# UPSC IAS Previous Year Mathematics Question Paper I 2022 (Optional Paper) | PDF Download

Question Details
 Exam UPSC Paper Optional Paper Subject Mathematics 1 Year 2022

Section: A

Question:01(a) Prove that any set of $$\mathrm{n}$$ linearly independent vectors in a vector space $$\mathrm{V}$$ of dimension $$\mathrm{n}$$ constitutes a basis for $$\mathrm{V}$$.

Question:01(b) Let $$\mathrm{T}: \mathbb{R}^2 \rightarrow \mathbb{R}^3$$ be a linear transformation such that $$\mathrm{T}\left(\begin{array}{l}1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$$ and $$\mathrm{T}\left(\begin{array}{l}1 \\ 1\end{array}\right)=\left(\begin{array}{r}-3 \\ 2 \\ 8\end{array}\right)$$. Find $$\mathrm{T}\left(\begin{array}{l}2 \\ 4\end{array}\right)$$.

Question:01(c) Evaluate $$\lim _{x \rightarrow \infty}\left(e^x+x\right)^{\frac{1}{x}}$$.

Question:-01(d) Examine the convergence of $$\int_0^2 \frac{d x}{\left(2 x-x^2\right)}$$.

Question:-01(e) A variable plane passes through a fixed point (a, b, c) and meets the axes at points A, B and C respectively. Find the locus of the centre of the sphere passing through the points $$\mathrm{O}, \mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}, \mathrm{O}$$ being the origin.

Question:-02(a) Find all solutions to the following system of equations by row-reduced method :
\begin{aligned} &\mathrm{x}_1+2 \mathrm{x}_2-\mathrm{x}_3=2 \\ &2 \mathrm{x}_1+3 \mathrm{x}_2+5 \mathrm{x}_3=5 \\ &-\mathrm{x}_1-3 \mathrm{x}_2+8 \mathrm{x}_3=-1 \end{aligned}

Question:-02(b) A wire of length $$l$$ is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange’s method of undetermined multipliers, find the least value of the sum of the areas so formed.

Question:-02(c) If $$\mathrm{P}, \mathrm{Q}, \mathrm{R} ; \mathrm{P}^{\prime}, \mathrm{Q}^{\prime}, \mathrm{R}^{\prime}$$ are feet of the six normals drawn from a point to the ellipsoid $$\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}+\frac{\mathrm{z}^2}{\mathrm{c}^2}=1$$, and the plane $$\mathrm{PQR}$$ is represented by $$l x+m y+n z=p$$, show that the plane $$P^{\prime} Q^{\prime} R^{\prime}$$ is given by $$\frac{\mathrm{x}}{\mathrm{a}^2 l}+\frac{\mathrm{y}}{\mathrm{b}^2 \mathrm{~m}}+\frac{\mathrm{z}}{\mathrm{c}^2 \mathrm{n}}+\frac{1}{\mathrm{p}}=0$$.

Question:-03(a) Let the set $$\left.P=\left\{\begin{array}{c}x \\ y \\ z\end{array}\right) \mid \begin{array}{c}x-y-z=0 \text { and } \\ 2 x-y+z=0\end{array}\right\}$$ be the collection of vectors of a vector space $$\mathbb{R}^3(\mathbb{R})$$. Then
(i) prove that $$\mathrm{P}$$ is a subspace of $$\mathbb{R}^3$$.
(ii) find a basis and dimension of $$P$$.

Question:-03(b) Use double integration to calculate the area common to the circle $$x^2+y^2=4$$ and the parabola $$y^2=3 x$$.

Question:-03(c) Find the equation of the sphere of smallest possible radius which touches the straight lines : $$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$$ and $$\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$$.

Question:-04(a) Find a linear map $$\mathrm{T}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ which rotates each vector of $$\mathbb{R}^2$$ by an angle $$\theta$$. Also, prove that for $$\theta=\frac{\pi}{2}, \mathrm{~T}$$ has no eigenvalue in $$\mathbb{R}$$.

Question:-04(b) Trace the curve $$y^2 x^2=x^2-a^2$$, where $$a$$ is a real constant.

Question:-04(c) If the plane $$u x+v y+w z=0$$ cuts the cone $$a x^2+b y^2+c z^2=0$$ in perpendicular generators, then prove that $$(b+c) u^2+(c+a) v^2+(a+b) w^2=0$$.

Question:-05(a) Show that the general solution of the differential equation $$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{Py}=\mathrm{Q}$$ can be written in the form $$y=\frac{Q}{P}-e^{-\int P d x}\left\{C+\int e^{\int P d x} d\left(\frac{Q}{P}\right)\right\}$$, where $$\mathrm{P}, \mathrm{Q}$$ are non-zero functions of $$\mathrm{x}$$ and $$\mathrm{C}$$, an arbitrary constant.

