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 \(z=0, z=1, x^2+y^2=4\).