Multiple Choice

1. Simplify: $12 - [4 - (6 - 9)] \div 5$

   $11$

   $\frac{59}{5}$

   $\frac{53}{5}$

   $\frac{57}{5}$

2. Simplify: $\frac{18}{3} - \left( 5 - \frac{14}{7} \right) \times 2$

   $0$

   $10$

   $-1$

   $6$

3. Evaluate: $8 - \left[ 2 - \left( \frac{9}{3} - 5 \right) \right]$

   $4$

   $8$

   $-4$

   $0$

4. Simplify: $20 \div (4 + 1) \times (6 - 2)$

   $20$

   $1$

   $16$

   $4$

5. Simplify: $7 - \left( \frac{15}{3} + 2^3 \right)$

   $10$

   $-4$

   $-6$

   $6$

6. Simplify: $\frac{48}{12} - \Big( 6 - \frac{9}{3} \Big)^2$

   $5$

   $-2$

   $-23$

   $-5$

7. Evaluate: $\left( \frac{2}{5} - \frac{1}{10} \right) \times 20$

   $4$

   $10$

   $2$

   $6$

8. Simplify: $\frac{3}{4} \div \left( \frac{5}{8} - \frac{1}{2} \right)$

   $\frac{3}{32}$

   $2$

   $6$

   $\frac{1}{6}$

9. Solve: $5x - 7 = 2x + 8$

   $x = \frac{15}{7}$

   $x = 5$

   $x = 1$

   $x = 3$

10. Solve: $\frac{2x}{3} + 4 = 10$

    $x = 21$

    $x = 3$

    $x = 9$

    $x = 6$

11. Solve: $7(x - 2) = 3(x + 4) + 2$

    $x = 7$

    $x = 6$

    $x = 8$

    $x = 2$

12. Solve: $\frac{x - 5}{4} = \frac{2x + 1}{6}$

    $x = 13$

    $x = -13$

    $x = -17$

    $x = -3$

13. Solve: $3x - \frac{5}{2} = \frac{x}{4} + \frac{7}{2}$

    $x = 2$

    $x = \frac{6}{11}$

    $x = \frac{4}{11}$

    $x = \frac{24}{11}$

14. Solve: $2(3x - 4) = 5(2x - 7) + 6$

    $x = \frac{27}{4}$

    $x = \frac{21}{4}$

    $x = \frac{33}{4}$

    $x = -\frac{3}{4}$

15. Solve: $\frac{5x}{2} - 7 = \frac{3x}{4} + 2$

    $x = -\frac{20}{7}$

    $x = \frac{20}{7}$

    $x = \frac{36}{7}$

    $x = \frac{9}{7}$

16. Solve: $\frac{2(x+3)}{5} - \frac{x-1}{3} = 1$

    $x = -8$

    $x = -22$

    $x = 8$

    $x = 2$

17. What is the solution to the inequality $3x - 7 < 2x + 5$?

    $x < 2$

    $x < 12$

    $x > 2$

    $x > 12$

18. What is the solution to the inequality $\frac{2x - 5}{3} \geq \frac{x + 7}{2}$?

    $x \leq 31$

    $x \geq 11$

    $x \leq 11$

    $x \geq 31$

19. What is the solution to the inequality $5 - 2x \leq 3(x + 1)$?

    $x \leq \frac{2}{5}$

    $x \geq 2$

    $x \leq 2$

    $x \geq \frac{2}{5}$

20. What is the solution to the inequality $\frac{x+4}{2} - \frac{x-2}{3} < 5$?

    $x < 10$

    $x > 14$

    $x > 10$

    $x < 14$

21. What is the solution to the inequality $2(x - 3) + 5 \geq x + 12$?

    $x \geq 17$

    $x \leq 17$

    $x \geq 13$

    $x \leq 13$

22. What is the solution to the inequality $\frac{7 - 2x}{4} > \frac{x}{2} - 1$?

    $x < \frac{11}{4}$

    $x > \frac{11}{4}$

    $x < \frac{3}{4}$

    $x > \frac{3}{4}$

23. Simplify: $\left(\frac{3x^{-1}y^2}{2x^2 y^{-3}}\right)^4$

    $\frac{12y^{20}}{8x^{12}}$

    $\frac{81y^{20}}{16x^{4}}$

    $\frac{81y^{10}}{16x^{12}}$

    $\frac{81y^{20}}{16x^{12}}$

24. Simplify: $\frac{2^{-3} \cdot 4^2}{8^{-1}}$

    $1/16$

    $1/4$

    $2$

    $16$

25. Simplify: $\frac{(x^{-2}y)^3}{(x^3 y^{-2})^2}$

    $\frac{y}{x^{12}}$

    $\frac{y^7}{x^{12}}$

    $\frac{y^7}{x^{6}}$

