1. Two circles touch each other externally at a point P and a direct common tangent touches the circles at A and B. Prove that (i) the common tangent at P bisects AB and (ii) AB subtends a right angle at P.

2.  Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisectors.

3. The bisectors of angles B and C of a triangle ABC meet the opposite sides at D and E respectively. If DE || BC, prove that the triangle is isosceles.

4. Solve the matrix equation:

5. If

, find the matrix X such that 3A + 5B + 2X = 0.

6. The number of ice cream cones bought by men, women, boys, girls, and children on a day at the Trade fair was 40, 42, 46, 48, and 44 respectively. Find the Standard Deviation.

7. A number is selected at random from the first 100 natural numbers. what is the probability that the number is either a multiple of 11 or a multiple of 13?

8. Show that

.

9. Find the values of angles and sides not given in the right triangle ABC in which angle C = 90°, angle A = 30° and AB = 8 centimeters.

10. Find the length of a side of a regular polygon inscribed in a circle of radius 1 meter if it has 24 sides. State the answer in centimeters.

11. Find the area of the quadrilateral formed by the points A (3,4), B(-1,6), C(-3,-4), and D(6,1).

12. If the line joining the points (-4,6) and (-1,-3) is perpendicular to the line joining the
points (0,-4) and (3,a), find a.

13. Show that x + y - 2 = 0 is perpendicular bisector of the line joining the origin and (2,2).

14. Find the equation of the line which is concurrent with the lines y=x and y = 2 - x and perpendicular to the line y = 4x + 5.