Ramana Sri IAS provides complete and updated solutions for the 2017 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 2017 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 2017 IFoS Maths Optional Paper I Solutions
These 2017 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 2017 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 2017 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 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 solutions for all questions are available in the full PYQ course. To purchase the full solutions,
please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
2017 IFoS Maths Optional Paper I Solutions: Table of Contents
Let \(A\) be a square matrix of order \(3\) whose diagonal elements are \(a\) and off-diagonal elements are \(1\). If \(B=bA\) is orthogonal, determine \(a\) and \(b\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 \(V\) be the vector space of all \(2\times2\) matrices over \(\mathbb R\). Show that \(W\) is not a subspace when (i) \(W\) contains all \(2\times2\) matrices with zero determinant, and (ii) \(W\) consists of all \(2\times2\) matrices \(A\) such that \(A^2=A\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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.
Using the Mean Value Theorem, show that (i) \(f(x)\) is constant in \([a,b]\) if \(f^{\prime}(x)=0\) in \([a,b]\), and (ii) \(f(x)\) is decreasing in \((a,b)\) if \(f^{\prime}(x)\lt 0\) everywhere in \((a,b)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 \(u=ax^2+2hxy+by^2\) and \(v=Ax^2+2Hxy+By^2\). Find \(J=\frac{\partial(u,v)}{\partial(x,y)}\), and hence show that \(u,v\) are independent unless \(\frac aA=\frac bB=\frac hH\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 planes parallel to \(3x-2y+6z+8=0\) and at a distance \(2\) from it.
2Diagram
Question 1(e): Parallel Planes at a Fixed Distance
3Concept Related to the Question
Planes parallel to \(3x-2y+6z+8=0\) have the same normal vector, so their equations are \(3x-2y+6z+d=0\).
4Detailed Solution
The distance between \(3x-2y+6z+8=0\) and \(3x-2y+6z+d=0\) is \(\frac{|d-8|}{\sqrt{3^2+(-2)^2+6^2}}=\frac{|d-8|}{7}\). This distance is given as \(2\). Hence \(\frac{|d-8|}{7}=2\), so \(|d-8|=14\). Therefore \(d=22\) or \(d=-6\).
5Final Answer
The required planes are \(3x-2y+6z+22=0\) and \(3x-2y+6z-6=0\).
Question 2(a)
Cayley-Hamilton Theorem
1Question
State and verify Cayley-Hamilton theorem for \(A=\begin{pmatrix}1&0&2\\0&-1&1\\0&1&0\end{pmatrix}\). Hence find \(A^{-1}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 \(\int_0^{\pi/2}\sin^p\theta\cos^q\theta\,d\theta=\frac12\frac{\Gamma\left(\frac{p+1}{2}\right)\Gamma\left(\frac{q+1}{2}\right)}{\Gamma\left(\frac{p+q+2}{2}\right)}\), where \(p,q\gt -1\). Hence evaluate \(\int_0^{\pi/2}\sin^4 x\cos^5 x\,dx\), \(\int_0^1 x^3(1-x^2)^{5/2}\,dx\), and \(\int_0^1 x^4(1-x)^3\,dx\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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.
Evaluate \(\int_0^\infty\int_0^\infty e^{-(x^2+y^2)}\,dx\,dy\) by changing to polar coordinates. Hence show that \(\int_0^\infty e^{-x^2}\,dx=\frac{\sqrt\pi}{2}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 eigenvalues and eigenvectors of \(A=\begin{pmatrix}0&-2\\1&3\end{pmatrix}\). Examine whether \(A\) is diagonalizable and obtain \(D=P^{-1}AP\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 right circular cone with vertex at the origin, axis making equal angles with the coordinate axes, and generator through the origin with direction ratios \((1,-2,2)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 and the line of shortest distance between \(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\) and \(\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 performs SHM of period \(T\) about centre \(O\). It passes through \(P\), where \(OP=b\), with velocity \(v\) in the direction \(OP\). Prove that the time before it returns to \(P\) is \(\frac T\pi\tan^{-1}\left(\frac{vT}{2\pi b}\right)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 heavy uniform cube balances on the highest point of a sphere whose radius is \(r\). If the sphere is rough enough to prevent sliding and if the side of the cube is \(\frac{\pi r}{2}\), prove that the total angle through which the cube can swing without falling is \(90^\circ\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 string of length \(a\) forms the shorter diagonal of a rhombus made of four uniform rods, each of length \(b\) and weight \(W\), hinged together. If one rod is horizontal, prove that the string tension is \(\frac{2W(2b^2-a^2)}{b\sqrt{4b^2-a^2}}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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.
Using Stokes theorem, evaluate \(\oint_C[(x+y)dx+(2x-z)dy+(y+z)dz]\), where \(C\) is the boundary of the triangle with vertices \((2,0,0)\), \((0,3,0)\), and \((0,0,6)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 planet describes an ellipse about the Sun as a focus. Show that its velocity away from the Sun is greatest when the radius vector is at right angle to the major axis, and the greatest value is \(\frac{2\pi ae}{T\sqrt{1-e^2}}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 semi-ellipse bounded by its minor axis is just immersed in a liquid whose density varies as depth. If the minor axis lies on the surface, find the eccentricity so that the focus is the centre of pressure.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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.
Evaluate \(\iint_S(\nabla\times\vec{f})\cdot\hat n\,dS\), where \(S\) is the cone \(z=2-\sqrt{x^2+y^2}\) above the \(xy\)-plane and \(\vec{f}=(x-z)\hat{i}+(x^3+yz)\hat{j}-3xy^2\hat{k}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 straight line towards a fixed point \(O\), with acceleration \(\mu\left(\frac{a^5}{x^2}\right)^{1/3}\) at distance \(x\). If it starts from rest at distance \(a\), prove that it arrives at \(O\) with velocity \(a\sqrt{6\mu}\) after time \(\frac8{15}\sqrt{\frac6\mu}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 curvature and torsion of the circular helix \(\vec{r}=a(\cos\theta,\sin\theta,\theta\cot\beta)\), where \(\beta\) is the constant angle at which it cuts its generators.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 tangent to a curve makes a constant angle \(\alpha\) with a fixed line, prove that \(\kappa\cos\alpha\pm\tau\sin\alpha=0\). Conversely, if \(\frac\kappa\tau\) is constant, show that the tangent makes a constant angle with a fixed direction.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2017 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 2017 IFoS Maths Optional Paper I 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(e) is given as the free sample solution on this page.
How should I use these 2017 IFoS Maths Optional Paper I 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 I?
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.
