Applied Mathematics-II-Nagpur-Summer-2015

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

Time: 3 hours
Maximum marks: 80

1. (a) Evaluate $latex \int\limits_0^\infty\frac{x^7}{7^x}dx$ [6M]

(b) If $latex I=\int\limits_0^{a^2}tan^{-1}\left(\frac xa\right)dx$ show that : $latex \frac{dI}{da}=2a\;tan^{-1}a-\frac12log\left(1+a^2\right)$. [6M]

OR

2. (a) Evaluate $latex \int\limits_0^{2a}x\sqrt{2ax-x^2}dx$. [6M]

(b) Show that the mean value of $latex kx\left(l-x\right)$ between 0 to l is two-third of its maximum value. [6M]

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

(b) Find the area enclosed between the curvey $latex ^2left(2a-xright)=x^3$ and ana its asymptote. [6M]

OR

4. (a) Find the length of an arc of the curve x = a(t + sin t), y = a(1 – cos t). [6M]

(b) The cardioid r = a(1 + cos θ) revolves about the initial line. Find the volume generated. [6M]

5. (a) Evaluate $latex \iint\limits_Axy\;dxdy,$ where A is the domain bounded A by x-axis, ordinate x = 2a and the curve x² = 4ay. [6M]

(b) Change the order of integration in $latex \int_0^a\int_{\sqrt{ax}}^{a}\frac{y^2}{\sqrt{y^4-a^2x^2}}dydx$ and hence evaluate. [6M]

(c) Evaluate $latex \iint\sqrt{a^2-x^2-y^2}\;dy\;dx,$ over the semicircle x² + y² = ax in the positive quadrant by changing into polar coordinates. [6M]

OR

6. (a) Evaluate $latex \iint\limits_Rr^2\;sin\theta\;dr\;d\theta,$ where R is the semi circle r=2a cosθ above the initial line. [6M]

b) Find by double integration, the area lying between the parabola y = 4x – x² and the line
y = x. [6M]

(c) Evaluate $latex \int_1^3\int_\frac1x^1\int_{0}^{\sqrt{xy}}xyzdzdydx$ [6M]

7. (a) Show that if $latex \overline a,\;\overline b,\;\overline c$ are non coplanar then $latex \overline a+\overline b,\;\overline b+\overline c;$ and $latex \overline c+\overline a$ are also non coplanar. [5M]

(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 tangential and normal components of its acceleration at t = 2. [7M]

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

OR

8. (a) Prove that $latex \left[\overline b\times\overline c\;\;\overline c\times\overline a\;\;\overline a\times\overline b\right]=\left[\overline a\overline b\overline c\right]^2.$ [4M]

(b) Find the Directional Derivative of $latex \phi=x^2y+y^2z+z^2x$ (2,2,2) in the direction of the normal to the surface $latex 4x^{2y}y+2z^2=2$ at the point (2,-1,3). [7M]

(c) Prove that : $latex \overline F=\left(y^2\;cosx+z^3\right)i+\left(2y\;sinx-4\right)j+\left(3xz^2+2\right)k$ is irrotational vector field. Hence find the scalar potential
$latex \phi$ such that $latex \overline F=\nabla\phi.$ [7M]

9. (a) If $latex \overline F=2yi-zj+xk,$ evaluate $latex \int\limits_C\overline F\times d\overline r$ along the curve x=cost, y=sint, z=2cost from t=0 and t=$latex \frac\pi2$. [7M]

OR

10. Applying Green’s theorem, evaluate $latex \int\limits_C\left[\left(y-sinx\right)dx+cosx\;dy\right],$ where C is the plane triangle enclosed by the lines y = 0, $latex x=\frac\pi2$ and $latex y=\frac\pi2x.$ [7M]

11. (a) Using method of least squares, fit a relation of the type $latex y=ab^x$ to the following data : [7M]

(b) Apply Lagrange’s formula to find fix) from the following data : [6M]

OR

12. (a) Calculate the coefficient of correlation and obtain the lines of regression for the following data : [7M]

(b) Solve : $latex y_{n+3}-5y_{n+2}+3y_{n+1}+9y_n=2^n+3n$ [6M]