Ramana Sri IAS - 2021 UPSC Maths Optional Paper II Solutions
2021 UPSC Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2021 UPSC Maths Optional Paper II. Aspirants preparing for the UPSC Mains Examination with Mathematics as their optional subject should solve all these questions carefully at least 5 to 10 times before the mains examination.
Ramana Sri IAS presents complete solutions for UPSC/IAS/CSE-Civil Service Examination 2021 Mathematics Optional Paper II. Each answer follows the same format: Question, Diagram where needed, Concept Related to the Question, Detailed Solution, and Final Answer.
About 2021 UPSC Maths Optional Paper II Solutions
These 2021 UPSC Maths Optional Paper II Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the UPSC Mains examination. This public page gives one full sample solution, while the complete paper-wise solutions are available in the full PYQ course.
Students can use these 2021 UPSC Maths Optional Paper II Solutions to understand the expected answer-writing method, diagram presentation, concept application, and final-answer format used by Ramana Sri IAS.
For the official examination source, students may also refer to the
UPSC previous year question papers page .
These 2021 UPSC Maths Optional Paper II Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the UPSC Mathematics optional paper.
Sample Full Solution
We are giving one question from 2021 UPSC Maths Optional Paper II Solutions as a free sample solution below:
Question 1(d) . This free sample includes all five sections:
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Detailed Solution , and Final Answer .
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2021 UPSC Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Bezout identity for several integers
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(m_1,m_2,\ldots,m_k\) be positive integers and \(d\gt0\) the greatest common divisor of \(m_1,m_2,\ldots,m_k\). Show that there exist integers \(x_1,x_2,\ldots,x_k\) such that \(d=x_1m_1+x_2m_2+\cdots+x_km_k\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 1(b) Uniform convergence of a series
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionTest 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 \([0,1]\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 1(c) Monotone functions and Riemann integrability
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf a function \(f\) is monotonic in the interval \([a,b]\), then prove that \(f\) is Riemann integrable in \([a,b]\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 1(d) Contour integral on a circle
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(c:[0,1]\to C\) be the curve, where \(c(t)=e^{4\pi it}\), \(0\leq t\leq1\). Evaluate the contour integral \(\int_c\frac{dz}{2z^2-5z+2}\).
2 DiagramQuestion 1(d): Contour integral on the unit circle
3 Concept Related to the QuestionThe curve \(c(t)=e^{4\pi it}\) describes the unit circle twice in the counterclockwise direction. We locate the poles and apply the residue theorem with winding number two.
4 Detailed SolutionFactor the denominator:
\[2z^2-5z+2=(2z-1)(z-2).\]
So the poles are
\[z=\frac12,\\qquad z=2.\]
The curve is the unit circle travelled twice counterclockwise. Thus \(z=\frac12\) lies inside the curve and \(z=2\) lies outside.
The residue at \(z=\frac12\) is
\[\operatorname{Res}_{z=1/2}\frac{1}{2z^2-5z+2}=\frac{1}{4(1/2)-5}=-\frac13.\]
Because the circle is traversed twice, the integral is
\[2\cdot2\pi i\left(-\frac13\right)=-\frac{4\pi i}{3}.\]
5 Final AnswerThe value of the contour integral is \(-\frac{4\pi i}{3}\).
Question 1(e) Assignment problem
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA department of a company has five employees with five jobs to be performed. The time, in hours, that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time:
Job I II III IV V A 10 5 13 15 16 B 3 9 18 13 6 C 10 7 2 2 2 D 7 11 9 7 12 E 7 9 10 4 12
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 2(a) Maximum and minimum on a closed interval
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the maximum and minimum values of \(f(x)=x^3-9x^2+26x-24\) for \(0\leq x\leq1\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 2(b) Field extension containing a root
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(F\) be a field and \(f(x)\in F[x]\) a polynomial of degree \(\gt0\) over \(F\). Show that there is a field \(F'\) and an imbedding \(q:F\to F'\) s.t. the polynomial \(f^q\in F'[x]\) has a root in \(F'\), where \(f^q\) is obtained by replacing each coefficient \(a\) of \(f\) by \(q(a)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 2(c) Laurent series about a shifted centre
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the Laurent series expansion of \(f(z)=\frac{z^2-z+1}{z(z^2-3z+2)}\) in the powers of \((z+1)\) in the region \(\lvert z+1\rvert\gt3\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(a) Essential singularity at the origin
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(f\) be an entire function whose Taylor series expansion with centre \(z=0\) has infinitely many terms. Show that \(z=0\) is an essential singularity of \(f\left(\frac1z\right)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(b) Stationary values with two constraints
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the stationary values of \(x^2+y^2+z^2\) subject to the conditions \(ax^2+by^2+cz^2=1\) and \(lx+my+nz=0\). Interpret the result geometrically.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(c) Dual of a mixed LPP
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionConvert the following LPP into dual LPP: Minimize \(Z=x_1-3x_2-2x_3\), subject to \(3x_1-x_2+2x_3\leq7\), \(2x_1-4x_2\geq12\), \(-4x_1+3x_2+8x_3=10\), where \(x_1,x_2\geq0\) and \(x_3\) is unrestricted in sign.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(a) Subgroups of the additive group of rationals
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that there are infinitely many subgroups of the additive group \(Q\) of rational numbers.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(b) Contour integral with sine integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionUsing contour integration, evaluate the integral \(\int_{-\infty}^{\infty}\frac{\sin x\,dx}{x(x^2+a^2)}\), \(a\gt0\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(c) Big M method in linear programming
