1. If α and β are the zeros of the quadratic polynomial f(x) = x2−px+q, then find the value of α²+β²
2. If f(x) = 2×3−13×2+17x+12, then find its value at x .
3. An unbiased die is thrown. What is the probability of getting a number greater than 3?
4. When do you say two triangles are similar?
5. How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius 8 cm?
6. Find the value of x, if the mode of the following data is 25:
15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20, 25, 20, x, 18
7. Sum of two numbers is 35 and their difference is 13. Find the numbers
8. Find θ, if sin(θ+36°)=cosθ, where (θ+36°)is an acute angle.
9. Define secant of a circle.
10. If the perimeter of a semi-circular protractor is 66 cm, find the diameter of the protractor.
12. If α and β are the zeros of the quadratic polynomial f(x)= ax2+bx+c, then evaluate α2/β + β2/α
13. Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is a multiple of 3 or 7?
14. In the given figure, DE ∥ BC. If AD = x, DB = x − 2, AE = x + 2 and EC = x− 1, find the value of x.
15. ABC is an isosceles triangle with AC = BC. If AB2=2AC2, prove that △ABC is right triangle.
16. Find the values of a and b so that x4+x3+8×2+ax+b is divisible by x2+1
17. Prove that √2+√5 is irrational.
18. Students of a class are made to stand in rows. If one student is increased in each row, then the number of rows is decreased by two. If however one student is less in each row then there would be three rows more. Find the number of students in the class.
19. By applying division algorithm, prove that the polynomial g(x)= x2+3x+1is a factor of the polynomial f(x)=3×4+5×3−7×2+2x+2
20. Prove that (4, −1), (6, 0), (7, 2) and (5, 1) are the vertices of a rhombus. Is it a square?
22. Find the coordinates of the centre of the circle passing through the points (0, 0), (−2, 1) and (−3, 2). Also, find its radius.
23. Take a point O on the plane of the paper. With O as centre draw a circle of radius 3 cm. Take a point P on this circle and draw a tangent at P
24. The perimeter of a sector of a circle of radius 5.4 cm is 17.4 cm. Find the area of the sector.
25. From an external point P, two tangents PA and PB are drawn to the circle with centre O. Prove that OP is the perpendicular bisector of AB.
26 Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 30° and at a distance 10 km further off from the mountain, along the same line, the angle of elevation is 15°
27. Water is flowing at the rate of 5 km / hr through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Determine the time in which the level of the water in the tank will rise by 7 cm.
28. ABC is a right-angled triangle right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6 cm and 8 cm. Find the radius of the circle.
29. A plane left 30 minutes later then the scheduled time and in order to reach its destination 1500 km away in time it has to increase its speed by 250 km / hr from its usual speed. Find its usual speed.
30. Compute the median from the following data: