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

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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Details For UPSC Maths Optional Solved Papers (2018-2022)

## UPSC Maths Optional Question Papers

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Section:- A
Question:-01 (a) Show that the multiplicative group $G=\left\{1,-1,i,-i\right\}$$G=\left\{1,-1,i,-i\right\}$G={1,-1,i,-i}G=\{1,-1, i,-i\}$G=\left\{1,-1,i,-i\right\}$, where $i=\sqrt{\left(-1\right)}$$i=\sqrt{\left(-1\right)}$i=sqrt((-1))i=\sqrt{(-1)}$i=\sqrt{\left(-1\right)}$, is isomorphic to the group ${G}^{\mathrm{\prime }}=\left(\left\{0,1,2,3\right\},+{}_{4}\right)$${G}^{\mathrm{\prime }}=\left(\left\{0,1,2,3\right\},+{}_{4}\right)$G^(‘)=({0,1,2,3},+_(4))G^{\prime}=\left(\{0,1,2,3\},+{ }_{4}\right)${G}^{\mathrm{\prime }}=\left(\left\{0,1,2,3\right\},+{}_{4}\right)$.
Question:-01 (b) If $f\left(z\right)=u+iv$$f\left(z\right)=u+iv$f(z)=u+ivf(z)=u+i v$f\left(z\right)=u+iv$ is an analytic function of $z$$z$zz$z$, and $u-v=\frac{\mathrm{cos}x+\mathrm{sin}x-{e}^{-y}}{2\mathrm{cos}x-{e}^{y}-{e}^{-y}}$$u-v=\frac{\mathrm{cos}x+\mathrm{sin}x-{e}^{-y}}{2\mathrm{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{\mathrm{cos}x+\mathrm{sin}x-{e}^{-y}}{2\mathrm{cos}x-{e}^{y}-{e}^{-y}}$, then find $f\left(z\right)$$f\left(z\right)$f(z)f(z)$f\left(z\right)$ subject to the condition $f\left(\frac{\pi }{2}\right)=0$$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{\mathrm{cos}x}{1+{x}^{2}}dx$${\int }_{0}^{\mathrm{\infty }} \frac{\mathrm{cos}x}{1+{x}^{2}}dx$int_(0)^(oo)(cos x)/(1+x^(2))dx\int_{0}^{\infty} \frac{\cos x}{1+x^{2}} d x${\int }_{0}^{\mathrm{\infty }}\frac{\mathrm{cos}x}{1+{x}^{2}}dx$.
Question:-01 (d) Expand $f\left(z\right)=\frac{1}{\left(z-1{\right)}^{2}\left(z-3\right)}$$f\left(z\right)=\frac{1}{\left(z-1{\right)}^{2}\left(z-3\right)}$f(z)=(1)/((z-1)^(2)(z-3))f(z)=\frac{1}{(z-1)^{2}(z-3)}$f\left(z\right)=\frac{1}{\left(z-1{\right)}^{2}\left(z-3\right)}$ 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 :
$\begin{array}{r}\text{Minimize}Z={x}_{1}+{x}_{2}\\ \text{subject to}\\ 2{x}_{1}+{x}_{2}\ge 4\\ {x}_{1}+7{x}_{2}\ge 7\\ {x}_{1},{x}_{2}\ge 0\end{array}${:[” Minimize “Z=x_(1)+x_(2)],[” subject to “],[2x_(1)+x_(2) >= 4],[x_(1)+7x_(2) >= 7],[x_(1)”,”x_(2) >= 0]:}\begin{array}{r} \text { Minimize } Z=x_{1}+x_{2} \\ \text { subject to } \\ 2 x_{1}+x_{2} \geq 4 \\ x_{1}+7 x_{2} \geq 7 \\ x_{1}, x_{2} \geq 0 \end{array}

Question:-02 (a) Let $f\left(x\right)={x}^{2}$$f\left(x\right)={x}^{2}$f(x)=x^(2)f(x)=x^{2}$f\left(x\right)={x}^{2}$ on $\left[0,k\right],k>0$$\left[0,k\right],k>0$[0,k],k > 0[0, k], k>0$\left[0,k\right],k>0$. Show that $f$$f$ff$f$ is Riemann integrable on the closed interval $\left[0,k\right]$$\left[0,k\right]$[0,k][0, k]$\left[0,k\right]$ 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$$G$GG$G$.
