1. If the mean and median of a statistical distribution are 35 and 33, respectively, determine the mode of the distribution.

2. If the combined mean of \(n\) quantities is \(\bar{x}\), and the sum of the first \((n-1)\) quantities is \(k\), then the \(n\)th quantity will be —

(a) \(k+n\) (b) \(\bar{x}+k\) (c) \(\bar{x}-k\) (d) \(n\bar{x}-k\)

3. From a statistical frequency distribution table, the given values are: \(n = 84,\) \(L = 30,\) \(f = 30,\) \(h = 30,\) \(cf = 40\) Therefore, the median of the distribution will be —

(a) 30 (b) 32 (c) 34 (d) 36

4. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

5. When the combined average (mean), median, and mode are given — they are called measures of central tendency.

6. 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

7. If the assumed mean is 22, class width is 10, total frequency is 80, and the value of \(\sum{f_iu_i}\) is 16, then the actual (or true) mean will be —

(a) 23 (b) 24 (c) 25 (d) 26

8. In a frequency distribution table, if \(\sum f_i d_i = 1000\) and \(\sum f_i = 30\), then the mean deviation will be —

(a) 33.33(approximately) (b) 33 (c) 3.33(approximately) (d) 30.33(approximately)

9. 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

10. If the combined average of the expressions ???? − 3 , ???? − 1 , 7 , ???? , 2 ???? − 1 , and 3 ???? − 5 is 3.5, then what will be the median of these expressions?

(a) 3.5 (b) 4.5 (c) 5 (d) 5.5

11. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?

12. If the area of a square is \(A_1\) square units, and the area of the square drawn on its diagonal is \(A_2\) square units, then the ratio \(A_1 : A_2\) will be —

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

13. If the interest rates for the first, second, and third years are 5%, 8%, and 10%, respectively, then the compound principal at the end of 3 years on ₹16,000 would be—

(a) ₹ 16,958 (b) ₹ 19,958.40 (c) ₹ 17,958 (d) ₹ 18,958.40

14. 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) \(2y:5x\)

15. If the capital ratio of A, B, and C is \(\frac{1}{6}:\frac{1}{5}:\frac{1}{4}\), then the profit distribution will be in the ratio 4:5:6.

16. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:4, then their radius ratio will be—.

(a) \(2:1\) (b) \(\sqrt2:1\) (c) \(1:2\) (d) \(1:\sqrt2\)

17. From the following cumulative frequency distribution table, prepare a frequency distribution table and determine the mode of the data:

18. 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) ₹\(\cfrac{xyz}{3x+y}\)

19. 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: \[ P\left(1 + \cfrac{r_1}{100}\right)\left(1 + \cfrac{r_2}{100}\right)\left(1 + \cfrac{r_3}{100}\right) \]

20. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —

(a) >0 (b) =0 (c) <0 (d) none of the above

21. 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.

22. If three real numbers a, b, and c are in continued geometric progression, then the value of the mean proportional b will be undefined when a and c have opposite signs.

23. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

24. If the roots of the quadratic equation \( ax^2 + 2bx - c = 0 \) (where \( a \ne 0 \)) are real and equal, then \( b^2 \) will be ——.

25. 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.

26. 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 |

27. If the base radius of a right circular cone is 3 cm and the height is 4 cm, then the lateral surface area of the cone will be.

(a) \(10\pi \, cm^2\) (b) \(15\pi \, cm^2\) (c) \(12\pi \, cm^2\) (d) \(18\pi \, cm^2\)

28. 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

29. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

30. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –

(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9