Ramana Sri IAS - 2016 UPSC Maths Optional Paper I Solutions
2016 UPSC Maths Optional Paper I Solutions Ramana Sri IAS provides complete and updated solutions for the 2016 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 2016 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 2016 UPSC Maths Optional Paper I Solutions
These 2016 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 2016 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 2016 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(d) . This free sample includes all five sections:
Question , Diagram ,
Concept Related to the Question ,
Detailed Solution , and Final Answer .
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2016 UPSC Maths Optional Paper I Solutions: Table of Contents
Question 1(a)(i) Inverse of a matrix by elementary row operations
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question Using elementary row operations, find the inverse of \(A={\large \left[\begin{smallmatrix}1&2&1\\1&3&2\\1&0&1\end{smallmatrix}\right]}\).
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 1(a)(ii) Finding \(A^{14}+3A-2I\)
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(A={\large \left[\begin{smallmatrix}1&1&3\\5&2&6\\-2&-1&-3\end{smallmatrix}\right]}\), then find \(A^{14}+3A-2I\).
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 1(b)(i) Condition for consistency of linear equations
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionUsing elementary row operations, find the condition that the linear equations \(x-2y+z=a\), \(2x+7y-3z=b\), \(3x+5y-2z=c\) have a solution.
2 Diagram
Full Solution Access The Diagram section for this question is available in the full 2016 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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5 Final Answer
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Question 1(b)(ii) Dimensions of subspace intersection and sum
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(W_1=\{(x,y,z)\mid x+y-z=0\}\), \(W_2=\{(x,y,z)\mid 3x+y-2z=0\}\), and \(W_3=\{(x,y,z)\mid x-7y+3z=0\}\), then find \(\dim(W_1\cap W_2\cap W_3)\) and \(\dim(W_1+W_2)\).
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 1(c) Evaluation of an improper integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question Evaluate \(I=\int_0^1 \sqrt[3]{x\log\left(\frac{1}{x}\right)}\,dx\).
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 1(d) Sphere through a given circle
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the equation of the sphere which passes through the circle \(x^2+y^2=4,\ z=0\) and is cut by the plane \(x+2y+2z=0\) in a circle of radius \(3\).
2 DiagramQuestion 1(d): Sphere cut by a plane
3 Concept Related to the QuestionA family of spheres passing through a fixed circle is obtained by combining the equation of the sphere through the circle with the equation of the plane of the circle. Then the radius of the circular section made by a plane is found from \(r^2=R^2-d^2\), where \(R\) is the sphere radius and \(d\) is the distance of the centre from the cutting plane.
4 Detailed SolutionThe circle is given by
\[
x^2+y^2=4,\qquad z=0.
\]
Every sphere passing through this circle can be written as
\[
x^2+y^2+z^2+\lambda z-4=0.
\]
This is because putting \(z=0\) gives \(x^2+y^2-4=0\), which is exactly the given circle. Now complete the square in \(z\):
\[
x^2+y^2+\left(z+\frac{\lambda}{2}\right)^2=4+\frac{\lambda^2}{4}.
\]
Hence the centre is \(C=(0,0,-\lambda/2)\), and
\[
R^2=4+\frac{\lambda^2}{4}.
\]
The distance of \(C\) from the plane \(x+2y+2z=0\) is
\[
d=\frac{|0+0+2(-\lambda/2)|}{\sqrt{1^2+2^2+2^2}}=\frac{|\lambda|}{3}.
\]
The plane cuts the sphere in a circle of radius \(3\). Therefore
\[
3^2=R^2-d^2=4+\frac{\lambda^2}{4}-\frac{\lambda^2}{9}.
\]
So
\[
9=4+\frac{5\lambda^2}{36}\quad\Rightarrow\quad \lambda^2=36.
\]
Thus \(\lambda=\pm6\).
5 Final AnswerThe required spheres are \(x^2+y^2+z^2+6z-4=0\) and \(x^2+y^2+z^2-6z-4=0\).
Question 1(e) Shortest distance between two lines
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{4}=z-3\) and \(y-mx=z=0\). For what value of \(m\) will the two lines intersect?
