Ramana Sri IAS - 2018 IFoS Maths Optional Paper I Solutions 2018 IFoS Maths Optional Paper I Solutions Ramana Sri IAS provides complete and updated solutions for the 2018 IFoS Maths Optional Paper I. 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 2018 Mathematics Optional Paper I. Each answer follows the same format: Question, Diagram where needed, Concept Related to the Question, Detailed Solution, and Final Answer.
About 2018 IFoS Maths Optional Paper I Solutions These 2018 IFoS Maths Optional Paper I 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 2018 IFoS Maths Optional Paper I 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 2018 IFoS Maths Optional Paper I Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the IFoS Mathematics optional paper.
Sample Full Solution
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Question 1(a) . This free sample includes all five sections:
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2018 IFoS Maths Optional Paper I Solutions: Table of Contents
Question 1(a) Maxima and Geometry 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that the maximum rectangle inscribed in a circle is a square.
2 DiagramQuestion 1(a): Maximum Rectangle Inscribed in a Circle
3 Concept Related to the QuestionThis question belongs to Maxima and Geometry . We use the standard theorem/formula and then simplify step by step.
4 Detailed SolutionLet the half-sides of the rectangle be \(x\) and \(y\), and let the circle radius be \(r\). Since the diagonal of the rectangle is the diameter of the circle, \(x^2+y^2=r^2\). The area is \(A=4xy\). Now \((x-y)^2\ge0\) gives \(2xy\le x^2+y^2=r^2\). Hence \(A=4xy\le2r^2\). Equality is possible only when \(x=y\).
5 Final AnswerThe maximum rectangle is a square.
Question 1(b) Adjoint Matrix 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionGiven \(\operatorname{Adj}A=\begin{pmatrix}2&2&0\\2&5&1\\0&1&1\end{pmatrix}\) and \(\det A=2\). Find \(A\).
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 1(c) Mean Value Theorem 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(f:[a,b]\to\mathbb R\) is continuous on \([a,b]\) and derivable on \((a,b)\), where \(0\lt a\lt b\), show that \(f(b)-f(a)=cf^{\prime}(c)\log(b/a)\) for some \(c\in(a,b)\).
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) Tangent Planes 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind tangent planes to \(2x^2+6y^2+3z^2=27\) passing through \(x-y-z=0=x-y+2z-9\).
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(e) Hermitian Matrix 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionProve that eigenvalues of a Hermitian matrix are real.
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) Cylinder 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the cylinder with generators parallel to \(\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}\) and guiding curve \(x^2+y^2=4,z=2\).
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 2(b) Congruent Matrices 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that \(A=\begin{pmatrix}1&1&-1\\1&2&1\\-1&1&3\end{pmatrix}\) and \(B=\begin{pmatrix}1&0&3\\0&2&2\\3&2&0\end{pmatrix}\) are congruent.
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) Separation of Zeros 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(\phi\psi'-\psi\phi'>0\), prove that between two consecutive roots of \(\phi=0\), there is exactly one root of \(\psi=0\).
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 2(d) Basis 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that the vectors \(\alpha_1=(1,0,-1)\), \(\alpha_2=(1,2,1)\), \(\alpha_3=(0,-3,2)\) form a basis for \(\mathbb R^3\). Express each of the standard basis vectors as a linear combination of \(\alpha_1,\alpha_2,\alpha_3\).
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) Sphere Tangent Plane 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind tangent plane to \(x^2+y^2+z^2-2x+6y+2z+8=0\) through \(3x-4y-8=0=y-3z+2\).
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 3(b) Chain Rule 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(f=f(u,v)\), \(u=e^x\cos y\), \(v=e^x\sin y\), show \(f_{xx}+f_{yy}=(u^2+v^2)(f_{uu}+f_{vv})\).
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) Linear Transformation Matrix 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor \(T(a,b)=(a,a+b)\), find the matrix using domain basis \(\{e_1,e_2\}\) and range basis \(\{(1,1),(1,-1)\}\).
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(d) Double Integral 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionEvaluate \(\iint_R(x^2+xy)dxdy\), where \(R\) is bounded by \(xy=1\), \(y=0\), \(y=x\), \(x=2\).
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) Cone and Plane 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the lines in which \(2x+y-z=0\) cuts \(4x^2-y^2+3z^2=0\), and find their angle.
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) Symmetric Functions 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor \(u=x+y+z\), \(v=xy+yz+zx\), \(w=x^3+y^3+z^3-3xyz\), find the relation.
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) Hyperbolic Paraboloid 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the locus of intersection of perpendicular generators of \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=2z\).
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) Linear Dependence 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(\alpha_1,\ldots,\alpha_n,\alpha\) are linearly dependent and \(\alpha_1,\ldots,\alpha_n\) are independent, prove that \(\alpha\) is their linear combination.
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) ODE 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFind the complementary function and particular integral for the equation \(\frac{d^2y}{dx^2}-y=xe^x+\cos^2x\), and hence the general solution of the equation.
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) Variation of Parameters 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve \(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=xe^x\log x\), \(x\gt0\), by the method of variation of parameters.
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) SHM 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf the velocities in a simple harmonic motion at distances \(a,b,c\) from a fixed point on the straight line which is not the centre of force are \(u,v,w\), respectively, show that the periodic time \(T\) is given by \(\frac{4\pi^2}{T^2}(b-c)(c-a)(a-b)=\begin{vmatrix}u^2&v^2&w^2\\a&b&c\\1&1&1\end{vmatrix}\).
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) Centre of Pressure 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFrom a semi-circle whose diameter is in the surface of a liquid, a circle is cut out, whose diameter is the vertical radius of the semi-circle. Find the depth of the centre of pressure of the remainder part.
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(e) Vector Analysis 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionIf \(\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\) and \(f(r)\) is differentiable, show that \(\operatorname{div}[f(r)\vec{r}]=rf^{\prime}(r)+3f(r)\). Hence or otherwise show that \(\operatorname{div}\left(\frac{\vec{r}}{r^3}\right)=0\).
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(a) Differential Equation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve \((y^2+2x^2y)dx+(2x^3-xy)dy=0\).
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(b) Elastic String 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(T_1\) and \(T_2\) be the periods of vertical oscillations of two different weights suspended by an elastic string, and \(C_1\) and \(C_2\) are the statical extensions due to these weights and \(g\) is the acceleration due to gravity. Show that \(g=\frac{4\pi^2(C_1-C_2)}{T_1^2-T_2^2}\).
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 6(c) Conservative Force 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionShow that \(\vec F=(2xy+z^3)\hat{i}+x^2\hat{j}+3xz^2\hat{k}\) is a conservative force. Hence, find the scalar potential. Also find the work done in moving a particle of unit mass in the force field from \((1,-2,1)\) to \((3,1,4)\).
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) Catenary 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionThe end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is \(\mu\log\frac{1+\sqrt{1+\mu^2}}{\mu}\), where \(\mu\) is the coefficient of friction.
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) Differential Equation 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionSolve \(\frac{dy}{dx}=\frac{4x+6y+5}{3y+2x+4}\).
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) Statics 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA frame \(ABC\) consists of three light rods, of which \(AB\), \(AC\) are each of length \(a\), \(BC\) of length \(\frac32a\), freely jointed together. It rests with \(BC\) horizontal, \(A\) below \(BC\), and the rods \(AB\), \(AC\) over two smooth pegs \(E\) and \(F\), in the same horizontal line, at a distance \(2b\) apart. A weight \(W\) is suspended from \(A\). Find the thrust in the rod \(BC\).
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) Curvature 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionLet \(\alpha\) be a unit-speed curve in \(\mathbb R^3\) with constant curvature and zero torsion. Show that \(\alpha\) is part of a circle.
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) Hydrostatics 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA solid hemisphere floating in a liquid is completely immersed with a point of the rim joined to a fixed point by means of a string. Find the inclination of the base to the vertical and tension of the string.
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) Melting Snowball 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionA snowball of radius \(r(t)\) melts at a uniform rate. If half of the mass of the snowball melts in one hour, how much time will it take for the entire mass of the snowball to melt, correct to two decimal places? Conditions remain unchanged for the entire process.
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) Curves on Sphere 1. Question 2. Diagram 3. Concept Related to the Question 4. Detailed Solution 5. Final Answer
1 QuestionFor a curve lying on a sphere of radius \(a\) and such that the torsion is never \(0\), show that \(\left(\frac1\kappa\right)^2+\left(\frac{\kappa^{\prime}}{\kappa^2\tau}\right)^2=a^2\).
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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2018 IFoS Maths Optional Paper I Solutions FAQs
Are these 2018 IFoS Maths Optional Paper I Solutions complete? This public page gives one full sample solution. Complete paper-wise solutions for all questions are available in the full PYQ course by Ramana Sri IAS.
Which question is given as a free sample solution on this page? Question 1(a) is given as the free sample solution on this page, including the question, diagram, concept, detailed solution, and final answer.
How should I use these 2018 IFoS Maths Optional Paper I Solutions for preparation? Use these solutions for answer-writing practice, concept revision, diagram understanding, and final-answer presentation before the IFoS Mains examination.
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