1. If \(x, 2x, 3\), and \(y\) are in continued proportion, then what will be the value of \(y\)?

(a) \(4x\) (b) \(6x\) (c) 4 (d) 6

2. If \(x, 12, y, 27\) are in continued proportion, find the positive values of \(x\) and \(y\).

3. If \(x, y, z\) are in continued proportion, then what is the value of \[ x^2y^2z^2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)? \]

(a) \(x+y+z\) (b) \(x^2+y^2+z^2\) (c) \(x^3+y^3+z^3\) (d) None of the above

4. If \(x, 12, y, 27\) are in continuous proportion, find the positive values of \(x\) and \(y\).

(a) 8,18 (b) 19,20 (c) 16,18 (d) 2,8

5. In a business, if two individuals invest \(x\) rupees and \(y\) rupees for 9 months and 6 months respectively, then their profit share ratio will be?

(a) \(2x:3y\) (b) \(3x:2y\) (c) \(x:y\) (d) \(2y:5x\)

6. If two individuals invest \(x\) rupees and \(y\) rupees in a business for \(z\) and \(w\) units of time, respectively, then their profit ratio will be:

(a) \(xw:yz\) (b) \(xz:wy\) (c) \(xy:wz\) (d) none of the above

7. If \( x, 12, y, 27 \) are in continuous proportion, find the positive values of \( x \) and \( y \).

8. If \(x, 12, y, 27\) are in continuous proportion, determine the positive values of \(x\) and \(y\).

9. In compound interest, if the interest rates for the first, second, and third years are \(r_1\%\), \(r_2\%\), and \(2r_3\%\) respectively, then the amount after 3 years on a principal of ₹\(P\) will be: \[ P\left(1 + \cfrac{r_1}{100}\right)\left(1 + \cfrac{r_2}{100}\right)\left(1 + \cfrac{r_3}{100}\right) \]

10. Puja, Uttam, and Meher started a partnership business by investing ₹5,000, ₹7,000, and ₹10,000 respectively, under the following conditions: (i) Monthly business operating expenses are ₹125. (ii) Puja and Uttam will each receive ₹200 per month for bookkeeping duties. At the end of the year, if the total profit is ₹6,960, calculate how much profit each person will receive.

11. If \(x, y\) are in direct proportion, and \(y, z\) are in inverse proportion, then \(x, z\) will be in inverse proportion.

12. Four quantities are in continuous proportion. If the first and second quantities are 3 and 2 respectively, then the fourth quantity will be:

(a) \(\cfrac{2}{3}\) (b) \(\cfrac{8}{9}\) (c) \(\cfrac{4}{3}\) (d) \(\cfrac{10}{12}\)

13. If the mode and the combined mean of a statistical distribution are ₹12.30 and ₹18.48 respectively, then the median of the distribution will be —

(a) ₹ 16 (b) ₹ 17 (c) ₹ 16.42 (d) ₹ 17.42

14. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

15. If \(x\), 12, \(y\), and 27 are in continued proportion, find the negative values of \(x\) and \(y\).

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

17. In triangle ABC, the midpoints of sides BC, CA, and AB are D, E, and F respectively. BE and DF intersect at point X, and CF and DE intersect at point Y. Then, the length of XY is equal.

(a) \(\cfrac{1}{2}\)BC (b) \(\cfrac{1}{4}\)BC (c) \(\cfrac{1}{3}\)BC (d) \(\cfrac{1}{8}\)BC

18. Sudip Babu deposited ₹7,800 in the bank at compound interest. If the compound interest rates for the first, second, third, and fourth years are 5%, 4%, 10%, and 12% respectively, then what will be the compound principal after 4 years?

(a) ₹ 10,493.68 (b) ₹ 10,000 (c) ₹ 14,936 (d) none of the above

19. The radii of two circles are 5 cm and 3 cm, respectively. If the circles touch each other externally, then the distance between their centers will be –

(a) 2 cm (b) 3 cm (c) 8 cm (d) 1.5 cm

20. Three friends—A, B, and C—started a business by investing \(x\), \(2x\), and \(y\) taka respectively. If the profit at the end of the term is \(z\) taka, then what will be A’s share of the profit?

(a) ₹ \(\cfrac{xz}{3x+y}\) (b) ₹ \(\cfrac{2xz}{3x+y}\) (c) ₹ \(\cfrac{z}{2x+y}\) (d) ₹\(\cfrac{xyz}{3x+y}\)

21. In a circle, triangle ABC is inscribed. AX, BY, and CZ are the angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) respectively, and they meet the circle at points X, Y, and Z respectively. Prove that AX is perpendicular to YZ.

22. If the heights of two right circular cones are in the ratio 1 : 3 and the lengths of their radii are in the ratio 3 : 1 respectively, then show by calculation that the ratio of their volumes will be 3 : 1.

23. If a planet's gravitational force on its satellite is directly proportional to the planet's mass (M) and inversely proportional to the square of their distance (D), and the square of the satellite's orbital period (T) is directly proportional to the cube of the distance and inversely proportional to the gravitational force, then for two sets of values \(m_1, d_1, t_1\) and \(m_2, d_2, t_2\) corresponding to M, D, and T respectively, prove that: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]