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 2(a)(i) Matrix representation and null space of a linear transformation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(M_2(R)\) is space of real matrices of order \(2\times2\) and \(P_2(x)\) is the space of real polynomials of degree at most \(2\), then find the matrix representation of \(T:M_2(R)\to P_2(x)\), such that \(T\!\left({\large \left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]}\right)=a+c+(a-d)x+(b+c)x^2\), with respect to the standard bases of \(M_2(R)\) and \(P_2(x)\). Further find the null space of \(T\).
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 2(a)(ii) Matrix of a polynomial transformation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(T:P_2(x)\to P_3(x)\) is such that \(T(f(x))=f(x)+5\int_0^x f(t)\,dt\), then choosing \(\{1,1+x,1-x^2\}\) and \(\{1,x,x^2,x^3\}\) as bases of \(P_2(x)\) and \(P_3(x)\) respectively, find the matrix of \(T\).
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 2(b)(i) Eigenvalues and eigenvectors
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionIf \(A={\large \left[\begin{smallmatrix}1&1&0\\1&1&0\\0&0&1\end{smallmatrix}\right]}\), then find the eigenvalues and eigenvectors of \(A\).
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 2(b)(ii) Eigenvalues of a Hermitian matrix are real
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that eigenvalues of a Hermitian matrix are all real.
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 2(c) Finding a linear transformation from its matrix representation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question If \(A={\large \left[\begin{smallmatrix}1&-1&2\\-2&1&-1\\1&2&3\end{smallmatrix}\right]}\) is the matrix representation of a linear transformation \(T:P_2(x)\to P_2(x)\) with respect to the bases \(\{1-x,x(1-x),x(1+x)\}\) and \(\{1,1+x,1+x^2\}\), find \(T\).
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(a) Maximum and minimum under two constraints
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the maximum and minimum values of \(x^2+y^2+z^2\) subject to the conditions \(\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}=1\) and \(x+y-z=0\).
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 3(b) Delta condition near the origin
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet
\[
f(x,y)=
\begin{array}{ll}
\dfrac{2x^4y-5x^2y^2+y^5}{(x^2+y^2)^2}, & (x,y)\ne(0,0),\\[6pt]
0, & (x,y)=(0,0).
\end{array}
\]
Find \(\delta>0\) such that \(|f(x,y)-f(0,0)|<.01\), whenever \(\sqrt{x^2+y^2}<\delta\).
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) Surface area cut from a plane
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question Find the surface area of the plane \(x+2y+2z=12\) cut off by \(x=0\), \(y=0\), and \(x^2+y^2=16\).
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) Surface generated by a moving line
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the surface generated by a line which intersects the lines \(y=a=z\), \(x+3z=a=y+z\) and parallel to the plane \(x+y=0\).
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 4(b) Mutually perpendicular generators of a cone
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that the cone \(3yz-2zx-2xy=0\) has an infinite set of three mutually perpendicular generators. If \(\frac{x}{1}=\frac{y}{1}=\frac{z}{2}\) is a generator belonging to one such set, find the other two.
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(c) Double integral over a rectangle
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionEvaluate \(\displaystyle \iint_R f(x,y)\,dx\,dy\) over the rectangle \(R=[0,1;0,1]\) where
\[
f(x,y)=
\begin{array}{ll}
x+y, & x^2\lt y\lt 2x^2,\\[4pt]
0, & \text{elsewhere.}
\end{array}
\]
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) Locus of intersection of three mutually perpendicular tangent planes
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid \(ax^2+by^2+cz^2=1\).
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) Particular integral
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 Question Find a particular integral of \(\dfrac{d^2y}{dx^2}+y=e^{\dfrac{x}{2}}\sin\dfrac{\sqrt3x}{2}\).
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) Triangle sides and medians
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionProve that the vectors \(\vec a=3\hat i+\hat j-2\hat k\), \(\vec b=-\hat i+3\hat j+4\hat k\), and \(\vec c=4\hat i-2\hat j-6\hat k\) can form the sides of a triangle. Find the lengths of the medians of the triangle.
2 Diagram
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Question 5(c) First-order differential equation
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1 Question Solve \(\dfrac{dy}{dx}=\dfrac{1}{1+x^2}\left(e^{\tan^{-1}x}-y\right)\).
