Applied Mathematics-II-Nagpur-Winter-2016

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

Time: 3 hours
Maximum marks: 80

1. a) Evaluate $latex \int_0^1\frac x{\sqrt{1-x^4}}dx$ [6M]

b) By differentiation under the integral sign evaluate $latex \int_0^\infty\frac{e^{-ax}sinx}xdx$ [6M]

OR

2. a) Evaluate $latex \int_0^\frac\pi2\sqrt{\tan\theta}\;d\theta$ [6M]

b) A rod of length ‘a’ is divided into two parts at random. Prove that the mean value of the sum of squares on these two segments is $latex \frac23a^2$. [6M]

3. a) Trace the curve $latex a^2x^2=y^3\left(2a-y\right)$ and show that its area is equal to $latex \pi a^2$. [6M]

b) Find the perimeter of the asteroid $latex x^\frac23+y^\frac23=a^\frac23.$. [6M]

OR

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

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

5. a) Evaluate $latex \iint\left(x^2+y^2\right)dx\;dy$ over the region in the positive quadrant for which $latex x+y\leq1.$ [6M]

b) Evaluate $latex \int_0^a\int_y^a\frac{x^2}{\left(x^2+y^2\right)^{\displaystyle\frac32}}dy\;dx$ by changing into polar form. [6M]

c) Evaluate by changing the order of integration $latex \int_0^\infty\int_y^\infty\frac{e^{-y}}y\;dy\;dx$ [6M]

OR

6. a) Evaluate $latex \int_0^1\int_0^{1-x}\int_0^{1-x-y}xyz\;dz\;dy\;dx$ [6M]

b) Find the mass of area bounded by the curves $latex y=x^2\;and\;x=y^2;$, if the density at any point is $latex \rho=\lambda\left(x^2+y^2\right)$. [6M]

c) Evaluate $latex \iint\frac{rdrd\theta}{\sqrt{a^2+r^2}}$ over one loop of the lemniscate $latex r^2=a^2cos2\theta$. [6M]

7. a) Show that.
$latex \left(\overrightarrow a\times\overrightarrow d\right)\times\left(\overrightarrow c\times\overrightarrow d\right)+\left(\overrightarrow a\times\overrightarrow c\right)\times\left(\overrightarrow d\times\overrightarrow b\right)+\left(\overrightarrow a\times\overrightarrow d\right)\times\left(\overrightarrow b\times\overrightarrow c\right)$
is paralled to the vector $latex \overrightarrow a.$. [6M]

b) Find the directional derivative of $latex \phi\;\left(x,\;y,\;z\right)=x^2-2y^2+4z^2$ at the point (1, 1, -1) in the direction
2i + j – k. In what direction will the directional derivative be maximum and what is its magnitude? [6M]

c) Prove that $latex \overline A=\left(6xy+z^3\right)i+\left(3x^2-z\right)j+\left(3xz^2-y\right)k$ is irrotational. Find the scaler potential $latex \phi$ such that $latex A=\triangle\phi.$ [6M]

OR

8. a) A particle moves so that its position rector is given by $latex \overrightarrow r=cos\omega ti+sin\omega tj$ where $latex \omega$ is constant, prove that. [6]
i) Velocity $latex \overrightarrow v$ of the particle is perpendicular to $latex \overrightarrow r$
ii) $latex \overrightarrow r\times\overrightarrow v=$ constant vector and.
iii) The acceleration $latex\overrightarrow a$ is directed towards the origin.

b) A particle moves along the curve $latex \overline r=\left(t^3-4t\right)i+\left(t^2+4t\right)j+\left(8t^2-3t^3\right)k$ where t is the time. Find the magnitude of the tangential and normal component of its acceleration at t = 2. [6M]

c) Find the value of ‘n’ for which the vector field $latex r^n\overrightarrow r$ will be solenoidal. Find also whether the vector field $latex r^n\overline r$ is irrotational or not. [6M]

9. If $latex \overline A=\left(y-2xy\right)i+\left(3x+2y\right)j,$ find the circulation of $latex \overline A$ about the circle C in the XY plane with Centre at origin and radius 2, C is traversed in the positive direction. [7M]

OR

10. Use Green’s theorem in the plane, evaluate $latex \int_c\left[\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)dy\right]$ Where C is the boundary of the region bounded by $latex y=\sqrt x\;and\;y=\sqrt x$ [7M]

11. a) Fit a curve $latex y=ab^x$ to the following data. [7M]

b) Find the function whose first order forward difference is $latex x^3-3x^2+9.$ [6M]

OR

12. a) In a partially distributed laboratory analysis of a correlation data, the following results only are eligible: $latex \sigma_x^2=9$ Regression equations: 8x – 10y + 66 = 0,
40x – 18y = 214 what were. [7M]
i) The mean values of x and y.
ii) Coefficient of correlation between x and y.
iii) Standard Deviation of y.

b) Solve the difference equation. $latex y_{n+2}-2y_n=2^n$ [6M]