1. In the given figure, LM || AB and AL = (x - 3) units, AC = 2x units, BM = (x - 2) units, and BC = (2x + 3) units. Determine the value of \(x\).

2. In the given figure, \( LM \parallel AB \) and \( AL = (x - 3) \) units, \( AC = 2x \) units, \( BM = (x - 2) \) units, and \( BC = (2x + 3) \) units. Find the value of \( x \).

3. In the adjacent figure, if \(LM \parallel AB\), and \(AL = (x - 3)\) units, \(AC = 2x\) units, \(BM = (x - 2)\) units, and \(BC = (2x + 3)\) units, then find the value of \(x\).

4. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

5. 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 \(x\).

6. In the adjacent figure, within triangle \(ABC\), if \(DE \parallel PQ \parallel BC\) and \(AD = 3\) cm, \(DP = x\) cm, \(PB = 4\) cm, \(AE = 4\) cm, \(EQ = 5\) cm, and \(QC = y\) cm, then determine the values of \(x\) and \(y\).

7. 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 \(n\).

8. Without solving, determine all values of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) has real and equal roots.

9. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

10. For what value of \(k\) will the system of equations \(x + (k - 2)y = 5\) and \((k - 3)x + 2y = 7\) have no common solution?

(a) 2 (b) 3 (c) 4 (d) 5

11. 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 \(y\).

12. A line parallel to side \(BC\) of triangle \(∆ABC\) intersects sides \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. If \(AX = 2.4\) cm, \(AY = 3.2\) cm and \(YC = 4.8\) cm, find the length of side \(AB\).

(a) 3.6 cm (b) 6 cm (c) 6.4 cm (d) 7.2 cm

13. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

14. Determine the value of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) will have equal and real roots.

15. 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) \(\bar{x}=8, n=6\)

16. If \(\sum\limits_{i=1}^5 x_i = 5\) and \(\sum\limits_{i=1}^5 x_i^2 = 14\), then the value of \(\sum\limits_{i=1}^5 2x_i(x_i - 3)\) will be —

(a) 2 (b) -2 (c) 0 (d) 4

17. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

18. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

19. 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) \(a =-4, p = 7\)

20. If \(\sum(x_i - 3) = 0\) and \(\sum(x_i + 3) = 66\), then find the values of \(\bar{x}\) (mean) and \(n\) (number of observations).

21. In \( \triangle PQR \), ST is parallel to QR and intersects PQ at point S and PR at point T. Given that \( PQ = (X+1) \) units, \( PS = 3 \) units, \( PR = (X+6) \) units, and \( PT = 6 \) units, determine the value of \( x \).

22. If \((5x - 3y) : (2x + 4y) = 11:12\), determine the value of \(x:y\).

23. 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 \(q\).

24. If \( BC \parallel DE \) in \( \triangle ABC \), and \( \frac{AD}{DB} = \frac{2}{5} \), with \( AC = 21 \) cm, determine the value of \( AE \).

25. If \(x = 3 + \sqrt{5}\) and \(xy = 4\), determine the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\).

26. If \(x^2 + ax + b = (x+2)(x-2)\), find the values of \(a\) and \(b\).

(a) 1,2 (b) 2,1 (c) 0,4 (d) 0,-4

27. If \(α\) and \(β\) are the roots of the equation \(x^2 - 22x + 105 = 0\), find the value of \((α - β)\).

28. If \((5x - 3y) : (2x + 4y) = 11 : 12\), then determine the value of \(x : y\).

29. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

30. If the median of the following data is 32 and \(x + y = 100\), determine the values of \(x\) and \(y\). | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|-------|-------| | Frequency | 10 | x | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | y | 10 | ---