1. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

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

2. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.

3. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).

4. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).

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

6. If the roots of the quadratic equation \(5x^2+13x+k=0\) are reciprocals of each other, then the value of \(k\) is:

(a) 3 (b) 4 (c) 5 (d) -5

7. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

8. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

9. If one root of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) is double the other, show that \(2b^2 = 9ac\).

10. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).

11. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocal to each other, then \(c =\) _____________

12. 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

13. 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

14. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

15. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is zero.

(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of the above

16. If the roots of the equation \(ax^2 + bx + c = 0\) are equal (where \(a \ne 0\)) —

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

17. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).

18. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

19. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

20. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

21. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —

(a) >0 (b) =0 (c) <0 (d) none of the above

22. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are equal, then —

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

23. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative method—that is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.

24. If the roots of the quadratic equation \(5x^2 + 13x + k = 0\) are the reciprocals of each other, then find the value of \(k\).

25. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

26. If the roots of the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), are in the ratio 1:2, then show that \(2b^2 = 9ac\).

27. Show that if one root of the quadratic equation \( ax^2 + bx + c = 0 \) is double the other, then \( 2a^2 = 9ac \).

28. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

29. Under what condition will one root of the quadratic equation \(ax^2 + bx + c = 0\) be zero?

(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these

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