1. If \(3x = 4y = 5z\), then \(x:y:z =\) _____.

2. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?

(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1

3. If \(4x = 5y = 6z\), what is the value of the ratio \(x : y : z\)?

(a) 12:10:15 (b) 10:12:15 (c) 15:12:10 (d) 15:10:12

4. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

5. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

6. 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

7. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

8. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

9. If \( (3x - 2y) : (3x + 2y) = 4 : 5 \), then what is the value of \(x : y\)?

(a) 1:6 (b) 1:1 (c) 2:1 (d) 6:1

10. If \(\cfrac{x-3y}{2y}=\cfrac{6x-5y}{5x}\), then what is the value of \(x:y\)?

(a) 5:1 or 2:5 (b) 1:5 or 2:5 (c) 1:5 or 5:2 (d) None of the above

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

12. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

13. If \(x : y = 3 : 4\), then what is the value of \((3y - x) : (2x + y)\)?

14. If \(x \propto y\), then – **x is directly proportional to y.** This means: \[ x = ky \quad \text{for some constant } k \] Or, the ratio \(\frac{x}{y}\) remains constant.

(a) \(x^2\propto y^3\) (b) \(x^3\propto y^2\) (c) \(x \propto y^3\) (d) \(x^2\propto y^2\)

15. If \(x : y = 2 : 3\), then what is the value of \((2x + 3y) : (2x + 5y)\)?

(a) 12:15 (b) 11:16 (c) 13:19 (d) 10:17

16. If \(a : b = 2 : 1\) and \(x : y = 3 : 4\), then what is the value of \((2ax - by) : (2by - ax)\)?

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

17. If \((4x - 3y) : (2x - y) = 2 : 3\), then what is the value of \(x : y\)?

(a) 15:14 (b) 8:7 (c) 7:8 (d) 14:15

18. If \(p + q : \sqrt{pq} = 2 : 1\), then what is the ratio \(p : q =\)?

19. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

20. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

21. Here’s the English translation of your question: **"If \( x = \frac{4ab}{a + b} \), then what is the value of \( \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \)?"** Would you like me to solve it step by step?

(a) 1 (b) -2 (c) 2 (d) -1

22. If \(a + b : \sqrt{ab} = 4 : 1\), then what is the ratio of \(a : b\)?

23. If \(x : a = y : b = z : c\), then show that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

24. If \((3x - 2y) : (x + 3y) = 5 : 6\), then find the ratio \((2x + 5y) : (3x + 4y)\).

25. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

26. If A:B = 2:3 and B:C = 4:5, then what is the ratio A:B:C?

27. If A : B = 3 : 2 and B : C = 3 : 5, then what is the ratio A : B : C?

28. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

29. If A : B = 2 : 3, B : C = 5 : 8, and C : D = 6 : 7, then what is the ratio A : D?

(a) 2:7 (b) 7:2 (c) 5:8 (d) 5:14

30. If \((2x - 3y) : (5x - 2y) = 1 : 19\), then what is the value of \((2x + 3y) : (3x - 2y)\)?

(a) 9:19 (b) 16:9 (c) 9:16 (d) 19:9