Ramana Sri IAS • IFoS Mathematics Optional 2017 IFoS Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2017 IFoS Maths Optional Paper II. Aspirants preparing for the IFoS Mains Examination with Mathematics as their optional subject should solve all these questions carefully before the mains examination.
Ramana Sri IAS presents complete solutions for Indian Forest Service Examination 2017 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 2017 IFoS Maths Optional Paper II Solutions
These 2017 IFoS Maths Optional Paper II Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the IFoS 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 2017 IFoS 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
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These 2017 IFoS Maths Optional Paper II Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the IFoS Mathematics optional paper.
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2017 IFoS Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Groups of Order Four
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that every group of order \(4\) is Abelian.
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) Continuity of Rational-Irrational Function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA function \(f:\mathbb R\to\mathbb R\) is defined as below:
\(f(x)=\begin{cases}x,&\text{if }x\text{ is rational},\\1-x,&\text{if }x\text{ is irrational}.\end{cases}\)
Prove that \(f\) is continuous at \(x=\frac12\) but discontinuous at all other points in \(\mathbb R\).
2 DiagramQuestion 1(b): Continuity of Rational-Irrational Function
3 Concept Related to the QuestionBetween any two real numbers there are rational and irrational numbers. Therefore the limit must be the same along rational and irrational sequences.
4 Detailed SolutionLet \(x_n\to c\) through rational values. Then \(f(x_n)=x_n\to c\). Let \(y_n\to c\) through irrational values. Then \(f(y_n)=1-y_n\to1-c\). For the limit to exist, we must have \(c=1-c\), so \(c=\frac12\).
At \(c=\frac12\), both rational and irrational limiting values are \(\frac12\), and \(f(\frac12)=\frac12\) because \(\frac12\) is rational. Hence \(f\) is continuous at \(\frac12\). At every other point, the two path limits are different, so \(f\) is discontinuous.
5 Final Answer\(f\) is continuous only at \(x=\frac12\), and discontinuous at every other real number.
Question 1(c) Analytic Function from Linear Combination
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(f(z)=u(x,y)+iv(x,y)\) is analytic and \(u+2v=x^3-2y^3+3xy(2x-y)\), find \(f(z)\).
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) Simplex Method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve by simplex method the following LPP:
Minimize \(Z=x_1-3x_2+2x_3\)
subject to the constraints \(3x_1-x_2+2x_3\le7\), \(-2x_1+4x_2\le12\), \(-4x_1+3x_2+8x_3\le0\), and \(x_1,x_2,x_3\ge0\).
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) Abelian Group Under Star Operation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(G\) be the set of all real numbers except \(-1\) and define \(a*b=a+b+ab\), \(\forall\,a,b\in G\). Examine if \(G\) is an Abelian group under \(*\).
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) Normal Subgroups of Coprime Orders
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(H\) and \(K\) be finite normal subgroups of coprime order in a group \(G\). Prove that \(hk=kh\) for all \(h\in H\), \(k\in K\).
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 2(c) Radical of an Ideal
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(A\) be an ideal of a commutative ring \(R\), and \(B=\{x\in R:x^n\in A\text{ for some positive integer }n\}\). Is \(B\) an ideal of \(R\)? Justify your answer.
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(d) Gaussian Integers
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that the ring \(\mathbb Z[i]=\{a+ib:a,b\in\mathbb Z,\ i=\sqrt{-1}\}\) of Gaussian integers is a Euclidean domain.
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) Unequal Mixed Partials
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(f_{xy}(0,0)\) and \(f_{yx}(0,0)\), given that
\(f(x,y)=\begin{cases}x^2\tan^{-1}\left(\frac yx\right)-y^2\tan^{-1}\left(\frac xy\right),&xy\ne 0,\\0,&\text{otherwise}.\end{cases}\)
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) Constrained Maxima and Minima
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the maximum and minimum values of \(x^2+y^2+z^2\) subject to \(\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}=1\).
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 3(c) Dirichlet Integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that \(\int_0^\infty\frac{\sin x}{x}\,dx\) is convergent but not absolutely convergent.
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(d) Common Volume of Two Cylinders
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the volume of the region common to the cylinders \(x^2+y^2=a^2\) and \(x^2+z^2=a^2\).
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) Definite Integral by Contour Method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that \(\int_0^\pi\frac{1+2\cos\theta}{5+4\cos\theta}\,d\theta=0\).
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 4(b) Residues of Tangent Function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the sum of residues of \(f(z)=\frac{\sin z}{\cos z}\) at its poles inside \(|z|=2\).
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) Change of Order of Integration
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate
\(\displaystyle \int_{x=0}^{\infty}\int_{y=0}^{x}x e^{-x^2/y}\,dy\,dx\).
