1. 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.

2. If the mean of a frequency distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then find the value of \(k\).

3. If the mean of a frequency distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then find the value of \(k\).

4. If the mean of a frequency distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then find the value of \(k\).

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

6. The mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\). Find the value of \(k\).

7. The mean of a statistical distribution is 4.1. Given that \(∑f_i.x_i = 132 + 5k\) and \(∑f_i = 20\), find the value of \(k\).

8. If the mean of a frequency distribution is 8.1, \(\sum f_i x_i = 132 + 5k\) and \(\sum f_i = 20\), then find the value of \(k\).

9. The mean of a statistical distribution is \( 4.1 \), given \( ∑f_i.x_i = 132 + 5k \) and \( ∑f_i = 20 \). Find the value of \( k \).

10. **"In a statistical distribution, if the mean is 8.1, \(\sum f_i x_i = 132 + 5k\), and \(\sum f_i = 20\), then \(k =\) _____"**

11. In a statistical distribution, the mean is 8.1, \( \sum f_i x_i = 132 + k \), and \( \sum f_i = 20 \). Find the value of \( k \).

12. 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

13. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?

14. 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

15. If \(u_i = \frac{x_i - 30}{10}\), \(∑f_i = 50\), and \(∑u_i f_i = 25\), then what is the value of \(\bar{x}\)?

16. For a frequency distribution, the mean is given as 8.1; \(\sum f_i x_i = 132+5k\) and \(\sum f_i = 20\). Find the value of \(k\).

17. If \(u_i = \cfrac{x_i - 35}{10}\), \(∑f_i u_i = 30\), and \(∑f_i = 60\), then the value of \(\bar{x}\) is –

(a) 40 (b) 20 (c) 80 (d) None of these

18. If \(∑f_i d_i = 400\), \(∑f_i = 50\), and \(a =\) assumed mean \(= 52\), then the value of the combined mean is –

(a) 52 (b) 60 (c) 80 (d) 55

19. If the equation \(kx^2 + 6x + 4k = 0\) has equal values for the sum and product of its roots, then what is the value of \(k\)?

(a) \(-\cfrac{3}{2}\) (b) \(\cfrac{3}{2}\) (c) \(\cfrac{2}{3}\) (d) \(-\cfrac{2}{3}\)

20. If \(u_i = \cfrac{x_i - 25}{10}\), \(\sum f_i u_i = 20\), and \(\sum f_i = 100\), then what is the value of \(\bar{x}\)?

21. In a statistical distribution, the average (mean) is 7 and \(\sum f_i x_i = 140\). Find the value of \(\sum f_i\).

22. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

(a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

23. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

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

24. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

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

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

26. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

27. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?

(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)

28. If a cuboid has number of faces = x, number of edges = y, number of vertices = z, and number of diagonals = p, then what is the value of (x − y + z + p)?

29. If \( u_i = \frac{x_i - 45}{10} \), \( ∑f_i u_i = -16 \), and \( ∑f_i = 200 \), then what is the value of \( \bar{x} \)?

30. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]