31) If \(x^4 + x^{-4} = 194, x > 0\), then the value of \((x - 2)^2\) is:

• (a) 1
• (b) 6
• (c) 2
• (d) 3

32) If \(16x^2 + 9y^2 + 4z^2 = 24(x - y + z) - 61\), then the value of \((xy + 2z)\) is:

• (a) 1
• (b) 2
• (c) 3
• (d) 5

33) If \(x + y + z = 19\), \(xy + yz + zx = 114\), then the value of \(\sqrt{x^3 + y^3 + z^3 - 3xyz}\) is:

• (a) 21
• (b) 17
• (c) 19
• (d) 13

34) If \(x + y = 1\) and \(xy(xy - 2) = 12\), then the value of \(x^4 + y^4\) is:

• (a) 19
• (b) 25
• (c) 20
• (d) 23

35) If \(ab + bc + ca = 8\) and \(a^2 + b^2 + c^2 = 20\), then a possible value of \[ \frac{1}{2}(a + b + c)[(a - b)^2 + (b - c)^2 + (c - a)^2] \] is:

• (a) 72
• (b) 56
• (c) 84
• (d) 80

36) If \(x = a + \frac{1}{a}\) and \(y = a - \frac{1}{a}\), then \[ \sqrt{x^4 + y^4 - 2x^2y^2} \] is equal to:

• (a) \(16a^2\)
• (b) 8
• (c) \(\frac{8}{a^2}\)
• (d) 4

37) If \(2x^2 + y^2 + 6x - 2xy + 9 = 0\), then the value of \(4x^3 - y^3 + x^2y^2\) is:

• (a) 0
• (b) 9
• (c) -3
• (d) -9

38) If \(a + \frac{1}{a} = 3\), then \[ \left(a^4 + \frac{1}{a^4}\right) \] is equal to:

• (a) 77
• (b) 47
• (c) 81
• (d) 27

39) If \(x^4 - 6x^2 - 1 = 0\), then the value of \[ x^6 - 5x^2 + \frac{5}{x^2} - \frac{1}{x^6} + 5 \] is:

• (a) 219
• (b) 209
• (c) 204
• (d) 239

40) If \(x = 2 - p\), then \(x^3 + 6xp + p^3\) is equal to:

• (a) 12
• (b) 6
• (c) 8
• (d) 4

41) If \(x^2 - 3x - 1 = 0\), then the value of \[ (x^2 + 8x - 1)\left(x^3 + x^{-1}\right)^{-1} \] is:

• (a) \(\frac{3}{8}\)
• (b) 8
• (c) 1
• (d) 3

42) If \(x + y + z = 0\), then the value of \[ \frac{x^2 + y^2 + z^2}{z - xy} \] is:

• (a) 1
• (b) 2
• (c) -2
• (d) -1

43) If \(\sqrt{x} - \frac{1}{\sqrt{x}} = 4\), then the value of \[ x^2 + \frac{1}{x^2} \] is:

• (a) 192
• (b) 326
• (c) 322
• (d) 256

44) If \[ (3x - 1)^3 + (4x - 3)^3 + (2x + 1)^3 = 3(3x - 1)(4x - 3)(2x + 1) \] and \(x \ne \frac{1}{3}\), then \(x =\):

• (a) \(\frac{1}{2}\)
• (b) 2
• (c) \(\frac{1}{4}\)
• (d) 1

45) If \(x + \frac{1}{x} = 4\), then the value of \[ x^5 + \frac{1}{x^5} \] is:

• (a) 736
• (b) 776
• (c) 684
• (d) 724

46) If \(x + y = 4\) and \(\frac{1}{x} + \frac{1}{y} = \frac{16}{15}\), then what is the value of \((x^3 + y^3)\)?

• (a) 18
• (b) 19
• (c) 21
• (d) 16

47) If \[ 8(x + y)^3 - 27(x - y)^3 = (5y - x)(Ax^2 + By^2 + Cxy), \] then what is the value of \((A + B - C)\)?

• (a) 16
• (b) -26
• (c) 36
• (d) -16

48) If \(x - y = 11\) and \[ \frac{1}{x} - \frac{1}{y} = \frac{11}{24}, \] then what is the value of \(x^3 - y^3 + x^2y^2\)?

• (a) 1331
• (b) 1115
• (c) 1105
• (d) 1307

49) If \(16x^2 + y^2 = 48\) and \(xy = 2\), with \(x, y > 0\), then the value of \(64x^3 + y^3\) is:

• (a) 320
• (b) 340
• (c) 300
• (d) 240

50) If \(x - \frac{1}{x} = 5, x \ne 0\), then what is the value of \[ \frac{x^6 - 5x^3 - 1}{x^6 + 7x^3 - 1} \]?

• (a) \(\frac{41}{45}\)
• (b) \(\frac{45}{41}\)
• (c) \(\frac{45}{49}\)
• (d) \(\frac{49}{45}\)