Question:-05(b) Show that the orthogonal trajectories of the system of parabolas : $$\mathrm{x}^2=4 \mathrm{a}(\mathrm{y}+\mathrm{a})$$ belong to the same system.

Question:-05(c) A body of weight $$w$$ rests on a rough inclined plane of inclination $$\theta$$, the coefficient of friction, $$\mu$$, being greater than $$\tan \theta$$. Find the work done in slowly dragging the body a distance ‘b’ up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane.

Question:-05(d) A projectile is fired from a point $$\mathrm{O}$$ with velocity $$\sqrt{2 \mathrm{gh}}$$ and hits a tangent at the point $$\mathrm{P}(\mathrm{x}, \mathrm{y})$$ in the plane, the axes $$\mathrm{OX}$$ and $$\mathrm{OY}$$ being horizontal and vertically downward lines through the point $$\mathrm{O}$$, respectively. Show that if the two possible directions of projection be at right angles, then $$\mathrm{x}^2=2 \mathrm{hy}$$ and then one of the possible directions of projection bisects the angle POX.

Question:-05(e) Show that $$\overrightarrow{\mathrm{A}}=\left(6 x y+z^3\right) \hat{i}+\left(3 x^2-z\right) \hat{j}+\left(3 x z^2-y\right) \hat{k}$$ is irrotational. Also find $$\phi$$ such that $$\overrightarrow{\mathrm{A}}=\nabla \phi$$.

Question:-06(a) A cable of weight w per unit length and length $$2 l$$ hangs from two points $$\mathrm{P}$$ and $$\mathrm{Q}$$ in the same horizontal line. Show that the span of the cable is $$2 l\left(1-\frac{2 h^2}{3 l^2}\right)$$, where $$h$$ is the sag in the middle of the tightly stretched position.

Question:-06(b) Solve the following differential equation by using the method of variation of parameters : $$\left(x^2-1\right) \frac{d^2 y}{d x^2}-2 x \frac{d y}{d x}+2 y=\left(x^2-1\right)^2$$, given that $$\mathrm{y}=\mathrm{x}$$ is one solution of the reduced equation.

Question:-06(c) Verify Green’s theorem in the plane for $$\oint_C\left(3 x^2-8 y^2\right) d x+(4 y-6 x y) d y$$, where $$\mathrm{C}$$ is the boundary curve of the region defined by $$\mathrm{x}=0, \mathrm{y}=0$$, $$x+y=1$$.

Question:-07(a) Verify Stokes’ theorem for $$\vec{F}=x \hat{i}+z^2 \hat{j}+y^2 \hat{k}$$ over the plane surface : $$x+y+z=1$$ lying in the first octant.

Question:-07(b) Solve the following initial value problem by using Laplace’s transformation $$\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dt}^2}-3 \frac{\mathrm{dy}}{\mathrm{dt}}+2 \mathrm{y}=\mathrm{h}(\mathrm{t})$$, where
$\mathrm{h}(\mathrm{t})=\left\{\begin{array}{cc} 2, & 0<\mathrm{t}<4, \\ 0, & \mathrm{t}>4, \end{array} \quad \mathrm{y}(0)=0, \quad \mathrm{y}^{\prime}(0)=0\right.$.

Question:-07(c) Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let $$\rho$$ and $$\rho^{\prime}$$ be the radii of curvature of the two cylinders at the point of contact and $$h$$ be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if $$\mathrm{h}<\frac{\rho \rho^{\prime}}{\rho+\rho^{\prime}}$$.

Question8(a) (i) Find the general and singular solutions of the differential equation: $$\left(x^2-a^2\right) p^2-2 x y p+y^2+a^2=0$$, where $$p=\frac{d y}{d x}$$. Also give the geometric relation between the general and singular solutions.

Question:-08(a) (ii) Solve the following differential equation :
$(3 x+2)^2 \frac{d^2 y}{d x^2}+5(3 x+2) \frac{d y}{d x}-3 y=x^2+x+1$.

Question:-08(b) A chain of $$\mathrm{n}$$ equal uniform rods is smoothly jointed together and suspended from its one end $$\mathrm{A}_1$$. A horizontal force $$\overrightarrow{\mathrm{P}}$$ is applied to the other end $$\mathrm{A}_{\mathrm{n}+1}$$ of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration.

Question:-08(c) Using Gauss’ divergence theorem, evaluate $$\iint_S \vec{F} \cdot \vec{n} d S$$, where $$\vec{F}=x \hat{i}-y \hat{j}+\left(z^2-1\right) \hat{k}$$ and $$S$$ is the cylinder formed by the surfaces $$z=0, z=1, x^2+y^2=4$$.

Class 11th NCERT Mathematics Book Solution Exercise 1.1

Class 11 Maths NCERT Solutions Chapter 1 Exercises

Class 11th NCERT Mathematics Book Chapterwise Solution

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