x,y)=x^4+y^4-2x^2+4xy-2y^2.$ [2016/15m/3b]

10. Find the absolute maximum and minimum values of the function $f(x,y)=x^2+3y^2-y$ over the region $x^2+2y^2\le 1$. [2015/15m/4b]

11. Obtain $\frac{{\partial }^{2}f(0,0)}{\partial x\partial y}$ and $\frac{{\partial }^{2}f(0,0)}{\partial y\partial x}$ for the function $f(x,y)=\{\begin{array}{cc}\hfill \frac{xy(3x^2-2y^2)}{x^2+y^2},\hfill & \hfill (x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill (x,y)=(0,0)\hfill \end{array}$ Also, discuss the continuity of $\frac{{\partial }^{2}f}{\partial x\partial y}$ and $\frac{{\partial }^{2}f}{\partial y\partial x}$ at $(0,0)$. [2014/15m/3b]

12. Find the minimum value of $x^2+y^2+z^2$ subject to the condition $xyz=a^3$ by the method of Lagrange multipliers. [2014/15m/4b]

13. Let $f(x,y)=y^2+4xy+3x^2+x^3+1$. At what points will $f(x,y)$ have a maximum or minimum ? [2013/10m/3c]

14. Let $f(x,y)=\{\begin{array}{cc}\hfill \frac{{(x+y)}^2}{x^2+y^2},\hfill & \hfill \mathrm{\text{if}}(x,y)\ne (0,0)\hfill \\ \hfill 1,\hfill & \hfill \mathrm{\text{if}}(x,y)=(0,0).\hfill \end{array}$ Show that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist at $(0,0)$ though $f(x,y)$ is not continuous at $(0,0)$. [2012/15m/2b]

15. Find the minimum distance of the line given by the planes $3x+4y+5z=7$ and $x-z=9$ from the origin, by the method of Lagrange’s multipliers. [2012/15m/2d]

16. Find the shortest distance from the origin $(0,0)$ to the hyperbola $x^2+8xy+7y^2=225$ [2011/15m/2d]

17. Find the maxima, minima and saddle points of the surface $Z=(x^2-y^2)e^{(-x^2-y^2)/2}$. [2010/15m/3d]

« Previous: Algebra  |  Next: Complex Analysis »

Back to: Mathematics Optional PYQs – Topic-wise | Also see: Mathematics Optional year-wise question papers

Last updated: 30 October 2026