BE Computer Engineering (IOE, TU) Engineering Mathematics II (IOE, SH 451) Question Paper 2078

(a) Evaluate the double integral $\displaystyle\int_{0}^{a}\int_{y}^{a}\frac{x}{x^{2}+y^{2}}\,dx\,dy$ by changing the order of integration. [6]

(b) Find the volume of the solid bounded by the cylinder $x^{2}+y^{2}=4$ and the planes $y+z=4$ and $z=0$ using a double integral in polar coordinates. [6]

(a) Solve the differential equation $\dfrac{d^{2}y}{dx^{2}}-3\dfrac{dy}{dx}+2y=x\,e^{3x}$ using the method of undetermined coefficients (or operator method), stating the complementary function and particular integral clearly. [7]

(b) Using the method of variation of parameters, solve $\dfrac{d^{2}y}{dx^{2}}+y=\sec x$. [5]

(a) Define the gradient of a scalar field, and the divergence and curl of a vector field. Show that for any twice-differentiable scalar field $\phi$, $\nabla\times(\nabla\phi)=\vec{0}$. [6]

(b) For the vector field $\vec{F}=(x^{2}-yz)\,\hat{i}+(y^{2}-zx)\,\hat{j}+(z^{2}-xy)\,\hat{k}$, compute $\operatorname{div}\vec{F}$ and $\operatorname{curl}\vec{F}$, and hence determine whether $\vec{F}$ is irrotational. If so, find a scalar potential $\phi$ such that $\vec{F}=\nabla\phi$. [6]

(a) State the Cayley–Hamilton theorem. Verify it for the matrix $A=\begin{pmatrix}1 & 2\\ 2 & -1\end{pmatrix}$ and hence find $A^{-1}$. [6]

(b) Find the eigenvalues and the corresponding eigenvectors of the matrix $A=\begin{pmatrix}2 & 0 & 1\\ 0 & 2 & 0\\ 1 & 0 & 2\end{pmatrix}$. [6]

Using De Moivre's theorem, express $\cos 4\theta$ in terms of powers of $\cos\theta$.

Find all the values of $(1+i)^{1/3}$ and represent them on the Argand diagram.

Test the convergence of the series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{2^{n}}$ stating clearly the test used.

Find the radius of convergence and the interval of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\frac{(x-2)^{n}}{n\,3^{n}}$.

Find the shortest distance between the lines $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$.

Find the equation of the plane passing through the point $(1,-2,3)$ and perpendicular to the line of intersection of the planes $x+2y+3z=4$ and $2x-y+z=5$.

Show that the vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-4\hat{j}-4\hat{k}$ form the sides of a right-angled triangle, using the scalar product.

Evaluate the triple integral $\displaystyle\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{1-x-y}(x+y+z)\,dz\,dy\,dx$.

For what value of $\lambda$ does the system of equations $x+y+z=6$, $x+2y+3z=10$, $x+2y+\lambda z=\mu$ have (i) no solution, (ii) a unique solution, and (iii) infinitely many solutions? Determine $\mu$ in each case where relevant.

Solve the Cauchy–Euler equation $x^{2}\dfrac{d^{2}y}{dx^{2}}-2x\dfrac{dy}{dx}+2y=0$.

Find the directional derivative of $\phi=x^{2}yz+4xz^{2}$ at the point $(1,-2,-1)$ in the direction of the vector $2\hat{i}-\hat{j}-2\hat{k}$.