Ramana Sri IAS - 2023 UPSC Maths Optional Paper I Solutions
2023 UPSC Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2023 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 2023 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 2023 UPSC Maths Optional Paper I Solutions
These 2023 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 2023 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 2023 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 2023 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 2023 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. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
2023 UPSC Maths Optional Paper I Solutions: Table of Contents
Let \(V_1=(2,-1,3,2)\), \(V_2=(-1,1,1,-3)\) and \(V_3=(1,1,9,-5)\) be three vectors of the space \(\mathbb R^4\). Does \((3,-1,0,-1)\in \operatorname{span}\{V_1,V_2,V_3\}\)? Justify your answer.
2Diagram
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The Diagram section for this question is available in the full 2023 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 which at a constant distance \(3p\) from the origin \(O\) cuts the axes in the points \(A\), \(B\), \(C\) respectively. Show that the locus of the centroid of the tetrahedron \(OABC\) is \(\displaystyle 9\left(\frac1{x^2}+\frac1{y^2}+\frac1{z^2}\right)=\frac{16}{p^2}\).
2Diagram
Question 1(e): Centroid locus of tetrahedron
3Concept Related to the Question
Write the plane in intercept form. Then express the intercepts in terms of the centroid coordinates.
4Detailed Solution
Let the plane cut the axes at \((a,0,0)\), \((0,b,0)\), and \((0,0,c)\). Its equation is
The locus of the centroid is \(9\left(\frac1{x^2}+\frac1{y^2}+\frac1{z^2}\right)=\frac{16}{p^2}\).
Question 2(a)
Change of basis matrix
1Question
If the matrix of a linear transformation \(T:\mathbb R^3\to\mathbb R^3\) relative to the basis \(\{(1,0,0),(0,1,0),(0,0,1)\}\) is \(\begin{bmatrix}1&1&2\\-1&2&1\\0&1&3\end{bmatrix}\), then find the matrix of \(T\) relative to the basis \(\{(1,1,1),(0,1,1),(0,0,1)\}\).
2Diagram
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The Diagram section for this question is available in the full 2023 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.
The plane \(\frac xa+\frac yb+\frac zc=1\) meets the coordinate axes in \(A\), \(B\), \(C\) respectively. Prove that the equation of the cone generated by the lines drawn from the origin \(O\) to meet the circle \(ABC\) is \(yz\left(\frac bc+\frac cb\right)+zx\left(\frac ca+\frac ac\right)+xy\left(\frac ba+\frac ab\right)=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 \(A={\large \left[\begin{smallmatrix}1&0&0\\1&0&1\\0&1&0\end{smallmatrix}\right]}\). (i) Verify the Cayley-Hamilton theorem for the matrix \(A\). (ii) Show that \(A^n=A^{n-2}+A^2-I\) for \(n\ge3\), where \(I\) is the identity matrix of order \(3\). Hence, find \(A^{40}\).
2Diagram
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The Diagram section for this question is available in the full 2023 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 through the circle \(x^2+y^2+z^2-4x-6y+2z-16=0\), \(3x+y+3z-4=0\) in the following two cases: (i) the point \((1,0,-3)\) lies on the sphere, (ii) the given circle is a great circle of the sphere.
2Diagram
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The Diagram section for this question is available in the full 2023 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.
Prove that the locus of a line which meets the lines \(y=mx,\ z=c\), \(y=-mx,\ z=-c\) and the circle \(x^2+y^2=a^2,\ z=0\) is \(c^2m^2(cy-mzx)^2+c^2(yz-cmx)^2=a^2m^2(z^2-c^2)^2\).
2Diagram
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The Diagram section for this question is available in the full 2023 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.
Obtain the solution of the initial-value problem \(\frac{dy}{dx}-2xy=2,\ y(0)=1\) in the form \(y=e^{x^2}\left[1+\sqrt\pi\,\operatorname{erf}(x)\right]\).
2Diagram
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The Diagram section for this question is available in the full 2023 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.
Given that \(L\{f(t);p\}=F(p)\). Show that \(\displaystyle \int_0^\infty\frac{f(t)}{t}\,dt=\int_0^\infty F(x)\,dx\). Hence evaluate the integral \(\displaystyle \int_0^\infty\frac{e^{-t}-e^{-3t}}{t}\,dt\).
