Ramana Sri IAS - 2019 UPSC Maths Optional Paper I Solutions
2019 UPSC Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2019 UPSC Maths Optional Paper I Solutions. 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 2019 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 2019 UPSC Maths Optional Paper I Solutions
These 2019 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 2019 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 2019 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 2019 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 2019 UPSC Maths Optional Paper I Solutions for all questions are available in the full PYQ course. To purchase the full solution,
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2019 UPSC Maths Optional Paper I Solutions: Table of Contents
Let \(f:[0,\dfrac{\pi}{2}]\to R\) be a continuous function such that \(f(x)=\dfrac{\cos^2 x}{4x^2-\pi^2}\), \(0<x<\dfrac{\pi}{2}\). Find the value of \(f\left(\dfrac{\pi}{2}\right)\).
2Diagram
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The Diagram section for this question is available in the full 2019 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 \(f:D(\subset R^2)\to R\) be a function and \((a,b)\in D\). If \(f(x,y)\) is continuous at \((a,b)\), then show that the functions \(f(x,b)\) and \(f(a,y)\) are continuous at \(x=a\) and at \(y=b\) respectively.
2Diagram
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The Diagram section for this question is available in the full 2019 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 \(T:R^2\to R^2\) be a linear map such that \(T(2,1)=(5,7)\) and \(T(1,2)=(3,3)\). If \(A\) is the matrix corresponding to \(T\) with respect to the standard bases \(e_1,e_2\), then find \(\operatorname{Rank}(A)\).
2Diagram
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The Diagram section for this question is available in the full 2019 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 \(A=\left[\begin{smallmatrix}1&2&1\\ 1&-4&1\\ 3&0&-3\end{smallmatrix}\right]\) and \(B=\left[\begin{smallmatrix}2&1&1\\ 1&-1&0\\ 2&1&-1\end{smallmatrix}\right]\), then show that \(AB=6I_3\). Use this result to solve the following system of equations: \(2x+y+z=5,\ x-y=0,\ 2x+y-z=1\).
2Diagram
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The Diagram section for this question is available in the full 2019 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 lines \(\dfrac{x+1}{-3}=\dfrac{y-3}{2}=\dfrac{z+2}{1}\) and \(\dfrac{x}{1}=\dfrac{y-7}{-3}=\dfrac{z+7}{2}\) intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.
2Diagram
Question 1(e): Intersecting lines and containing plane
3Concept Related to the Question
To check whether two lines intersect, write both lines in parametric form and solve for their parameters. If a common point exists, the lines intersect.
The plane containing two intersecting lines is found by using the common point and the two direction vectors of the lines.
At the point of intersection, the corresponding coordinates must be equal. Hence
\[-1-3r=s,\qquad 3+2r=7-3s,\qquad -2+r=-7+2s.\]
Solving these equations gives
\[r=-1,\qquad s=2.\]
Substituting \(r=-1\) in the first line gives the common point
\[(x,y,z)=(2,1,-3).\]
Therefore the two lines intersect at \((2,1,-3)\).
The direction ratios of the two lines are
\[\vec d_1=(-3,2,1),\qquad \vec d_2=(1,-3,2).\]
A normal vector to the required plane is obtained from their cross product:
\[\vec n=\vec d_1\times\vec d_2=(7,7,7).\]
Thus the required plane through \((2,1,-3)\) is
\[7(x-2)+7(y-1)+7(z+3)=0.\]
Dividing by \(7\),
\[x+y+z=0.\]
5Final Answer
The point of intersection is
\[(2,1,-3),\]
and the plane containing the two lines is
\[x+y+z=0.\]
Question 2(a)
Differentiability of absolute value function
1Question
Is \(f(x)=|\cos x|+|\sin x|\) differentiable at \(x=\dfrac{\pi}{2}\)? If yes, then find its derivative at \(x=\dfrac{\pi}{2}\). If no, then give a proof of it.
2Diagram
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The Diagram section for this question is available in the full 2019 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 plane \(z=0\) cuts the enveloping cone of the sphere \(x^2+y^2+z^2=11\) which has the vertex at \((2,4,1)\), in a rectangular hyperbola.
2Diagram
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The Diagram section for this question is available in the full 2019 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, in general, three normals can be drawn from a given point to the paraboloid \(x^2+y^2=2az\), but if the point lies on the surface \(27a(x^2+y^2)+8(a-z)^3=0\), then two of the three normals coincide.
2Diagram
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The Diagram section for this question is available in the full 2019 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 \(A={\large \left(\begin{smallmatrix}5&7&2&1\\1&1&-8&1\\2&3&5&0\\3&4&-3&1\end{smallmatrix}\right)}\). (i) Find the rank of matrix \(A\). (ii) Find the dimension of the subspace \(V=\{(x_1,x_2,x_3,x_4)\in R^4\mid A{\large \left(\begin{smallmatrix}x_1\\x_2\\x_3\\x_4\end{smallmatrix}\right)}=0\}\).
2Diagram
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The Diagram section for this question is available in the full 2019 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.
State the Cayley-Hamilton theorem. Use this theorem to find \(A^{100}\), where \(A=\left[\begin{smallmatrix}1&0&0\\ 1&0&1\\ 0&1&0\end{smallmatrix}\right]\).
