1. **Problem:** Determine which implication is valid if $x$ is a positive real number. - (a) $x$ is prime $\implies$ $x$ is even - (b) $x$ is whole number $\implies$ $x \in \{1, 2, 3, 4, ...\}$ **Step:** Recall that prime numbers are positive integers greater than 1 with only two divisors: 1 and itself. Most primes are odd except 2. - (a) is false because primes like 3, 5, 7 are not even. - (b) is true by definition of whole numbers. 2. **Problem:** Evaluate $\frac{450}{(0.003)^2}$ in standard form. **Step:** Calculate denominator: $(0.003)^2 = (3 \times 10^{-3})^2 = 9 \times 10^{-6}$. Then, $$\frac{450}{9 \times 10^{-6}} = 450 \times \frac{1}{9} \times 10^{6} = 50 \times 10^{6} = 5.0 \times 10^{7}.$$ 3. **Problem:** Find the percentage equivalent of the difference between 1.5 and $\frac{3}{4}$. **Step:** Difference = $1.5 - \frac{3}{4} = 1.5 - 0.75 = 0.75$. Percentage = $0.75 \times 100 = 75\%$. 4. **Problem:** Simplify $(1 + \frac{2}{5}) + 2 \frac{3}{5} - 3$. **Step:** Convert mixed number: $2 \frac{3}{5} = \frac{13}{5}$. Sum: $1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}$. Expression: $\frac{7}{5} + \frac{13}{5} - 3 = \frac{20}{5} - 3 = 4 - 3 = 1$. 5. **Problem:** Simplify $0.04 + 98.478 - (-8.2 + 4.09)$ to two decimal places. **Step:** Calculate inside parentheses: $-8.2 + 4.09 = -4.11$. Expression: $0.04 + 98.478 - (-4.11) = 0.04 + 98.478 + 4.11 = 102.628$. Rounded to two decimals: $102.63$. 6. **Problem:** Express sets as intervals. (a) $\{x : x \leq -4\} = (-\infty, -4]$ (b) $\{x : -3 < x \leq 8\} = (-3, 8]$ 7. **Problem:** Express interval $A = [-4, 10]$ as a set and mark elements $-2, -\frac{1}{4}, -\frac{1}{8}, 9$. **Step:** Set notation: $A = \{x : -4 \leq x \leq 10\}$. All given elements lie within $A$. 8. **Problem:** Make $w$ the subject of $z = \frac{v^2 + w^2}{t}$. **Step:** Multiply both sides by $t$: $zt = v^2 + w^2$. Isolate $w^2$: $w^2 = zt - v^2$. Take square root: $w = \pm \sqrt{zt - v^2}$. 9. **Problem:** Evaluate $32^{\frac{6}{5}} \times 2^{-6} \times 9^0$. **Step:** $9^0 = 1$. Rewrite $32 = 2^5$. So, $32^{\frac{6}{5}} = (2^5)^{\frac{6}{5}} = 2^{5 \times \frac{6}{5}} = 2^6 = 64$. Expression: $64 \times 2^{-6} = 64 \times \frac{1}{64} = 1$. 10. **Problem:** If $p = (2x - y)^3$, evaluate $p$ for $x = -1$, $y = -2$. **Step:** Calculate inside parentheses: $2(-1) - (-2) = -2 + 2 = 0$. Then, $p = 0^3 = 0$. 11. **Problem:** Find what must be added to $(4x + 2y)$ to get $(-2x + y)$. **Step:** Let $A$ be the expression to add. $4x + 2y + A = -2x + y$. Solve for $A$: $A = -2x + y - 4x - 2y = -6x - y$. 12. **Problem:** Rationalize $\frac{2}{4 - \sqrt{3}}$. **Step:** Multiply numerator and denominator by conjugate $4 + \sqrt{3}$: $$\frac{2}{4 - \sqrt{3}} \times \frac{4 + \sqrt{3}}{4 + \sqrt{3}} = \frac{2(4 + \sqrt{3})}{4^2 - (\sqrt{3})^2} = \frac{8 + 2\sqrt{3}}{16 - 3} = \frac{8 + 2\sqrt{3}}{13}.$$ 13. **Problem:** Express $\sqrt{13}(\sqrt{52} - \frac{27}{117})$ as a single surd. **Step:** Simplify $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$. Expression: $\sqrt{13} \times (2\sqrt{13} - \frac{27}{117}) = \sqrt{13} \times 2\sqrt{13} - \sqrt{13} \times \frac{27}{117}$. Calculate each term: $\sqrt{13} \times 2\sqrt{13} = 2 \times 13 = 26$. $\sqrt{13} \times \frac{27}{117} = \frac{27}{117} \sqrt{13}$. Final: $26 - \frac{27}{117} \sqrt{13}$. 