2022 IFoS Maths Optional Paper I Solutions | Ramana Sri IAS
2022 IFoS Maths Optional Paper I Solutions
2022 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2022 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 2022 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 2022 IFoS Maths Optional Paper I Solutions
These 2022 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 2022 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 2022 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 2022 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 2022 IFoS Maths Optional Paper I Solutions for all questions are available in the full PYQ course.
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Question 1(a)
Vector Spaces – Cosets of Subspaces
1Question
Let \(U\) and \(W\) be subspaces of a vector space \(V\) and \(x,y\in V\). Then prove that \(x+U\subseteq y+W\) iff \(U\subseteq W\) and \(x-y\in W\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 \(v_1=(1,1,-1)\), \(v_2=(4,1,1)\), \(v_3=(1,-1,2)\) be a basis of \(\mathbb R^3\) and let \(T:\mathbb R^3\to\mathbb R^2\) be the linear transformation such that \(Tv_1=(1,0)\), \(Tv_2=(0,1)\), \(Tv_3=(1,1)\). Describe the linear transformation \(T\).
2Diagram
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The Diagram section for this question is available in the full 2022 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 at a constant distance of \(6\) units from the origin and meets the axes in \(A(a,0,0)\), \(B(0,b,0)\) and \(C(0,0,c)\). Find the locus of the centroid of triangle \(ABC\).
2Diagram
Question 1(e): Locus of the Centroid in 3D
3Concept Related to the Question
This question belongs to Analytical Geometry – Plane Intercepts. The method is to first write the definition or standard formula, then substitute the given data, and finally simplify each step without skipping the reason behind the step.
4Detailed Solution
The plane in intercept form is \(\frac xa+\frac yb+\frac zc=1\). Its distance from the origin is \(\frac1{\sqrt{1/a^2+1/b^2+1/c^2}}\). This distance is given as \(6\), so \(\frac1{a^2}+\frac1{b^2}+\frac1{c^2}=\frac1{36}\).
If the centroid is \((X,Y,Z)\), then \(X=a/3\), \(Y=b/3\), and \(Z=c/3\). Substituting \(a=3X\), \(b=3Y\), and \(c=3Z\) gives the required locus.
5Final Answer
The locus is \(\frac1{X^2}+\frac1{Y^2}+\frac1{Z^2}=\frac14\).
Question 2(a)
Matrices – Similarity
1Question
Are the matrices \(A=\begin{pmatrix}2&4\\0&4\end{pmatrix}\) and \(B=\begin{pmatrix}3&1\\1&3\end{pmatrix}\) similar? Justify your answer.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 Lagrange's undetermined multipliers method, find the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
Analytical Geometry – Shortest Distance Between Lines
1Question
Obtain the coordinates of the points where the shortest distance line between the straight lines \(\displaystyle \frac{x-3}{-1}=\frac{y-2}{2}=\frac{z-2}{-1}\), \(\displaystyle \frac{x-2}{2}=\frac{y+3}{3}=\frac{z+2}{2}\) meets them. Also find the magnitude of the shortest distance and the equation of the shortest distance line between the straight lines mentioned above.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 centre of mass of a solid bounded below by \(x^2+y^2\le4,\; z=0\) and above by the paraboloid \(z=4-x^2-y^2\). Take the density of the solid as uniform.
2Diagram
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The Diagram section for this question is available in the full 2022 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 complex vector space of \(3\times3\) skew-symmetric matrices with complex entries, i.e. \(V=\{A\in M_{3\times3}(\mathbb C)\mid A^t=-A\}\). Let \(B=\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}\). Define a linear transformation \(T:V\to V\) by \(T(A)=BA-AB\). Find the eigenvalues and eigenvectors of \(T\).
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Three forces \(P,Q,R\) act along the sides \(BC,CA,AB\) of \(\triangle ABC\) in order to keep the system in equilibrium. If the resultant force touches the inscribed circle, then prove that \(\displaystyle \frac{1+\cos\alpha}{P}+\frac{1+\cos\beta}{Q}+\frac{1+\cos\gamma}{R}=0\), where \(\alpha,\beta,\gamma\) are the interior angles subtended at \(A,B,C\) respectively.
