2024 IFoS Maths Optional Paper I Solutions | Ramana Sri IAS
2024 IFoS Maths Optional Paper I Solutions
2024 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2024 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 2024 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 2024 IFoS Maths Optional Paper I Solutions
These 2024 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 2024 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 2024 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 2024 IFoS Maths Optional Paper I Solutions as a free sample solution below:
Question 1(c). This free sample includes all five sections:
Question, Diagram,
Concept Related to the Question,
Detailed Solution, and Final Answer.
Complete 2024 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.
2024 IFoS Maths Optional Paper I Solutions: Table of Contents
Find the relation between the radii of a right circular cylinder and a cone if the former with maximum possible curved surface area is inscribed in the latter.
2Diagram
Question 1(c): Maxima and Minima – Cylinder in a Cone
3Concept Related to the Question
This question belongs to Maxima and Minima – Cylinder in a Cone. The textbook method is to identify the relevant theorem or formula first, substitute the given data, and simplify each step clearly.
4Detailed Solution
Let the cone have base radius \(R\) and height \(H\). Let the inscribed cylinder have radius \(r\) and height \(x\). By similar triangles, \(r=R\left(1-\frac{x}{H}\right)\).
The curved surface area of the cylinder is \(S=2\pi rx=2\pi R x\left(1-\frac{x}{H}\right)\). Differentiating, \(\frac{dS}{dx}=2\pi R\left(1-\frac{2x}{H}\right)\). For maximum area, \(x=\frac H2\). Hence \(r=R\left(1-\frac12\right)=\frac R2\).
5Final Answer
The radius of the cylinder is half the radius of the cone: \(r=\frac R2\).
Question 1(d)
Limits – Logarithmic Method
1Question
Find the limit of \(\displaystyle (\cot x-\tan x)^{\frac{1}{\log_e x}}\), when \(x\to0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 \(W_1=\left\{\begin{pmatrix}x&y\\z&0\end{pmatrix}:x,y,z\in\mathbb C\right\}\) and \(W_2=\left\{\begin{pmatrix}x&0\\0&y\end{pmatrix}:x,y\in\mathbb C\right\}\) be two subspaces of the vector space of all \(2\times2\) matrices over the complex field \(\mathbb C\). Show that \(\displaystyle \dim\left(\frac{W_1+W_2}{W_2}\right)=\dim\left(\frac{W_1}{W_1\cap W_2}\right)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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.
Reduce the equation \((c^2+d^2)(x^2+y^2)=(cx+dy+2f)^2\) to its canonical form and show that it represents a parabola. Find the latus rectum of the parabola.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 variable sphere passes through the points \((0,0,\pm c)\) and cuts the lines \(y-x\tan\theta=0,\;z-c=0\) and \(y+x\tan\theta=0,\;z+c=0\) in the points \(P\) and \(Q\). If \(|PQ|=2a\), where \(a\) is a positive number, then show that the centre of all such spheres lies on the circle \(x^2+y^2=(a^2-c^2)\,\csc^2(2\theta),\;z=0\).
2Diagram
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The Diagram section for this question is available in the full 2024 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 \(\displaystyle u=\exp\left\{\sin^{-1}\left(\frac{x+y}{\sqrt{x}-\sqrt{y}}\right)\right\}\), then show that \(\displaystyle x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\frac12u\tan(\log_e u)\).
2Diagram
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The Diagram section for this question is available in the full 2024 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.
Reduce the matrix \(\displaystyle A=\begin{pmatrix}2&2&-1&6&4\\4&4&1&10&13\\8&8&-1&26&23\end{pmatrix}\) to echelon form and then to row canonical form.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 equation of the plane containing the line \(\displaystyle \frac{y}{b}+\frac{z}{c}=1,\;x=0\) and parallel to the line \(\displaystyle \frac{x}{a}-\frac{z}{c}=1,\;y=0\) is \(\displaystyle \frac{x}{a}-\frac{y}{b}-\frac{z}{c}+1=0\). Further show that if \(2d\) is the shortest distance between the given lines, then \(\displaystyle \frac1{a^2}+\frac1{b^2}+\frac1{c^2}=\frac1{d^2}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 variable plane is parallel to the plane \(\displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0\) and meets the coordinate axes in \(A,B,C\) respectively. Prove that the circle \(ABC\) lies on the cone \(\displaystyle yz\left(\frac bc+\frac cb\right)+zx\left(\frac ca+\frac ac\right)+xy\left(\frac ab+\frac ba\right)=0\).
