1 (a). The difference between the mean and the variance of a binomial distribution is 1 and the difference between their squares is 11. Find n.
  (b) Show that the total probability is 1 for a Poisson distribution.

2. Discuss the solutions of the system of equations for all values of

. (Use rank method).

3. Find the vector and Cartesian equations of the plane passing through the point (-1,-2,1) and perpendicular to two planes x+2y+4z+7 = 0 and 2x-y+3z+3 = 0.

4. Solve the equation:-

.

5. A cable of a suspension bridge is in the form of a parabola whose span is 40 meters. The roadway is 5 meters below the lowest point of the cable. An extra support is provided across the cable 30 meters above the ground level. Find the length of the support if the heights of the pillars are 55 meters.

6. Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is

where

is the volume of the sphere.

7. Find the intervals of concavity and the points of inflection of the function

.

8. Verify Euler's theorem for

.

9. Evaluate

.

10. Find the perimeter of a circle with radius a, using integration method.