1. The number of bacteria in a yeast culture grows at a rate which is proportional to the number present. If the bacteria triple in 1 hour, show that the number of bacteria at the end of five hours is 3⁵ times of the population at initial time.

2. Solve

when x = log 2, y = 0 and when x = 0, y = 0.

3. Show that the set G of all positive rationals form a group under the composition *, defined by

for all

.

4. The number of accidents in a year involving taxi drivers in a city follows a Poisson distribution with mean equal to 3. Out of 1000 taxi drivers, find approximately the number of drivers with
(i) no accident in a year  (ii) more than 3 accidents in a year (e^-3 = 0.0498).

5. Find the equation of the hyperbola if its asymptotes are parallel to x + 2y - 12 = 0 and
x - 2y + 8 = 0 respectively, (2,4) is the center of the hyperbola and the hyperbola passes through (2,0).

6. Show that the lines

and

intersect and find their point of intersection.

7. Solve the following non-homogeneous system of linear equations by determinant method:
4x + 5y = 9; 8x + 10y = 18.

8. Show that the vectors

form a right angled triangle.

9. A force of magnitude 5 units acting parallel to

displaces the point of application from (1,2,3) to (5,3,7). Find the work done.

10. Show that

.