1. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

2. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

3. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

4. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?

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

6. If the equation \(kx^2 + 6x + 4k = 0\) has equal values for the sum and product of its roots, then what is the value of \(k\)?

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

7. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), then the value of \(\left(\cfrac{1}{α} + \cfrac{1}{β}\right)\) is —

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

8. If the roots of the quadratic equation \(2x^2 + 5x + k - 3 = 0\) are reciprocals of each other, then what is the value of \(k\)?

(a) 2 (b) 1 (c) 5 (d) 3

9. If the roots of the equation ( ???? + 2 ) ???? 2 − ( ???? − 3 ) ???? + 3 ???? − 1 = 0 are equal in magnitude but opposite in sign, what is the value of ???? ?

(a) -3 (b) 1 (c) 3 (d) None of the above

10. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone?

(a) 10 units (b) 11 units (c) 12 units (d) 13 units

11. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

12. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?

13. If the base area of a cone is \(x\) square units, its height is \(y\) units, and its volume is \(z\) cubic units, then what is the value of \(\frac{xy}{z}\)?

14. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

15. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

16. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

17. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is 4, then what is the value of \(b\)?

18. What is the value of \(k\) if the sum and product of the roots of the equation \(kx^2 + 2x + 3k = 0\) \((k \ne 0)\) are equal?

19. If 8, 14, \(7x + 2\), and 28 are in continued proportion, then what is the value of \(x\)?

(a) 1 (b) 3 (c) 2 (d) 4

20. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

(a) 1 (b) 2 (c) 4 (d) 3

21. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?

(a) 4 (b) 3 (c) 2 (d) 1

22. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

23. If the sum of the roots of the quadratic equation \[ x^2 - x = k(2x - 1) \] is zero, then what is the value of \(k\)?

24. 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 \(q\).

25. The dimensions of two cuboids are 4, 6, 4 units and 8, (2h − 1), 2 units respectively. If the volumes of the two cuboids are equal, then what is the value of \(h\)?

26. If the equation \(ax^2 + bx + c = 0\) has equal roots, then what is the value of \(c\)?

(a) \(\cfrac{-b}{2a}\) (b) \(\cfrac{b}{2a}\) (c) \(\cfrac{-b^2}{4a}\) (d) \(\cfrac{b}{4a}\)

27. 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) \(-\cfrac{1}{2}\)

28. If the equation \(ax^2 + 2bx + c = 0\) has equal roots, then what is the value of \(c\)?

(a) \(\cfrac{b^2}{a}\) (b) \(\cfrac{b^2}{4a}\) (c) \(\cfrac{a^2}{b}\) (d) \(\cfrac{a^2}{4b}\)

29. 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) \(-\cfrac{3}{2}, \cfrac{1}{3}\)

30. 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