2Diagram
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The Diagram section for this question is available in the full 2019 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 length of the normal chord through a point \(P\) of the ellipsoid \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\) and prove that if it is equal to \(4PG_3\), where \(G_3\) is the point where the normal chord through \(P\) meets the \(xy\)-plane, then \(P\) lies on the cone \(\dfrac{x^2}{a^6}(2c^2-a^2)+\dfrac{y^2}{b^6}(2c^2-b^2)+\dfrac{z^2}{c^4}=0\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 \(u=\sin^{-1}\sqrt{\dfrac{x^{\frac{1}{3}}+y^{\frac{1}{3}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}}\), then show that \(\sin^2u\) is a homogeneous function of \(x\) and \(y\) of degree \(-\dfrac{1}{6}\). Hence show that \(x^2\dfrac{\partial^2u}{\partial x^2}+2xy\dfrac{\partial^2u}{\partial x\partial y}+y^2\dfrac{\partial^2u}{\partial y^2}=\dfrac{\tan u}{12}\left(\dfrac{13}{12}+\dfrac{\tan^2u}{12}\right)\).
2Diagram
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The Diagram section for this question is available in the full 2019 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.
One end of a heavy uniform rod \(AB\) can slide along a rough horizontal rod \(AC\), to which it is attached by a ring. \(B\) and \(C\) are joined by a string. When the rod is on the point of sliding, then \(AC^2-AB^2=BC^2\). If \(\theta\) is the angle between \(AB\) and the horizontal line, then prove that the coefficient of friction is \(\dfrac{\cot\theta}{2+\cot^2\theta}\).
2Diagram
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The Diagram section for this question is available in the full 2019 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 force of attraction of a particle by the earth is inversely proportional to the square of its distance from the earth's centre. A particle, whose weight on the surface of the earth is \(W\), falls to the surface of the earth from a height \(3h\) above it. Show that the magnitude of work done by the earth's attraction force is \(\dfrac{3}{4}hW\), where \(h\) is the radius of the earth.
2Diagram
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The Diagram section for this question is available in the full 2019 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 consists of a cone and underlying hemisphere. The base of the cone and the top of the hemisphere have same radius \(a\). The whole body rests on a rough horizontal table with hemisphere in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is \(\sqrt3a\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 circulation of \(\vec F\) round the curve \(C\), where \(\vec F=(2x+y^2)\vec i+(3y-4x)\vec j\) and \(C\) is the curve \(y=x^2\) from \((0,0)\) to \((1,1)\) and the curve \(y^2=x\) from \((1,1)\) to \((0,0)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 Laplace transforms of \(t^{-\frac{1}{2}}\) and \(t^{\frac{1}{2}}\). Prove that the Laplace transform of \(t^{n+\dfrac{1}{2}}\), where \(n\in N\), is \(\dfrac{\Gamma\left(n+1+\dfrac{1}{2}\right)}{s^{n+1+\dfrac{1}{2}}}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 linearly independent solutions of the corresponding homogeneous differential equation of the equation \(x^2y''-2xy'+2y=x^3\sin x\) and then find the general solution of the given equation by the method of variation of parameters.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 moving along the \(y\)-axis has an acceleration \(Fy\) towards the origin, where \(F\) is positive and even function of \(y\). The periodic time, when the particle vibrates between \(y=-a\) and \(y=a\), is \(T\). Show that \(\dfrac{2\pi}{\sqrt{F_1}}<T<\dfrac{2\pi}{\sqrt{F_2}}\), where \(F_1\) and \(F_2\) are the greatest and the least values of \(F\) within the range \([-a,a]\). Further, show that when a simple pendulum of length \(l\) oscillates through \(30^\circ\) on either side of the vertical line, \(T\) lies between \(2\pi\sqrt{\dfrac{l}{g}}\) and \(2\pi\sqrt{\dfrac{l}{g}}\sqrt{\dfrac{\pi}{3}}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 singular solution of the differential equation \(\left(\dfrac{dy}{dx}\right)^2\left(\dfrac{y}{x}\right)^2\cot^2\alpha-2\left(\dfrac{dy}{dx}\right)\left(\dfrac{y}{x}\right)+\left(\dfrac{y}{x}\right)^2cosec^2\alpha=1\). Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 path of a planet, which is moving so that its acceleration is always directed to a fixed point (star) and is equal to \(\dfrac{\mu}{(\text{distance})^2}\), is a conic section. Find the conditions under which the path becomes (i) ellipse, (ii) parabola and (iii) hyperbola.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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.
State Gauss divergence theorem. Verify this theorem for \(\vec F=4x\vec i-y^2\vec j+z^2\vec k\), taken over the region bounded by \(x^2+y^2=4\), \(z=0\) and \(z=3\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2019 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 2019 UPSC Maths Optional Paper I Solutions complete?
This public page gives one full sample solution from 2019 UPSC Maths Optional Paper I Solutions. Complete question-wise solutions for the full paper are available in the full PYQ course by Ramana Sri IAS.
Which question is given as a free sample solution on this page?
The free sample solution on this page is Question 1(e). It includes Question, Diagram, Concept Related to the Question, Detailed Solution, and Final Answer.
How should I use these 2019 UPSC Maths Optional Paper I Solutions for preparation?
Students should first solve the question independently, then compare their approach with the step-by-step solution, diagram, concept explanation, and final answer format.
Do these solutions include diagrams and detailed solutions?
Yes. The full PYQ course includes diagrams where needed, concept explanations, detailed solutions, and final answers for the questions in the paper.
How can I get complete solutions for all questions in 2019 UPSC Maths Optional Paper I Solutions?
To get complete solutions for all questions, fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
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