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve the following linear programming problem using Big M method: Maximize \(Z=4x_1+5x_2+2x_3\), subject to \(2x_1+x_2+x_3\geq10\), \(x_1+3x_2+x_3\leq12\), \(x_1+x_2+x_3=6\), \(x_1,x_2,x_3\geq0\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(a) Formation of first order PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionObtain the partial differential equation by eliminating arbitrary function \(f\) from the equation \(f(x+y+z,x^2+y^2+z^2)=0\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(b) Newton-Raphson numerical root
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind a positive root of the equation \(3x=1+\cos x\) by a numerical technique using initial values \(0,\frac{\pi}{2}\), and further improve the result using Newton-Raphson method correct to \(8\) significant figures.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(c)(i) Number system conversion
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionConvert \((3798\cdot3875)_{10}\) into octal and hexadecimal equivalents.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(c)(ii) Principal conjunctive normal form
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
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1 QuestionObtain the principal conjunctive normal form of \((\neg P\to R)\land(Q\Leftrightarrow P)\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(d) D’Alembert principle on a circle
1. Question
2. Diagram
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4. Detailed Solution
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1 QuestionA particle is constrained to move along a circle lying in the vertical \(xy\)-plane. With the help of D’Alembert’s principle, show that its equation of motion is \(x\ddot y-y\ddot x-gx=0\), where \(g\) is the acceleration due to gravity.
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 5(e) Sources, sinks and streamlines
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionWhat arrangements of sources and sinks can have the velocity potential \(w=\log_e\left(z-\frac{a^2}{z}\right)\)? Draw the corresponding sketch of the streamlines and prove that two of them subdivide into the circle \(r=a\) and the axis of \(y\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(a) Wave equation with fixed ends
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
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1 QuestionSolve the wave equation \(a^2\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2}\), \(0\lt x\lt L\), \(t\gt0\), subject to the conditions \(u(0,t)=0\), \(u(L,t)=0\), \(u(x,0)=\frac14x(L-x)\), and \(\frac{\partial u}{\partial t}(x,0)=0\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(b) Boolean function and gate network
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionObtain the Boolean function \(F(x,y,z)\) based on the table given below. Then simplify \(F(x,y,z)\) and draw the corresponding GATE network:
x y z F(x,y,z) 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 6(c) Lagrangian equation for two masses over a pulley
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionObtain the Lagrangian equation for the motion of a system of two particles of unequal masses connected by an inextensible string passing over a small smooth pulley.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(a) Linear PDE with constant coefficients
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the general solution of the partial differential equation \((D^2-D'^2-3D+3D')z=xy+e^{x+2y}\), where \(D=\frac{\partial}{\partial x}\) and \(D'=\frac{\partial}{\partial y}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(b) Gauss-Seidel method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve the system of equations \(3x_1+9x_2-2x_3=11\), \(4x_1+2x_2+13x_3=24\), \(4x_1-2x_2+x_3=-8\), correct up to \(4\) significant figures by using Gauss-Seidel method after verifying whether the method is applicable in your transformed form of the system.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 7(c) Incompressible circular flow
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that \(\vec q=\frac{\lambda(-y\hat i+x\hat j)}{x^2+y^2}\), where \(\lambda\) is constant, is a possible incompressible fluid motion. Determine the streamlines. Is the kind of motion potential? If yes, then find the velocity potential.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 8(a) Complete integral by Charpit method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind a complete integral of the partial differential equation \(p=(z+qy)^2\) by using Charpit’s method.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 8(b) Newton backward interpolation formula
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionDerive Newton’s backward difference interpolation formula and also do error analysis.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 8(c) Complex potential and circular streamlines
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that for the complex potential \(\tan^{-1}z\), the streamlines and equipotential curves are circles. Find the velocity at any point and check the singularities at \(z=\pm i\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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2021 UPSC Maths Optional Paper II Solutions FAQs
Are these 2021 UPSC Maths Optional Paper II Solutions complete? This public page gives one full sample solution for 2021 UPSC Maths Optional Paper II Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course.
Which question is given as a free sample solution on this page? Question 1(d) is given as the free sample solution on this 2021 UPSC Maths Optional Paper II Solutions page.
How should I use these 2021 UPSC Maths Optional Paper II Solutions for preparation? Students should first solve the question independently, then compare their method with the solution format, diagram presentation, concept explanation, detailed solution, and final answer.
Do these solutions include diagrams and detailed solutions? Yes. The complete PYQ course includes question-wise diagrams where needed, concept related to the question, detailed solutions, and final answers.
How can I get complete solutions for all questions in 2021 UPSC Maths Optional Paper II? To get complete solutions for all questions, students can fill out the admission form. The Ramana Sri IAS admission team will guide students through WhatsApp, email, or call.
