UPSC Previous Years Maths Optional Papers with Solution | Paper-02

Section:- A

Question:-01 (a) Show that the multiplicative group $G=\{1,-1,i,-i\}$$G=\{1,-1,i,-i\}$G={1,-1,i,-i}G=\{1,-1, i,-i\}$G=\{1,-1,i,-i\}$, where $i=\sqrt{(-1)}$$i=\sqrt{(-1)}$i=sqrt((-1))i=\sqrt{(-1)}$i=\sqrt{(-1)}$, is isomorphic to the group $G^{\mathrm{\prime }}=(\{0,1,2,3\},+{}_4)$$G^{\mathrm{\prime }}=\left(\{0,1,2,3\},+{}_4\right)$G^(‘)=({0,1,2,3},+_(4))G^{\prime}=\left(\{0,1,2,3\},+{ }_{4}\right)$G^{\mathrm{\prime }}=(\{0,1,2,3\},+{}_4)$.

Question:-01 (b) If $f(z)=u+iv$$f(z)=u+iv$f(z)=u+ivf(z)=u+i v$f(z)=u+iv$ is an analytic function of $z$$z$zz$z$, and $u-v=\frac{\cos x+\sin x-e^{-y}}{2\cos x-e^y-e^{-y}}$$u-v=\frac{\cos x+\sin x-e^{-y}}{2\cos x-e^y-e^{-y}}$u-v=(cos x+sin x-e^(-y))/(2cos x-e^(y)-e^(-y))u-v=\frac{\cos x+\sin x-e^{-y}}{2 \cos x-e^{y}-e^{-y}}$u-v=\frac{\cos x+\sin x-e^{-y}}{2\cos x-e^y-e^{-y}}$, then find $f(z)$$f(z)$f(z)f(z)$f(z)$ subject to the condition $f\left(\frac{\pi }{2}\right)=0$$f\left(\frac{\pi }{2}\right)=0$f((pi)/(2))=0f\left(\frac{\pi}{2}\right)=0$f\left(\frac{\pi }{2}\right)=0$.

Question:-01 (c) Test the convergence of ${\int }_0^{\mathrm{\infty }}\frac{\cos x}{1+x^2}dx$${\int }_0^{\mathrm{\infty }} \frac{\cos x}{1+x^2}dx$int_(0)^(oo)(cos x)/(1+x^(2))dx\int_{0}^{\infty} \frac{\cos x}{1+x^{2}} d x${\int }_0^{\mathrm{\infty }}\frac{\cos x}{1+x^2}dx$.

Question:-01 (d) Expand $f(z)=\frac{1}{(z-1{)}^2(z-3)}$$f(z)=\frac{1}{(z-1{)}^2(z-3)}$f(z)=(1)/((z-1)^(2)(z-3))f(z)=\frac{1}{(z-1)^{2}(z-3)}$f(z)=\frac{1}{(z-1{)}^2(z-3)}$ in a Laurent series valid for the regions

(i) $0<|z-1|<2$$0<|z-1|<2$0 < |z-1| < 20<|z-1|<2$0<|z-1|<2$ and (ii) $0<|z-3|<2$$0<|z-3|<2$0 < |z-3| < 20<|z-3|<2$0<|z-3|<2$.

Question:-01 (e) Use two-phase method to solve the following linear programming problem :

---

Question:-02 (a) Let $f(x)=x^2$$f(x)=x^2$f(x)=x^(2)f(x)=x^{2}$f(x)=x^2$ on $[0,k],k>0$$[0,k],k>0$[0,k],k > 0[0, k], k>0$[0,k],k>0$. Show that $f$$f$ff$f$ is Riemann integrable on the closed interval $[0,k]$$[0,k]$[0,k][0, k]$[0,k]$ and ${\int }_0^{k}fdx=\frac{k^3}{3}$${\int }_0^k fdx=\frac{k^3}{3}$int_(0)^(k)fdx=(k^(3))/(3)\int_{0}^{k} f d x=\frac{k^{3}}{3}${\int }_0^{k}fdx=\frac{k^3}{3}$.

Question:-02 (b) Prove that every homomorphic image of a group $G$$G$GG$G$ is isomorphic to some quotient group of $G$$G$GG$G$.

Question:-02 (c) Apply the calculus of residues to evaluate ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\cos xdx}{(x^2+a^2)(x^2+b^2)},a>b>0$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} \frac{\cos xdx}{\left(x^2+a^2\right)\left(x^2+b^2\right)},a>b>0$int_(-oo)^(oo)(cos xdx)/((x^(2)+a^(2))(x^(2)+b^(2))),a > b > 0\int_{-\infty}^{\infty} \frac{\cos x d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}, a>b>0${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\cos xdx}{(x^2+a^2)(x^2+b^2)},a>b>0$.

