1. Find the value of k for kx2 + 2x - 1 = 0, so that it has two equal roots
2. Find the value of k for k x2 - 2√ 5 x + 4 = 0, so that it has two equal roots.
3. If the roots of the equation (b - c) x2 + (c - c) x + (a - b) = 0 are equal, accordingly prove that 2b = a + c.
4. Find the discriminant of the quadratic equation 3x2– 4 √3 x + 4 = 0, and hence find the nature of its roots.
5. Find the value of k for 2 x2 + k x + 3 = 0, so that it has two equal roots.
6. Find the value of k for k x (x – 2) + 6 = 0, so that it has two equal roots.
7. Find the value of k for which the equation x2 + 5kx + 16 = 0 has no real roots.
8 Find the discriminant of the quadratic equation 2x2– 6x + 3 = 0, and hence find the nature of its roots.
9. Find the value of k for k2 x2 – 2 (2 k - 1) x + 4 = 0, so that it has two equal roots.
10. Find the value of k for (k + 1) x2 – 2 ( k - 1) x + 1= 0, so that it has two equal roots.
11. Determine the positive value of k for which the equation x2 + k x + 64 + 0 and x2 – 8x + k = 0 will both have real roots.
12. Find the discriminant of the quadratic equation 2x2 – 3 x + 5 = 0, and hence find the nature of its roots.
13. Find the value of k for x2 – 2(k + 10x + k2 ) = 0, so that it has two equal roots.
14. If -4 is a root of the quadratic equation x2 + px–4=0 and the quadratic equation
x2+ px +k=0 has equal roots, find the value of k.
15. Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, and hence find the nature of its roots.
16. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
17. In a class test, the sum of Amit’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 Marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
18. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
19. A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in
such a way that the differences of its distances from two diametrically opposite fixed gates A
and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the
two gates should the pole be erected?
20. Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
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