Ramana Sri IAS - 2016 IFoS Maths Optional Paper I Solutions
2016 IFoS Maths Optional Paper I Solutions
Ramana Sri IAS provides complete and updated solutions for the 2016 IFoS Maths Optional Paper I. Aspirants preparing for the IFoS Mains Examination with Mathematics as their optional subject should solve all these questions carefully before the mains examination.
Ramana Sri IAS presents complete solutions for Indian Forest 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 IFoS Maths Optional Paper I Solutions
These 2016 IFoS Maths Optional Paper I Solutions are prepared by Ramana Sri IAS for aspirants who want question-wise clarity before the IFoS 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 IFoS 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 2016 IFoS Maths Optional Paper I Solutions are useful for revision, answer-writing practice, and understanding the step-by-step method expected in the IFoS 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.
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.
2016 IFoS Maths Optional Paper I Solutions: Table of Contents
Let \(T:\mathbb R^3\to\mathbb R^4\) be given by \(T(x,y,z)=(2x-y,2x+z,x+2z,x+y+z)\). Find the matrix of \(T\) with respect to standard basis of \(\mathbb R^3\) and \(\mathbb R^4\), i.e., \((1,0,0),(0,1,0)\), etc. Examine whether \(T\) is a linear map.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Examine whether \(f(x,y)=\dfrac{xy}{x^2+y^2}\), \((x,y)\ne(0,0)\), and \(f(0,0)=0\), is continuous at \((0,0)\). Find \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) at points other than the origin.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 point \((2,3)\) is the midpoint of a chord of the parabola \(y^2=4x\), obtain the equation of the chord.
2Diagram
Question 1(d): Midpoint of a chord of the parabola
3Concept Related to the Question
For the parabola \(y^2=4ax\), the chord whose midpoint is \((x_1,y_1)\) is obtained by the midpoint formula \(T=S_1\), where \(T=yy_1-2a(x+x_1)\) and \(S_1=y_1^2-4ax_1\).
4Detailed Solution
Here \(y^2=4x\), so \(a=1\), and the midpoint is \((x_1,y_1)=(2,3)\). Therefore \(T=3y-2(x+2)\), and \(S_1=3^2-4(2)=1\).
Putting \(T=S_1\), we get \(3y-2x-4=1\). Hence \(3y-2x-5=0\).
5Final Answer
The required chord is \(3y-2x-5=0\).
Question 1(e)
Eigenvalues and Polynomial of a Matrix
1Question
For \(A=\begin{pmatrix}-1&2&2\\2&-1&2\\2&2&-1\end{pmatrix}\), obtain the eigenvalues and get the value of \(A^4+3A^3-9A^2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Changing Order of Integration and Dirichlet Integral
1Question
After changing the order of integration of \(\int_0^\infty\int_0^\infty e^{-xy}\sin nx\,dx\,dy\), show that \(\int_0^\infty \dfrac{\sin nx}{x}\,dx=\dfrac{\pi}{2}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Locus of Foot of Perpendicular to Tangent of Ellipse
1Question
A perpendicular is drawn from the centre of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) to any tangent. Prove that the locus of the foot of the perpendicular is \((x^2+y^2)^2=a^2x^2+b^2y^2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Using the Mean Value Theorem, find a point on the curve \(y=\sqrt{x-2}\), defined on \([2,3]\), where the tangent is parallel to the chord joining the endpoints of the curve.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 \(T:V_3\to V_2\) be defined by \(T(e_1)=2f_1-f_2\), \(T(e_2)=f_1+2f_2\), and \(T(e_3)=0f_1+0f_2\). Find the matrix of \(T\) relative to the standard bases. Further take \(B_1=[(1,1,0),(1,0,1),(0,1,1)]\) and \(B_2=[(1,1),(1,-1)]\). Obtain the matrix \(T_1\) relative to \(B_1\) and \(B_2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 \(A=\begin{pmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{pmatrix}\), find two non-singular matrices \(P\) and \(Q\) such that \(PAQ=I\). Hence find \(A^{-1}\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 equation of the sphere on which the intersection of the plane \(5x-2y+4z+7=0\) with the sphere having \((0,1,0)\) and \((3,-5,2)\) as endpoints of its diameter is a great circle.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Examine whether the real quadratic form \(4x^2-y^2+2z^2+2xy-2yz-4xz\) is positive definite or not. Reduce it to diagonal form and determine its signature.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 integral \(\int_0^\infty e^{-x}x^{\alpha-1}\,dx\), where \(\alpha>0\), exists by separately taking the cases \(\alpha\ge1\) and \(0<\alpha<1\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 plane \(\dfrac xa+\dfrac yb+\dfrac zc=1\) cuts the coordinate axes at \(A,B,C\). Find the equation of the cone with vertex at the origin and guiding curve as the circle passing through \(A,B,C\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 weight \(W\) is hanging with the help of two strings of length \(l\) and \(2l\) in such a way that the other ends \(A\) and \(B\) of those strings lie on a horizontal line at a distance \(2l\). Obtain the tensions in the two strings.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