---

Question:-03 (a) Evaluate ${\int }_C\frac{z+4}{z^2+2z+5}dz$${\int }_C \frac{z+4}{z^2+2z+5}dz$int_(C)(z+4)/(z^(2)+2z+5)dz\int_{C} \frac{z+4}{z^{2}+2 z+5} d z${\int }_C\frac{z+4}{z^2+2z+5}dz$, where $C$$C$CC$C$ is $|z+1-i|=2$$|z+1-i|=2$|z+1-i|=2|z+1-i|=2$|z+1-i|=2$

Question:-03 (b) Find the maximum and minimum values of $\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$$\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$(x^(2))/(a^(4))+(y^(2))/(b^(4))+(z^(2))/(c^(4))\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}$\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$, when $lx+my+nz=0$$lx+my+nz=0$lx+my+nz=0l x+m y+n z=0$lx+my+nz=0$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$(x^(2))/(a^(2))+(y^(2))/(b^(2))+(z^(2))/(c^(2))=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. Interpret the result geometrically.

Question:-03 (c) Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem :

---

Question:-04 (a) Let $R$$R$RR$R$ be a field of real numbers and $S$$S$SS$S$, the field of all those polynomials $f(x)\in R[x]$$f(x)\in R[x]$f(x)in R[x]f(x) \in R[x]$f(x)\in R[x]$ such that $f(0)=0=f(1)$$f(0)=0=f(1)$f(0)=0=f(1)f(0)=0=f(1)$f(0)=0=f(1)$. Prove that $S$$S$SS$S$ is an ideal of $R[x]$$R[x]$R[x]R[x]$R[x]$. Is the residue class ring $R[x]/S$$R[x]/S$R[x]//SR[x] / S$R[x]/S$ an integral domain? Give justification for your answer.

Question:-04 (b) Test for convergence or divergence of the series

Question:-04 (c) Find the initial basic feasible solution of the following transportation problem by Vogel’s approximation method and use it to find the optimal solution and the transportation cost of the problem :

---

Section:- B

Question:-05 (a) It is given that the equation of any cone with vertex at $(a,b,c)$$(a,b,c)$(a,b,c)(a, b, c)$(a,b,c)$ is $f(\frac{x-a}{z-c},\frac{y-b}{z-c})=0$$f\left(\frac{x-a}{z-c},\frac{y-b}{z-c}\right)=0$f((x-a)/(z-c),(y-b)/(z-c))=0f\left(\frac{x-a}{z-c}, \frac{y-b}{z-c}\right)=0$f(\frac{x-a}{z-c},\frac{y-b}{z-c})=0$. Find the differential equation of the cone.

Question:-05(b) Solve, by Gauss elimination method, the system of equations

Question:-05 (c) (i) Convert the number $(1093.21875{)}_{10}$$(1093.21875{)}_{10}$(1093.21875)_(10)(1093.21875)_{10}$(1093.21875{)}_{10}$ into octal and the number $(1693\cdot 0628{)}_{10}$$(1693\cdot 0628{)}_{10}$(1693*0628)_(10)(1693 \cdot 0628)_{10}$(1693\cdot 0628{)}_{10}$ into hexadecimal systems.

(ii) Express the Boolean function $F(x,y,z)=xy+x^{\mathrm{\prime }}z$$F(x,y,z)=xy+x^{\mathrm{\prime }}z$F(x,y,z)=xy+x^(‘)zF(x, y, z)=x y+x^{\prime} z$F(x,y,z)=xy+x^{\mathrm{\prime }}z$ in a product of maxterms form.

Question:-05 (d) A particle at a distance $r$$r$rr$r$ from the centre of force moves under the influence of the central force $F=-\frac{k}{r^2}$$F=-\frac{k}{r^2}$F=-(k)/(r^(2))F=-\frac{k}{r^{2}}$F=-\frac{k}{r^2}$, where $k$$k$kk$k$ is a constant. Obtain the Lagrangian and derive the equations of motion.

