2023 IFoS Maths Optional Paper I Solutions | Ramana Sri IAS
2023 IFoS Maths Optional Paper I Solutions
2023 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2023 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 2023 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 2023 IFoS Maths Optional Paper I Solutions
These 2023 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 2023 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.
These 2023 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
We are giving one question from 2023 IFoS Maths Optional Paper I Solutions as a free sample solution below:
Question 1(e). This free sample includes all five sections:
Question, Diagram,
Concept Related to the Question,
Detailed Solution, and Final Answer.
Complete 2023 IFoS Maths Optional Paper I Solutions for all questions are available in the full PYQ course. To purchase the full solution,
please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
2023 IFoS Maths Optional Paper I Solutions: Table of Contents
If \(u=z\sin\left(\frac yx\right)\), where \(x=3r^2+3s\), \(y=4r-2s^3\), and \(z=2r^2-3s^2\), then find \(\frac{\partial u}{\partial r}\) and \(\frac{\partial u}{\partial s}\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
If the equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) represents two intersecting straight lines, then show that the square of the distance of the point of intersection from the origin is \(\displaystyle \frac{c(a+b)-(f^2+g^2)}{ab-h^2}\).
2Diagram
Question 1(e): Pair of Intersecting Lines and Distance from Origin
3Concept Related to the Question
This question belongs to Conic Sections – Pair of Lines. The standard method is to first write the definition, theorem, or formula, then substitute the given data carefully, and finally simplify step by step so that every line has a clear reason.
4Detailed Solution
The point of intersection of the two straight lines is obtained from \(ax+hy+g=0\) and \(hx+by+f=0\). Solving, \(x=\frac{hf-bg}{ab-h^2}\) and \(y=\frac{gh-af}{ab-h^2}\).
Now compute \(x^2+y^2\) and use the pair-of-lines condition \(abc+2fgh-af^2-bg^2-ch^2=0\). The expression simplifies to the required formula.
Let the function \(f:\mathbb R^2\to\mathbb R\) be defined by \(\displaystyle f(x,y)=\begin{cases}\frac{xy^2}{x^2+y^2},&\text{if }(x,y)\ne(0,0),\\0,&\text{if }(x,y)=(0,0).\end{cases}\) Then show that:
(i) \(D_1f(0,0)\) and \(D_2f(0,0)\) exist.
(ii) \(f(x,y)\) is continuous at \((0,0)\) by \(\varepsilon\)-\(\delta\) method.
(iii) \(f(x,y)\) is not differentiable at \((0,0)\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Let \(T:\mathbb R^3\to\mathbb R^3\) be a linear transformation defined by \(T(x,y,z)=(2x,4x-y,2x+3y-z)\). Prove that \(T\) is invertible and find \(T^{-1}\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
(i) Find the coordinates of the vertex, focus and the length of the latus rectum of the principal sections of the paraboloid given by the equation \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{2z}{c}\).
(ii) Find the nature of the quadric surface given by the equation \(2x^2+5y^2+3z^2-4x+20y-6z=5\). Also find its associated characteristics, principal axes and principal planes.
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Prove that the straight lines \(\displaystyle \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}\) and \(\displaystyle \frac{x-4}{1}=\frac{y+3}{-4}=\frac{z+1}{7}\) intersect and find the equation of the plane containing them. Also find their point of intersection.
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Show that the matrix \(\displaystyle A=\begin{pmatrix}7&-6&6\\2&0&4\\1&-2&6\end{pmatrix}\) is diagonalizable and find a spectral decomposition of the matrix \(A\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Find the solution of the differential equation \(\displaystyle 2x^2y\left(\frac{dy}{dx}\right)=\tan(x^2y^2)-2xy^2\). What will be the definite value of the arbitrary constant, appearing in the solution, on coordinate axes?
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Identify that one solution of the equation \(xy''+(x-1)y'-y=0\) which is of the form \(ce^{\pm ax}\) and then find the other solution by method of reduction of order.
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Four bars are joined together to form a rhombus. The bars are uniform and each bar is of weight \(W\). A rhombus is suspended vertically from one of the joints and a spherical ball of weight \(S\) is balanced inside the rhombus so as to keep the system intact. If \(2\theta\) is the angle at a fixed joint in the state of equilibrium, then find the ratio of weight of the rhombus to that of the spherical ball in terms of the radius of the sphere, the length of a bar and the angle \(\theta\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
In a central orbit, the central force is given as \(\mu u^3(3+2a^2u^2)\). If a particle is projected at a distance \(a\) with velocity \(\displaystyle \sqrt{\frac{5\mu}{a^2}}\) in a direction making an angle \(\tan^{-1}\frac12\) with the radius, then show that equation of its path can be written as \(r=a\tan\theta\).
