1. If \(2a = 3b = 4c\), then \(a : b : c\) will be.
(a) 2:3:4 (b) 6:4:3 (c) 4:3:2 (d) 3:4:6
2. If \(2A=3B=4C\), then \(A:B:C=6:4:______\).
3. If \(2a=3b=4c\), then \(a:b:c=6:4:3\).
4. 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
5. 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) \(=\cfrac{3}{5}\)
6. If \[ 2a = 3b = 4c, \] then what will be the ratio \(a : b : c\)?
7. If 2a=3b=4c, then what is 2a:5b:7c ?
(a) 2:3:4 (b) 4:15:28 (c) 12:20:21 (d) None of these
8. 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
9. 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
10. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]
11. If \(a : b = 2 : 3\) and \(b : c = 2 : 3\), then what is \((a + b) : (b + c)\)?
12. If \(a:b=3:2\) and \(b:c=3:2\), then \((a+b):(b+c) =\)_____.
13. If \( x = \cfrac{6ab}{a+b} \), then prove that: \[ \cfrac{x+3a}{x-3a}+\cfrac{x+3b}{x-3b}=2 \]
14. If \(a:b = b:c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]
15. If \(a:b = 3:2\) and \(b:c = 3:2\), then what is \(a+b : b+c\)?
16. If \(A : B = 2 : 3\), \(B : C = 4 : 5\), and \(C : D = 6 : 7\), then find the value of \(A : D\).
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 \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).
(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)
19. If \(a : b = 3 : 2\) and \(b : c = 3 : 2\), then find the value of the ratio \((a + b) : (b + c)\).
20. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.
(a) undefined (b) positive (c) negative (d) zero
21. If 2A = 3B = 4C, then A : B : C = ------------
22. If, tanθ=cot3θ, then the value of sin2θ be:
(a) \(\cfrac{1}{√2}\) (b) \(\cfrac{√3}{2}\) (c) \(\cfrac{1}{2}\) (d) 0
23. if a: \(\cfrac{27}{64}=\cfrac{3}{4}\):a then the value of a be-
24. A, B, and C made a joint business profit of ₹1,500. If A's capital : B's capital = 2 : 3 and B's capital : C's capital = 2 : 5, then C's share of the profit is...
(a) ₹ 900 (b) ₹ 1000 (c) ₹ 1100 (d) ₹ 1200
25. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:
(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13
26. The base length and width of a rectangular reservoir are 15 meters and 12 meters respectively. Water is filled into the reservoir from a nearby pond using a pump. If the pump can fill 36,000 liters of water per hour, then how long will it take for the pump to fill the reservoir up to a height of 7.2 decimeters? [Note: 1 liter = 1 cubic decimeter]
27. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).
28. 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 \(a \ne 0\).
29. 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).
30. If \(a : b = b : c\), then prove that \[ (a + b)^2 : (b + c)^2 = a : c \]