Question:-02 (c) Apply the calculus of residues to evaluate ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{cos}xdx}{\left({x}^{2}+{a}^{2}\right)\left({x}^{2}+{b}^{2}\right)},a>b>0$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} \frac{\mathrm{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{\mathrm{cos}xdx}{\left({x}^{2}+{a}^{2}\right)\left({x}^{2}+{b}^{2}\right)},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 :
$\begin{array}{r}\text{Maximize}Z={x}_{1}+{x}_{2}+{x}_{3}\\ \text{subject to}\\ 2{x}_{1}+{x}_{2}+{x}_{3}\le 2\\ 4{x}_{1}+2{x}_{2}+{x}_{3}\le 2\\ {x}_{1},{x}_{2},{x}_{3}\ge 0\end{array}${:[” Maximize “Z=x_(1)+x_(2)+x_(3)],[” subject to “],[2x_(1)+x_(2)+x_(3) <= 2],[4x_(1)+2x_(2)+x_(3) <= 2],[x_(1)”,”x_(2)”,”x_(3) >= 0]:}\begin{array}{r} \text { Maximize } Z=x_{1}+x_{2}+x_{3} \\ \text { subject to } \\ 2 x_{1}+x_{2}+x_{3} \leq 2 \\ 4 x_{1}+2 x_{2}+x_{3} \leq 2 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array}

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\left(x\right)\in R\left[x\right]$$f\left(x\right)\in R\left[x\right]$f(x)in R[x]f(x) \in R[x]$f\left(x\right)\in R\left[x\right]$ such that $f\left(0\right)=0=f\left(1\right)$$f\left(0\right)=0=f\left(1\right)$f(0)=0=f(1)f(0)=0=f(1)$f\left(0\right)=0=f\left(1\right)$. Prove that $S$$S$SS$S$ is an ideal of $R\left[x\right]$$R\left[x\right]$R[x]R[x]$R\left[x\right]$. Is the residue class ring $R\left[x\right]/S$$R\left[x\right]/S$R[x]//SR[x] / S$R\left[x\right]/S$ an integral domain? Give justification for your answer.
Question:-04 (b) Test for convergence or divergence of the series
$x+\frac{{2}^{2}{x}^{2}}{2!}+\frac{{3}^{3}{x}^{3}}{3!}+\frac{{4}^{4}{x}^{4}}{4!}+\frac{{5}^{5}{x}^{5}}{5!}+\cdots \phantom{\rule{1em}{0ex}}\left(x>0\right)$$x+\frac{{2}^{2}{x}^{2}}{2!}+\frac{{3}^{3}{x}^{3}}{3!}+\frac{{4}^{4}{x}^{4}}{4!}+\frac{{5}^{5}{x}^{5}}{5!}+\cdots \phantom{\rule{1em}{0ex}}\left(x>0\right)$x+(2^(2)x^(2))/(2!)+(3^(3)x^(3))/(3!)+(4^(4)x^(4))/(4!)+(5^(5)x^(5))/(5!)+cdotsquad(x > 0)x+\frac{2^{2} x^{2}}{2 !}+\frac{3^{3} x^{3}}{3 !}+\frac{4^{4} x^{4}}{4 !}+\frac{5^{5} x^{5}}{5 !}+\cdots \quad(x>0)$x+\frac{{2}^{2}{x}^{2}}{2!}+\frac{{3}^{3}{x}^{3}}{3!}+\frac{{4}^{4}{x}^{4}}{4!}+\frac{{5}^{5}{x}^{5}}{5!}+\cdots \phantom{\rule{1em}{0ex}}\left(x>0\right)$
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 :
 $A$$A$AA$A$ $B$$B$BB$B$ $C$$C$CC$C$ $D$$D$DD$D$ ${S}_{1}$${S}_{1}$S_(1)S_1${S}_{1}$ 21 16 25 13 11 ${S}_{2}$${S}_{2}$S_(2)S_2${S}_{2}$ 17 18 14 23 13 ${S}_{3}$${S}_{3}$S_(3)S_3${S}_{3}$ 32 27 18 41 19 6 10 12 15 43
A B C D S_(1) 21 16 25 13 11 S_(2) 17 18 14 23 13 S_(3) 32 27 18 41 19 6 10 12 15 43| | $A$ | $B$ | $C$ | $D$ | | | :— | :—: | :—: | :—: | :—: | :—: | | $S_1$ | 21 | 16 | 25 | 13 | 11 | | $S_2$ | 17 | 18 | 14 | 23 | 13 | | $S_3$ | 32 | 27 | 18 | 41 | 19 | | | 6 | 10 | 12 | 15 | 43 |

Section:- B
Question:-05 (a) It is given that the equation of any cone with vertex at $\left(a,b,c\right)$$\left(a,b,c\right)$(a,b,c)(a, b, c)$\left(a,b,c\right)$ is $f\left(\frac{x-a}{z-c},\frac{y-b}{z-c}\right)=0$$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\left(\frac{x-a}{z-c},\frac{y-b}{z-c}\right)=0$. Find the differential equation of the cone.
