BE Computer Engineering (IOE, TU) Engineering Mathematics I (IOE, SH 401) Question Paper 2079

(a) State and prove Euler's theorem on homogeneous functions of two variables. If $u = \tan^{-1}\!\left(\dfrac{x^3 + y^3}{x - y}\right)$, show that $x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} = \sin 2u$. (8)

(b) If $z = f(x, y)$ where $x = r\cos\theta$ and $y = r\sin\theta$, prove that $\left(\dfrac{\partial z}{\partial x}\right)^2 + \left(\dfrac{\partial z}{\partial y}\right)^2 = \left(\dfrac{\partial z}{\partial r}\right)^2 + \dfrac{1}{r^2}\left(\dfrac{\partial z}{\partial \theta}\right)^2$. (8)

(a) Find all the asymptotes of the curve $x^3 + 2x^2 y - xy^2 - 2y^3 + 4y^2 + 2xy + y - 1 = 0$. (8)

(b) Derive the formula for the radius of curvature in Cartesian form $\rho = \dfrac{\left[1 + (y_1)^2\right]^{3/2}}{y_2}$, and hence find the radius of curvature of the curve $y = x^3 - 6x^2 + 3x + 1$ at the point $(1, -1)$. (8)

(a) Obtain a reduction formula for $\displaystyle\int_0^{\pi/2} \sin^n x\, dx$ and hence evaluate $\displaystyle\int_0^{\pi/2} \sin^6 x\, dx$. (8)

(b) Establish a reduction formula for $I_{m,n} = \displaystyle\int \cos^m x \sin^n x\, dx$ in terms of $I_{m-2, n}$, and use it to evaluate $\displaystyle\int_0^{\pi/2} \cos^4 x \sin^2 x\, dx$. (8)

State Leibnitz's theorem for the $n$th derivative of a product of two functions. If $y = \sin^{-1} x$, prove that $(1 - x^2)y_{n+2} - (2n+1)x\, y_{n+1} - n^2 y_n = 0$, and hence find $(y_n)_0$, the value of the $n$th derivative at $x = 0$.

(a) Test the convergence of the improper integral $\displaystyle\int_1^{\infty} \dfrac{dx}{x^p}$ and state for what values of $p$ it converges. (4)

(b) Evaluate $\displaystyle\int_0^{1} \dfrac{dx}{\sqrt{1 - x^2}}$ as an improper integral and discuss its convergence. (4)

(a) Solve the differential equation $(x^2 - 4xy - 2y^2)\,dx + (y^2 - 4xy - 2x^2)\,dy = 0$ by testing for exactness. (4)

(b) Solve the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$, $x > 0$. (4)

Find the volume of the solid generated by revolving the region bounded by the curve $y^2 = 4x$ and the line $x = 4$ about the $x$-axis. Also find the area of the surface generated by the revolution of this arc.

Reduce the conic $3x^2 + 2xy + 3y^2 - 16y + 23 = 0$ to its standard form by removing the $xy$ term through rotation of axes. Identify the type of conic and state the length of its semi-axes.

(a) Find the angle of intersection between the curves $r = a(1 + \cos\theta)$ and $r = b(1 - \cos\theta)$. (4)

(b) Find the pedal equation of the cardioid $r = a(1 + \cos\theta)$. (4)

Examine the function $f(x, y) = x^3 + y^3 - 3axy$ for maxima and minima. Find the stationary points and classify them using the second-derivative test for functions of two variables.

Find the orthogonal trajectories of the family of curves $x^2 + y^2 = 2cx$, where $c$ is an arbitrary parameter.