14. **Problem:** Write numbers with absolute values. (a) $| -2 - 10| = |-12| = 12$. (b) $2| -6| + |2| - | -10| = 2 \times 6 + 2 - 10 = 12 + 2 - 10 = 4$. 15. **Problem:** Evaluate $\frac{6 \frac{3}{8} \times 1 \frac{7}{8}}{3 + \frac{3}{10}}$ without calculator. **Step:** Convert mixed numbers: $6 \frac{3}{8} = \frac{51}{8}$, $1 \frac{7}{8} = \frac{15}{8}$, $3 + \frac{3}{10} = \frac{33}{10}$. Multiply numerator: $\frac{51}{8} \times \frac{15}{8} = \frac{765}{64}$. Divide by denominator: $\frac{765}{64} \div \frac{33}{10} = \frac{765}{64} \times \frac{10}{33} = \frac{7650}{2112}$. Simplify numerator and denominator by 3: $\frac{2550}{704}$. Further simplification possible but this is exact fraction. 16. **Problem:** Sum and product of two real numbers is 16. (a) Equation: Let numbers be $a$ and $b$. $$a + b = 16$$ $$ab = 16$$ (b) Find $a$ and $b$. Use quadratic: $x^2 - 16x + 16 = 0$. Discriminant: $\Delta = 16^2 - 4 \times 16 = 256 - 64 = 192$. Roots: $$x = \frac{16 \pm \sqrt{192}}{2} = 8 \pm 4\sqrt{3}.$$ So numbers are $8 + 4\sqrt{3}$ and $8 - 4\sqrt{3}$. 17. **Problem:** For $x=4$, $y=3$, find $4x^2 y - 18x^2 y^3$. **Step:** Calculate powers: $x^2 = 16$, $y^3 = 27$. Expression: $$4 \times 16 \times 3 - 18 \times 16 \times 27 = 192 - 7776 = -7584.$$ 18. **Problem:** Solve for $x$ in $\frac{4(1+3x)}{7} = 2x$. **Step:** Multiply both sides by 7: $4(1+3x) = 14x$. Expand: $4 + 12x = 14x$. Rearranged: $4 = 14x - 12x = 2x$. Divide: $x = 2$. 19. **Problem:** Given sets: (a) $E = \{2,4,6,8,10,12,14,16,18,20,22,24,26,28\}$ $H = \{3,6,9,12,15,18,21,24,27\}$ (b) $I = \{1,3,5,7,9,11,13,15,17,19\}$ $V = \{2,3,5,7,11,13,17,19\}$ (i) Universal sets: - For (a): $U = \{1,2,3,...,29\}$ - For (b): $U = \{1,2,3,...,19\}$ (ii) Intersections: - $E \cap H = \{6,12,18,24\}$, number of elements = 4 - $I \cap V = \{3,5,7,11,13,17,19\}$, number of elements = 7 (iii) Unions: - $E \cup H = \{2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24,26,27,28\}$ - $I \cup V = \{1,2,3,5,7,9,11,13,15,17,19\}$ 20. **Problem:** Bacteria population starts at $10^6$. Every 10 minutes, one-tenth of remaining bacteria die. (a) After 10 minutes: Remaining = $\frac{9}{10} \times 10^6 = 9 \times 10^5$. (b) After 30 minutes (3 intervals): Remaining = $\left(\frac{9}{10}\right)^3 \times 10^6 = \frac{729}{1000} \times 10^6 = 7.29 \times 10^5$. (c) After 50 minutes (5 intervals): Remaining = $\left(\frac{9}{10}\right)^5 \times 10^6 = \frac{59049}{100000} \times 10^6 = 5.9049 \times 10^5$. **Final answers:** 1. (b) is valid implication. 2. $5.0 \times 10^{7}$ 3. $75\%$ 4. $1$ 5. $102.63$ 6. (a) $(-\infty, -4]$, (b) $(-3, 8]$ 7. $A = \{x : -4 \leq x \leq 10\}$ with elements $-2, -\frac{1}{4}, -\frac{1}{8}, 9$ inside. 8. $w = \pm \sqrt{zt - v^2}$ 9. $1$ 10. $0$ 11. $-6x - y$ 12. $\frac{8 + 2\sqrt{3}}{13}$ 13. $26 - \frac{27}{117} \sqrt{13}$ 14. (a) $12$, (b) $4$ 15. $\frac{2550}{704}$ 16. (a) $a+b=16$, $ab=16$ (b) $8 + 4\sqrt{3}$ and $8 - 4\sqrt{3}$ 17. $-7584$ 18. $x=2$ 19. (a) $U=\{1,...,29\}$, $E \cap H=\{6,12,18,24\}$ (4 elements), $E \cup H=\{2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24,26,27,28\}$ (b) $U=\{1,...,19\}$, $I \cap V=\{3,5,7,11,13,17,19\}$ (7 elements), $I \cup V=\{1,2,3,5,7,9,11,13,15,17,19\}$ 20. (a) $9 \times 10^5$, (b) $7.29 \times 10^5$, (c) $5.9049 \times 10^5$