Ramana Sri IAS - 2018 UPSC Maths Optional Paper I Solutions
2018 UPSC Maths Optional Paper I Solutions Ramana Sri IAS provides complete and updated solutions for the 2018 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 2018 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 2018 UPSC Maths Optional Paper I Solutions These 2018 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 2018 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.
For the official examination source, students may also refer to the UPSC previous year question papers page .
These 2018 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 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.
2018 UPSC Maths Optional Paper I Solutions: Table of Contents
Question 1(a) Singular matrix by rank argument
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(A\) be a \(3\times2\) matrix and \(B\) a \(2\times3\) matrix. Show that \(C=A\cdot B\) is a singular matrix.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2018 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.
3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
Full Solution Access The Detailed Solution section for this question is available in the full 2018 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.
5 Final Answer
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Question 1(b) Change of basis in \(\mathbb{R}^2\)
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionExpress basis vectors \(e_1=(1,0)\) and \(e_2=(0,1)\) as linear combinations of \(\alpha_1=(2,-1)\) and \(\alpha_2=(1,3)\).
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2018 UPSC Maths Optional Paper I PYQ Solutions course.
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 1(c) Limit involving tangent
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionDetermine if \(\displaystyle \lim_{z\to1}(1-z)\tan\frac{\pi z}{2}\) exists or not. If the limit exists, then find its value.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2018 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.
3 Concept Related to the Question
Full Solution Access The Concept Related to the Question section for this question is available in the full 2018 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.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
Question 1(d) Limit as a Riemann sum
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind \(\displaystyle \lim_{n\to\infty}\frac{1}{n^2}\sum_{r=0}^{n-1}\sqrt{n^2-r^2}\).
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2018 UPSC Maths Optional Paper I PYQ Solutions course.
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 1(e) Projection of a line on a plane
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the projection of the straight line \(\frac{x-1}{2}=\frac{y-1}{3}=\frac{z+1}{-1}\) on the plane \(x+y+2z=6\).
2 DiagramQuestion 1(e): Projection of the line on the plane x + y + 2z = 6
3 Concept Related to the QuestionThe projection of a point on a plane can be obtained by moving it along the normal direction of the plane. This gives the projected point in coordinate form.
4 Detailed SolutionLet the parameter on the given line be \(t\). Then a general point on the line is
\[P(t)=(1+2t,\,1+3t,\,-1-t).\]
The normal direction of the plane \(x+y+2z=6\) is \((1,1,2)\). A point on the projected line is therefore of the form
\[Q(t)=P(t)+\lambda(1,1,2).\]
To make \(Q(t)\) lie on the plane, substitute into \(x+y+2z=6\). For \(P(t)\), we have
\[(1+2t)+(1+3t)+2(-1-t)=3t.\]
Hence \(3t+6\lambda=6\), so \(\lambda=1-\frac{t}{2}\). Therefore
\[Q(t)=\left(2+\frac{3t}{2},\,2+\frac{5t}{2},\,1-2t\right).\]
Putting \(s=\frac{t}{2}\), the projected line becomes
\[x=2+3s,\qquad y=2+5s,\qquad z=1-4s.\]
5 Final Answer\[\frac{x-2}{3}=\frac{y-2}{5}=\frac{z-1}{-4}.\]
Question 2(a) Similar matrices have same eigenvalues
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that if \(A\) and \(B\) are similar \(n\times n\) matrices, then they have the same eigenvalues.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2018 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.
3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 2(b) Shortest distance from a point to a parabola
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the shortest distance from the point \((1,0)\) to the parabola \(y^2=4x\).
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 2(c) Volume of ellipsoid of revolution
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionThe ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) revolves about the \(x\)-axis. Find the volume of the solid of revolution.
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 2(d) Shortest distance from an intersection line to the z-axis
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the shortest distance between the lines \(a_1x+b_1y+c_1z+d_1=0\), \(a_2x+b_2y+c_2z+d_2=0\), and the \(z\)-axis.
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 3(a) Consistency of a system of linear equations
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFor the system of linear equations
\[x+3y-2z=-1\]
\[5y+3z=-8\]
\[x-2y-5z=7\]
determine which of the following statements are true and which are false:
(i) The system has no solution.
(ii) The system has a unique solution.
(iii) The system has infinitely many solutions.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
5 Final Answer
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Question 3(b) Existence of partial derivatives
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet
\[f(x,y)=\begin{cases}xy^2, & \text{if } y\gt0,\\-xy^2, & \text{if } y\leq0.\end{cases}\]
Determine which of \(\dfrac{\partial f}{\partial x}(0,1)\) and \(\dfrac{\partial f}{\partial y}(0,1)\) exists and which does not exist.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(c) Generating lines of a ruled surface
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the equations to the generating lines of the paraboloid \((x+y+z)(2x+y-z)=6\) which pass through the point \((1,1,1)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 3(d) Sphere through four points
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the equation of the sphere in \(xyz\)-plane passing through the points \((0,0,0)\), \((0,1,-1)\), \((-1,2,0)\) and \((1,2,3)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(a) Maximum and minimum on an interval
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the maximum and the minimum values of \(x^4-5x^2+4\) on the interval \([2,3]\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(b) Changing order of integration
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(\displaystyle \int_0^a\int_x^a \frac{x}{x^2+y^2}\,dy\,dx\).
2 Diagram
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3 Concept Related to the Question
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Complete diagrams, concepts, detailed solutions and final answers for all questions are available in the full PYQ course.