2Diagram
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The Diagram section for this question is available in the full 2023 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 cylinder of radius ‘\(a\)’ touches a vertical wall along a generating line. Axis of the cylinder is fixed horizontally. A uniform flat beam of length ‘\(l\)’ and weight ‘\(W\)’ rests with its extremities in contact with the wall and the cylinder, making an angle of \(45^\circ\) with the vertical. If frictional forces are neglected, then show that \(\displaystyle \frac al=\frac{\sqrt5+5}{4\sqrt2}\). Also, find the reactions of the cylinder and wall.
2Diagram
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The Diagram section for this question is available in the full 2023 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 particle is moving under Simple Harmonic Motion of period \(T\) about a centre \(O\). It passes through the point \(P\) with velocity \(v\) along the direction \(OP\) and \(OP=p\). Find the time that elapses before the particle returns to the point \(P\). What will be the value of \(p\) when the elapsed time is \(\frac T2\)?
2Diagram
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The Diagram section for this question is available in the full 2023 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 \(\vec a=\sin\theta\,\hat i+\cos\theta\,\hat j+0\hat k\), \(\vec b=\cos\theta\,\hat i-\sin\theta\,\hat j-3\hat k\), and \(\vec c=2\hat i+3\hat j-3\hat k\), then find the values of the derivative of the vector function \(\vec a\times(\vec b\times\vec c)\) w.r.t. \(\theta\) at \(\theta=\frac\pi2\) and \(\theta=\pi\).
2Diagram
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The Diagram section for this question is available in the full 2023 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.
When a particle is projected from a point \(O_1\) on the sea level with a velocity \(v\) and angle of projection \(\theta\) with the horizon in a vertical plane, its horizontal range is \(R_1\). If it is further projected from a point \(O_2\), which is vertically above \(O_1\) at a height \(h\) in the same vertical plane, with the same velocity \(v\) and same angle \(\theta\) with the horizon, its horizontal range is \(R_2\). Prove that \(R_2>R_1\) and \((R_2-R_1):R_1\) is equal to \(\frac12\left\{\sqrt{1+\frac{2gh}{v^2\sin^2\theta}}-1\right\}:1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
Evaluate the integral \(\displaystyle \iint_S(3y^2z^2\hat i+4z^2x^2\hat j+zy^2\hat k)\cdot\hat n\,dS\), where \(S\) is the upper part of the surface \(4x^2+4y^2+4z^2=1\) above the plane \(z=0\) and bounded by the \(xy\)-plane. Hence, verify Gauss-Divergence theorem.
2Diagram
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The Diagram section for this question is available in the full 2023 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.
Reduce the equation \(x^2p^2+y(2x+y)p+y^2=0\) to Clairaut’s form by the substitution \(y=u\) and \(xy=v\). Hence solve the equation and show that \(y+4x=0\) is a singular solution of the differential equation.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface is in contact. If \(\theta\) is the angle of inclination of the string with vertical and \(\phi\) is the angle of inclination of the plane base of the hemisphere to the vertical, then find the value of \((\tan\phi-\tan\theta)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 the tangent to a curve makes a constant angle \(\theta\) with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to \(\tan\theta\). Further prove that if this ratio is constant, then 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 2023 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 transform technique : \(\frac{d^2y}{dt^2}-4\frac{dy}{dt}+3y(t)=f(t)\), \(y(0)=1\), \(y'(0)=0\) and \(f(t)\) is a given function of \(t\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 particle is projected from an apse at a distance \(\sqrt c\) from the centre of force with a velocity \(\sqrt{\frac{2\lambda}{3}c^3}\) and is moving with central acceleration \(\lambda(r^5-c^2r)\). Find the path of motion of this particle. Will that be the curve \(x^4+y^4=c^2\)?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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.
For a scalar point function \(\phi\) and vector point function \(\vec f\), prove the identity \(\nabla\cdot(\phi\vec f)=\nabla\phi\cdot\vec f+\phi(\nabla\cdot\vec f)\). Also find the value of \(\nabla\cdot\left(\frac{f(r)}r\vec r\right)\) and then verify stated identity.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2023 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 2023 UPSC Maths Optional Paper I Solutions complete?
This public page gives one full sample solution from 2023 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 page. It includes the Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer sections.
How should I use these 2023 UPSC Maths Optional Paper I Solutions for preparation?
Use these solutions for revision, answer-writing practice, concept clarity, diagram presentation, and final-answer formatting before the UPSC Mains Mathematics optional paper.
Do these solutions include diagrams and detailed solutions?
Yes. The full PYQ course includes diagrams where needed, concepts, detailed solutions, and final answers for the questions in 2023 UPSC Maths Optional Paper I Solutions.
How can I get complete solutions for all questions in 2023 UPSC Maths Optional Paper I?
To get complete solutions for all questions, fill out the admission form. The Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
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