2Diagram
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The Diagram section for this question is available in the full 2022 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 person is drawing water from a well with a light bucket which leaks uniformly. The bucket weighs \(50\) kg when it is full. When it arrives at the top, half of the water remains inside. If the depth of the water level in the well from the top is \(30\) m, then find the work done in raising the bucket to the top from the water level.
2Diagram
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The Diagram section for this question is available in the full 2022 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.
Determine constants \(a,b,c\) so that the directional derivative of \(\phi(x,y,z)=axy^2+byz+cz^2x^3\) at \((1,2,-1)\) has a maximum magnitude \(88\) in a direction parallel to \(z\)-axis.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 of mass \(m\), which is attached to one end of a light string whose other end is fixed at a point \(O\), describes a circular motion in a horizontal plane about the vertical axis through \(O\). Prove that the particle moves in a conical pendulum only if \(g\lt l\omega^2\), where \(l\) is the length of the string and \(\omega\) being angular velocity.
Further, a particle of mass \(m\) is attached to the middle of a light string of length \(2l\), one end of which is fastened to a fixed point and the other end to a smooth ring of mass \(M\) which slides on a smooth vertical rod. If the particle describes a horizontal circle with uniform angular velocity \(\omega\) about the rod, then prove that the inclination of both portions of the string to the vertical is \(\displaystyle \cos^{-1}\left\{\frac{(m+2M)g}{ml\omega^2}\right\}\).
2Diagram
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The Diagram section for this question is available in the full 2022 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 \(C\) is a curve of the intersection of the cylinder \(x^2+y^2=4\) and the plane \(x+y+z=2\) and \(C\) is described counterclockwise. Verify Stokes' theorem for the line integral \(\displaystyle \int_C -y^3\,dx+x^3\,dy-z^3\,dz\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
Derive vector identity for divergence of cross product of two vector point functions. Given a relation between linear and angular velocity as \(\vec v=\vec\omega\times\vec r\). If \(\vec\omega\) is constant, then show that \(\operatorname{curl}\vec v=2\vec\omega\) and \(\operatorname{div}\vec v=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 \(y_1=x^2\) is a solution of the differential equation \(\displaystyle x^3\frac{d^2y}{dx^2}-(x^2+3x)\frac{dy}{dx}+6y=0\), find the other linearly independent solution of the above differential equation and write down the general solution of the differential equation.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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.
\(PR\) and \(QR\) are two equal heavy strings tied together at \(R\) and carrying a weight \(W\) at \(R\). \(P\) and \(Q\) are two points in the same horizontal line and \(2a\) is the distance between them. \(l\) is the length of each string and \(h\) is the depth of \(R\) below \(PQ\). Prove that \(l^2-h^2=2c^2\left(\cosh\frac ac-1\right)\) and tension at \(P\) or \(Q=\dfrac1{2h}\{lW+(l^2+h^2)w\}\), where \(c\) is the parameter of the catenary and \(w\) is the line density of the string.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 bucket is in the form of a frustum of a cone and is filled with water of density \(\rho\). If the bottom and top ends of the bucket have radii \(a\) and \(b\) respectively and \(h\) is the height of the bucket, then find the resultant vertical thrust on the curved surface of the bucket. Is that thrust equal to \(\displaystyle \frac13\pi\rho gh(b-a)(b+2a)\)?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 a curve in a space is represented by \(\vec r=\vec r(t)\), then derive expressions of its torsion and curvature in terms of \(\dot{\vec r},\ddot{\vec r}\) and \(\dddot{\vec r}\). Find the curvature and torsion of the curve given by \(\vec r=(at-a\sin t,\;a-a\cos t,\;bt)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 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 2022 IFoS Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2022 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(e) is given as the free sample solution on this 2022 IFoS Maths Optional Paper I Solutions page.
How should I use these 2022 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 2022 IFoS Maths Optional Paper I?
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