1. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0

2. The product of the roots of the equation \(x^2−7x+3=0\) is—?

(a) 7 (b) -7 (c) 3 (d) -3

3. If \(x\) is a real positive number and \(\sin x = \frac{2}{3}\), then what is the value of \(\tan x\)?

(a) \(\cfrac{2}{\sqrt5}\) (b) \(\cfrac{\sqrt5}{2}\) (c) \(\sqrt{\cfrac{5}{3}}\) (d) \(\cfrac{\sqrt5}{\sqrt2}\)

4. What is the value of \(x\) in the equation \(x^2 + x - 6 = 0\)?

(a) -3,2 (b) 2,3 (c) -2,3 (d) -2,-3

5. If one root of the equation \(x^2 - ax - 15 = 0\) is 3, then what is the value of \(a\)?

(a) -2 (b) 2 (c) 3 (d) 0

6. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

7. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?

8. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.

9. The number of real roots of the equation \((x - 1)^2 + (x - 2)^2 + (x - 3)^2 = 0\) is ______.

10. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

11. The common root of the equations \(x^2 + ax + b = 0\) and \(x^2 + bx + a = 0\) is _____.

12. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

13. If the product of the roots of the equation \(x^2 – 5x + k = 12\) is \(–3\), find the value of \(k\).

14. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

15. If one root of the equation \(x^2 - ax - 15 = 0\) is \(-3\), then the value of \(a\) is -

(a) 1 (b) -1 (c) 2 (d) -2

16. If the equation \(x^2 - x = k(2x - 1)\) has a sum of roots equal to \(0\), then what is the value of \(k\)?

(a) -2 (b) 2 (c) \(\cfrac{1}{2}\) (d) \(-\cfrac{1}{2}\)

17. If the equation \(x^2 + k(4x + k - 1) + 2 = 0\) has equal roots, then what is the value of \(k\)?

(a) \(-\cfrac{2}{3}, 1\) (b) \(\cfrac{2}{3}, 1\) (c) \(\cfrac{2}{3}, \cfrac{1}{3}\) (d) \(-\cfrac{3}{2}, \cfrac{1}{3}\)

18. If the sum of the roots of the equation \(x^2 - (k + 6)x + 2(2k - 1) = 0\) is half of their product, then what is the value of \(k\)?

(a) 6 (b) 7 (c) 1 (d) 5

19. If \(2\) is a root of both equations \(x^2 + bx + 12 = 0\) and \(x^2 - bx + q = 0\), what is the value of \(q\)?

20. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), determine the value of \(k\).

(a) -2 (b) -8 (c) 8 (d) 12

21. If one of the roots of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), determine the value of \(q\).

22. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is zero, determine the product of the roots.

23. If the product of the roots of the equation \(x^2 - 2x + k = 8\) is \(-2\), then the value of \(k\) is—

(a) -2 (b) 6 (c) 1 (d) -6

24. In the equation \(x^2+4x+3=(x+3)\), the coefficient of \(x°\) is ____.

25. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

26. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is zero, determine the value of \(k\).

27. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is 2, find the value of \(k\).

28. Let us calculate the value of \(k\) for which \(-a\) is a root of the quadratic equation \(x^2 + 3ax + k = 0\).

29. If one root of the equation \(x^2 + ax + 3 = 0\) is 1, find the value of \(a\).

30. If one root of the equation \(x^2 - (2 + b)x + 6 = 0\) is 2, write the value of the other root.