1. If \(2a = 3b = 4c\), the ratio \(a:b:c\) will be –

(a) 3:4:6 (b) 4:3:6 (c) 3:6:4 (d) 6:4:3

2. If \(2a = 3b = 4c\), then \(a : b : c = 2 : 3 : 4\).

3. If \(2A=3B=4C\), then \(A:B:C=6:4:______\).

4. If \(2a=3b=4c\), then \(a:b:c=6:4:3\).

5. If \[ 2a = 3b = 4c, \] then what will be the ratio \(a : b : c\)?

(a) 3:4:6 (b) 4:3:6 (c) 3:6:4 (d) 6:4:3

6. If \(3a = 4b = Sc\), then \(a:b:c\) will be _____.

7. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:

(a) >0 (b) <0 (c) 0 (d) None of these

8. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

9. If \(a : b = b : c\), then prove that \[ (a + b)^2 : (b + c)^2 = a : c \]

10. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

11. If \(a : 2 = b : 5 = c : 4\), then 50% of \(a\) = 20% of \(b\) = ______% of \(c\).

12. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

13. If \(A = \frac{4}{3}B\) and \(B = \frac{5}{4}C\), then find the ratio \(A : B : C\).

(a) 20 : 15 : 16 (b) 15 : 20: 16 (c) 16: 15 : 20 (d) 15 : 16 : 20

14. If \(a : b = 2 : 3\) and \(b : c = 4 : 5\), then what is the value of \(a^2 : b^2 : bc\)?

(a) 8:18:21 (b) 16:36:45 (c) 16:20:36 (d) 8:15:18

15. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

16. If \(a:b = 2:3\), then the value of \(5a:6b\) will be \(1:1\).

17. If \(a:b = b:c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

18. If \(a:b = 3:2\) and \(b:c = 3:2\), then what is \(a+b : b+c\)?

19. If \((a+b) : \sqrt{ab} = 2:1\), then the value of \(a:b\) will be 1:1.

20. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

21. If \(a : b = b : c\), then prove that \(\cfrac{abc(a+b+c)^3}{(ab+bc+ca)^3} = 1\).

22. If \(A : B = 6 : 7\) and \(B : C = 8 : 7\), then find the value of \(A : C\).

23. If \(a : b = 8 : 7\), then show that \((7a - 3b) : (11a - 9b) = 7 : 5\).

24. If \((3a + 7b) : (5a - 3b) = 5 : 3\), then determine the value of \(a : b\).

25. If \(a : b = b : c\), then prove that \[ \left(\cfrac{a + b}{b + c}\right)^2 = \cfrac{a^2 + b^2}{b^2 + c^2} \]

26. If \(a : b = b : c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3} + \cfrac{1}{b^3} + \cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

27. If \(a : b = b : c\), then prove that \[ \cfrac{abc(a + b + c)^3}{(ab + bc + ca)^3} = 1 \]

28. If \(a : 2 = b : 5 = c : 8\), then 50% of \(a\) = 20% of \(b\) = ________% of \(c\).

29. If \(A = B = 45^\circ\), then let us verify that: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\)

30. If \(A = B = 45^\circ\), then let us verify that: \(\cos (A + B) = \cos A \cos B - \sin A \sin B\)