Ramana Sri IAS - 2017 UPSC Maths Optional Paper II Solutions
2017 UPSC Maths Optional Paper II Solutions Ramana Sri IAS provides complete and updated solutions for the 2017 UPSC Maths Optional Paper II. 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 2017 Mathematics Optional Paper II. Each answer follows the same format: Question, Diagram where needed, Concept Related to the Question, Detailed Solution, and Final Answer.
About 2017 UPSC Maths Optional Paper II Solutions
These 2017 UPSC Maths Optional Paper II 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 2017 UPSC Maths Optional Paper II 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 2017 UPSC Maths Optional Paper II 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
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Question 1(e) . This free sample includes all five sections:
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2017 UPSC Maths Optional Paper II Solutions: Table of Contents
Question 1(a) Convergence of a recursive sequence
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(x_1=2\) and \(x_{n+1}=\sqrt{x_n+20}\), \(n=1,2,3,\ldots\). Show that sequence \(x_1,x_2,x_3,\ldots\) is convergent.
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(b) Cayley theorem
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(G\) be a group of order \(n\). Show that \(G\) is isomorphic to a subgroup of the permutation group \(S_n\).
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) Supremum and infimum
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the supremum and the infimum of \(\dfrac{x}{\sin x}\) on the interval \(\left(0,\dfrac{\pi}{2}\right]\).
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(d) Entire functions and removable singularity
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionDetermine all entire functions \(f(z)\) such that \(0\) is a removable singularity of \(f\left(\dfrac{1}{z}\right)\).
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(e) Graphical linear programming
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionUsing graphical method, find the maximum value of
\[2x+y\]
subject to
\[4x+3y\leq12\]
\[4x+y\leq8\]
\[4x-y\leq8\]
\[x,y\geq0.\]
2 DiagramQuestion 1(e): Graphical method for linear programming
3 Concept Related to the QuestionIn graphical linear programming, each inequality represents a half-plane. Their common part is called the feasible region. A linear objective function reaches its maximum or minimum at a corner point of the feasible region.
4 Detailed SolutionWe have to maximize
\[Z=2x+y.\]
The constraints are
\[4x+3y\leq12,\qquad 4x+y\leq8,\qquad 4x-y\leq8,\qquad x\geq0,\quad y\geq0.\]
First find the corner points of the feasible region.
On the \(y\)-axis, put \(x=0\). The strongest restriction is \(3y\leq12\), so \(y\leq4\). Hence one corner point is \((0,4)\).
On the \(x\)-axis, put \(y=0\). The restrictions give \(4x\leq12\), \(4x\leq8\), and \(4x\leq8\). Hence \(x\leq2\), so one corner point is \((2,0)\).
The point of intersection of \(4x+3y=12\) and \(4x+y=8\) is found by subtracting the second equation from the first:
\[2y=4,\qquad y=2.\]
Substituting \(y=2\) in \(4x+y=8\),
\[4x+2=8,\qquad x=\frac32.\]
Thus another corner point is \(\left(\frac32,2\right)\). The origin \((0,0)\) is also a corner point.
Now evaluate \(Z=2x+y\) at all corner points.
Corner point Value of \(Z=2x+y\) \((0,0)\) \(0\) \((0,4)\) \(4\) \(\left(\frac32,2\right)\) \(2\cdot\frac32+2=5\) \((2,0)\) \(4\)
The largest value is \(5\).
5 Final AnswerThe maximum value of \(2x+y\) is \(5\), attained at \(x=\frac32\), \(y=2\).
Question 2(a) Differentiability of an integral with greatest-integer function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet
\[f(t)=\int_0^t [x]\,dx,\]
where \([x]\) denote the largest integer less than or equal to \(x\).
(i) Determine all real numbers \(t\) at which \(f\) is differentiable.
(ii) Determine all real numbers \(t\) at which \(f\) is continuous but not differentiable.
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) Contour integral evaluation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionUsing contour integral method, prove that \(\displaystyle\int_0^\infty \frac{x\sin mx}{a^2+x^2}\,dx=\frac{\pi}{2}e^{-ma}\).
