2025 IFoS Maths Optional Paper I Solutions | Ramana Sri IAS
2025 IFoS Maths Optional Paper I Solutions
2025 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2025 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 2025 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 2025 IFoS Maths Optional Paper I Solutions
These 2025 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 2025 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 2025 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 2025 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 2025 IFoS Maths Optional Paper I Solutions for all questions are available in the full PYQ course.
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2025 IFoS Maths Optional Paper I Solutions: Table of Contents
If a subspace \(W\) of \(\mathbb R^4\) is generated by the vectors \((3,8,-3,-5)\), \((1,-2,5,-3)\) and \((2,3,1,-4)\), then find a basis and dimension of \(W\). Extend that basis to get a basis of \(\mathbb R^4\).
2Diagram
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The Diagram section for this question is available in the full 2025 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 a row echelon matrix which is row equivalent to \(A=\begin{bmatrix}0&0&-2&3&1\\2&4&1&4&3\\1&2&-3&1&2\\4&8&2&3&5\end{bmatrix}\) and find the rank of \(A\).
2Diagram
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The Diagram section for this question is available in the full 2025 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.
Amreek has \(n\) number of children by his first wife. Shaina has \((n+1)\) children by her first husband. They marry and have children of their own also. The whole family now has \(12\) children. It is assumed that children of Amreek from his first wife do not fight among themselves, and likewise, children of Shaina by her first husband do not fight among themselves. Find the maximum possible number of fights between children that can take place.
2Diagram
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The Diagram section for this question is available in the full 2025 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 equation of the plane passing through the points \((2,2,1)\) and \((9,3,6)\) and perpendicular to the plane \(2x+6y+6z=9\).
2Diagram
Question 1(e): Plane through two points and perpendicular to a plane
3Concept Related to the Question
This question belongs to \(Three-Dimensional\) Geometry – Plane. The textbook method is to identify the formula or theorem first, write all given data clearly, and then simplify step by step without skipping algebra.
4Detailed Solution
The required plane passes through \(A(2,2,1)\) and \(B(9,3,6)\), so it contains the direction vector \(\overrightarrow{AB}=(7,1,5)\). The normal to the given plane is \((2,6,6)\). Since the required plane is perpendicular to the given plane, its normal must be perpendicular to both \((7,1,5)\) and \((2,6,6)\).
Hence a normal to the required plane is \((7,1,5)\times(2,6,6)=(-24,-32,40)\), proportional to \((3,4,-5)\). Using the point \((2,2,1)\), we get \(3(x-2)+4(y-2)-5(z-1)=0\).
5Final Answer
Required plane: \(3x+4y-5z-9=0\).
Question 2(a)
Matrix Theory – Companion Matrix
1Question
For the companion matrix \(C=\begin{bmatrix}0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&\cdots&1\\-a_0&-a_1&-a_2&\cdots&-a_{n-1}\end{bmatrix}\) of a \(n^{\text{th}}\) degree polynomial \(\phi(\lambda)=\lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0\), prove that: (i) the characteristic polynomial is \(\phi(\lambda)\); (ii) if \(\lambda_i\) is an eigenvalue of \(C\), then \(x_i=[1\;\lambda_i\;\lambda_i^2\;\cdots\;\lambda_i^{n-1}]^T\) is the associated eigenvector; (iii) if \(\lambda_1,\lambda_2,\ldots,\lambda_n\) are distinct eigenvalues of \(C\), then \(V^{-1}CV=\operatorname{diag}(\lambda_1,\lambda_2,\ldots,\lambda_n)\), where \(V=(\lambda_j^{i-1})\) is the Vandermonde matrix.
2Diagram
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The Diagram section for this question is available in the full 2025 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 shortest distance between the straight lines \(\displaystyle \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\displaystyle \frac{x-5}{4}=\frac{y-4}{4}=\frac{z-5}{5}\). Also show that the lines are coplanar.
2Diagram
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The Diagram section for this question is available in the full 2025 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 enveloping cylinder of the ellipsoid \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\), whose generators are parallel to the line \(\displaystyle \frac{x}{0}=\frac{y}{\pm\sqrt{a^2-b^2}}=\frac{z}{c}\), meets the plane \(z=0\) in circles.
2Diagram
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The Diagram section for this question is available in the full 2025 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 \(\displaystyle \frac{y-x}{1+y^2}<\tan^{-1}y-\tan^{-1}x<\frac{y-x}{1+x^2}\), \(0<x<y\). Hence or otherwise, show that \(\displaystyle \frac\pi4+\frac3{25}<\tan^{-1}\left(\frac43\right)<\frac\pi4+\frac16\).
2Diagram
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The Diagram section for this question is available in the full 2025 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.
