1. Solve by matrix inversion method:-

2x - y + 3z = 9

x + y + z = 6

x - y + z = 2.

2. Prove by vector method that the altitudes of a traingle are concurrent.

3. Find the vector and Cartesian equations of the plane through the points (1,2,3) and (2,3,1) and perpendicular to the plane

3x - 2y + 4z - 5 = 0.

4. Find the axis, vertex, focus, equation of the directrix, equation of the latus rectum, and length of the latus rectum of the parabola

y^{2} + 8x - 6y + 1 = 0 and hence sketch their graph.

5. An arch is in the form of a semi-ellipse whose span is 48meters wide. the height of the arch is 20 meters. How wide is the arch at a heaight of 10 meters above the ground?

6. Find the equation of the rectangular hyperbola which has one of its asymptotes x + 2y - 5 = 0 and passes through the points (6,0), and

(-3,0).

7. Show that the equation of the normal to the curve

x = a cos^{3}θ, y = a sin^{3}θ at 'θ'

is x cos θ - y sin θ = a cos 2θ.

8. Find the area between the curve y = x^{2} - x - 2, x-axis and the lines x = -2 and x = 4.

9. Prove that the curved surface area of a sphere of radius r intecepted between two parallel planes at a distance a and b from the center of the sphere is

and hence deuce the surface area of the sphere. (b>a).

10. Solve:

.