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) Polynomial division algorithm
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(F\) be a field and \(F[X]\) denote the ring of polynomials over \(F\) in a single variable \(X\) and let \(f(X),g(X)\in F[X]\) with \(g(X)\neq0\). Show that there exist \(q(X),r(X)\in F[X]\) such that degree \(r(X)\lt\) degree \(g(X)\) and \(f(X)=q(X)\cdot g(X)+r(X)\).
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 3(a) Isomorphism of cyclic groups
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that the groups \(\mathbb Z_5\times\mathbb Z_7\) and \(\mathbb Z_{35}\) are isomorphic.
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) Analytic functions and harmonic functions
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(f=u+iv\) be an analytic function on the unit disc \(D=\{z\in\mathbb C:|z|\lt1\}\). Show that \(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0=\dfrac{\partial^2v}{\partial x^2}+\dfrac{\partial^2v}{\partial y^2}\) at all points of \(D\).
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) Simplex method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve the following linear programming problem by simplex method :
Maximize
\[z=3x_1+5x_2+4x_3\]
subject to
\[2x_1+3x_2\leq8\]
\[2x_2+5x_3\leq10\]
\[3x_1+2x_2+4x_3\leq15\]
\[x_1,x_2,x_3\geq0.\]
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) Zeros of derivatives of an entire function
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFor a function \(f:\mathbb C\to\mathbb C\) and \(n\geq1\), let \(f^{(n)}\) denote the \(n^{\text{th}}\) derivative of \(f\), and \(f^{(0)}=f\). Let \(f\) be an entire function such that for some \(n\geq1\), \(f^{(n)}\left(\dfrac{1}{k}\right)=0\) for all \(k=1,2,3,\ldots\). Show that \(f\) is a polynomial.
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) Vogel approximation method
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind the initial basic feasible solution of the following transportation problem using Vogel’s approximation method and find the cost.
Origins D1 D2 D3 D4 D5 Supply O1 4 7 0 3 6 14 O2 1 2 -3 3 8 9 O3 3 -1 4 0 5 17 Demand 8 3 8 13 8
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) Riemann rearrangement theorem
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(\displaystyle\sum_{n=1}^{\infty}x_n\) be a conditionally convergent series of real numbers. Show that there is a rearrangement \(\displaystyle\sum_{n=1}^{\infty}x_{\pi(n)}\) of the series \(\displaystyle\sum_{n=1}^{\infty}x_n\) that converges to \(100\).
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 partial differential equation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionSolve \((D^2-2DD'+D'^2)z=e^{x+2y}+x^3+\sin2x\), where \(D=\dfrac{\partial}{\partial x}\), \(D'=\dfrac{\partial}{\partial y}\), \(D^2=\dfrac{\partial^2}{\partial x^2}\), and \(D'^2=\dfrac{\partial^2}{\partial y^2}\).
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) Gauss-Jordan inverse of a matrix
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionExplain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix \( {\large \left[\begin{smallmatrix}2&6&6\\2&8&6\\2&6&8\end{smallmatrix}\right]} \).
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 5(c) Boolean simplification
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionWrite the Boolean expression \(z(y+z)(x+y+z)\) in its simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.
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) Uniqueness for Poisson equation
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionLet \(\Gamma\) be a closed curve in \(xy\)-plane and let \(S\) denote the region bounded by the curve \(\Gamma\). Let \(\dfrac{\partial^2w}{\partial x^2}+\dfrac{\partial^2w}{\partial y^2}=f(x,y)\), \(\forall(x,y)\in S\). If \(f\) is prescribed at each point \((x,y)\) of \(S\) and \(w\) is prescribed on the boundary \(\Gamma\) of \(S\), then prove that any solution \(w=w(x,y)\), satisfying these conditions, is unique.
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) Moment of inertia of elliptic area
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionShow that the moment of inertia of an elliptic area of mass \(M\) and semi-axis \(a\) and \(b\) about a semi-diameter of length \(r\) is \(\dfrac14 M\dfrac{a^2b^2}{r^2}\). Further, prove that the moment of inertia about a tangent is \(\dfrac{5M}{4}p^2\), where \(p\) is the perpendicular distance from the centre of the ellipse to the tangent.