Question:-05(b) Solve, by Gauss elimination method, the system of equations
$\begin{array}{r}2x+2y+4z=18\\ x+3y+2z=13.\\ 3x+y+3z=14\end{array}$$\begin{array}{r}2x+2y+4z=18\\ x+3y+2z=13.\\ 3x+y+3z=14\end{array}${:[2x+2y+4z=18],[x+3y+2z=13.],[3x+y+3z=14]:}\begin{array}{r} 2 x+2 y+4 z=18 \\ x+3 y+2 z=13 . \\ 3 x+y+3 z=14 \end{array}$\begin{array}{r}2x+2y+4z=18\\ x+3y+2z=13.\\ 3x+y+3z=14\end{array}$
Question:-05 (c) (i) Convert the number $\left(1093.21875{\right)}_{10}$$\left(1093.21875{\right)}_{10}$(1093.21875)_(10)(1093.21875)_{10}$\left(1093.21875{\right)}_{10}$ into octal and the number $\left(1693\cdot 0628{\right)}_{10}$$\left(1693\cdot 0628{\right)}_{10}$(1693*0628)_(10)(1693 \cdot 0628)_{10}$\left(1693\cdot 0628{\right)}_{10}$ into hexadecimal systems.
(ii) Express the Boolean function $F\left(x,y,z\right)=xy+{x}^{\mathrm{\prime }}z$$F\left(x,y,z\right)=xy+{x}^{\mathrm{\prime }}z$F(x,y,z)=xy+x^(‘)zF(x, y, z)=x y+x^{\prime} z$F\left(x,y,z\right)=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$$k$kk$k$ is a constant. Obtain the Lagrangian and derive the equations of motion.
Question:-05 (e) The velocity components of an incompressible fluid in spherical polar coordinates $\left(r,\theta ,\psi \right)$$\left(r,\theta ,\psi \right)$(r,theta,psi)(r, \theta, \psi)$\left(r,\theta ,\psi \right)$ are $\left(2M{r}^{-3}\mathrm{cos}\theta ,M{r}^{-2}\mathrm{sin}\theta ,0\right)$$\left(2M{r}^{-3}\mathrm{cos}\theta ,M{r}^{-2}\mathrm{sin}\theta ,0\right)$(2Mr^(-3)cos theta,Mr^(-2)sin theta,0)\left(2 M r^{-3} \cos \theta, M r^{-2} \sin \theta, 0\right)$\left(2M{r}^{-3}\mathrm{cos}\theta ,M{r}^{-2}\mathrm{sin}\theta ,0\right)$, where $M$$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}},00$$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}},00$(del u)/(del t)=(del^(2)u)/(delx^(2)),0 < x < l,t > 0\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, 0<x<l, t>0$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}},00$ subject to the conditions
$\begin{array}{rl}& u\left(0,t\right)=u\left(l,t\right)=0\\ & u\left(x,0\right)=x\left(l-x\right),\phantom{\rule{1em}{0ex}}0\le x\le l\end{array}$$\begin{array}{rl}& u\left(0,t\right)=u\left(l,t\right)=0\\ & u\left(x,0\right)=x\left(l-x\right),\phantom{\rule{1em}{0ex}}0\le x\le l\end{array}${:[u(0″,”t)=u(l”,”t)=0],[u(x”,”0)=x(l-x)”,”quad0 <= x <= l]:}\begin{aligned} &u(0, t)=u(l, t)=0 \\ &u(x, 0)=x(l-x), \quad 0 \leq x \leq l \end{aligned}$\begin{array}{rl}& u\left(0,t\right)=u\left(l,t\right)=0\\ & u\left(x,0\right)=x\left(l-x\right),\phantom{\rule{1em}{0ex}}0\le x\le l\end{array}$
Question:-06 (b) Find a combinatorial circuit corresponding to the Boolean function
$f\left(x,y,z\right)=\left[x\cdot \left(\overline{y}+z\right)\right]+y$$f\left(x,y,z\right)=\left[x\cdot \left(\overline{y}+z\right)\right]+y$f(x,y,z)=[x*( bar(y)+z)]+yf(x, y, z)=[x \cdot(\bar{y}+z)]+y$f\left(x,y,z\right)=\left[x\cdot \left(\overline{y}+z\right)\right]+y$
and write the input/output table for the circuit.
