BE Computer Engineering (IOE, TU) Engineering Mathematics II (IOE, SH 451) Question Paper 2079

(a) Define the rank of a matrix. Reduce the following matrix to its echelon form and hence find its rank:

(b) Investigate for what values of $\lambda$ and $\mu$ the system of equations

has (i) no solution, (ii) a unique solution, and (iii) infinitely many solutions. Solve the system completely in the case of infinitely many solutions.

(a) Evaluate the double integral $\displaystyle\int_0^1 \int_{x^2}^{x} (x^2 + y^2)\, dy\, dx$ and sketch the region of integration. (b) Change the order of integration in $\displaystyle\int_0^a \int_{x^2/a}^{\,2a-x} xy\, dy\, dx$ and hence evaluate it. (c) Using a triple integral, find the volume of the region bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z = 0$ and $z = 3 - x$.

(a) Solve the differential equation $(D^2 - 4D + 4)y = e^{2x} + x^3$, where $D \equiv \dfrac{d}{dx}$. (b) Using the method of variation of parameters, solve $\dfrac{d^2 y}{dx^2} + y = \sec x$. (c) Solve the Cauchy–Euler equation $x^2 \dfrac{d^2 y}{dx^2} - 3x\dfrac{dy}{dx} + 4y = x^2 \ln x$.

(a) Define the gradient of a scalar field and the divergence and curl of a vector field. Show that $\nabla \times (\nabla \phi) = \vec{0}$ for any twice-differentiable scalar field $\phi$. (b) A vector field is given by $\vec{F} = (x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k}$. Show that $\vec{F}$ is irrotational and find a scalar potential $\phi$ such that $\vec{F} = \nabla \phi$. (c) Verify Green's theorem in the plane for $\oint_C \big[(xy + y^2)\,dx + x^2\,dy\big]$, where $C$ is the closed curve bounded by $y = x$ and $y = x^2$.

Find the eigenvalues and the corresponding eigenvectors of the matrix

Hence verify the Cayley–Hamilton theorem for $A$.

(a) If $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{c} = 3\hat{i} - 4\hat{j} + 5\hat{k}$, evaluate the scalar triple product $[\vec{a}\ \vec{b}\ \vec{c}]$ and state whether the three vectors are coplanar. (b) Prove, using vector methods, that the diagonals of a rhombus bisect each other at right angles.

(a) State De Moivre's theorem. Use it to express $\cos 4\theta$ and $\sin 4\theta$ in terms of $\cos\theta$ and $\sin\theta$. (b) Find all the values of $(1 + i)^{1/3}$ and represent them on the Argand diagram.

(a) State the ratio test (D'Alembert's test) for the convergence of a series of positive terms. Test the convergence of the series $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}$. (b) Examine the convergence of the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}$ for different values of $p$ using the integral test.

(a) Find the equation of the plane passing through the points $(1, 1, 0)$, $(-2, 2, -1)$ and $(1, 2, 1)$. (b) Find the shortest distance between the lines

Compute the work done by the force field $\vec{F} = (3x^2 + 6y)\hat{i} - 14yz\,\hat{j} + 20xz^2\,\hat{k}$ in moving a particle from the point $(0,0,0)$ to the point $(1,1,1)$ along the curve $x = t$, $y = t^2$, $z = t^3$.

By changing to polar coordinates, evaluate $\displaystyle\int_0^{\infty}\int_0^{\infty} e^{-(x^2 + y^2)}\, dx\, dy$ and hence deduce the value of $\displaystyle\int_0^{\infty} e^{-x^2}\, dx$.