Mathematics-I (3110014)

Winter-2019

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 $(1, 1, 1)$ .

(b) Evaluate $\lim_{x \to 0} \frac{{xe}^{x} - \log (1 + x)}{x^{2}}$

(c) Using Gauss Elimination method solve the following system

$- x + 3 y + 4 z = 30$
$3 x + 2 y - z = 9$
$2 x - y + 2 z = 10$

Q2 (a) Test the convergence of the series $(\frac{1}{3}) + {(\frac{2}{5})}^{2} + {(\frac{3}{7})}^{3} + . . . + {(\frac{n}{2 n + 1})}^{n} + . . .$

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

Unit : Functions of several variables

Q2 (c) Find the fourier series of $f (x) = \frac{π - x}{2}$ .

Unit : Fourier Series

OR

Q2 (c) Change the order of integration and evaluate $\int_{0}^{1} \int_{x}^{1} \sin (y^{2}) dy dx$ .

Unit : Integral

Q3 (a) Find the value of $β (\frac{7}{2}, \frac{5}{2})$ .

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

Unit : Fourier Series

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

Unit : Functions of several variables

OR

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

Q3 (b) Evaluate $\iint (x^{2} - y^{2}) dx dy$ over the triangle with the vertices $(0, 1), (1, 1), (1, 2)$ .

Unit : Integral

Q3 (c) Find the volume of the solid generated by rotating the plane region bounded by $y = \frac{1}{x}, x = 1, x = 3$ about the X axis.

Q4 (a) Evaluate $\int_{0}^{π} \int_{0}^{\sin (θ)} r d r d θ$ .

Unit : Integral

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

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

OR

Q4 (a) Using Cayley-Hamilton Theorem find $A^{- 1}$ for $[\begin{matrix} 1 & - 1 \\ 2 & 3 \end{matrix}]$ .

Q4 (b) Evaluate $\int_{0}^{\infty} \frac{dx}{1 + x^{2}}$ .

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

Q5 (a) Evaluate $\int_{0}^{1} \int_{1}^{2} xy d y d x$ .

Q5 (b) Find the eigen values and eigenvectors of the matrix $[\begin{matrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & - 3 & 3 \end{matrix}]$ .

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

OR

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

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

Q5 (c) Evaluate $\int \int \int xyz dx dy dz$ over the positive octant of the sphere $x^{2} + y^{2} + z^{2} = 4$ .

Mathematics-I (3110014)

Q1 (a) Find the equations of the tenagent plane and normal line to the surface x2+y2+z2=3 at the point (1,1,1).

(b) Evaluate limx→0xex-log1+xx2

(c) Using Gauss Elimination method solve the following system -x+3y+4z =30 3x+2y-z =9 2x-y+2z =10

Q2 (a) Test the convergence of the series13 +252+373+...+n2n+1n+...

Q2 (b) Discuss the Maxima and Minima of the function 3x2-y2+x3.

Q2 (c) Find the fourier series of fx=π-x2.

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Q2 (c) Change the order of integration and evaluate ∫01∫x1siny2 dy dx.

Q3 (a) Find the value of β72,52.

Q3 (b) Obtain the fourier cosine series of the function fx=ex.

Q3 (c) Find the maximum and minimum distance from the point 1,2,2 to the sphere x2+y2+z2=36.

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Q3 (a) Test the convergence of the series 112-122+132-142+....

Q3 (b) Evaluate ∬x2-y2 dx dy over the triangle with the vertices 0,1, 1,1, 1,2.

Q3 (c) Find the volume of the solid generated by rotating the plane region bounded by y=1x, x=1, x=3 about the X axis.

Q4 (a) Evaluate ∫0π∫0sinθrdr dθ.

Q4 (b) Express fx=2x3+3x2-8x+7 in terms of x-2.

Q4 (c) Using Gauss-Jordan method find 123253108.

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Q4 (a) Using Cayley-Hamilton Theorem find A-1 for 1-12 3.

Q4 (b) Evaluate ∫0∞dx1+x2.

Q4 (c) Test the convergence of the series x1·2+x23·4+x35·6+x47·8+....

Q5 (a) Evaluate ∫01∫12xy dy dx.

Q5 (b) Find the eigen values and eigenvectors of the matrix 0 1 10 0 11-3 3.

Q5 (c) If u=fx-y, y-z, z-x then show that ∂u∂x+∂u∂y+∂u∂z=0.

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Q5 (a) Find the directional derivatives of f=xy2+yz2 at the point 2,-1,1, in the direction of i+2j+2k.

Q5 (b) Test the convergence of the series ∑n=1∞nn2+1.

Q5 (c) Evaluate ∫∫∫xyz dx dy dz over the positive octant of the sphere x2+y2+z2=4.

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 $(1, 1, 1)$ .

(b) Evaluate $\lim_{x \to 0} \frac{{xe}^{x} - \log (1 + x)}{x^{2}}$

(c) Using Gauss Elimination method solve the following system

$- x + 3 y + 4 z = 30$
$3 x + 2 y - z = 9$
$2 x - y + 2 z = 10$

Q2 (a) Test the convergence of the series $(\frac{1}{3}) + {(\frac{2}{5})}^{2} + {(\frac{3}{7})}^{3} + . . . + {(\frac{n}{2 n + 1})}^{n} + . . .$

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

Q2 (c) Find the fourier series of $f (x) = \frac{π - x}{2}$ .

OR

Q2 (c) Change the order of integration and evaluate $\int_{0}^{1} \int_{x}^{1} \sin (y^{2}) dy dx$ .

Q3 (a) Find the value of $β (\frac{7}{2}, \frac{5}{2})$ .

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

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

OR

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

Q3 (b) Evaluate $\iint (x^{2} - y^{2}) dx dy$ over the triangle with the vertices $(0, 1), (1, 1), (1, 2)$ .

Q3 (c) Find the volume of the solid generated by rotating the plane region bounded by $y = \frac{1}{x}, x = 1, x = 3$ about the X axis.

Q4 (a) Evaluate $\int_{0}^{π} \int_{0}^{\sin (θ)} r d r d θ$ .

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

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

OR

Q4 (a) Using Cayley-Hamilton Theorem find $A^{- 1}$ for $[\begin{matrix} 1 & - 1 \\ 2 & 3 \end{matrix}]$ .

Q4 (b) Evaluate $\int_{0}^{\infty} \frac{dx}{1 + x^{2}}$ .

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

Q5 (a) Evaluate $\int_{0}^{1} \int_{1}^{2} xy d y d x$ .

Q5 (b) Find the eigen values and eigenvectors of the matrix $[\begin{matrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & - 3 & 3 \end{matrix}]$ .

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

OR

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

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

Q5 (c) Evaluate $\int \int \int xyz dx dy dz$ over the positive octant of the sphere $x^{2} + y^{2} + z^{2} = 4$ .