Engineering Mathematics-I 2022 Question Paper — Government College of…

Instructions

• There will be eight questions of 20 marks each, four from each section. Students are required to attempt five questions selecting at least two questions from each section. Use of calculator is allowed.
• Assume missing data if any.
• Use of scientific calculator is allowed.

Questions

Section A

• Q1. (a) State Euler's Theorem for homogeneous functions and verify it for the function $u(x,y) = \sqrt{x^2+y^2}$. (b) If $u = (x-y)z^2 + (y-z)x^2 + (z-x)y^2$, then prove that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$. [20 marks]
• Q2. (a) Using Taylor series expansion, express $f(x,y) = xy^2 + 3x - 2$ in powers of $x + 2$ and $y - 1$. (b) Find the extreme values of the function $f(x,y) = x^3 + y^3 - 3x - 12y + 20$. [20 marks]
• Q3. (a) Evaluate the following integral: $$\int\limits_{0}^{1} \frac{\log x \, dx}{\sqrt{1-x^2}}$$ (b) Define beta function and prove that $$\beta(p,q) = \int\limits_{0}^{\infty} \frac{y^{q-1} \, dy}{(1+y)^{p+q}}$$ [20 marks]
• Q4. (a) Prove that $$\int\limits_{0}^{t} \operatorname{erf}_c(ax) \, dx = t \operatorname{erf}_c(at) - \frac{1}{a\sqrt{\pi}}(e^{-a^2t^2}-1).$$ (b) Evaluate $\iint_{R} xydxdy$ over the region R in the positive quadrant bounded by the circle $x^2 + y^2 = a^2$. [20 marks]

Section B

• Q5. (a) If $\tan(\alpha + i\beta) = x + iy$, then prove that (i) $x^2 + y^2 + 2x \cot 2\alpha = 1$, (ii) $x^2 + y^2 - 2y \operatorname{coth} 2\beta = -1$. (b) Prove that $\log \left[ \frac{\sin(x+iy)}{\sin(x-iy)} \right] = 2i \tan^{-1}(\cot x \tanh y)$. [20 marks]
• Q6. (a) If $a + ib = c \tan(x + iy)$ then prove that $\tan 2x = \frac{2ac}{c^2 - a^2 - b^2}$. (b) Sum the following series: $$1 + \frac{\cos \alpha}{\cos \alpha} + \frac{\cos 2\alpha}{2! \cos^2 \alpha} + \frac{\cos 3\alpha}{3! \cos^3 \alpha} + \dots$$ [20 marks]
• Q7. (a) Find the rank of the following matrix A for various values of k $$A = \begin{bmatrix} 1 & k & -1 & 2 \\ 2 & -1 & -k & 5 \\ 1 & 10 & -6 & 1 \end{bmatrix}$$ (b) Using elementary column transformations only, find the inverse of the matrix $$\begin{bmatrix} 1 & 2 & 5 \\ 3 & 1 & 4 \\ 1 & 1 & 2 \end{bmatrix}$$ [20 marks]
• Q8. (a) Verify Cayley-Hamilton theorem for the matrix $\begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$. (b) Diagonalize the matrix $\begin{bmatrix} 2 & -2 & 3 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix}$. [20 marks]

Related pages

• Engineering Mathematics-I syllabus
• All Engineering Mathematics-I papers from 2022

Other years

• Engineering Mathematics-I 2022 question paper — end-term
• Engineering Mathematics-I 2022 question paper — end-term
• Engineering Mathematics-I 2018 question paper — end-term
• Engineering Mathematics-I 2018 question paper — end-term