Ramana Sri IAS - 2022 UPSC Maths Optional Paper I Solutions
2022 UPSC Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2022 UPSC Maths Optional Paper I. Aspirants preparing for the UPSC Mains Examination with Mathematics as their optional subject should solve all these questions carefully at least 5 to 10 times before the mains examination.
Ramana Sri IAS presents complete solutions for UPSC/IAS/CSE-Civil 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 UPSC Maths Optional Paper I Solutions
These 2022 UPSC Maths Optional Paper I Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the UPSC 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 UPSC 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 UPSC Maths Optional Paper I Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the UPSC Mathematics optional paper.
Sample Full Solution
We are giving one question from 2022 UPSC 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 UPSC 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.
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2022 UPSC Maths Optional Paper I Solutions: Table of Contents
Let \(T:\mathbb R^2\to\mathbb R^3\) be a linear transformation such that \(T{\large \left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)}={\large \left(\begin{smallmatrix}1\\2\\3\end{smallmatrix}\right)}\) and \(T{\large \left(\begin{smallmatrix}1\\1\end{smallmatrix}\right)}={\large \left(\begin{smallmatrix}-3\\2\\8\end{smallmatrix}\right)}\). Find \(T{\large \left(\begin{smallmatrix}2\\4\end{smallmatrix}\right)}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 passes through a fixed point \((a,b,c)\) and meets the axes at points \(A,B\) and \(C\) respectively. Find the locus of the centre of the sphere passing through the points \(O,A,B\) and \(C\), \(O\) being the origin.
2Diagram
Question 1(e): Sphere through intercepts of a variable plane
3Concept Related to the Question
Use the intercept form of a plane and the standard equation of a sphere passing through the origin and the three intercept points.
A wire of length \(l\) is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange’s method of undetermined multipliers, find the least value of the sum of the areas so formed.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 \(P,Q,R;P',Q',R'\) are feet of the six normals drawn from a point to the ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\), and the plane \(PQR\) is represented by \(lx+my+nz=p\), show that the plane \(P'Q'R'\) is given by \(\frac{x}{a^2l}+\frac{y}{b^2m}+\frac{z}{c^2n}+\frac1p=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 the set \(P=\left\{{\large \left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)}:x-y-z=0\text{ and }2x-y+z=0\right\}\) be the collection of vectors of a vector space \(\mathbb R^3(\mathbb R)\). Then (i) prove that \(P\) is a subspace of \(\mathbb R^3\), and (ii) find a basis and dimension of \(P\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 sphere of smallest possible radius which touches the straight lines \(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\) 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 2022 UPSC 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 linear map \(T:\mathbb R^2\to\mathbb R^2\) which rotates each vector of \(\mathbb R^2\) by an angle \(\theta\). Also, prove that for \(\theta=\frac{\pi}{2}\), \(T\) has no eigenvalue in \(\mathbb R\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 general solution of the differential equation \(\frac{dy}{dx}+Py=Q\) can be written in the form \(y=\frac QP-e^{-\int P\,dx}\left\{C+\int e^{\int P\,dx}\,d\left(\frac QP\right)\right\}\), where \(P,Q\) are non-zero functions of \(x\) and \(C\), an arbitrary constant.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 body of weight \(w\) rests on a rough inclined plane of inclination \(\theta\), the coefficient of friction, \(\mu\), being greater than \(\tan\theta\). Find the work done in slowly dragging the body a distance \(b\) up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 fired from a point \(O\) with velocity \(\sqrt{2gh}\) and hits a target at the point \(P(x,y)\) in the plane, the axes \(OX\) and \(OY\) being horizontal and vertically downward lines through the point \(O\), respectively. Show that if the two possible directions of projection be at right angles, then \(x^2=2hy\) and then one of the possible directions of projection bisects the angle \(POX\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 cable of weight \(w\) per unit length and length \(2l\) hangs from two points \(P\) and \(Q\) in the same horizontal line. Show that the span of the cable is \(2l\left(1-\frac{2h^2}{3l^2}\right)\), where \(h\) is the sag in the middle of the tightly stretched position.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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.
Solve the following differential equation by using the method of variation of parameters: \((x^2-1)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2y=(x^2-1)^2\), given that \(y=x\) is one solution of the reduced equation.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 Green’s theorem in the plane for \(\displaystyle \oint_C(3x^2-8y^2)\,dx+(4y-6xy)\,dy\), where \(C\) is the boundary curve of the region defined by \(x=0\), \(y=0\), and \(x+y=1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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.
Solve the following initial value problem by using Laplace's transformation \(\dfrac{d^2y}{dt^2}-3\dfrac{dy}{dt}+2y=h(t)\), where \(h(t)=2\), \(0<t<4\), \(h(t)=0\), \(t\gt4\), \(y(0)=0\), \(y'(0)=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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.
Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let \(\rho\) and \(\rho'\) be the radii of curvature of the two cylinders at the point of contact and \(h\) be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if \(h<\dfrac{\rho\rho'}{\rho+\rho'}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 general and singular solutions of the differential equation \((x^2-a^2)p^2-2xyp+y^2+a^2=0\), where \(p=\frac{dy}{dx}\). Also give the geometric relation between the general and singular solutions.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 chain of \(n\) equal uniform rods is smoothly jointed together and suspended from its one end \(A_1\). A horizontal force \(\vec P\) is applied to the other end \(A_{n+1}\) of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 Gauss’ divergence theorem, evaluate \(\displaystyle \iint_S\vec F\cdot\vec n\,dS\), where \(\vec F=x\hat i-y\hat j+(z^2-1)\hat k\) and \(S\) is the cylinder formed by the surfaces \(z=0\), \(z=1\), and \(x^2+y^2=4\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2022 UPSC 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 UPSC Maths Optional Paper I Solutions complete?
This public page gives one full sample solution for 2022 UPSC 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 UPSC Maths Optional Paper I Solutions page.
How should I use these 2022 UPSC 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 UPSC Maths Optional Paper I?
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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