2Diagram
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The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
If \(\nabla\cdot\vec E=0\), \(\nabla\cdot\vec H=0\), \(\nabla\times\vec E=-\frac{\partial\vec H}{\partial t}\) and \(\nabla\times\vec H=\frac{\partial\vec E}{\partial t}\), then show that \(\nabla^2\vec H=\frac{\partial^2\vec H}{\partial t^2}\) and \(\nabla^2\vec E=\frac{\partial^2\vec E}{\partial t^2}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
A particle slides down the smooth curve \(y=a\sinh\left(\frac xa\right)\), the axis of \(x\) being horizontal and the axis of \(y\) downwards, starting from rest at the point where the tangent is inclined at \(\alpha\) to the horizon. Show that the particle will leave the curve when it has fallen through a vertical distance \(a\sec\alpha\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
(i) Show that the area bounded by a simple closed curve \(C\) is given by \(\displaystyle \frac12\oint_C(x\,dy-y\,dx)\). Hence obtain the area of an ellipse.
(ii) Evaluate \(\displaystyle \oint_\Gamma(e^x\,dx+2y\,dy-dz)\) by using Stokes' theorem, where \(\Gamma\) is the curve \(x^2+y^2=a^2,\;z=h\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
A uniform ladder of \(10\) m length and of \(10\) kg weight rests with its foot on the rough ground and its upper end against a smooth wall, the inclination to the vertical being \(\alpha\). A force \(P\) is applied horizontally to the ladder at a point distant \(3\) m from the foot, so as to make the foot approach the wall. Prove that the force \(P\) must exceed \(\displaystyle \frac{100}{7}\left(\mu+\frac12\tan\alpha\right)\), where \(\mu\) is the coefficient of friction at the foot.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
(i) Given that \(u=x+y+z\), \(v=x^2+y^2+z^2\) and \(w=xy+yz+zx\). Show that the vectors \(\operatorname{grad}u\), \(\operatorname{grad}v\) and \(\operatorname{grad}w\) are coplanar.
(ii) For the curve given by \(\displaystyle \vec r=\left(2t,t^2,\frac{t^3}{3}\right)\), find the curvature and torsion at \(t=1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
(i) Given that the vectors \(\vec f\) and \(\vec g\) are irrotational. Show that the vector \(\vec f\times\vec g\) is solenoidal.
(ii) Show that the vector \(\vec q=(6xy+z^3)\hat i+(3x^2-z)\hat j+(3xz^2-y)\hat k\) is irrotational and find a scalar function \(\phi\) such that \(\vec q=\operatorname{grad}\phi\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Find singular solution of the differential equation \(\displaystyle y^2\left(\frac{dy}{dx}\right)^2-2xy\left(\frac{dy}{dx}\right)\tan^2\beta+y^2\sec^2\beta-x^2\tan^2\beta=0\) directly and from its complete primitive. Determine tac-locus. Show that the envelope of family of curves, which is represented by the given equation, is \(y=\pm x\tan\beta\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
An elliptic lamina is completely immersed in water with its plane vertical. Its minor axis is horizontal and is at a depth \(h\). Determine the centre of pressure.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 IFoS Maths Optional Paper I PYQ Solutions course.
Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Are these 2023 IFoS Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2023 IFoS Maths Optional Paper I Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course by Ramana Sri IAS.
Which question is given as a free sample solution on this page?
The free sample solution on this page is Question 1(e). It includes all five sections: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2023 IFoS Maths Optional Paper I Solutions for preparation?
Students should first solve the question independently, then compare their method with the solution format. This helps improve concept clarity, diagram presentation, answer-writing, and final-answer accuracy.
Do these solutions include diagrams and detailed solutions?
Yes. The sample question shows the complete format. The full PYQ course includes question-wise guidance, diagrams where needed, concepts, detailed solutions, and final answers.
How can I get complete solutions for all questions in 2023 IFoS Maths Optional Paper I?
Complete solutions are available in the full PYQ course. Students can fill out the admission form, and the Ramana Sri IAS admission team will guide them through WhatsApp, email, or call.
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