FINAL EXAMINATION | |

DECEMBER 1995 | TIME: 3 HOURS |

MARKS | |||

(6) | 1. | If and ,
evaluate .
y = 2st | |

(6) | 2. | Let . Find | |

(a) | the rate of change of at in the direction of
;
f | ||

(b) | the direction in which changed most rapidly at f;
P | ||

(c) | the maximum rate of change of at f.P
| ||

(5) | 3. | Find the equation of the tangent line to the curve of intersection of and at . | |

(5) | 4. | Find and classify the extreme points of the function . | |

(5) | 5. | Use Lagrange multipliers to find the
maximum value of subject to the constraint .
f = xyz | |

(6) | 6. | Change the order of integration and evaluate the integral . | |

(5) | 7. | Find the volume of the solid under the cone
and above the ring . | |

(5) | 8. | Use triple integrals to find the volume of the tetrahedron bounded by the planes | |

(6) | 9. | Use cylindrical coordinates to find the volume of the solid bounded by | |

(6) | 10. | (a) | Let and . Evaluate, if possible, and .
BA |

(b) | Let be a non-singular matrix such that
M | ||

| |||

Find the value(s) of . | |||

(c) | Let be a 3 x 3 matrix such that . Find
the value of .
X | ||

(5) | 11. | Solve the linear system: | |

(5) | 12. | Use the inverse method to solve the system: | |

(5) | 13. | Show that 2 is an eigenvalue of | |

Find the associated eigenvector. |