1. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?

(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)

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 \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

4. 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) \(a^2+ac\)

5. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

6. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

7. 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

8. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .

9. 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

10. 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

11. 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

12. If \(x^2 - px + 12 = (x - 3)(x - a)\) is an identity, then the values of \(a\) and \(p\) respectively are?

(a) \(a = 4, p = 7\) (b) \(a = 7, p = 4\) (c) \(a = 4, p =-7\) (d) \(a =-4, p = 7\)

13. 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) \(-\cfrac{2}{3}\)

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

15. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

16. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).

17. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

18. If the quadratic equation \(ax^2 + 7x + b = 0\) has two roots \(\frac{2}{3}\) and \(-3\), then let us find the values of \(a\) and \(b\).

19. If the roots of the equation \(ax^2 + bx + 35 = 0\) are \(-5\) and \(-7\), find the values of \(a\) and \(b\).

20. 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

21. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

22. If the roots of the equation \(x^2 + ax + b = 0\) are \(-2\) and \(3\), then find the values of \(a\) and \(b\).

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