1. Problem Q1: Express $3\sqrt{11}$, $2\sqrt{23}$, $7\sqrt{2}$, $10$, and $4\sqrt{6}$ as entire surds and arrange in ascending order. 2. Convert each term to decimal approximation: - $3\sqrt{11} = 3 \times \sqrt{11} \approx 3 \times 3.317 = 9.951$ - $2\sqrt{23} = 2 \times \sqrt{23} \approx 2 \times 4.796 = 9.592$ - $7\sqrt{2} = 7 \times \sqrt{2} \approx 7 \times 1.414 = 9.898$ - $10$ is already a whole number. - $4\sqrt{6} = 4 \times \sqrt{6} \approx 4 \times 2.449 = 9.796$ 3. Arrange decimal values ascending: $9.592 (2\sqrt{23}), 9.796 (4\sqrt{6}), 9.898 (7\sqrt{2}), 9.951 (3\sqrt{11}), 10$ Therefore ascending order: $$2\sqrt{23} < 4\sqrt{6} < 7\sqrt{2} < 3\sqrt{11} < 10$$ --- 4. Problem Q2a: Simplify $\sqrt{63} - \sqrt{7} + \sqrt{28}$ 5. Express inside radicals with perfect squares: - $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$ - $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$ 6. Substitute and combine: $$3\sqrt{7} - \sqrt{7} + 2\sqrt{7} = (3 - 1 + 2)\sqrt{7} = 4\sqrt{7}$$ --- 7. Problem Q2b: Simplify $(2\sqrt{3} - 1)(3\sqrt{3} - 1)$ 8. Use distributive property: $$(2\sqrt{3})(3\sqrt{3}) - 2\sqrt{3} \times 1 - 1 \times 3\sqrt{3} + 1$$ 9. Calculate: - $(2\sqrt{3})(3\sqrt{3}) = 6 \times 3 = 18$ - $-2\sqrt{3} = -2\sqrt{3}$ - $-3\sqrt{3} = -3\sqrt{3}$ - $+1 = 1$ 10. Combine like terms: $$18 + 1 - (2\sqrt{3} + 3\sqrt{3}) = 19 - 5\sqrt{3}$$ --- 11. Problem Q3: Express $$\frac{5}{\sqrt{3} - 1} - \frac{1}{2 + \sqrt{3}}$$ with rational denominators. 12. Rationalize denominators: - For $$\frac{5}{\sqrt{3} - 1}$$ multiply numerator and denominator by $$\sqrt{3} + 1$$: $$\frac{5(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{5(\sqrt{3} + 1)}{3 - 1} = \frac{5(\sqrt{3} + 1)}{2}$$ - For $$\frac{1}{2 + \sqrt{3}}$$ multiply numerator and denominator by $$2 - \sqrt{3}$$: $$\frac{1(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 -3} = 2 - \sqrt{3}$$ 13. Subtract: $$\frac{5(\sqrt{3} + 1)}{2} - (2 - \sqrt{3}) = \frac{5\sqrt{3} + 5}{2} - 2 + \sqrt{3}$$ 14. Get common denominator 2: $$\frac{5\sqrt{3} + 5}{2} - \frac{4}{2} + \frac{2\sqrt{3}}{2} = \frac{5\sqrt{3} + 5 - 4 + 2\sqrt{3}}{2} = \frac{7\sqrt{3} + 1}{2}$$ Final answer: $$\frac{7\sqrt{3} + 1}{2}$$ --- 15. Problem Q4a: Solve $2x - 11 = |3x - 4|$ Case 1: $3x - 4 \geq 0 \implies x \geq \frac{4}{3}$, then $$2x - 11 = 3x - 4 \implies -x = 7 \implies x = -7$$ (contradicts $x \geq \frac{4}{3}$) Case 2: $3x -4 < 0 \implies x < \frac{4}{3}$, then $$2x - 11 = -(3x - 4) = -3x + 4 \implies 5x = 15 \implies x = 3$$ (contradicts $x < \frac{4}{3}$) No solution. --- 16. Problem Q4b: Solve $|5 - 3x| = |x + 3|$ Check intervals: 1) $x \geq -3$ and $5 - 3x \geq 0 \Rightarrow x \leq \frac{5}{3}$ Then $|5 - 3x| = 5 - 3x$, $|x + 3| = x + 3$, equation: $$5 - 3x = x + 3 \implies 4x = 2 \implies x = \frac{1}{2}$$ (satisfies interval) 2) $x \geq -3$ and $5 - 3x < 0 \Rightarrow x > \frac{5}{3}$ Equation: $$3x - 5 = x + 3 \implies 2x = 8 \implies x = 4$$ (satisfies interval) 3) $x < -3$ Equation: $$3x - 5 = -x - 3 \implies 4x = 