Applied Mathematics-II-Nagpur-Summer-2017

B.E. Second Semester All Branches (C.B.S.)

Time: 3 hours
Maximum marks: 80

1. a) Prove that $latex \int_2^1x^{n-1}\left(\log\frac12\right)^{m-1}dx=\frac{\left|\overline m\right.}{n^m}$ [6M]

b) Evaluate $latex \int_0^1\frac{x^\alpha-1}{\log x}dx,\;\alpha\geq0$ by differentiating under integral sign. [6M]

OR

2. a) Evaluate $latex \int_0^1x^4\left(1-\sqrt x\right)^5dx$ [6M]

b) Find Root Mean square value of $latex \log_e^x$ over the range x = 1 to
x = e. [6M]

3. a) Trace the curve $latex y^2\left(2a-x\right)=x^3$. [6M]

b) Find the area enclosed by two parabolas y²=4x and $latex y^2=-4\left(x-2\right).$ [6M]

OR

4. a) Trace the curve $latex r=a\left(1+cos\theta\right)$ and find the perimeter of the curve. [6M]

b)Find the volume of the solid obtained by revolving the ellipse
$latex \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ about x axis. [6M]

5. a) Evaluate $latex \iint\limits_Rydx\;dy$ where R is the region bounded by parabolas y²=4x and x²=4y. [6M]

b) Evaluate by changing order of integration. [6M]
$latex \int_0^4\int_y^4\frac x{x^2+y^2}dydx$

c) Evaluate $latex \int_{0}^{2}\int_{0}^{\sqrt{2x-x^{2}}}\frac{x}{\sqrt{x^{2}}+y^{2}}dy dx dz$ by changing into polar coordinates. [6M]

OR

6. a) Evaluate $latex \int_{-1}^{1}\int_{0}^{z}\int_{x-z}^{x+z}\left ( x+y+z \right )dy dx dz$ [6M]

b) Evaluate $latex \iint r^3drd\theta$ over the area bounded by circles $latex r=2cos\theta\;and\;r=4cos\theta.$ [6M]

c) Find the area lying between the parabola $latex y=4x-x^2$ and the line y = x. [6M]

7. a) Prove that [6M]
i) $latex \left[\overline b-\overline c\;\;\overline c-\overline a\;\;\overline a-\overline b\right]=0$
ii) $latex \bar{b}\times \bar{c}\cdot \left \{ \left ( \bar{c}\times \bar{a} \right )\times \left ( \bar{a}\times \bar{b} \right ) \right \}=\left \{ \left ( \bar{a}\times \bar{b} \right )\cdot \bar{c} \right \}^{2}$

b) A particle moves along a curve $latex x=t^3+1,\;y=t^2,\;z=2t+5,$ where t is the time. Find the component of its velocity and acceleration at t = 1 in the direction i + j + 3k. [6M]

c) Find the angle between the tangents to the curve $latex \overline r=t^2i-2tj+t^3k$ at the points t = 1 and t = 2. [6M]

OR

8.    a) Find the directional derivative of $latex \phi=x^2-y^2+2z^2$ at the point P(1, 2, 3) in the direction of line PQ where Q is the point (5, 0, 4). In what direction will it be maximum. [6M]

b) A vector field is given by -$latex \overline A=\left(x^2+xy^2\right)i+\left(y^2+x^2y\right)j$ Show that field is irrotational and find its scalar potential. [6M]

c) If $latex \overline r=xi+yj+zk$ show that [6M]
i) grad $latex r=\frac{\overrightarrow r}r$
ii) $latex \nabla r^n=n\;r^{n-2}\overrightarrow r$

9. Find the total work done in moving a particle in a field of force given by $latex \overline F=3xyi-5zj+10xk$ along the curve $latex x=t^2+1,\;y=2t^2,\;z=t^3$ from t=1 and t=2. [7M]

OR

10. Verify Greens’ theorem in the plane for $latex \int_C\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)dy$ where C is the boundary of the region defined by $latex y=\sqrt x,\;y=x^2$. [7M]

11. a) Fit a curve $latex y=a+bx^2$ for the following data : [7M]

b) Using Lagrange’s interpolation formula, find the value of y when x = 10 from the
following table. [6M]

OR

12. a) The two lines of regressions are $latex 8x-10y+66=0\;:\;40x-18y=214$ If $latex \sigma_x^2=9$ Find : [7M]
i) Mean values of x and y
ii) Coefficient of correlation, and
iii) $latex \sigma_y$, the standard deviation of y.

b) Solve $latex u_{n+2}-2u_{n+1}+u_n=n^2\cdot2^n$ [6M]