From a point on a smooth horizontal plane, a particle is projected with velocity \(u\) at angle \(\alpha\) to the horizontal from the foot of a plane inclined at an angle \(\beta\) to the horizon. Show that it will strike the plane at right angles if \(\cot\beta=2\tan(\alpha-\beta)\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 \(E\) be the solid bounded by the \(xy\)-plane and the paraboloid \(z=4-x^2-y^2\), then evaluate \(\iint_S\vec F\cdot d\vec S\), where \(S\) is the surface bounding the volume \(E\) and \(\vec F=(zx\sin yz+x^3)\hat i+\cos yz\hat j+(3zy^2-e^{x^2+y^2})\hat k\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 stone is thrown vertically with the velocity which would just carry it to a height of \(40\) m. Two seconds later another stone is projected vertically from the same place with the same velocity. When and where will they meet?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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.
Water is flowing through a pipe of \(80\) mm diameter under a gauge pressure of \(60\) kPa with mean velocity \(2\) m/s. Find the total head if the pipe is \(7\) m above the datum line.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 \(\iint_S(\nabla\times\vec f)\cdot\hat n\,dS\) for \(\vec f=(2x-y)\hat i-yz^2\hat j-y^2z\hat k\), where \(S\) is the upper half surface of the sphere \(x^2+y^2+z^2=1\) bounded by its projection on the \(xy\)-plane.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 Stokes’ theorem. Verify it for \(\vec f=x\hat i+z\hat j+2y\hat k\), where \(C\) is the curve obtained by the intersection of the plane \(z=x\) and the cylinder \(x^2+y^2=1\), and \(S\) is the surface inside the intersected curve.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 uniform rod of weight \(W\) is resting against an equally rough horizontal plane and a wall, at an angle \(\alpha\) with the wall. At this condition, a horizontal force \(P\) is stopping it from sliding, implemented at the mid-point of the rod. Prove that \(P=W\tan(\alpha-2\lambda)\), where \(\lambda\) is the angle of friction. Is there any condition on \(\lambda\) and \(\alpha\)?
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 immersed in a liquid is balanced by a weight \(P\) attached by a thread passing over a fixed pulley, and when half immersed it is balanced in the same manner by weight \(2P\). Prove that the density of the body and the liquid are in the ratio \(3:2\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 \(\vec a\times(\vec b\times\vec c)=(\vec a\times\vec b)\times\vec c\) if and only if either \(\vec b=\vec0\), or \(\vec c\) is collinear with \(\vec a\), or \(\vec b\) is perpendicular to both \(\vec a\) and \(\vec c\).
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 acted on by a force parallel to the axis of \(y\), whose acceleration is \(\lambda y\). It is initially projected with velocity \(a\sqrt\lambda\) parallel to the \(x\)-axis from the point where \(y=a\). Prove that it describes a catenary.
2Diagram
Full Solution Access
The Diagram section for this question is available in the full 2016 IFoS 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 2016 IFoS 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.
How should I use these 2016 IFoS Maths Optional Paper I Solutions for preparation?
Students should first read the question carefully, understand the concept, study the diagram where required, and then practise writing the detailed solution and final answer.
Do these solutions include diagrams and detailed solutions?
Yes. The full PYQ course includes diagrams where needed, concept explanation, detailed solution, and final answer for each question.
How can I get complete solutions for all questions in 2016 IFoS Maths Optional Paper I?
To get complete solutions for all questions, please fill out the admission form first. Our Ramana Sri IAS admission team will guide you through WhatsApp, email, or call.