Diagonalize the quadratic form \(5x_1^2+26x_2^2+10x_3^2+4x_2x_3+14x_3x_1+6x_1x_2\). Show that it is positive semi-definite and find a non-zero set of values of \(x_1,x_2,x_3\) which makes the diagonalized form zero.
2Diagram
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The Diagram section for this question is available in the full 2025 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 \(W\) is a subspace of a finite dimensional vector space \(V(F)\), then prove that \(W\) is finite dimensional and \(\dim W\le\dim V\). Also, prove that \(\dim W=\dim V\) if and only if \(W=V\).
2Diagram
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The Diagram section for this question is available in the full 2025 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.
For the linear operator \(T:\mathbb R^3\to\mathbb R^3\) defined by \(T(x,y,z)=(x+y+z,\;2y+z,\;2y+3z)\), find the eigenvalues and the basis for eigenspace.
2Diagram
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The Diagram section for this question is available in the full 2025 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.
The displacement of a particle in a straight line is given by the equation \(x=a\cos nt+b\sin nt\). Show that the particle describes a simple harmonic motion whose amplitude is \(\sqrt{a^2+b^2}\) and period is \(\dfrac{2\pi}{n}\).
2Diagram
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The Diagram section for this question is available in the full 2025 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.
Given that \(y=x+\dfrac1x\) is a solution of the differential equation \(\displaystyle x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0\). Find the other linearly independent solution and write down the general solution of the given differential equation.
2Diagram
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The Diagram section for this question is available in the full 2025 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 projectile is launched with initial speed \(u\) making an angle \((90-\theta)\) with the vertical. An air resistance force \(-\beta\vec v\), \((\beta>0)\), acts upon the projectile, where \(\vec v\) is the instantaneous velocity. Find the velocity \(\vec v\), and show that the position vector \(\vec r\) at any time \(t\) is \(\displaystyle \vec r=\frac{mu}{\beta}(\cos\theta\,\hat j+\sin\theta\,\hat k)\left(1-e^{-\beta t/m}\right)-\frac{mg}{\beta}\left(t+\frac m\beta e^{-\beta t/m}-\frac m\beta\right)\hat k\), where \(m\) is the mass of the projectile.
2Diagram
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The Diagram section for this question is available in the full 2025 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 directional derivative of \(F(x,y,z)=xy^2-4x^2y+z^2\) at \((1,-1,2)\) in the direction of \(6\hat i+2\hat j+3\hat k\). Also find its maximum value.
2Diagram
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The Diagram section for this question is available in the full 2025 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 Stokes' theorem for \(\vec f=(x+y)\hat i+yz^2\hat j+y^2z\hat k\), where \(S\) is the upper surface of the sphere \(x^2+y^2+z^2=1\) over \(z=0\) and \(\Gamma\) is its boundary in the \(xy\)-plane.
2Diagram
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The Diagram section for this question is available in the full 2025 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 framework \(ABCD\) consists of four equal, light rods smoothly jointed together to form a square. It is suspended from a peg at \(A\), and a weight \(w\) is attached to \(C\), the framework being kept in shape by a light rod connecting \(B\) and \(D\). Find the thrust in this rod.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 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.
Use the method of variation of parameters to show that the solution of \(\displaystyle \frac{d^2y}{dx^2}+k^2y=\phi(x)\), satisfying the conditions \(y(0)=0=\left.\frac{dy}{dx}\right|_{x=0}\), is given by \(\displaystyle y(x)=\frac1k\int_0^x\phi(t)\sin k(x-t)\,dt\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 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 curve in a space is represented by \(\vec r=e^t\cos t\,\hat i+e^t\sin t\,\hat j+e^t\hat k\). Find the curvature and principal normal of this curve at \(t=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 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 values of \(\alpha\) and \(\beta\) such that the vectors \(\vec f=(\alpha x+y)\hat i+(y-3z)\hat j+(x+\alpha z)\hat k\) and \(\vec g=\dfrac{x\hat i+y\hat j}{x^2+y^2}+\hat k\) are solenoidal.
2Diagram
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The Diagram section for this question is available in the full 2025 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 vector \(\vec f=(2x-yz)\hat i+(2y-zx)\hat j+(2z-xy)\hat k\) is irrotational and find a scalar function \(\phi\) such that \(\vec f=\operatorname{grad}\phi\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 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 solid hemisphere is placed with its base inclined to the surface of a liquid, in which it is completely immersed, at a given angle \(\alpha\). Show that if the resultant thrust on the curved portion of the surface is equal to twice the weight of the liquid displaced, then \(\tan\alpha=2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2025 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 2025 IFoS Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2025 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(e) 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 2025 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 2025 IFoS Maths Optional Paper I Solutions?
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