|TIME: 3 HOURS|
CALCULATORS ARE NOT ALLOWED
|(10)||1.||(a)||Calculate the angle between the vectors|
|(b)||Given , find the vector projection of onto .|
|(c)||Find the area of a triangle with vertices at .|
|(d)||Find the work done by the force , as its point of application moves from the point A = (2,1,3) to B = (4,-1,5).|
|(20)||2.||(a)||Find the equation of the plane passing through the point A = (6,5,-2) and parallel to the plane x + y - z + 1 = 0.|
|(b)||Find the (i) parametric and (ii) symmetric equations of the line through the point A = (1,-2,3) and perpendicular to the plane 2x - y + 3z = 5.|
|(c)||Find the point at which the line intersects the plane 4x + 5y - 2z = 18.|
|(d)||Find the equation of the plane that passes through the point (1,6,-4) and contains the line .|
|(e)||Find the point of intersection of the lines L1 and L2, where|
|(7)||3.||(a)||If z = f(x,y), where ,|
||Find the equation of the tangent plane and the
equations of the normal line to the surface
z = x2 + xy - y2 at
the point where x = 2 and y = -1.
||Use differentials to find the approximate value of
||Given the function
f(x,y,z) = xy + yz2 + xz3
and the point
P = (2,0,3), find
||the directional derivative of the function at P in the
direction of the vector
||the maximum rate of change at P and the direction in which it
use the chain rule to find
||Find the local maximum and minimum values and
saddle points of
f(x,y) = 2x3 - 24xy + 16y2.
||Use Lagrange multipliers to find the point on the plane
2x + 3y
- z = 1, which is closest to the origin.
||Sketch the region of integration and evaluate
|by reversing the order of integration.
||Find the volume of the solid bounded by the planes
||Find the volume of the solid under the paraboloid
z = x2 +
y2 and above the region y = x2 and x = y2.
||Sketch the solid whose volume is given by the
and evaluate the integral.
||Use cylindrical co-ordinates to find the volume of the region
E bounded by the paraboloids
z = x2 + y2 and
z = 36 - 3x2 -
||Use spherical co-ordinates to evaluate
where E lies between the spheres