Ramana Sri IAS - 2021 IFoS Maths Optional Paper I Solutions
2021 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2021 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 2021 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 2021 IFoS Maths Optional Paper I Solutions
These 2021 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 2021 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 2021 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 2021 IFoS Maths Optional Paper I Solutions as a free sample solution below:
Question 1(d). This free sample includes all five sections:
Question, Diagram,
Concept Related to the Question,
Detailed Solution, and Final Answer.
Complete 2021 IFoS Maths Optional Paper I Solutions for all questions are available in the full PYQ course.
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Question 1(a)
Quadratic form under orthonormal change of basis
1Question
Consider the following quadratic form: \(q(x,y,z)=2x^2+2y^2+6z^2+2xy-6yz-6zx\), where \((x,y,z)\) are the coordinates of the vector \(X\) with respect to the standard basis \(\{(1,0,0),(0,1,0),(0,0,1)\}\) of \(\mathbb R^3\). Find the expression of \(q(x,y,z)\) with respect to the basis \(B=\left\{\left(\frac1{\sqrt6},\frac1{\sqrt6},-\frac2{\sqrt6}\right),\left(\frac1{\sqrt2},-\frac1{\sqrt2},0\right),\left(\frac1{\sqrt3},\frac1{\sqrt3},\frac1{\sqrt3}\right)\right\}\). Is \(q\) positive definite? Justify your answer.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 product of two Hermitian matrices \(A,B\) is Hermitian if and only if \(A\) and \(B\) commute. Give an example of a pair of \(3\times3\) symmetric matrices such that their product is again symmetric, not considering only diagonal matrices, and also check whether they commute or not.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 \(\iint_R x^2\,dx\,dy\), where \(R\) is the region in the first quadrant bounded by the hyperbola \(xy=16\) and the lines \(y=x\), \(y=0\) and \(x=8\).
2Diagram
Question 1(d): Region bounded by xy = 16, y = x, y = 0 and x = 8
3Concept Related to the Question
This question belongs to Double integration and region splitting. The main idea is to translate the given condition into the standard formula, apply the formula carefully, and simplify without changing the mathematical meaning.
4Detailed Solution
The curves \(xy=16\) and \(y=x\) meet at \(x=4\) in the first quadrant. Therefore the region must be split into two vertical strips.
For \(0\le x\le4\), the upper boundary is \(y=x\). For \(4\le x\le8\), the upper boundary is \(y=16/x\). Hence \(\iint_Rx^2\,dx\,dy=\int_0^4\int_0^x x^2\,dy\,dx+\int_4^8\int_0^{16/x}x^2\,dy\,dx\).
This is \(\int_0^4x^3\,dx+16\int_4^8x\,dx=64+384=448\).
5Final Answer
The required value is \(448\).
Question 1(e)
Plane through a line and perpendicular to a plane
1Question
Find the equation of the plane passing through the points \((1,-1,1)\) and \((-2,1,-1)\) and perpendicular to the plane \(2x+y+z+5=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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.
Express the polynomial \(f(x)=x^2+4x-3\) over \(\mathbb R\) as a linear combination of the polynomials \(e_1=x^2-2x+5\), \(e_2=2x^2-3x\), \(e_3=x+3\). Also, show that the set \(\{e_1,e_2,e_3\}\) forms a basis of all quadratic polynomials over \(\mathbb R\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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.
Does \(f(x)=x+\frac1x\) in \(\left[\frac12,3\right]\) satisfy the conditions of the mean value theorem? If yes, then justify your answer and find \(c\in(a,b)\) such that \(f'(c)=\frac{f(b)-f(a)}{b-a}\), where \(a=\frac12\) and \(b=3\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 straight lines whose direction cosines are given by the equations \(al+bm+cn=0\) and \(ul^2+vm^2+wn^2=0\), where \(a,b,c,u,v,w\) are constants, are parallel if \(\frac{a^2}{u}+\frac{b^2}{v}+\frac{c^2}{w}=0\) and perpendicular if \(a^2(v+w)+b^2(w+u)+c^2(u+v)=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 is projected in a direction making an angle \(\alpha\) with the horizon. It passes through the two points \((x_1,y_1)\) and \((x_2,y_2)\). Prove that \(\tan\alpha=\frac{y_1R}{Rx_1-x_1^2}=\frac{x_2^2y_1-x_1^2y_2}{x_1x_2(x_2-x_1)}\), where \(R\) denotes the horizontal range.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 light rods are joined smoothly to form a quadrilateral \(ABCD\). Let \(P\) and \(Q\) be the mid-points of an opposite pair of rods and these points are connected by a string in a state of tension \(T\). Let \(R\) and \(S\) be the mid-points of the other opposite pair of rods and these points are connected by a light rod in a state of thrust \(X\). Show that \(T\cdot(RS)=X\cdot(PQ)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 is moving in a medium with central acceleration \(P\). The medium is a resisting medium in which resistance \(=kv^2\), \(v\) being the velocity. Let \(s\) be the arc-length, \((r,\theta)\) be plane polar coordinates, \(u=\frac1r\) and \(M_0\) be the initial moment of momentum about the centre of force. Show that the equation of the path of the particle is \(Pe^{2ks}=M_0^2u^2\left(u+\frac{d^2u}{d\theta^2}\right)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 \(\vec a\) and \(\vec b\) be any two vector point functions defined on Euclidean space \(\mathbb R^3\). Derive the vector identity for \(\nabla(\vec a\cdot\vec b)\). Verify that identity for \(\operatorname{grad}(\operatorname{grad}\phi\cdot\operatorname{grad}\psi)\), where \(\phi=3x^2y\), \(\psi=xz^2-2y\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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.
State Gauss' Divergence Theorem completely. Verify the theorem for a field vector \(\vec f=4x\hat i-2y^2\hat j+z^2\hat k\) taken over the region bounded by the cylinder \(x^2+y^2=9\), \(z=0\), \(z=4\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 a solid in the shape of a double cone bounded by two equal circular ends. The solid floats in a liquid, whose density is twice that of the cone, with its axis horizontal. Prove that the equilibrium is stable or unstable according as the semi-vertical angle is less than or greater than \(60^\circ\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 mass density at any point of a cord varies as the radius of curvature of the curve in which it hangs freely under gravity, then prove that this curve is the catenary of uniform strength.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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: (i) Principal normals at consecutive points on a curve in a space do not intersect unless its torsion is zero. (ii) Principal normal of a curve in a space will be binormal of another curve if the curvature of the given curve is proportional to \((k^2+\tau^2)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2021 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 2021 IFoS Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2021 IFoS Maths Optional Paper I Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course.
Which question is given as a free sample solution on this page?
Question 1(d) is given as the free sample solution on this 2021 IFoS Maths Optional Paper I Solutions page.
How should I use these 2021 IFoS Maths Optional Paper I Solutions for preparation?
Students should first solve the question independently, then compare their method with the solution format, diagram presentation, concept explanation, detailed solution, and final answer.
Do these solutions include diagrams and detailed solutions?
Yes. The complete PYQ course includes question-wise diagrams where needed, concept related to the question, detailed solutions, and final answers.
How can I get complete solutions for all questions in 2021 IFoS Maths Optional Paper I Solutions?
To get complete solutions for all questions, students can fill out the admission form. The Ramana Sri IAS admission team will guide students through WhatsApp, email, or call.
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