Question:-05 (e) The velocity components of an incompressible fluid in spherical polar coordinates $(r,\theta ,\psi )$$(r,\theta ,\psi )$(r,theta,psi)(r, \theta, \psi)$(r,\theta ,\psi )$ are $(2M{r}^{-3}\cos \theta ,M{r}^{-2}\sin \theta ,0)$$\left(2M{r}^{-3}\cos \theta ,M{r}^{-2}\sin \theta ,0\right)$(2Mr^(-3)cos theta,Mr^(-2)sin theta,0)\left(2 M r^{-3} \cos \theta, M r^{-2} \sin \theta, 0\right)$(2M{r}^{-3}\cos \theta ,M{r}^{-2}\sin \theta ,0)$, where $M$$M$MM$M$ is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the streamlines.

---

Question:-06 (a) Solve the heat equation $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2},0<x<l,t>0$$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2},0<x<l,t>0$(del u)/(del t)=(del^(2)u)/(delx^(2)),0 < x < l,t > 0\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, 0<x<l, t>0$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2},0<x<l,t>0$ subject to the conditions

Question:-06 (b) Find a combinatorial circuit corresponding to the Boolean function

and write the input/output table for the circuit.

Question:-06 (c) Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass $M$$M$MM$M$, height $h$$h$hh$h$ and the radius of base as $a$$a$aa$a$.

---

Question:-07 (a) Find the general solution of the partial differential equation

where $D\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }x}$$D\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }x}$D-=(del)/(del x)D \equiv \frac{\partial}{\partial x}$D\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }x}$ and $D^{\mathrm{\prime }}\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }y}$$D^{\mathrm{\prime }}\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }y}$D^(‘)-=(del)/(del y)D^{\prime} \equiv \frac{\partial}{\partial y}$D^{\mathrm{\prime }}\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }y}$.

Question:-07 (b) The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in $\mathrm{k}\mathrm{m}/$$\mathrm{k}\mathrm{m}/$km//\mathrm{km} /$\mathrm{k}\mathrm{m}/$ hour :

Using Simpson’s $\frac{1}{3}\mathrm{r}\mathrm{d}$$\frac{1}{3}\mathrm{r}\mathrm{d}$(1)/(3)rd\frac{1}{3} \mathrm{rd}$\frac{1}{3}\mathrm{r}\mathrm{d}$ rule, estimate approximately in $\mathrm{k}\mathrm{m}$$\mathrm{k}\mathrm{m}$km\mathrm{km}$\mathrm{k}\mathrm{m}$ the total distance run in 20 minutes.

Question:-07 (c) Two point vortices each of strength $k$$k$kk$k$ are situated at $(\pm a,0)$$(\pm a,0)$(+-a,0)(\pm a, 0)$(\pm a,0)$ and a point vortex of strength $-\frac{k}{2}$$-\frac{k}{2}$-(k)/(2)-\frac{k}{2}$-\frac{k}{2}$ is situated at the origin. Show that the fluid motion is stationary and also find the equations of streamlines. If the streamlines, which pass through the stagnation points, meet the $x$$x$xx$x$-axis at $(\pm b,0)$$(\pm b,0)$(+-b,0)(\pm b, 0)$(\pm b,0)$, then show that $3\sqrt{3}{(b^2-a^2)}^2=16a^{3}b$$3\sqrt{3}{\left(b^2-a^2\right)}^2=16a^{3}b$3sqrt3(b^(2)-a^(2))^(2)=16a^(3)b3 \sqrt{3}\left(b^{2}-a^{2}\right)^{2}=16 a^{3} b$3\sqrt{3}{(b^2-a^2)}^2=16a^{3}b$

---

Question:-08 (a) Reduce the following partial differential equation to a canonical form and hence solve it :

Question:-08 (b) Using Runge-Kutta method of fourth order, solve the differential equation $\frac{dy}{dx}=x+y^2$$\frac{dy}{dx}=x+y^2$(dy)/(dx)=x+y^(2)\frac{d y}{d x}=x+y^{2}$\frac{dy}{dx}=x+y^2$ with $y(0)=1$$y(0)=1$y(0)=1y(0)=1$y(0)=1$, at $x=0\cdot 2$$x=0\cdot 2$x=0*2x=0 \cdot 2$x=0\cdot 2$. Use four decimal places for calculation and step length $0\cdot 1$$0\cdot 1$0*10 \cdot 1$0\cdot 1$.

Question:-08 (c) Verify that $w=ik\log \{(z-ia)/(z+ia)\}$$w=ik\log \{(z-ia)/(z+ia)\}$w=ik log{(z-ia)//(z+ia)}w=i k \log \{(z-i a) /(z+i a)\}$w=ik\log \{(z-ia)/(z+ia)\}$ is the complex potential of a steady flow of fluid about a circular cylinder, where the plane $y=0$$y=0$y=0y=0$y=0$ is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder.