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

Instructions

• Attempt five questions in all, selecting at least two questions from each section.
• Each question carries 20 marks.

Questions

Section A

• Q1. (a) Define a homogeneous function. State and prove Euler's theorem for homogeneous functions. (b) State Lagrange's mean value theorem. Discuss the applicability of Lagrange's mean value theorem for the function $f(x) = \begin{cases} 2x + 1, & x < 0 \\ 1 - 2x, & x \ge 0 \end{cases}$ in $[-1, 2]$. [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 $f(x,y) = x^3 + y^3 + 3xy$. [20 marks]
• Q3. (a) Evaluate $\int\limits_{0}^{\pi} \frac{xdx}{1 + \sin x}$. (b) Define gamma and beta functions. Prove that $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$. [20 marks]
• Q4. (a) Prove that $B(m,n) = B(m, n+1) + B(m+1, n)$. (b) Evaluate the following integral by changing order of integration: $$\int\limits_{3}^{5} \int\limits_{0}^{4/x} xy \, dxdy$$ [20 marks]

Section B

• Q5. (a) Separate $\tanh(x + iy)$ into real and imaginary parts. (b) If $\tan(\alpha + i\beta) = x + iy$, then prove that (i) $x^2 + y^2 + 2x \cot 2\alpha = 1$, (ii) $x^2 + y^2 - 2y \coth 2\beta = -1$ [20 marks]
• Q6. (a) Prove that $\tan\left[i \log \left(\frac{a-ib}{a+ib}\right)\right] = \frac{2ab}{a^2 - b^2}$. (b) Sum to $n$ terms and to infinity the following series: $$1 + x \cos \alpha + x^2 \cos 2\alpha + x^3 \cos 3\alpha + \dots$$ where $|x| < 1$. [20 marks]
• Q7. (a) Using elementary row and column transformations reduce the following matrix to normal form and hence find its rank $$\begin{bmatrix} 1 & 4 & -2 & 1 \\ 5 & 2 & 4 & 3 \\ 2 & -1 & 3 & 1 \end{bmatrix}$$ (b) Find the eigen values and their corresponding eigen vectors for the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 2 & 3 & 4 \\ -1 & -1 & -2 \end{bmatrix}$. [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 $\begin{bmatrix} 1 & 2 & 2 \\ 0 & 2 & 1 \\ -1 & 2 & 2 \end{bmatrix}$. [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