1. 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 \(k\)?

2. 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 \(k\)?

3. 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.

4. 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) -\(\cfrac{1}{3}\)

5. 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

6. 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

7. 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

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

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

9. 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) \(-r^2\)

10. If \(\frac{a}{2} = \frac{b}{3} = \frac{c}{4} = \frac{3a - 2b + 4c}{p}\), then what is the value of \(p\)?

(a) 12 (b) 13 (c) 16 (d) 18

11. 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)?

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

13. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then what is the value of \(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\)?

14. If \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4} = \cfrac{2a - 3b + 4c}{p}\), then what is the value of \(p\)?

15. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

16. If \(x = \cfrac{1}{2 + \sqrt{3}}\) and \(y = \cfrac{1}{2 - \sqrt{3}}\), then what is the value of \(\cfrac{1}{1 + x} + \cfrac{1}{1 + y}\)?

17. If \(m = \sqrt{\cfrac{n}{n + \cfrac{1}{2}}}\) and \(m = \cfrac{1}{2}\), then what is the value of \(n\)?

18. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

19. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

20. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

21. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then what is the value of \(\frac{x^2 + y^2 + xy}{x^2 + y^2 - xy}\)?

(a) \(\cfrac{6}{7}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{11}\) (d) None of the above

22. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

23. 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) \(-\cfrac{2}{3}\)

24. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

25. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?

26. If \(a = \cfrac{1}{2+\sqrt3}\) and \(b = \cfrac{1}{2-\sqrt3}\), then what is the value of \(\cfrac{1}{a+1}+\cfrac{1}{b+1}\)?

27. If \(x = \cfrac{\sqrt3 - \sqrt2}{\sqrt3 + \sqrt2}\) and \(xy = 1\), then what is the value of \(3x^2 - 5xy + 3y^2\)?

28. If \(a = \cfrac{\sqrt5+1}{\sqrt5-1}\) and \(ab = 1\), then what is the value of \(\cfrac{3a^2+5ab+3b^2}{3a^2-5ab+3b^2}\)?

29. 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 \(k\).

30. 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 \(k\).