1. 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) \(\cfrac{16}{9}\)

2. If \(a\), \(b\), and \(c\) are in continuous geometric progression and both \(a\) and \(c\) have negative signs, then \(b\) will also be positive in sign. Here’s the full English translation of your sentence: If \(a\), \(b\), and \(c\) are in continuous geometric proportion and both \(a\) and \(c\) have negative signs, then \(b\) will have a ——— sign.

3. \(a\) is a positive number, and given \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), determine the value of \(a\).

4. If \[ a : 2 = b : 5 \] then, what percentage of \(a\) and \(b\) will be equal?

(a) 20 (b) 40 (c) 30 (d) 50

5. If a right-angled quadrilateral has \(a\) number of vertices, \(b\) number of edges, and \(c\) number of faces, then what is the value of \(2a - b + 3c\)?

(a) 16 (b) 18 (c) 20 (d) 22

6. Let \(a\) be a positive number, and given: \[ a : \frac{27}{64} = \frac{3}{4} : a \] Find the value of \(a\).

7. Translate: If the average of \(p\) numbers is \(b\), and the average of \((p + q)\) numbers is \(a\), then the average of the \(q\) numbers is _____

8. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?

(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)

9. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

10. If the median of the following frequency distribution table is 27, then find the value of \(a\): | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |----------------|------|--------|--------|--------|--------| | Frequency | 3 | a | 20 | 12 | 7 |

11. 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

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

13. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

14. If 75% of \(A\) equals 40% of \(B\), then the ratio \(A:B\) is –?

15. If the volume of a right circular cone is \(V\) cubic units, the area of the base is \(A\) square units, and the height is \(H\), then what is the value of \(\frac{AH}{V}\)?

16. If the combined mean of the following data is 20.6, then find the value of \(a\):

17. If \(b ∝ a^3\) and the increase in \(a\) is in the ratio \(2 : 3\), then determine the ratio of increase in \(b\).

18. 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

19. If a cuboid has \(x\) edges and \(y\) faces, what is the minimum value of \(a\) such that \((x + y + a)\) becomes a perfect square?

20. If \(∑f_i(x_i - a) = 400\), \(∑f_i = 50\), and \(a\) (assumed mean) = 52, then the value of the combined mean \(\bar{x}\) is –

(a) 52 (b) 60 (c) 80 (d) 90

21. If one root of the equation \(x^2 - ax - 15 = 0\) is 3, then what is the value of \(a\)?

(a) -2 (b) 2 (c) 3 (d) 0

22. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

23. If the surface area of a sphere is \(A\) and its volume is \(V\), then find the value of \(\cfrac{A^3}{V^2}\).

24. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).

25. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?

26. If the volume of a right circular cone is \(V\), the area of its base is \(A\), and its height is \(H\), then find the value of \(\left(\frac{AH}{V}\right)^2\).

(a) 1 (b) 4 (c) 9 (d) 3

27. 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

28. 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

29. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

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

30. A, B, and C made a total profit of ₹15,000 in a joint business. If the capital ratio between A and B is 2:3 and between B and C is 2:5, then how much profit will B receive?

(a) 4500 (b) 9000 (c) 2400 (d) 3600