1. If the roots of the equation ( ???? + 2 ) ???? 2 − ( ???? − 3 ) ???? + 3 ???? − 1 = 0 are equal in magnitude but opposite in sign, what is the value of ???? ?

(a) -3 (b) 1 (c) 3 (d) None of the above

2. If ???? = 3 + 2 and ???? = 3 − 2 , then what is the value of 8 ???? ???? ( ???? 2 + ???? 2 ) ?

(a) 24 (b) 80 (c) 16 (d) 8

3. If ???? 3 = ???? 5 = ???? 8 , then what is the value of 3 ???? − 5 ???? + 2 ???? ???? ?"

(a) \(\frac{1}{5}\) (b) 5 (c) 0 (d) 10

4. If 3 cos ⁡ ???? − 4 sin ⁡ ???? = 5 , then what is the value of 3 sin ⁡ ???? + 4 cos ⁡ ???? ?

(a) 1 (b) 2 (c) 0 (d) \(\cfrac{1}{2}\)

5. The fourth proportional of ( ???? 2 − ???? ???? + ???? 2 ) , ( ???? 3 + ???? 3 ) , and ( ???? − ???? ) is

(a) \(x^2+y^2\) (b) \(x+y\) (c) \(x^2-y^2\) (d) \(x-y\)

6. If \(n\) is an even number, then the median will be the average of the \(\left(\cfrac{n}{2}\right)\)-th and \(\left(\cfrac{n}{2}−1\right)\)-th observations.

7. "If ???? sin ⁡ ???? = 7 2 and ???? cos ⁡ ???? = 7 3 2 , then what is the value of ???? ?

(a) \(49\) (b) \(7\) (c) \(\sqrt7\) (d) \(-7\)

8. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

9. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

10. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—

(a) 2 (b) 18 (c) 9 (d) 8

11. If sin ⁡ 3 ???? ⋅ sec ⁡ 6 ???? = 1 , then what is the value of tan ⁡ 6 ???? ?

(a) \(\sqrt{\cfrac{13}{5}}\) (b) \(\sqrt3\) (c) \(\cfrac{1}{\sqrt3}\) (d) 0

12. If the data arranged in ascending order is 6, 9, 11, 12, x+2, x+3, 17, 20, 21, 24 and the median is 21, then the value of x will be —

(a) 13.5 (b) 15.5 (c) 18.5 (d) 21.5

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. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

15. A low-lying plot of land measuring 48 meters in length and 31.5 meters in width is to be raised by 6.5 decimeters. To achieve this, a nearby pit measuring 27 meters in length and 18.2 meters in width is dug. What will be the depth of the pit?

16. If the average of the numbers \(7, x - 3, x + 3, 10, x - 5\) is \(15\), then their median will be \(10\).

17. In a business, the capital ratio of A and C is 2:1, and the capital ratio of A and B is 3:2. If the annual profit is ₹1,57,300, what will be B's share of the profit?

(a) ₹ 48,000 (b) ₹ 58,400 (c) ₹ 48,400 (d) ₹68,000

18. Three jars with equal diameter and height were partially filled with dilute sulfuric acid: - the first jar was filled to \(\frac{2}{3}\) of its height, - the second to \(\frac{5}{6}\), and - the third to \(\frac{7}{9}\). If the acid from these three jars is poured into a single jar with a diameter of 2.1 decimetres, and the height of the acid in this jar becomes 4.1 decimetres, then calculate and write the height of each of the original three jars, given that their diameter was 1.4 decimetres.

19. If the following data has a combined (weighted) average of 20.6, then what is the value of \(a\)?

20. Here’s the English translation of your math problem: If the ascending data set is \[ 27,\ 31,\ 46,\ 52,\ x,\ y + 2,\ 71,\ 79,\ 85,\ 90 \] and the median of the set is \(64\), then what is the value of \(x + y\)?

(a) 125 (b) 126 (c) 127 (d) 128

21. If the ratio of the capital investments of A, B, and C is \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{4}\) and the total profit is ₹520, then C's share of the profit will be

(a) ₹120 (b) ₹160 (c) ₹180 (d) ₹140

22. Asit and Dipak invested ₹22,500 and ₹35,000 respectively in a joint business. If the total profit is ₹13,800, then Dipak's share of the profit will be —

(a) ₹ 8400 (b) ₹ 8600 (c) ₹ 8200 (d) ₹ 8000

23. If the simple interest on ₹850 for 3 years and on ₹1250 for 4 years at the same rate amounts to ₹302 in total, then what will be the annual rate of interest?

(a) 3% (b) 5% (c) 7% (d) None of the above

24. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?

(a) 0 (b) 1 (c) 0.1 (d) 0.01

25. 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]

26. At the beginning of the year, Pradip Babu and Amina Bibi started a business with ₹24,000 and ₹30,000 respectively. After 5 months, Pradip Babu added ₹4,000 more to his capital. At the end of the year, the total profit was ₹27,716. How much profit will each of them receive?

27. If \(u_i = \frac{x_i − 35}{10}\), \(Σf_iu_i = 30\), and \(Σf_i = 60\), then find the value of \(\bar{x}\).

28. At the beginning of the year, Souvik and Rahul started a business by investing ₹24,000 and ₹30,000 respectively. After 5 months, Souvik added an additional ₹4,000 to his capital. At the end of the year, the total profit was ₹27,716. Calculate how much profit each of them will receive.

29. 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

30. To raise a plot of land measuring 25.5 meters in length and 20.2 meters in width by filling it with soil up to a height of 1.5 meters, a pit is dug in the adjacent plot with dimensions 15.3 meters in length and 10.1 meters in width. What will be the depth of the pit?

(a) 5 meters (b) 10 meters (c) 15 meters (d) 20 meters