2 \implies x = \frac{1}{2}$$ (doesn't satisfy $x < -3$) Solutions: $x=\frac{1}{2}, 4$ --- 17. Problem Q4c: Solve $|2x - 1| > 7$ Inequality splits to: $$2x - 1 > 7 \implies 2x > 8 \implies x > 4$$ Or $$2x - 1 < -7 \implies 2x < -6 \implies x < -3$$ Solution: $$x < -3 \text{ or } x > 4$$ --- 18. Problem Q5a: Sketch $y = (x+1)(2 - x) = -x^2 - x + 2$ - This is a downward-opening parabola. - Vertex at $$x = -\frac{b}{2a} = -\frac{-1}{2\times(-1)} = -\frac{1}{-2} = \frac{1}{2}$$ - Vertex value: $$y = - (\frac{1}{2})^2 - \frac{1}{2} + 2 = -\frac{1}{4} - \frac{1}{2} + 2 = \frac{5}{4}$$ - x-intercepts/find roots for $y=0$: $$(x+1)(2-x) = 0 \implies x = -1 \text{ or } x = 2$$ - y-intercept for $x=0$: $$y = (0+1)(2 - 0) = 2$$ --- 19. Problem Q5b: Sketch circle $x^2 + (y-1)^2 = 9$ - Center: $(0,1)$ - Radius: $3$ - x-intercepts: solve $y=0$ $$x^2 + (0-1)^2 = 9 \Rightarrow x^2 + 1 = 9 \Rightarrow x^2 = 8 \Rightarrow x = \pm 2\sqrt{2}$$ - y-intercepts: solve $x=0$ $$0 + (y-1)^2 = 9 \Rightarrow (y-1)^2=9 \Rightarrow y=1 \pm 3 $$ (i.e., $y=4$ and $y=-2$) --- 20. Problem Q5c: Sketch line $y - 2x -12 =0$ or $y=2x + 12$ - Slope = 2 - y-intercept = 12 - x-intercept: set $y=0$ $$0=2x + 12 \Rightarrow x = -6$$ --- 21. Problem Q5d: Sketch $y = |(x+1)(x+3)|$ - Inner quadratic roots: $x=-1$, $x=-3$ - Parabola opens upward, absolute value makes negative parts positive - y-intercepts: at $x=0$, $$y=|1 \times 3|=3$$ - Shape is a W-shaped graph touching x-axis at $-3$ and $-1$, with vertex minima of 0. --- 22. Problem Q6a: Inequality $x - 3y < -6$ Rewrite: $$-3y < -x -6 \Rightarrow y > \frac{x}{3} + 2$$ Region: above line $y = \frac{x}{3} + 2$ --- 23. Problem Q6b: Inequality $y \leq 2x - x^2$ This region is below or on the parabola $y=2x - x^2$ --- 24. Problem Q7a: Sets from $S = \{1,2,...,20\}$ - $A$ (prime numbers in S): $\{2,3,5,7,11,13,17,19\}$ - $B$ (factors of 30 less than 20): Factors of 30 are $1,2,3,5,6,10,15,30$ so $B=\{1,2,3,5,6,10,15\}$ - $C$ (subset $>1$ and $<9$): $\{2,3,4,5,6,7,8\}$ --- 25. Problem Q7b: Venn diagram showing relationship of sets A, B, C (visual representation omitted here as per instructions). --- 26. Problem Q7c: Find $(A \cup C)^c \cap B $ - $A \cup C = \{2,3,4,5,6,7,8,11,13,17,19\}$ - Complement relative to $S$ is all elements in $S$ not in $A \cup C$: $$ (A \cup C)^c = S \setminus (A \cup C) = \{1,9,10,12,14,15,16,18,20\}$$ - Now intersect with $B$: $$ (A \cup C)^c \cap B = \{1,9,10,12,14,15,16,18,20\} \cap \{1,2,3,5,6,10,15\} = \{1,10,15\} $$ --- 27. Problem Q7d: Verify $$ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) $$ Calculate: - $n(A) = 8$ - $n(B) = 7$ - $n(C) = 7$ - $A \cap B = \{2,3,5\}$ so $n=3$ - $A \cap C = \{2,3,5,7\}$ (primes in C) so $n=4$ - $B \cap C = \{2,3,5,6\}$ so $n=4$ - $A \cap B \cap C = \{2,3,5\}$ so $n=3$ Calculate LHS: - $A \cup B \cup C$ includes all unique elements in $A,B,C$ which are: $$\{1,2,3,4,5,6,7,8,10,11,13,15,17,19\}$$ - Count: 14 elements Calculate RHS: $$8 + 7 + 7 - 3 - 4 - 4 + 3 = 22 - 11 + 3 = 14$$ Verification true.