Engineering Mathematics-I 2018 Question Paper — Government College of…

Instructions

• Attempt Q.No. 1, which is compulsory and it consists of short answer questions of total 10 marks.
• From each unit, attempt one question of 15 marks each (Total marks: 15 × 6 = 90)

Questions

Questions

• Q1. Verify whether the following statements are true or false. (i) The curve $y^2 = x^3$ has a node at the origin. (ii) The curve $r = a \cos 2\theta$ has two leaves. (iii) $\sin z = \sin x \cdot \cosh y + i \cos x \cdot \sinh y$. (iv) $\sin x$ and $\sin 2x$ are independent solutions of $(D^4 + 1)y = 0$ (v) $(x^2 + y^2 + 2x) dx + 2y dy = 0$ is an exact differential equation. (vi) The function $z = x \cdot \sin^{-1}\left(\frac{y}{x}\right)$ is homogeneous of degree 2. (vii) The volume of a sphere of radius 'a' is $\frac{4}{3}\pi a^3$. (viii) The value of $\int_{0}^{\pi/2} \sin^7 x dx$ is $\frac{16}{35}$. (ix) $\cosh(x + y) = \cosh x \cdot \cosh y - \sinh x \cdot \sinh y$. (x) The angle between the vectors $4\hat{i} - 2\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - 2\hat{k}$ is $\frac{\pi}{2}$. (xi) $\text{div } \vec{r} = 3$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. [10 marks]

UNIT - I

• Q2. (a) Find all the asymptotes of $x^3 + 3x^2y - xy^2 - 3y^3 + x^2 - 2xy - 3y^2 + 4x + 5 = 0$. (b) Trace the curve: $r = a \sin 2\theta$. [15 marks]
• Q3. (a) Show that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$, where $u = \tan^{-1}\left(\frac{y}{x}\right)$. (b) Find the position and nature of double point on $a^2y^2 = a^2x^2 - 4x^3$. (c) Find the radius of curvature at any point of the cardioid $r = a(1 - \cos\theta)$. [15 marks]

UNIT - II

• Q4. (a) Verify Rolle's theorem for $f(x) = e^x(\sin x - \cos x)$ in $\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]$. (b) Examine the function $f = xy(a - x - y)$ for extreme values. [15 marks]
• Q5. (a) Determine $\lim_{x \to 0} \left(\frac{\tan x}{x}\right)^{1/x^2}$. (b) Expand $f(x, y) = e^x \log(1 + y)$ as a series in powers of $x$ and $y$ upto third degree terms. (c) State and prove mean value theorem. [15 marks]

UNIT - III

• Q6. (a) Find the length of the loop of the curve: $3ay^2 = x(x - a)^2$. (b) Evaluate $\iint_R (x^2 + y^2) dxdy$ over the annular region $R$ lying between $x^2 + y^2 = 4$ and $x^2 + y^2 = 9$. [15 marks]
• Q7. (a) Find the volume of the solid generated by revolving area about the x-axis, of the area included between $y^2 = x^3$ and $x^2 = y^3$. (b) Evaluate $\int_{0}^{\infty} x^3 \cdot e^{-\sqrt{x}} dx$. (c) Evaluate $\int_{-1}^{1} (1 + x)^{3/2} (1 - x)^{7/2} dx$ by using beta/gamma functions. [15 marks]

UNIT - IV

• Q8. (a) Verify Green's theorem in the xy-plane for $\oint_C (2xy - x^2) dx + (x^2 + y^2) dy$ where $C$ is the boundary of the region enclosed by curves $y = x^2$ and $x = y^2$. (b) Evaluate $\oint_C \vec{F} \cdot d\vec{r}$ around a triangle in the xy-plane with vertices at $(0, 0)$, $(3, 0)$ and $(3, 2)$ taken in counter clockwise direction, where $\vec{F} = (2x - y + 4)\hat{i} + (5y + 3x - 6)\hat{j}$. [15 marks]
• Q9. (a) Find the angle between the surfaces $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 - z = 3$ at the point $(2, -1, 2)$. (b) Show that $\text{div}(\text{curl } \vec{F}) = 0$. (c) Show that $\iint_S \nabla r^2 \cdot d\vec{S} = 6V$, where $r = |\vec{r}|$, $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $S$ is a closed surface enclosing a volume $V$. [15 marks]

UNIT - V

• Q10. (a) Express $\tan^{-1} Z$ into real and imaginary parts. (b) If $e^{i\theta} = \sinh(x + i y)$, show that $\sinh^4 x = \cos^3 \theta = \cos^4 y$. [15 marks]
• Q11. (a) Show that $\tan \left[i \log \left(\frac{a - ib}{a + ib}\right)\right] = \frac{2ab}{a^2 - b^2}$. (b) Show that $\sin(\log i) = -1$. (c) Sum the series to infinity. $1 + x \cos\theta + \frac{x^2}{2!}\cos 2\theta + \frac{x^3}{3!}\cos 3\theta + \dots$ [15 marks]

UNIT - VI

• Q12. (a) Solve: $x^2 y'' + 7xy' + 5y = 4(x^{-1} + x^3)$ (b) Solve: $y'' - y = \frac{2}{1 + e^x}$ [15 marks]
• Q13. (a) Solve: $(D^3 - D^2 - 6D)y = x^2$. (b) Solve: $y(e^x + 2xy) dx - e^x dy = 0$. (c) Solve: $\frac{dy}{dx} + 4xy + xy^3 = 0$ [15 marks]

Related pages

• Engineering Mathematics-I syllabus
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