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

Instructions

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

Questions

Questions

• Q1. Verify whether the following statements are true or false : (i) If : $z = \frac{x^2 + y^2}{x - y}$, then : $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$. (ii) The curve $x^2 y^2 - x^2 + y^2 = 1$ has three asymptotes. (iii) The area of a loop of the curve $r^2 = a^2 \cos 2\theta$ is $\frac{1}{2} a^2$. (iv) $\text{erf}_c(x) = 1 - \text{erf}(x)$ (v) $\sinh^2 z + \cosh^2 z = 1$ (vi) Two non-zero vectors are perpendicular if their dot product is zero. (vii) The gradient of a constant function is zero. (viii) The differential equation $(D^3 - D)y = 0$ has two only solutions $e^x$ amd $e^{-x}$. (ix) The P.I. of $(D - 3)^2 y = 2e^{3x}$ is $x^2 e^{3x}$. (x) $\int_{0}^{2} \int_{0}^{3} (x + y) \, dx \, dy = 15$. [10 marks]

Unit I

• Q2. (a) Trace the curve : $x(x^2 + y^2) = a(x^2 - y^2)$, where $a > 0$. (b) Find all the asymptotes of the curve : $x(y - 3)^3 = 4y(x - 1)^3$. [15 marks]
• Q3. (a) Verify Euler's theorem for the function : $z = 2x^2 + 3xy + 4y^2$. (b) Find the radius of curvature at any point of the curve : $xy = c^2$. (c) Find the nature of double points at the origin on the curve : $x^3 + y^3 - 4axy = 0$. [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) Find the extreme values of the function $\cos x \cdot \cos y \cdot \cos z$, where $x + y + z = \pi$. [15 marks]
• Q5. (a) State and prove mean value theorem. (b) Evaluate : $\lim_{x \to 0} \left(\frac{1}{x^2} - \frac{1}{x \sin x}\right)$. (c) Expand $f(x, y) = e^{x+y}$ as a series in powers of $x$ and $y$ upto third degree terms. [15 marks]

Unit III

• Q6. (a) The area between the curves $y^2 = 4ax$ and $x^2 = 4ay$ revolves about the $x$-axis. Find the volume of the solid thus generated. (b) Evaluate $\iint_R (x^2 + y^2) \, dx \, dy$ over the area $R$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ in the first quadrant by using the transformation $x = au$ and $y = bv$. [15 marks]
• Q7. (a) Evaluate : $\int_{0}^{1} (x \log x)^3 \, dx$. (b) Show that : $\text{erf}_c(x) + \text{erf}_c(-x) = 2 \quad \forall \, x$. (c) Find the perimeter of the curve : $r = a(1 + \sin \theta)$. [15 marks]

Unit IV

• Q8. (a) Verify Green's theorem for $\oint_C (3x^2 - 8y^2) \, dx + (4y - 6xy) \, dy$, where $C$ is the boundary of the region defined by $x = 0$, $y = 0$ and $x + y = 1$. (b) Verify divergence theorem for $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$ taken over the cube bounded by $x = 0$, $x = a$, $y = 0$, $y = a$, $z = 0$ and $z = a$. [15 marks]
• Q9. (a) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then show that : $\nabla^2(r^m) = m(m - 1)r^{m-2}$. (b) Find the divergence and curl of the vector function $\vec{F} = x^2 y\hat{i} + xy\hat{j} + 3yz\hat{k}$ at the point $(-1, 1, 1)$. (c) Find the circulation of $\vec{F}$ around the curve $C$, where $\vec{F} = y\hat{i} + z\hat{j} + x\hat{k}$ and $C$ is the circle $x^2 + y^2 = 1, z = 0$. [15 marks]

Unit V

• Q10. (a) If $\sin(\alpha + i\beta) = re^{i\theta}$, then show that $2r^2 = \cosh 2\beta - \cos 2\alpha$ and $\tan \theta = \tanh \beta \cot \alpha$. (b) If $\cosh x = \sec \theta$, then show that : $\theta = \frac{\pi}{2} - 2\tan^{-1}(e^{-x})$. [15 marks]
• Q11. (a) Express $\cosh z$ into real and imaginary parts. (b) Express $\log(\sqrt{3} + i)$ into real and imaginary parts. (c) Sum the series to infinity : $\sin \alpha \cdot \sin \beta + \sin 2\alpha \cdot \sin^2 \beta + \sin 3\alpha \cdot \sin^3 \beta + \dots$ [15 marks]

Unit VI

• Q12. Solve : (a) $(D^2 + 5D + 4)y = x^2 + 7x + 9$ (b) $(D^2 + 1)y = 4x \cos x$ (c) $(D^2 - 2D + 1)y = e^x \log x$ [15 marks]
• Q13. Solve : (a) $(x^3 + xy^4) \, dx + 2y^3 \, dy = 0$ (b) $(x^3 y^3 - xy) \, dx = \, dy$ (c) $dx = (x + y + 1) \, dy$. [15 marks]

Related pages

• Engineering Mathematics-I syllabus
• All Engineering Mathematics-I papers from 2018

Other years

• Engineering Mathematics-I 2022 question paper — end-term
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