2 Diagram
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Question 5(d) Self-orthogonal family of parabolas
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1 QuestionShow that the family of parabolas \(y^2=4cx+4c^2\) is self-orthogonal.
2 Diagram
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Question 5(e) Path under inverse-cube central acceleration
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1 QuestionA particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance \(a\) from the origin with a velocity which is \(\sqrt2\) times the velocity for a circle of radius \(a\), then find the equation to the path.
2 Diagram
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Question 6(a) Solving a first-order differential equation
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1 Question Solve \(\{y(1-x\tan x)+x^2\cos x\}\,dx-x\,dy=0\).
2 Diagram
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Question 6(b) Variation of parameters
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1 QuestionUsing the method of variation of parameters, solve the differential equation \((D^2+2D+1)y=e^{-x}\log(x)\), \(\left[D=\frac{d}{dx}\right]\).
2 Diagram
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Question 6(c) Cauchy-Euler differential equation
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1 Question Find the general solution of \(x^2\dfrac{d^3y}{dx^3}-4x\dfrac{d^2y}{dx^2}+6\dfrac{dy}{dx}=4\).
2 Diagram
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Question 6(d) Solution by Laplace transformation
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1 Question Using Laplace transformation, solve \(y''-2y'-8y=0\), \(y(0)=3\), \(y'(0)=6\).
2 Diagram
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Question 7(a) Reaction at the hinge of a rod
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1 QuestionA uniform rod \(AB\) of length \(2a\) movable about a hinge at \(A\) rests with other end against a smooth vertical wall. If \(\alpha\) is the inclination of the rod to the vertical, prove that the magnitude of reaction of the hinge is \(\frac{1}{2}W\sqrt{4+\tan^2\alpha}\), where \(W\) is the weight of the rod.
2 Diagram
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Question 7(b) Inclination of a light rod kept apart by strings
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1 QuestionTwo weights \(P\) and \(Q\) are suspended from a fixed point \(O\) by strings \(OA\), \(OB\) and are kept apart by light rod \(AB\). If the strings \(OA\) and \(OB\) make angles \(\alpha\) and \(\beta\) with the rod \(AB\), show that the angle \(\theta\) which the rod makes with the vertical is given by
\[
\tan\theta=\dfrac{P+Q}{P\cot\alpha-Q\cot\beta}.
\]
2 Diagram
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Question 7(c) Tension in an endless string passing over square pegs
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1 QuestionA square \(ABCD\), the length of whose sides is \(a\), is fixed in a vertical plane with two of its sides horizontal. An endless string of length \(l(>4a)\) passes over four pegs at the angles of the board and through a ring of weight \(W\) which is hanging vertically. Show that the tension of the string is \(\dfrac{W(l-3a)}{2\sqrt{l^2-6la+8a^2}}\).
2 Diagram
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Question 8(a) Finding a scalar function from its gradient
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1 Question Find \(f(r)\) such that \(\nabla f=\dfrac{\vec r}{r^5}\) and \(f(1)=0\).
2 Diagram
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Question 8(b) Vector integral identity
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1 Question Prove that \(\oint_C f\,d\vec r=\iint_S d\vec S\times\nabla f\).
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Question 8(c) Time of arrival under variable acceleration
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1 QuestionA particle moves in a straight line. Its acceleration is directed towards a fixed point \(O\) in the line and is always equal to \(\mu\left(\frac{a^5}{x^2}\right)^{\frac{1}{3}}\) when it is at a distance \(x\) from \(O\). It starts from rest at distance \(a\) from \(O\), then find the time, the particle will arrive at \(O\).
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Question 8(d) Radius of curvature of a cardioid
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1 QuestionFor the cardioid \(r=a(1+\cos\theta)\), show that the square of the radius of curvature at any point \((r,\theta)\) is proportional to \(r\). Also find the radius of curvature if \(\theta=0,\frac{\pi}{4},\frac{\pi}{2}\).
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2016 UPSC Maths Optional Paper I Solutions FAQs
Are these 2016 UPSC Maths Optional Paper I Solutions complete? This public page gives one complete sample solution. Complete solutions for all questions are available in the full PYQ course.
Which question is given as a free sample solution on this page? Question 1(d) is given as the free sample solution on this page.
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