4 Detailed Solution
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5 Final Answer
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Question 4(c) Cone with given vertex and guiding curve
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the equation of the cone with \((0,0,1)\) as the vertex and \(2x^2-y^2=4\), \(z=0\), as the guiding curve.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 4(d) Plane parallel to a given plane
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the equation of the plane parallel to \(3x-y+3z=8\) and passing through \((1,1,1)\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(a) Linear differential equation with constant coefficients
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve \(y''-y=x^2e^{2x}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(b) Angle between tangent and a line
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the angle between the tangent at a general point of the curve \(x=3t\), \(y=3t^2\), \(z=3t^3\), and the line \(y=z-x=0\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(c) Higher order linear differential equation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve \(y'''-6y''+12y'-8y=12e^{2x}+27e^{-x}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(d) Laplace transform and inverse Laplace transform
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question(i) Find the Laplace transform of \(f(t)=\dfrac{1}{\sqrt t}\).
(ii) Find the inverse Laplace transform of \(\dfrac{5s^2+3s-16}{(s-1)(s-2)(s+3)}\).
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 5(e) Projectile just clearing a wall
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionA particle projected from a given point on the ground just clears a wall of height \(h\) at a distance \(d\) from the point of projection. If the particle moves in a vertical plane and if the horizontal range is \(R\), find the elevation of the projection.
2 Diagram
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3 Concept Related to the Question
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4 Detailed Solution
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5 Final Answer
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Question 6(a) First order differential equation
1. Question
2. Diagram
3. Concept Related to the Question
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1 QuestionSolve
\[\left(\frac{dy}{dx}\right)^2y+2\frac{dy}{dx}x-y=0.\]
2 Diagram
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4 Detailed Solution
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Question 6(b) Simple harmonic motion period
1. Question
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1 QuestionA particle moving with simple harmonic motion in a straight line has velocities \(v_1\) and \(v_2\) at distances \(x_1\) and \(x_2\), respectively, from the centre of its path. Find the period of the motion.
2 Diagram
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Question 6(c) Second order differential equation
1. Question
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1 QuestionSolve \(y''+16y=32\sec2x\).
2 Diagram
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4 Detailed Solution
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Question 6(d) Gauss' divergence theorem
1. Question
2. Diagram
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1 QuestionIf \(S\) is the surface of the sphere \(x^2+y^2+z^2=a^2\), then evaluate
\[\iint_S[(x+z)\,dy\,dz+(y+z)\,dz\,dx+(x+y)\,dx\,dy]\]
using Gauss' divergence theorem.
2 Diagram
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4 Detailed Solution
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Question 7(a) Differential equation reducible by substitution
1. Question
2. Diagram
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1 QuestionSolve \((1+x)^2y''+(1+x)y'+y=4\cos(\log(1+x))\).
2 Diagram
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Question 7(b) Curvature and torsion of a space curve
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind curvature and torsion of the curve
\[\vec r=a(t-\sin t)\vec i+a(1-\cos t)\vec j+bt\vec k.\]
2 Diagram
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4 Detailed Solution
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Question 7(c) Initial value problem
1. Question
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4. Detailed Solution
5. Final Answer
1 QuestionSolve the initial value problem \(y''-5y'+4y=e^{2t}\), \(y(0)=\frac{19}{12}\), \(y'(0)=\frac{8}{3}\).
2 Diagram
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4 Detailed Solution
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Question 7(d) Integrating factor and exact equation
1. Question
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5. Final Answer
1 QuestionFind \(\alpha\) and \(\beta\) such that \(x^\alpha y^\beta\) is an integrating factor of \((4y^2+3xy)dx-(3xy+2x^2)dy=0\), and solve the equation.
2 Diagram
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4 Detailed Solution
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Question 8(a) Vector identity involving curl and divergence
1. Question
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1 QuestionLet \(\vec v=v_1\vec i+v_2\vec j+v_3\vec k\). Show that \(\operatorname{curl}(\operatorname{curl}\vec v)=\operatorname{grad}(\operatorname{div}\vec v)-\nabla^2\vec v\).
2 Diagram
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4 Detailed Solution
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Question 8(b) Line integral using Stokes' theorem
1. Question
2. Diagram
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4. Detailed Solution
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1 QuestionEvaluate the line integral
\[\oint_C y^3\,dx+x^3\,dy+z^3\,dz\]
using Stokes' theorem. Here \(C\) is the intersection of the cylinder \(x^2+y^2=1\) and the plane \(x+y+z=1\). The orientation on \(C\) corresponds to counterclockwise motion in the \(xy\)-plane.
2 Diagram
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4 Detailed Solution
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Question 8(c) Green's theorem over a bounded region
1. Question
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4. Detailed Solution
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1 QuestionLet \(\vec F=xy^2\vec i+(y+x)\vec j\). Integrate \((\nabla\times\vec F)\cdot\vec k\) over the region in the first quadrant bounded by the curves \(y=x^2\) and \(y=x\) using Green's theorem.
2 Diagram
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4 Detailed Solution
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Question 8(d) Exact differential equation
1. Question
2. Diagram
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4. Detailed Solution
5. Final Answer
1 QuestionFind \(f(y)\) such that \((2xe^y+3y^2)dy+(3x^2+f(y))dx=0\) is exact, and hence solve it.
2 Diagram
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4 Detailed Solution
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5 Final Answer
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2018 UPSC Maths Optional Paper I Solutions FAQs
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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 Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2018 UPSC Maths Optional Paper I Solutions for preparation? Use these solutions for revision, answer-writing practice, concept clarity, and checking the step-by-step presentation expected in the UPSC Mathematics optional paper.
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