# Mathematics-I(3110014)

BE | Semester 1
Winter - 2019 | 17-01-2020
Total Marks 70

## Q1 (a) Find the equations of the tenagent plane and normal line to the surface ${x}^{2}+{y}^{2}+{z}^{2}=3$ at the point $\left(1,1,1\right)$.

Unit : Functions of several variables |  Topic : tangent plane and normal line

## (b) Evaluate $\underset{\mathrm{x}\to 0}{\mathrm{lim}}\frac{{\mathrm{xe}}^{\mathrm{x}}-\mathrm{log}\left(1+\mathrm{x}\right)}{{\mathrm{x}}^{2}}$

Unit : Indeterminate Forms and L'Haspital's Rule, Improper Integrals, Applications of definite integral

## (c) Using Gauss Elimination method solve the following system

Unit : Elementary row operations in Matrix

## Q2 (a) Test the convergence of the series

Unit : Convergence and divergence of sequences

## Q2 (b) Discuss the Maxima and Minima of the function $3{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{x}}^{3}$.

Unit : Functions of several variables

## Q2 (c) Find the fourier series of $\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{\pi }-\mathrm{x}}{2}$.

Unit : Fourier Series

Unit : Integral

## Q3 (a) Find the value of $\mathrm{\beta }\left(\frac{7}{2},\frac{5}{2}\right)$.

Unit : Indeterminate Forms and L'Haspital's Rule, Improper Integrals, Applications of definite integral

## Q3 (b) Obtain the fourier cosine series of the function $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}}$.

Unit : Fourier Series

## Q3 (c) Find the maximum and minimum distance from the point $\left(1,2,2\right)$ to the sphere ${\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{z}}^{2}=36$.

Unit : Functions of several variables

## Q3 (a) Test the convergence of the series $\frac{1}{{1}^{2}}-\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}-\frac{1}{{4}^{2}}+...$.

Unit : Convergence and divergence of sequences

Unit : Integral

## Q3 (c) Find the volume of the solid generated by rotating the plane region bounded by about the X axis.

Unit : Indeterminate Forms and L'Haspital's Rule, Improper Integrals, Applications of definite integral

Unit : Integral

## Q4 (b) Express $\mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{x}}^{3}+3{\mathrm{x}}^{2}-8\mathrm{x}+7$ in terms of $\left(\mathrm{x}-2\right)$.

Unit : Convergence and divergence of sequences

## Q4 (c) Using Gauss-Jordan method find $\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 3\\ 1& 0& 8\end{array}\right]$.

Unit : Elementary row operations in Matrix

## Q4 (a) Using Cayley-Hamilton Theorem find ${\mathrm{A}}^{-1}$ for .

Unit : Elementary row operations in Matrix |  Topic : Cayley-Hamilton theorem

## Q4 (b) Evaluate ${\int }_{0}^{\infty }\frac{\mathrm{dx}}{1+{\mathrm{x}}^{2}}$.

Unit : Indeterminate Forms and L'Haspital's Rule, Improper Integrals, Applications of definite integral |  Topic : Improper Integrals

## Q4 (c) Test the convergence of the series $\frac{\mathrm{x}}{1·2}+\frac{{\mathrm{x}}^{2}}{3·4}+\frac{{\mathrm{x}}^{3}}{5·6}+\frac{{\mathrm{x}}^{4}}{7·8}+...$.

Unit : Convergence and divergence of sequences |  Topic : Ratio test

## Q5 (a) Evaluate .

Unit : Integral |  Topic : Multiple integral

## Q5 (b) Find the eigen values and eigenvectors of the matrix .

Unit : Elementary row operations in Matrix |  Topic : Eigen values and eigen vectors

## Q5 (c) If then show that $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0$.

Unit : Functions of several variables |  Topic : Chain rule

## Q5 (a) Find the directional derivatives of $\mathrm{f}={\mathrm{xy}}^{2}+{\mathrm{yz}}^{2}$ at the point $\left(2,-1,1\right)$, in the direction of $\mathrm{i}+2\mathrm{j}+2\mathrm{k}$.

Unit : Functions of several variables |  Topic : Gradient, Directional derivative

## Q5 (b) Test the convergence of the series $\sum _{\mathrm{n}=1}^{\infty }\frac{\sqrt{\mathrm{n}}}{{\mathrm{n}}^{2}+1}$.

Unit : Convergence and divergence of sequences |  Topic : The Comparison test

## Q5 (c) Evaluate over the positive octant of the sphere ${\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{z}}^{2}=4$.

Unit : Integral |  Topic : multiple integral by substitution