1. If

and

,
then verify that

.

2. Solve the following system of linear equations by determinant method:
2x = 3y = 8; 4x + 6y = 16.

3. Show that the points (3, -1, -1), (1, 0, -1), and (5, -2, -1) are collinear.

4. Find the vector and Cartesian equation of a sphere with center having position vector

and radius 4 units.

5. Solve

if 1 + i is one of the roots.

6. Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.

7. Find the absolute maximum and minimum values of the function

.

8. Verify Lagrange's law of mean for the function

on [1,3].

9. Find the approximate value of

using differentials.

10. Find the area of the region bounded by y = 2x + 4, y = 1, y = 3, and the y axis.

11. Solve:

given that y = 1 when x = 1.

12. If cosA + cosB + cosC = 0 = sinA + sinB +sinC, then prove that
cos2A + cos2B + cos2C = 0 and sin2A + sin2B + sin2C = 0.