Question:-06 (c) Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass $M$$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
$\left({D}^{2}+D{D}^{\mathrm{\prime }}-6{D}^{\mathrm{\prime }2}\right)z={x}^{2}\mathrm{sin}\left(x+y\right)$$\left({D}^{2}+D{D}^{\mathrm{\prime }}-6{D}^{\mathrm{\prime }2}\right)z={x}^{2}\mathrm{sin}\left(x+y\right)$(D^(2)+DD^(‘)-6D^(‘2))z=x^(2)sin(x+y)\left(D^{2}+D D^{\prime}-6 D^{\prime 2}\right) z=x^{2} \sin (x+y)$\left({D}^{2}+D{D}^{\mathrm{\prime }}-6{D}^{\mathrm{\prime }2}\right)z={x}^{2}\mathrm{sin}\left(x+y\right)$
where $D\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }x}$$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 :
 $t$$t$tt$t$ (minutes) 2 4 6 8 10 12 14 16 18 20 $v\left(\text{}\mathrm{k}\mathrm{m}/$$v\left(\text{}\mathrm{k}\mathrm{m}/$v(km//v(\mathrm{~km} / hour $\right)$$\right)$))$\right)$ 16 $28\cdot 8$$28\cdot 8$28*828 \cdot 8$28\cdot 8$ 40 $46\cdot 4$$46\cdot 4$46*446 \cdot 4$46\cdot 4$ $51\cdot 2$$51\cdot 2$51*251 \cdot 2$51\cdot 2$ 32 $17\cdot 6$$17\cdot 6$17*617 \cdot 6$17\cdot 6$ 8 $3\cdot 2$$3\cdot 2$3*23 \cdot 2$3\cdot 2$ 0
t (minutes) 2 4 6 8 10 12 14 16 18 20 v(km// hour ) 16 28*8 40 46*4 51*2 32 17*6 8 3*2 0| $t$ (minutes) | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | | :— | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | $v(\mathrm{~km} /$ hour $)$ | 16 | $28 \cdot 8$ | 40 | $46 \cdot 4$ | $51 \cdot 2$ | 32 | $17 \cdot 6$ | 8 | $3 \cdot 2$ | 0 |
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$$k$kk$k$ are situated at $\left(±a,0\right)$$\left(±a,0\right)$(+-a,0)(\pm a, 0)$\left(±a,0\right)$ 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 $\left(±b,0\right)$$\left(±b,0\right)$(+-b,0)(\pm b, 0)$\left(±b,0\right)$, then show that $3\sqrt{3}{\left({b}^{2}-{a}^{2}\right)}^{2}=16{a}^{3}b$$3\sqrt{3}{\left({b}^{2}-{a}^{2}\right)}^{2}=16{a}^{3}b$3sqrt3(b^(2)-a^(2))^(2)=16a^(3)b3 \sqrt{3}\left(b^{2}-a^{2}\right)^{2}=16 a^{3} b$3\sqrt{3}{\left({b}^{2}-{a}^{2}\right)}^{2}=16{a}^{3}b$

Question:-08 (a) Reduce the following partial differential equation to a canonical form and hence solve it :
$y{u}_{xx}+\left(x+y\right){u}_{xy}+x{u}_{yy}=0$$y{u}_{xx}+\left(x+y\right){u}_{xy}+x{u}_{yy}=0$yu_(xx)+(x+y)u_(xy)+xu_(yy)=0y u_{x x}+(x+y) u_{x y}+x u_{y y}=0$y{u}_{xx}+\left(x+y\right){u}_{xy}+x{u}_{yy}=0$
Question:-08 (b) Using Runge-Kutta method of fourth order, solve the differential equation $\frac{dy}{dx}=x+{y}^{2}$$\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\left(0\right)=1$$y\left(0\right)=1$y(0)=1y(0)=1$y\left(0\right)=1$, at $x=0\cdot 2$$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$$0\cdot 1$0*10 \cdot 1$0\cdot 1$.
