1. Solve: \[ \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}, \quad x \ne 0,\ -(a + b) \]

2. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).

3. If \(x = \cfrac{4ab}{a + b}\), show that \[ \cfrac{x + 2a}{x - 2a} + \cfrac{x + 2b}{x - 2b} = 2 \] [Given: \(a \ne 0\), \(b \ne 0\), and \(a \ne b\)]

4. 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 \(a ≠ 0\).

5. 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

6. 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).

7. Solve: \[ \frac{1}{x} - \frac{1}{x + b} = \frac{1}{a} - \frac{1}{a + b}, \quad x \ne 0,\ -b \]

8. Assume, \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) (where \(k \ne 0\)) So, \(a = bk,\; b = ck,\; c = dk\) \(\therefore b = dk \cdot k = dk^2\) and \(a = dk^2 \cdot k = dk^3\)

Left-hand side \[ = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = \{(dk^3)^2 + (dk^2)^2 + (dk)^2\} \cdot \{(dk^2)^2 + (dk)^2 + d^2\} \] \[ = \{d^2k^6 + d^2k^4 + d^2k^2\} \cdot \{d^2k^4 + d^2k^2 + d^2\} = d^2k^2(k^4 + k^2 + 1) \cdot d^2(k^4 + k^2 + 1) \] \[ = d^4k^2(k^4 + k^2 + 1)^2 \] Right-hand side \[ = (ab + bc + cd)^2 = (dk^3 \cdot dk^2 + dk^2 \cdot dk + dk \cdot d)^2 = (d^2k^5 + d^2k^3 + d^2k)^2 = \{d^2k(k^4 + k^2 + 1)\}^2 = d^4k^2(k^4 + k^2 + 1)^2 \] ∴ Left-hand side = Right-hand side (Hence Proved)

9. Solve: \[ \frac{1}{x - a + b} = \frac{1}{x} - \frac{1}{a} + \frac{1}{b} \]

10. \(\cfrac{x}{4 - x} = \cfrac{1}{3x},\ (x ≠ 0,\ x ≠ 4)\) — Let us express this in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and determine the coefficient of \(x\).

11. Solve: \[ \frac{1}{x - \frac{1}{p}} + \frac{1}{x - \frac{1}{q}} = p + q \] where \(x \ne \frac{1}{p}, \frac{1}{q}\)

12. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants and \(a \ne 0\), \(b \ne 0\)) is:

(a) Parallel to the x-axis. (b) Parallel to the y-axis. (c) Not parallel to any axis. (d) Passing through the origin.

13. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants, and \(a \ne 0\)) will be the equation of the x-axis when:

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

14. If \((a+b)^2 x^2 - (a+b)x - 6 = 0\) \([a+b \neq 0]\), then \(x =\)

(a) \(x = \cfrac{3}{a+b}\) (b) \(\cfrac{a+b}{3}\) (c) \(a+b\) (d) \(1\)

15. Prove that the quadratic equation \(x^2(a^2+b^2) + 2(ac+bd)x + (c^2+d^2) = 0\) has no real roots when \(ad \ne bc\).

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

17. Let us express \((x + 2)^3 = x(x^2 - 1)\) in the form \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\).

18. \(x - 1 + \cfrac{1}{x} = 6\), where \(x \ne 0\)

19. Given that \[ \frac{a + b - c}{a + b} = \frac{b + c - a}{b + c} = \frac{c + a - b}{c + a} \] and \[ a + b + c \ne 0 \] prove that \(a = b = c\).

20. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x - \frac{1}{x} \]

21. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

22. Solve: \[ \frac{ax + b}{a + bx} = \frac{cx + d}{c + dx}, \quad \text{where } a \ne b,\, c \ne d,\; x \ne -\frac{a}{b},\, -\frac{c}{d} \]

23. Solve: \(\frac{a}{b - x} + \frac{b}{a - x} + 2 = 0\), given that \(x \ne a, b\)." Let me know if you'd like me to solve it too!

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

25. Write whether the equation \( x + \frac{3}{x} = x^2 \), where \( x \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, c \) are real numbers and \( a \ne 0 \).

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

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. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

29. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

30. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]