2Diagram
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The Diagram section for this question is available in the full 2024 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 equations of the generating lines of the hyperboloid \(\displaystyle \frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{16}=1\) passing through the point \((2,3,-4)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 defined by \(T(x,y,z)=(5x-y+3z,\;-6x+4y-6z,\;-6x+2y-4z)\). Find all the eigenvalues and corresponding eigenvectors.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 moves from rest at a distance \(a\) from a centre of force where repulsion at distance \(x\) is \(\mu x^{-2}\). Show that its velocity at distance \(x\) is \(\displaystyle \sqrt{\frac{2\mu(x-a)}{ax}}\) and that the time it has taken is \(\displaystyle \sqrt{\frac{a}{2\mu}}\left[\sqrt{x^2-ax}+a\log_e\left(\sqrt{\frac xa}+\sqrt{\frac xa-1}\right)\right]\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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.
Two uniform steel rods of equal size \(l\) hang from their junction and rest on a symmetrically placed smooth vertical circular base of radius \(a\). If each of the rods subtends an angle \(\phi\) with the vertical line passing through the centre of the circular base, show that, applying the principle of virtual work, the relation obtained is \(l=2a\cot\phi\,\cosec^2\phi\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 \(\vec F\) is a solenoidal vector, then show that \(\operatorname{curl}\operatorname{curl}\operatorname{curl}\operatorname{curl}\vec F=\nabla^2\nabla^2\vec F=\nabla^4\vec F\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 moves in a path so that its acceleration is always directed to a fixed point and is equal to \(\displaystyle \frac{\mu}{(\text{distance})^2}\). Show that its path is a conic section and distinguish between the three cases that arise. Further show that the square of the periodic time varies as the cube of the major axis.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 near the surface of a celestial body having atmosphere, the gravity is almost constant and the absolute temperature in its atmosphere is given by \(\displaystyle T=T_0\sqrt{1-\frac{z^2}{n^2H^2}}\), \(H\) being the height of the homogeneous atmosphere and \(n\) a constant quantity, show that the pressure in the atmosphere will be given by \(\displaystyle p=p_0\exp\left(\sin^{-1}\frac{z}{nH}\right)\), where \(p_0\) is the pressure at \(z=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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.
Verify the divergence theorem for the function \(\vec F=(x^2-yz)\hat i+(y^2-zx)\hat j+(z^2-xy)\hat k\) taken over the parallelepiped \(0\le x\le a,\;0\le y\le b,\;0\le z\le c\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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.
Establish a stability criterion if a rigid body is lying on another rigid body at a point of contact, and also both have rough surfaces to prevent sliding and a small area around the point of contact of both of them is circular. A solid frustum of a paraboloid of revolution of height \(h\) and latus rectum \(2a\) rests with its vertex on that of a paraboloid of revolution of latus rectum \(2b\). Find the stability condition.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2024 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 2024 IFoS Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2024 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?
Question 1(c) is given as the free sample solution on this page. The sample includes all five sections: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2024 IFoS Maths Optional Paper I Solutions for preparation?
Students should first solve the questions independently, then compare their approach with the solution method. These solutions are useful for revision, answer-writing practice, diagram presentation, and concept clarity before the IFoS Mains examination.
Do these solutions include diagrams and detailed explanations?
Yes. The full PYQ course follows a structured format wherever required: Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer. This public page shows the format through one free sample solution.
How can I get complete solutions for all questions in 2024 IFoS Maths Optional Paper I Solutions?
To access 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.
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