Question:-08 (c) Verify that $w=ik\mathrm{log}\left\{\left(z-ia\right)/\left(z+ia\right)\right\}$$w=ik\mathrm{log}\left\{\left(z-ia\right)/\left(z+ia\right)\right\}$w=ik log{(z-ia)//(z+ia)}w=i k \log \{(z-i a) /(z+i a)\}$w=ik\mathrm{log}\left\{\left(z-ia\right)/\left(z+ia\right)\right\}$ is the complex potential of a steady flow of fluid about a circular cylinder, where the plane $y=0$$y=0$y=0y=0$y=0$ is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder.
upsc-m2021-2-4843690d-2e01-49d3-ba3c-bc046aac2819
खण्ड-A / SECTION-A
1(a) मान लीजिए कि ${m}_{1},{m}_{2},\cdots ,{m}_{k}$${m}_{1},{m}_{2},\cdots ,{m}_{k}$m_(1),m_(2),cdots,m_(k)m_1, m_2, \cdots, m_k${m}_{1},{m}_{2},\cdots ,{m}_{k}$ धनात्मक पूर्णांक हैं तथा $d>0,{m}_{1},{m}_{2},\cdots ,{m}_{k}$$d>0,{m}_{1},{m}_{2},\cdots ,{m}_{k}$d > 0,m_(1),m_(2),cdots,m_(k)d>0, m_1, m_2, \cdots, m_k$d>0,{m}_{1},{m}_{2},\cdots ,{m}_{k}$ का महत्तम समापवर्तक है। दर्शाइए कि ऐसे पूर्णांक ${x}_{1},{x}_{2},\cdots ,{x}_{k}$${x}_{1},{x}_{2},\cdots ,{x}_{k}$x_(1),x_(2),cdots,x_(k)x_1, x_2, \cdots, x_k${x}_{1},{x}_{2},\cdots ,{x}_{k}$ अस्तित्व में हैं ताकि
$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$d=x_(1)m_(1)+x_(2)m_(2)+cdots+x_(k)m_(k)d=x_1 m_1+x_2 m_2+\cdots+x_k m_k$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$
Let ${m}_{1},{m}_{2},\cdots ,{m}_{k}$${m}_{1},{m}_{2},\cdots ,{m}_{k}$m_(1),m_(2),cdots,m_(k)m_1, m_2, \cdots, m_k${m}_{1},{m}_{2},\cdots ,{m}_{k}$ be positive integers and $d>0$$d>0$d > 0d>0$d>0$ the greatest common divisor of ${m}_{1},{m}_{2},\cdots ,{m}_{k}$${m}_{1},{m}_{2},\cdots ,{m}_{k}$m_(1),m_(2),cdots,m_(k)m_1, m_2, \cdots, m_k${m}_{1},{m}_{2},\cdots ,{m}_{k}$. Show that there exist integers ${x}_{1},{x}_{2},\cdots ,{x}_{k}$${x}_{1},{x}_{2},\cdots ,{x}_{k}$x_(1),x_(2),cdots,x_(k)x_1, x_2, \cdots, x_k${x}_{1},{x}_{2},\cdots ,{x}_{k}$ such that
$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$d=x_(1)m_(1)+x_(2)m_(2)+cdots+x_(k)m_(k)d=x_1 m_1+x_2 m_2+\cdots+x_k m_k$d={x}_{1}{m}_{1}+{x}_{2}{m}_{2}+\cdots +{x}_{k}{m}_{k}$
(b) श्रेणी
${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$x^(4)+(x^(4))/(1+x^(4))+(x^(4))/((1+x^(4))^(2))+(x^(4))/((1+x^(4))^(3))+cdotsx^4+\frac{x^4}{1+x^4}+\frac{x^4}{\left(1+x^4\right)^2}+\frac{x^4}{\left(1+x^4\right)^3}+\cdots${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$
के $\left[0,1\right]$$\left[0,1\right]$[0,1][0,1]$\left[0,1\right]$ पर एकसमान अभिसरण की जाँच कीजिए।
Test the uniform convergence of the series
${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$x^(4)+(x^(4))/(1+x^(4))+(x^(4))/((1+x^(4))^(2))+(x^(4))/((1+x^(4))^(3))+cdotsx^4+\frac{x^4}{1+x^4}+\frac{x^4}{\left(1+x^4\right)^2}+\frac{x^4}{\left(1+x^4\right)^3}+\cdots${x}^{4}+\frac{{x}^{4}}{1+{x}^{4}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{2}}+\frac{{x}^{4}}{{\left(1+{x}^{4}\right)}^{3}}+\cdots$
on $\left[0,1\right]$$\left[0,1\right]$[0,1][0,1]$\left[0,1\right]$.