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) Complete integral of first-order PDE
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFind complete integral of the partial differential equation
\[2(pq+yp+qx)+x^2+y^2=0.\]
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 6(b) Lagrange interpolation formula
1. Question
2. Diagram
3. Concept Related to the Question
4. Detailed Solution
5. Final Answer
1 QuestionFor given equidistant values \(u_{-1},u_0,u_1\) and \(u_2\), a value is interpolated by Lagrange’s formula. Show that it may be written in the form
\[u_x=yu_0+xu_1+\frac{y(y^2-1)}{3!}\Delta^2u_{-1}+\frac{x(x^2-1)}{3!}\Delta^2u_0,\]
where \(x+y=1\).
2 Diagram
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Question 6(c) Kinetic energy and Lagrange equations
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1 QuestionTwo uniform rods \(AB,AC\), each of mass \(m\) and length \(2a\), are smoothly hinged together at \(A\) and move on a horizontal plane. At time \(t\) the mass centre of the rods is at the point \((\xi,\eta)\) referred to fixed perpendicular axes \(Ox, Oy\) in the plane, and the rods make angles \(\theta\pm\phi\) with \(Ox\). Prove that the kinetic energy of the system is
\[m\left[\dot{\xi}^{\,2}+\dot{\eta}^{\,2}+\left(\frac13+\sin^2\phi\right)a^2\dot{\theta}^{\,2}+\left(\frac13+\cos^2\phi\right)a^2\dot{\phi}^{\,2}\right].\]
Also derive Lagrange’s equations of motion for the system if an external force with components \([X,Y]\) along the axes acts at \(A\).
2 Diagram
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Question 7(a) Canonical form of second-order PDE
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1 QuestionReduce the equation
\[y^2\frac{\partial^2z}{\partial x^2}-2xy\frac{\partial^2z}{\partial x\partial y}+x^2\frac{\partial^2z}{\partial y^2}=\frac{y^2}{x}\frac{\partial z}{\partial x}+\frac{x^2}{y}\frac{\partial z}{\partial y}\]
to canonical form and hence solve it.
2 Diagram
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Question 7(b) Simpson's three-eighth rule
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1 QuestionDerive the formula
\[\int_a^b y\,dx=\frac{3h}{8}\left[(y_0+y_n)+3(y_1+y_2+y_4+y_5+\cdots+y_{n-1})+2(y_3+y_6+\cdots+y_{n-3})\right].\]
Is there any restriction on \(n\)? State that condition. What is the error bound in the case of Simpson’s \(\frac38\) rule?
2 Diagram
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Question 7(c) Steady flow through a conical pipe
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1 QuestionA stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are \(D\) and \(d\). If \(V\) and \(v\) be the corresponding velocities of the stream and if the motion is assumed to be steady and diverging from the vertex of the cone, then prove that
\[\frac{v}{V}=\frac{D^2}{d^2}e^{(v^2-V^2)/2K},\]
where \(K\) is the pressure divided by the density and is constant.
2 Diagram
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Question 8(a) One-dimensional wave equation
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1 QuestionGiven the one-dimensional wave equation
\[\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2},\qquad t\gt0,\]
where \(c^2=\dfrac{T}{m}\), \(T\) is the constant tension in the string and \(m\) is the mass per unit length of the string.
(i) Find the appropriate solution of the above wave equation.
(ii) Find also the solution under the conditions
\[y(0,t)=0,\qquad y(l,t)=0\quad\text{for all }t\]
and
\[\left[\frac{\partial y}{\partial t}\right]_{t=0}=0,\qquad y(x,0)=a\sin\frac{\pi x}{l},\qquad 0\lt x\lt l,\quad a\gt0.\]
2 Diagram
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Question 8(b) Newton-Raphson method flow chart and failures
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1 QuestionWrite an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.
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Question 8(c) Velocity potential and streamlines
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1 QuestionIf the velocity of an incompressible fluid at the point \((x,y,z)\) is given by \(\left(\dfrac{3xz}{r^5},\dfrac{3yz}{r^5},\dfrac{3z^2-r^2}{r^5}\right)\), \(r^2=x^2+y^2+z^2\), then prove that the liquid motion is possible and that the velocity potential is \(\dfrac{z}{r^3}\). Further, determine the streamlines.
2 Diagram
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2017 UPSC Maths Optional Paper II Solutions FAQs
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