Practice Sheet PDF

UNIT-III 1. A particle moves along the curve [\$\\overline{r} = \\left( t\^{3} - 4t \\right)\\widehat{i} + \\left( t\^{2} + 4t \\right)\\widehat{j} + (8t\^{2} - 3t\^{3})\\widehat{k}\$]{.math.inline} where t is the time. Find the magnitude of the tangential and normal components of...

UNIT-III 1. A particle moves along the curve [\$\\overline{r} = \\left( t\^{3} - 4t \\right)\\widehat{i} + \\left( t\^{2} + 4t \\right)\\widehat{j} + (8t\^{2} - 3t\^{3})\\widehat{k}\$]{.math.inline} where t is the time. Find the magnitude of the tangential and normal components of its acceleration at t = 2 2. A particle moves along the curve [*x* = *t*^2^ + 1, *y* = *t*^2^, *z* = 2*t* + 5]{.math.inline} where t is the time. Find the components of its velocity and acceleration at t = 1 in the direction [\$\\widehat{i} + \\ \\widehat{j} + 3\\widehat{\\text{\~k}}\$]{.math.inline}. 3. A particle moves along moves along the curve [*x* = 2*t*^2^]{.math.inline}, [*y* = *t*^2^ − 4*t*]{.math.inline}, [*z* = 3*t* − 5]{.math.inline}. Find the component of its velocity and acceleration at [*t* = 1]{.math.inline} in the direction [*i* − 3*j* + 2*k*]{.math.inline} 4. A particle moves so that its position vector is given by , where is constant. Show that (i) velocity of a particle is perpendicular to and (ii). 5. Find the Directional derivatives of [⌀ = 4*e*^2*x* − *y* + *z*^]{.math.inline} at the point (1, 1, -1) in the direction towards the point (-3,5,6). 6. Find the directional derivative of [*ϕ*(*x*,*y*,*z*) = *x*^2^ − 2*y*^2^ + 4*z*^2^]{.math.inline} at the point (1,1,-1) in the direction of [2*i* + *j* − *k*]{.math.inline}. In what direction will the directional derivative be maximum and what is its magnitude? 7. Find the angle between the tangents to the curve [\$\\overline{r} = t\^{2}i + 2tj - t\^{3}k\$]{.math.inline} at the points [*t* =  ± 1]{.math.inline}. 8. Find the values of a, b, c so that the directional derivative of at (1, 2, -1) has a maximum magnitude 64 in the direction parallel to z-axis. 9. If , then prove that (i) (ii) 10. Show that [\$\\overline{F} = \\left( ysinz\\ - sinx \\right)\\ \\widehat{i} + (xsinz + 2yz)\\ \\widehat{j} + (xycosz + y\^{2})\\widehat{\\text{\~k}}\$]{.math.inline} is a irrotational. Find its scalar potential. 11. Prove that [\$\\overline{A} = \\left( 6xy + z\^{3} \\right)i + \\left( 3x\^{2} - z \\right)j + \\left( 3xz\^{2} - y \\right)k\$]{.math.inline} is irrotational. Find scalar potential [*ϕ*]{.math.inline} such that [\$\\overline{A} = \\nabla\\phi\$]{.math.inline} 12. Show that the vector field is irrotational and hence find its scalar potential. UNIT-IV 1. Test the Convergence of Series [\$\\sum\_{}\^{}\\frac{\\sqrt{n}}{n\^{2} + 1}\$]{.math.inline} 2. Use D'Alembert's ratio test to check the convergence of the series [\$\\frac{2!}{3} + \\frac{3!}{3\^{2}} + \\frac{4!}{3\^{3}} + - - - - - -\$]{.math.inline} 3. Test the convergence of the series whose nth term is 4. Test the Convergence of the series [\$\\frac{1}{3} + \\left( \\frac{2}{5} \\right)\^{2} + \\left( \\frac{3}{7} \\right)\^{3} + - - - + \\left( \\frac{n}{2n + 1} \\right)\^{n} + - - -\$]{.math.inline} by Cauchy's root test. 5. Test the convergence of series by Cauchy's Root Test[\$\\sum\_{}\^{}\\frac{{(n + 1)}\^{n}x\^{n}}{{(n)}\^{n + 1}}\$]{.math.inline} 6. Test the convergence of series[\$1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\frac{1}{5} - \\frac{1}{6} - \\ldots\$]{.math.inline} 7. Test the Convergence of the series [\$1 - \\frac{2}{3} + \\frac{3}{3\^{2}} - \\frac{4}{3\^{3}} + - - - -\$]{.math.inline} 8. Test the convergence of the series 9. Test the convergence of the series: 10. Test the convergence of the series