(c) यदि एक फलन $f$$f$ff$f$, अन्तराल $\left[a,b\right]$$\left[a,b\right]$[a,b][a, b]$\left[a,b\right]$ में एकदिष्ट है, तब सिद्ध कीजिए कि $f,\left[a,b\right]$$f,\left[a,b\right]$f,[a,b]f,[a, b]$f,\left[a,b\right]$ में रीमान समाकलनीय है।
If a function $f$$f$ff$f$ is monotonic in the interval $\left[a,b\right]$$\left[a,b\right]$[a,b][a, b]$\left[a,b\right]$, then prove that $f$$f$ff$f$ is Riemann integrable in $\left[a,b\right]$$\left[a,b\right]$[a,b][a, b]$\left[a,b\right]$.
(d) मान लीजिए कि $c:\left[0,1\right]\to \mathbb{C},c\left(t\right)={e}^{4\pi it},0\le t\le 1$$c:\left[0,1\right]\to \mathbb{C},c\left(t\right)={e}^{4\pi it},0\le t\le 1$c:[0,1]rarrC,c(t)=e^(4pi it),0 <= t <= 1c:[0,1] \rightarrow \mathbb{C}, c(t)=e^{4 \pi i t}, 0 \leq t \leq 1$c:\left[0,1\right]\to \mathbb{C},c\left(t\right)={e}^{4\pi it},0\le t\le 1$ के द्वारा परिभाषित एक वक्र है। कन्दूर समाकल ${\int }_{c}\frac{dz}{2{z}^{2}-5z+2}$${\int }_{c} \frac{dz}{2{z}^{2}-5z+2}$int _(c)(dz)/(2z^(2)-5z+2)\int_c \frac{d z}{2 z^2-5 z+2}${\int }_{c}\frac{dz}{2{z}^{2}-5z+2}$ का मान निकालिए।
Let $c:\left[0,1\right]\to \mathbb{C}$$c:\left[0,1\right]\to \mathbb{C}$c:[0,1]rarrCc:[0,1] \rightarrow \mathbb{C}$c:\left[0,1\right]\to \mathbb{C}$ be the curve, where $c\left(t\right)={e}^{4\pi it},0\le t\le 1$$c\left(t\right)={e}^{4\pi it},0\le t\le 1$c(t)=e^(4pi it),0 <= t <= 1c(t)=e^{4 \pi i t}, 0 \leq t \leq 1$c\left(t\right)={e}^{4\pi it},0\le t\le 1$. Evaluate the contour integral ${\int }_{c}\frac{dz}{2{z}^{2}-5z+2}$${\int }_{c} \frac{dz}{2{z}^{2}-5z+2}$int _(c)(dz)/(2z^(2)-5z+2)\int_c \frac{d z}{2 z^2-5 z+2}${\int }_{c}\frac{dz}{2{z}^{2}-5z+2}$.
(e) एक कम्पनी के एक विभाग के पाँच कर्मचारियों को पाँच कार्य सम्पन्न करने हैं। जितना समय (घंटों में) एक व्यक्ति एक कार्य को सम्पन्न करने के लिए लेता है, वह प्रभाविता आव्यूह में दिया गया है। इन पाँच कर्मचारियों को इन सभी कार्यों को इस तरह निर्धारित कीजिए जिससे कि समस्त कार्य सम्पन्न करने का समय न्यूनतम हो :
A department of a company has five employees with five jobs to be performed. The time (in hours) that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time :

2(a) $f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$$f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$f(x)=x^(3)-9x^(2)+26 x-24f(x)=x^3-9 x^2+26 x-24$f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$ का, $0\le x\le 1$$0\le x\le 1$0 <= x <= 10 \leq x \leq 1$0\le x\le 1$ के लिए, अधिकतम तथा न्यूनतम मान निकालिए।
Find the maximum and minimum values of $f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$$f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$f(x)=x^(3)-9x^(2)+26 x-24f(x)=x^3-9 x^2+26 x-24$f\left(x\right)={x}^{3}-9{x}^{2}+26x-24$ for $0\le x\le 1$$0\le x\le 1$0 <= x <= 10 \leq x \leq 1$0\le x\le 1$
(b) मान लीजिए कि $F$$F$FF$F$ एक क्षेत्र है तथा