Engineering Mathematics-I 2022 Question Paper — Government College of…
The 2022 Engineering Mathematics-I previous year question paper (Government College of Engineering and Technology, Jammu), with every question and its marks.
- University: Government College of Engineering and Technology, Jammu
- Subject: Engineering Mathematics-I
- Course: Engineering Mathematics - I
- Branch: Regular & Private
- Year: 2022
- Paper: end-term
- Duration: 180 minutes
- Total marks: 100
Instructions
- Attempt five questions in all selecting at least two questions from each Section.
- Each question carries 20 marks.
Questions
Section A
Attempt at least two questions from this section.
- Q1. (a) If $u(x,y) = \log\left(\frac{x^2+y^2}{xy}\right)$, then prove that $\frac{\partial^2 u}{\partial x \partial y} = \frac{\partial^2 u}{\partial y \partial x}$. (b) State Lagrange's Mean Value theorem and verify it for the function $f(x) = \sin x$ in $\left[0, \frac{\pi}{2}\right]$. (c) Using Lagrange's method of undetermined multipliers, find the maximum value of $\cos x \cos y \cos z$ given that $x + y + z = \pi$. [20 marks]
- Q2. (a) State and prove Euler's theorem for homogeneous functions of two variables of degree $n$. (b) Find the Taylor's series expansion of $f(x,y) = e^x \sin y$ about $\left(0, \frac{\pi}{2}\right)$ upto third degree terms. (c) Find the extreme values of the function $f(x,y) = x^3 + 3x^2 + 4xy + y^2$. [20 marks]
- Q3. (a) Evaluate the following integral: $$\int_0^1 \frac{\log x \, dx}{\sqrt{1-x^2}}$$ (b) Define beta function and prove that $$B(p,q) = \int_0^\infty \frac{y^{q-1} \, dy}{(1+y)^{p+q}}$$ (c) Evaluate $\iint_R xy \, dxdy$ over the region $R$ in the positive quadrant bounded by the circle $x^2 + y^2 = a^2$. [20 marks]
- Q4. (a) Define Gamma function and prove that $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$. (b) Prove that $$\int_0^t \text{erf}_c(ax) \, dx = t \text{erf}_c(at) - \frac{1}{a\sqrt{\pi}}(e^{-a^2t^2} - 1)$$ (c) Evaluate $\iiint \cos(x+y+z) \, dxdydz$ over the region of the positive octant cut off by the plane $x + y + z = \pi$. [20 marks]
Section B
Attempt at least two questions from this section.
- Q5. (a) Prove that $\tan\left[i \log \left(\frac{a-ib}{a+ib}\right)\right] = \frac{2ab}{a^2-b^2}$. (b) Separate into real and imaginary parts $\cos^{-1}\left(\frac{3i}{4}\right)$. (c) Sum the following series: $$1 + \frac{\cos \alpha}{\cos \alpha} + \frac{\cos 2\alpha}{2! \cos^2 \alpha} + \frac{\cos 3\alpha}{3! \cos^3 \alpha} + \dots$ [20 marks]
- Q6. (a) If $\tan(\alpha + i\beta) = x + iy$, then prove that (i) $x^2 + y^2 + 2x \cot 2\alpha = 1$, (ii) $x^2 + y^2 - 2y \coth 2\beta = -1$ (b) If $a + ib = c \tan(x + iy)$ then prove that $\tan 2x = \frac{2ac}{c^2 - a^2 - b^2}$. (c) Sum the following series: $$\sin \alpha \sin \beta + \sin 2\alpha \sin^2 \beta + \sin 3\alpha \sin^3 \beta + \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) Find the eigen values and their corresponding eigen vectors for the matrix $$\begin{bmatrix} 8 & -8 & -2 \\ 4 & -3 & -2 \\ 3 & -4 & 1 \end{bmatrix}$$ (c) Using Cayley Hamilton Theorem, find the inverse of the matrix $$\begin{bmatrix} 1 & -1 & 1 \\ 4 & 1 & 0 \\ 8 & 1 & 1 \end{bmatrix}$$ [20 marks]
- Q8. (a) Reduce the matrix $A = \begin{bmatrix} 1 & 2 & 3 & 2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 & 5 \end{bmatrix}$ to normal form and hence find its rank. (b) Using elementary column transformations only, find the inverse of the matrix $$\begin{bmatrix} 1 & 2 & 5 \\ 3 & 1 & 4 \\ 1 & 1 & 2 \end{bmatrix}$$ (c) Diagonalize the matrix $$\begin{bmatrix} 1 & 2 & 2 \\ 0 & 2 & 1 \\ -1 & 2 & 2 \end{bmatrix}$$ [20 marks]
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