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(d) Assignment Problem
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA computer centre has four expert programmers. The centre needs four application programs to be developed. The head of the centre, after studying carefully the programs to be developed, estimates the computer times in hours required by the experts to the application programs as follows:
Programmer A B C D \(P_1\) 5 3 2 8 \(P_2\) 7 9 2 6 \(P_3\) 6 4 5 7 \(P_4\) 5 7 7 8
Assign the programs to the programmers in such a way that total computer time is least.
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) PDE by Eliminating Arbitrary Functions
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionForm the partial differential equation by eliminating arbitrary functions \(\phi\) and \(\psi\) from the relation \(z=\phi(x^2-y)+\psi(x^2+y)\).
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 5(b) BASIC Program for Matrix Inverse
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionWrite a BASIC program to compute the multiplicative inverse of a non-singular square matrix.
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) Moment of Inertia of Rectangular Parallelepiped
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA uniform rectangular parallelepiped of mass \(M\) has edges of lengths \(2a,2b,2c\). Find the moment of inertia of this rectangular parallelepiped about the line through its centre parallel to the edge of length \(2a\).
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(d) Composite Trapezoidal Rule
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(\displaystyle\int_0^1e^{-x^2}\,dx\) using the composite trapezoidal rule with four decimal precision, i.e., with the absolute value of the error not exceeding \(5\times10^{-5}\).
2 Diagram
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4 Detailed Solution
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5 Final Answer
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Question 6(a) Lagrange Partial Differential Equation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve \((x-y)\frac{\partial z}{\partial x}+(x+y)\frac{\partial z}{\partial y}=2xz\).
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(b) Orthogonal Surfaces
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the surface orthogonal to the family \(z(x+y)=c(3z+1)\) and passing through \(x^2+y^2=1, z=1\).
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 6(c) Complete Integral of First-Order PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the complete integral of \(xp-yq=xq f(z-px-qy)\), where \(p=z_x\), \(q=z_y\).
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 6(d) Vibrating String
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA tightly stretched string with fixed end points \(x=0\) and \(x=l\) is initially in a position given by \(y=y_0\sin^3\left(\frac{\pi x}{l}
ight)\). It is released from rest from this position. Find the displacement \(y(x,t)\).
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) Newton-Raphson Method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the real root of \(x^3+x^2+3x+4=0\) correct up to five decimal places using Newton-Raphson 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 7(b) Simpson One-Third Rule
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA river is \(80\) metre wide. The depth \(y\), in metre, of the river at a distance \(x\) from one bank is given by the following table:
\(x\) 0 10 20 30 40 50 60 70 80 \(y\) 0 4 7 9 12 15 14 8 3
Find the area of cross-section of the river using Simpson's \(\frac13\)rd rule.
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) Modified Euler Method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind \(y\) for \(x=0.2\), taking \(h=0.1\), by modified Euler method for \(\frac{dy}{dx}=x+y\), \(y(0)=1\). Compute the error.
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(d) Two’s Complement Addition
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionAssuming a \(32\)-bit signed integer representation using two’s complement, add \(-1\) and \(-1024\), and give the answer in two’s complement representation.
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) Lagrangian of Vertical Spring
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionConsider a mass \(m\) on the end of a spring of natural length \(l\) and spring constant \(k\). Let \(y\) be the vertical coordinate of the mass as measured from the top of the spring. Assume that the mass can only move up and down in the vertical direction. Show that \(L=\frac12m\dot y^2-\frac12k(y-l)^2+mgy\). Also determine and solve the corresponding Euler-Lagrange equations of motion.
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) Streamlines and Pathlines
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the streamlines and pathlines of the velocity field \(u=\frac{x}{1+t}\), \(v=y\), \(w=0\).
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 8(c) Vortex Lines
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionThe velocity vector in the flow field is given by \(\vec q=(az-by)\hat i+(bx-cz)\hat j+(cy-ax)\hat k\), where \(a,b,c\) are non-zero constants. Determine the equations of vortex lines.
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(d) Laplace Equation in Rectangle
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve Laplace's equation \(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0\), subject to the conditions \(u(0,y)=u(l,y)=u(x,0)=0\) and \(u(x,a)=\sin\left(\dfrac{n\pi x}{l}\right)\).
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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2017 IFoS Maths Optional Paper II Solutions FAQs
Are these 2017 IFoS Maths Optional Paper II Solutions complete? This public page gives one complete sample solution. Complete solutions for all questions are available in the full PYQ course.
Which question is given as a free sample solution on this page? Question 1(b) is given as the free sample solution on this page.
How should I use these 2017 IFoS Maths Optional Paper II Solutions for preparation? Students should first read the question carefully, understand the concept, study the diagram where required, and then practise writing the detailed solution and final answer.
Do these solutions include diagrams and detailed solutions? Yes. The full PYQ course includes diagrams where needed, concept explanation, detailed solution, and final answer for each question.
How can I get complete solutions for all questions in 2017 IFoS Maths Optional Paper II? To get complete solutions for all questions, please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