    $\frac{y^7}{x^{0}}$

26. Simplify: $\left(\frac{p^{-1}q^2}{r^{-3}}\right)^{-2}$

    $\frac{q^4 r^6}{p^2}$

    $\frac{p^2}{q^4 r^6}$

    $\frac{p^2 q^4}{r^6}$

    $\frac{p^2 r^6}{q^4}$

27. Simplify: $\frac{(a^2b^{-3})^{-2}(a^{-1}b)^3}{a^{-4}b^{-2}}$

    $\frac{b^{5}}{a^3}$

    $\frac{b^{11}}{a^{11}}$

    $\frac{b^{11}}{a^3}$

    $\frac{a^3}{b^{11}}$

28. Simplify: $\frac{(x^{-2}y^3z^{-1})^2 \cdot (x^3y^{-1}z)^4}{x^{-5}y^2z^{-3}}$

    $x^{3}z^5$

    $x^{13}z^5$

    $\frac{x^{13}z^5}{y^4}$

    $x^{13}y^4z^5$

29. Simplify: $\frac{(a^{-3}b^2c)^3 \cdot (a^2b^{-1})^{-2}}{a^{-4}b^3c^{-2}}$

    $\frac{a^9c^5}{b^5}$

    $\frac{b^5c^5}{a^9}$

    $\frac{b^5c}{a^9}$

    $\frac{b^5c^5}{a^{17}}$

30. Simplify: $\frac{(p^2q^{-3}r)^4}{(p^{-1}q^2r^{-2})^{-3}}$

    $\frac{p^{11}}{q^{18}r^{10}}$

    $\frac{p^5 q^6}{r^2}$

    $\frac{p^5}{q^6r^2}$

    $\frac{q^6r^2}{p^5}$

31. Simplify: $\frac{(x^{-1}y^2z^{-3})^{-2} \cdot (x^4yz^{-1})^3}{(x^2y^{-1}z^2)^{-1}}$

    $\frac{x^{16}y^2}{z^5}$

    $\frac{x^{16}z}{y^2}$

    $\frac{x^{12}z^5}{y^2}$

    $\frac{x^{16}z^5}{y^2}$

32. Simplify: $\sqrt{50}$

    $2\sqrt{5}$

    $5\sqrt{2}$

    $25$

    $5\sqrt{10}$

33. Simplify: $\sqrt{72} - \sqrt{32}$

    $4\sqrt{2}$

    $2\sqrt{2}$

    $\sqrt{40}$

    $10\sqrt{2}$

34. Simplify: $\sqrt{18} \cdot \sqrt{12}$

    $6\sqrt{2}$

    $\sqrt{216}$

    $5\sqrt{6}$

    $6\sqrt{6}$

35. Simplify: $\frac{\sqrt{45}}{\sqrt{5}}$

    $\sqrt{9}$

    $3$

    $\sqrt{50}$

    $5$

36. Rationalise: $\frac{5}{\sqrt{3}}$

    $\frac{5\sqrt{3}}{3}$

    $\frac{5}{3}$

    $5\sqrt{3}$

    $\frac{\sqrt{3}}{5}$

37. Rationalise: $\frac{7}{2 - \sqrt{3}}$

    $14 - 7\sqrt{3}$

    $2 + \sqrt{3}$

    $14 + 7\sqrt{3}$

    $7(2 - \sqrt{3})$

38. Simplify: $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$

    $\frac{7 + 2\sqrt{10}}{3}$

    $\frac{7 - 2\sqrt{10}}{3}$

    $7 + 2\sqrt{10}$

    $\frac{3}{7 + 2\sqrt{10}}$

39. Simplify: $\frac{1}{\sqrt{7} + \sqrt{3}}$

    $\frac{\sqrt{7} - \sqrt{3}}{4}$

    $\sqrt{7} - \sqrt{3}$

    $\frac{\sqrt{7} + \sqrt{3}}{4}$

    $\frac{4}{\sqrt{7} - \sqrt{3}}$

40. Solve the quadratic equation $2x^2 - 5x - 12 = 0$.

    $x=4,\; x=\frac{3}{2}$

    $x=3,\; x=-\frac{2}{3}$

    $x=4,\; x=-\frac{3}{2}$

    $x=-4,\; x=\frac{3}{2}$

41. Solve the quadratic equation $3x^2 + 2x - 8 = 0$.

    $x=\frac{3}{4},\; x=-2$

    $x=\frac{4}{3},\; x=-2$

    $x=-\frac{3}{4},\; x=2$

    $x=-\frac{4}{3},\; x=2$

42. Solve the quadratic equation $4x^2 - 12x + 9 = 0$.

    $x=\frac{2}{3}$

    $x=-\frac{3}{2}$

    $x=\frac{3}{2}$

    $x=-\frac{2}{3}$

43. Solve the equation $\frac{1}{2}x^2-\frac{3}{2}x+1=0$.

    $x=1,\; x=-2$

    $x=-1,\; x=-2$

    $x=-1,\; x=2$

    $x=1,\; x=2$

44. Solve the equation $5x^2 - 45 = 0$.

    $x=\pm 3$

    $x=\pm \sqrt{3}$

    $x=\pm 9$

    $x=3$

45. Solve the quadratic equation $6x^2 + 5x - 6 = 0$.

    $x=\frac{2}{3},\; x=-\frac{3}{2}$

    $x=\frac{3}{2},\; x=-\frac{2}{3}$

    $x=-\frac{2}{3},\; x=\frac{3}{2}$

