In each of the following questions, please circle only one answer. If you do not believe any of the answers to a given quesion, please provide your reason in the space given on this page.

• A1. If the vectors and are orthogonal, then t=

  (a) 1, (b) -1, (c) 0, (d) 2, (e) .

• A2. An equation for the plane passing through the points (1,0,-3), (0,-2,-4) and (4,10,6) is

  (a) 34x+87y-56y+12=0, (b) 23x-17y-11z-43=0,

  (c) 17x-6y-5z-32=0, (d) 4x-3y+2z+2=0, (e) x-y-2z-1=0.

• A3. The line passing through the point (5,1,3) and parallel to the vector can be described by the parametric equations

  (a) , (b) ,

  (c) \ (d) ,

  (e) None of the above.

• A4. The unit tangent vector to the curve at the point is

  (a) (1,0,0), (b) , (c) , \

  (d) , (e) .

• A5. The arc length of the circular helix with vector equation from the point (1,0,0) to the point is

  (a) 4, (b) , (c) , (d) , (e) .

• A6. The gradient of the function at the point P(1,-2,1) is

  (a) 4(1,-1,3), (b) 12, (c) (12, 3, 4), (d) (-2,2, 3), (e) None of the above.

• A7. The tangent plane to the surface at the point where x=2 and y=-1 is

  (a) z=3x+4y-1, (b) 2z-4x+6y+3=0, (c) z=x-y+1,

  (d) 23x-38y-57z+11=0, (e) None of the above.

• A8. If xy+yz+zx=0, then

  (a) , (b) , (c) , (d) , (e) .

• A9. Suppose that and x=st, , . By means of the chain rule, we find that, when s=1, t=0, ( ) is equal to

  (a) 0, (b) 1, (c) 2, (d) -1, (e) None of the above.

• A10. The volume of the region S in 3-space, bounded by z=0 (below), (on the side) and (on the outside) as a triple integral in spherical coordinates is given by

  (a) , \ (b)

  (c) , (d) ,

  (e) None of the above.

• A11. By using polar coordinates, we find that

  (a) (b) (c) (d) ,

  (e) None of the above.

• A12. Given the solid in the first octant, bounded by the cylinder and the planes y=z, x=0 and z=0 with density , the integral expression for the total mass is

  (a) , \

  (b) ,

  (c) ,

  (d) , (e) None of the above.

• B1. Evaluate the integral

  

• B2. Use the method of Lagrange multipliers to find, for x,y,z>0, the minimum of x+y+z subject to the constraint

  

• B3. Find the local maximum and minimum values and saddle points of the function

  

• B4. The density at any point on a semicircular lamina is proportional to the distance from the center of the circle. Find the centre of mass of the lamina.

• B5. Expand the period 2L function

  

  into a Fourier series.

• B6. Given , determine if there is a function f such that and, if there is, find it.

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