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

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

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

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

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

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

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

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

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

8. 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

9. 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) \(\bar{x}=8, n=6\)

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 \(\cos 52^\circ = \frac{x}{\sqrt{x^2 + y^2}}\), then what is the value of \(\tan 38^\circ\)?

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

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

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

16. 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

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

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

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

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

21. 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.

22. If \(\sin 3\theta = \frac{1}{2}\), then what is the value of \(\cos(80^\circ + \theta)\)?

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

24. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°?

(a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)

25. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?

(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

26. 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

27. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?

(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above

28. If \(x = \cfrac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then what is the value of \(\cfrac{x + \sqrt{5}}{x - \sqrt{5}} + \cfrac{x + \sqrt{3}}{x - \sqrt{3}}\)?

(a) -1 (b) 0 (c) 1 (d) None of the above

29. If \(a = 3 + 2\sqrt{2}\), then what is the value of \[ \frac{a^6 + a^4 + a^2 + 1}{a^3}? \]

(a) 100 (b) 200 (c) 204 (d) 250

30. If \( x = 2 + \sqrt{3} \), then what is the value of \( x + \frac{1}{x} \)?

(a) -4 (b) 2 (c) 4 (d) \(2+\sqrt3\)