    $x=\frac{1}{2},\; x=-3$

46. Solve the equation $(x+5)(x-2) = -10$.

    $x=-2,\; x=5$

    $x=0,\; x=3$

    $x=2,\; x=-5$

    $x=0,\; x=-3$

47. Solve the equation $(3x-2)^2-(x+1)^2=0$.

    $x=-\frac{3}{2},\; x=-\frac{1}{4}$

    $x=\frac{3}{2},\; x=\frac{1}{4}$

    $x=\frac{2}{3},\; x=-1$

    $x=0,\; x=1$

48. Classify the roots of $2x^2 - 7x + 3 = 0$.

    No real roots (complex conjugates).

    Exactly one real root (repeated).

    Two distinct real irrational roots.

    Two distinct real rational roots.

49. Classify the roots of $\tfrac12 x^2 - 5x + 6 = 0$.

    No real roots (complex conjugates).

    Exactly one real root (repeated).

    Two distinct real rational roots.

    Two distinct real irrational roots.

50. Classify the roots of $x^2 - 4x + 8 = 0$.

    Exactly one real root (repeated).

    No real roots (complex conjugates).

    Two distinct real irrational roots.

    Two distinct real rational roots.

51. Find all real $k$ for which $x^2 - (k+3)x + k = 0$ has equal roots.

    $k = -3$

    $k = -1$

    $k = 3$

    No real $k$ satisfies this condition.

52. For which $m$ does $2x^2 + (m-1)x + (m+4) = 0$ have real roots?

    $m \le 5 + 2\sqrt{14}$

    $m = 5 - 2\sqrt{14}$ or $m = 5 + 2\sqrt{14}$

    $m \le 5 - 2\sqrt{14}$ or $m \ge 5 + 2\sqrt{14}$.

    $5 - 2\sqrt{14} < m < 5 + 2\sqrt{14}$

53. Find $k$ so that $4x^2 + kx + 9 = 0$ has a repeated root.

    $k = 12$

    $k = \pm 12$.

    No real $k$

    $k = -12$

54. For which $t$ does $t x^2 - 6x + 9 = 0$ have exactly one solution as a quadratic?

    $t = -1$

    $t = 0$

    $t = 1$.

    $t = 4$

55. Determine all $p$ for which $(p+1)x^2 - 2(p-2)x + p = 0$ has no real roots.

    $p < \frac{4}{5}$

    $p > \frac{4}{5}$.

    $p = \frac{4}{5}$

    $p \le \frac{4}{5}$

56. Solve $2x^2 - 3x - 5 = 0$.

    $x = 5, -1$

    $x = -\frac{5}{2}, -1$

    $x = \frac{5}{2}, -1$

    $x = \frac{3}{2}, 1$

57. Solve $5x^2 + 2x + 7 = 0$.

    $x = \frac{-1 \pm \sqrt{34}}{5}$

    $x = \frac{1 \pm i\sqrt{34}}{5}$

    $x = \frac{-1 \pm i\sqrt{39}}{5}$

    $x = \frac{-1 \pm i\sqrt{34}}{5}$

58. Solve $x^2 - 6x + 7 = 0$.

    $x = 3 \pm \sqrt{2}$

    $x = 6 \pm \sqrt{2}$

    $x = 3 \pm \sqrt{8}$

    $x = -3 \pm \sqrt{2}$

59. Solve $\frac{1}{2}x^2 - 5x + 6 = 0$.

    $x = 5 \pm \sqrt{13}$

    $x = -5 \pm \sqrt{13}$

    $x = \frac{5 \pm \sqrt{13}}{2}$

    $x = 5 \pm \sqrt{7}$

60. Solve $4x^2 + x - 3 = 0$.

    $x = \frac{3}{2}, -1$

    $x = \frac{3}{4}, -1$

    $x = -\frac{3}{4}, 1$

    $x = \frac{1}{4}, -3$

61. Solve $7x^2 - 2x - 5 = 0$.

    $x = 1, -\frac{2}{7}$

    $x = \frac{2}{7}, -5$

    $x = 1, -\frac{5}{7}$

    $x = -1, \frac{5}{7}$

62. Solve $(x+1)(x-3)=7$.

    $x = 1 \pm \sqrt{10}$

    $x = 1 \pm \sqrt{11}$

    $x = -1 \pm \sqrt{11}$

    $x = 2 \pm \sqrt{11}$

63. Solve $\frac{3}{4}x^2-\frac{5}{2}x+1=0$.

    $x = \frac{2 \pm \sqrt{13}}{3}$

    $x = \frac{5 \pm \sqrt{17}}{3}$

    $x = \frac{5 \pm \sqrt{13}}{3}$

    $x = \frac{5 \pm \sqrt{13}}{4}$