Q16. Let \(x = \sqrt[6]{27} - \sqrt{6\frac{3}{4}}\) and \(y = \frac{\sqrt{45} + \sqrt{605} + \sqrt{245}}{\sqrt{80} + \sqrt{125}}\), then the value of \(x^2 + y^2\) is:

• (a) 223/36
• (b) 221/36
• (c) 221/9
• (d) 227/9

Q17. If \(x\) is real and \(x^4 - 5x^2 - 1 = 0\), then the value of \[ x^6 - 3x^2 + \frac{3}{x^2} - \frac{1}{x^6} + 1 \] is:

• (a) 126
• (b) 110
• (c) 116
• (d) 96

Q18. If \(3^\sqrt[4]{x} + 4^\sqrt[4]{x} = 5^\sqrt[4]{x}\), then the value of \(x\) is:

• (a) 4
• (b) 2
• (c) 8
• (d) 16

Q19. If \(\sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7}\), then \(x^3 + \frac{1}{x^3}\) is equal to:

• (a) 140
• (b) 130
• (c) 120
• (d) 110

Q20. If roots of \(x^2 - 4x + a = 0\) are equal, then \(a =\):

• (a) -4
• (b) 4
• (c) 8
• (d) -8

Q21. If \(x + y = 7\) and \(xy = 10\), then the value of \(\frac{1}{x^3} + \frac{1}{y^3}\) is:

• (a) 0.543
• (b) 0.131
• (c) 0.133
• (d) 0.453

Q22. If \(x \ne -1, 2, 5\), then the simplified value of \[ \left\{ \frac{2(x^3 - 8)}{x^2 - x - 2} \times \frac{x^2 + 2x + 1}{x^2 - 4x - 5} \right\} \div \frac{x^2 + 2x + 4}{3x - 15} \] is equal to:

• (a) \(\frac{1}{6}\)
• (b) 6
• (c) \(\frac{3}{2}\)
• (d) \(\frac{2}{3}\)

Q23. If \(5^\sqrt[4]{x} + 12^\sqrt[4]{x} = 13^\sqrt[4]{x}\), then the value of \(x\) is:

• (a) 2
• (b) 8
• (c) 1
• (d) 4

Q24. If \(x = 2 - \sqrt{3}\), then the value of \[ x^3 - x^{-3} \] is:

• (a) \(-30\sqrt{3}\)
• (b) \(30\sqrt{3}\)
• (c) \(-30\sqrt{2}\)
• (d) \(30\sqrt{2}\)

Q25. The value of the expression \[ \frac{1}{4} \left\{ \left(a + \frac{1}{a} \right)^2 - \left(a - \frac{1}{a} \right)^2 \right\} \] is:

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

Q26. If \[ (x + y)^{\frac{1}{3}} + (z + y)^{\frac{1}{3}} = -(x + z)^{\frac{1}{3}}, \] then \[ x^3 + y^3 + z^3 \] can be expressed as:

• (a) \(\frac{1}{8}xyz\)
• (b) \((x + y)(y + z)(z + x)\)
• (c) \(\frac{3}{8}(x + y)(y + z)(z + x)\)
• (d) \(3xyz\)

Q27. If \((a + b + 4)(ab + 4(a + b)) - 4ab = 0\), \(a \ne -4\), \(b \ne -4\), then \[ \left\{ \frac{1}{(a + b + 4)^{117^{-2^{-234}}}} \right\} \] is equal to:

• (a) \(\frac{1}{4^{117}}\)
• (b) \(\frac{1}{2^{117}}\)
• (c) -\(\frac{1}{2^{234}}\)
• (d) 0

Q28. If \(a = \sqrt{8} - \sqrt{7}\) and \(a = \frac{1}{b}\), then the value of \[ \frac{a^2 + b^2 - 3ab}{a^2 + ab + b^2} \] is:

• (a) \(\frac{27}{31}\)
• (b) \(\frac{27}{32}\)
• (c) \(\frac{29}{33}\)
• (d) \(\frac{29}{31}\)

Q29. If (\(5\sqrt5 *x^3 - 81\sqrt3*{y^3}) ÷ (\sqrt{5}*x - 3\sqrt3*y) = A x^2 + B y^2 + C xy\), then the value of \((6A + B - \sqrt{15}C)\) is:

• (a) 10
• (b) 9
• (c) 15
• (d) 12

Q30. If \(x + y + z = 19\), \(x^2 + y^2 + z^2 = 133\), and \(xz = y^2\), then the difference between \(z\) and \(x\) is:

• (a) 5
• (b) 3
• (c) 6
• (d) 4