BE Computer Engineering (IOE, TU) Engineering Mathematics I (IOE, SH 401) Question Paper 2078

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

(b) Find the radius of curvature at the origin for the curve $y - x = x^2 + 2xy + y^2$ using Newton's method (the method of expansion / limit of $\dfrac{x^2}{2y}$).

(a) If $u = \sin^{-1}\!\left(\dfrac{x^2 + y^2}{x + y}\right)$, show by Euler's theorem on homogeneous functions that $x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} = \tan u$.

(b) Find the extreme values (maxima and minima) of the function $f(x,y) = x^3 + y^3 - 3axy$, where $a > 0$.

(a) Find the area bounded by one loop of the curve $r = a\cos 2\theta$ (a four-leaved rose).

(b) The cardioid $r = a(1 + \cos\theta)$ revolves about the initial line. Find the volume of the solid of revolution so generated.

(a) Identify the conic represented by the equation $3x^2 + 2xy + 3y^2 - 16y + 23 = 0$ by removing the $xy$-term through a suitable rotation of axes, and reduce it to its standard form. State the nature of the conic and its eccentricity.

(b) Find the equation of the conic with focus at the pole, eccentricity $e = \tfrac{1}{2}$ and the directrix $r\cos\theta = -4$, and hence identify it.

If $y = \sin(m\sin^{-1} x)$, prove that $(1 - x^2)\,y_{n+2} - (2n+1)x\,y_{n+1} + (m^2 - n^2)\,y_n = 0$, and hence find $(y_n)_0$ when $x = 0$.

Evaluate the limit $\displaystyle \lim_{x \to 0}\left(\frac{1}{x^2} - \frac{\cot x}{x}\right)$ using L'Hospital's rule or series expansion.

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

Test the convergence of the improper integral $\displaystyle \int_0^{1} \frac{dx}{\sqrt{1 - x^4}}$, and express it in terms of Beta and Gamma functions.

Solve the differential equation $(x^2 - ay)\,dx + (y^2 - ax)\,dy = 0$ after testing it for exactness.

Solve the Bernoulli equation $\dfrac{dy}{dx} + \dfrac{y}{x} = y^2 x$ by reducing it to a linear differential equation.

Find the equations of the tangent plane and the normal line to the surface $x^2 + 2y^2 + 3z^2 = 12$ at the point $(1, 2, -1)$.

Find the radius of curvature at the point $\left(\tfrac{3a}{2}, \tfrac{3a}{2}\right)$ on the folium of Descartes $x^3 + y^3 = 3axy$.