1. 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

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

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

4. Determine the value of \(a\) if \(\sqrt2\) is one of the roots of the equation \(3x^2 + \sqrt2x + a = 0\).

(a) 7 (b) -8 (c) 9 (d) 8

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

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

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

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

9. If one root of the equation \(2x^2 + kx + 4 = 0\) is \(2\), write the value of the other root.

10. If one root of the quadratic equation \(ax^2 + abcx + bc = 0\) (\(a \ne 0\)) is the reciprocal of the other, then —

(a) abc=1 (b) b=ac (c) bc=1 (d) a=bc

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

12. 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

13. If the roots of a quadratic equation in one variable are 2 and 7, determine the equation.

14. 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

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

16. If the product of the roots of the quadratic equation \(3x^2 – 4x + k = 0\) is 5, then what will be the value of \(k\)?

(a) 5 (b) -12 (c) 15 (d) -20

17. 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

18. 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 \(q\).

19. In the quadratic equation \(ax^2 + bx + c = 0\), if one root is three times the other, prove that \(3b^2 = 16ac\).

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

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

22. Divide 42 into two parts such that one part is equal to the square of the other part. — Form a quadratic equation with one variable from the given statement.

23. If one root of the equation \[ ax^2 + bx + c = 0 \] is double the other, prove that \[ 2b^2 = 9ac \]

24. The length of a rectangular field is 36 meters more than its breadth. The area of the field is 460 square meters. Form a quadratic equation in one variable from this statement and determine the coefficients of \(x^2\), \(x\), and \(x^0\).

25. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

26. 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

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

28. In a quadratic equation with one variable, if the constant term (the term with zero power of the variable) is missing, then one root of the equation will always be zero.

(a) 3 (b) 2 (c) 1 (d) 0

29. A quadratic equation in one variable \(x\), with real coefficients, has equal coefficients for \(x^2\) and the constant term. Show with reasoning that any other quadratic equation whose roots are the reciprocals of the original equation's roots must be identical to the original equation.