1. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be –

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

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

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

3. 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

4. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

5. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

6. If \(x^2 - \frac{5}{6}x + \frac{1}{6} = 0\), then the value of \(x\) is -

(a) \(\cfrac{1}{2}, \cfrac{1}{3}\) (b) 2, 3 (c) 3, 4 (d) 3, 4

7. Solve: \(x^2 + x - 6 = 0\)

(a) x = 1,2 (b) x = -1,-2 (c) x = 2,2 (d) x = -3,2

8. Solve: \(x^2 - 5x + 6 = 0\)

(a) x = 2,3 (b) x = 3,5 (c) x = -1, 2 (d) x = 2,2

9. Solve: \(x^2 - 8x + 16 = 0\)

(a) x = 4, 4 (b) x = 5, 5 (c) x = 4, 2 (d) x = 2, 2

10. Solve: \(x^2 + x - a^2 - 5a - 6 = 0\)

(a) \(x=-(a+3), (a+2)\) (b) \(x=a+3, b\) (c) \(x = a+2, b\) (d) \(x = a, b\)

11. 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

12. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

13. \(x^2 - 6\sqrt{x} + 2 = 0\) — x squared minus 6 root x plus 2 equals 0

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

15. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\)

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

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

17. 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

18. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –

(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

19. The roots of the equation \(x^2 - 18x + 8 = 0\) are —

(a) Real , Rational , Unequal (b) equal,Rational (c) Real , Rational , equal (d) None of the above

20. What are the roots of the equation \(x^2 - 4x + 4 = 0\)?

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

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

(a) \(\pm 1\) (b) \(\pm \cfrac{1}{2}\) (c) \(\pm \cfrac{1}{\sqrt2}\) (d) \(\pm \cfrac{1}{\sqrt3}\)

22. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{x^3}{x^6 + 7x^3 + 1}\).

(a) \(\cfrac{1}{2737}\) (b) \(\cfrac{1}{2730}\) (c) \(\cfrac{1}{2710}\) (d) \(\cfrac{1}{2709}\)

23. Let radius \(r\), slant height \(l = 7\) m, and total canvas area \(= 77\) m² Lateral surface area \(= \pi r l = 7\pi r\) Base area \(= \pi r^2\) Total area \(= 7\pi r + \pi r^2 = 77\) \(\pi r(r + 7) = 77\) \(r(r + 7) = \frac{77}{\pi} \approx 24.52\) \(r^2 + 7r - 24.52 = 0\) \(r = \frac{-7 \pm \sqrt{147.08}}{2} \approx \frac{-7 \pm 12.13}{2}\) \(r \approx 2.565\) Base area \(= \pi r^2 \approx 3.1416 \times 6.58 \approx 20.67\) m²

(a) 38.5 square meters (b) 39.5 square meters (c) 36.5 square meters (d) 37.5 square meters

24. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

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

26. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

27. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

28. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

29. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).

30. Check whether 0 and -2 are roots of the quadratic equation \(8x^2 + 7x = 0\).