1. Convert the number systems: 1.1 Convert binary $10101110_2$ to denary: $$10101110_2 = 1\times2^7 + 0\times2^6 + 1\times2^5 + 0\times2^4 + 1\times2^3 + 1\times2^2 + 1\times2^1 + 0\times2^0$$ $$= 128 + 0 + 32 + 0 + 8 + 4 + 2 + 0 = 174_{10}$$ 1.2 Convert binary $1111011011_2$ to hexadecimal: Group bits in 4 from right: $11\ 1101\ 1011_2$ Add leading zeros: $0011\ 1101\ 1011_2$ Convert each group: $0011_2=3$, $1101_2=D$, $1011_2=B$ So, $1111011011_2 = 3DB_{16}$ 1.3 Convert denary $(10.25)_{10}$ to binary: Integer part 10: $10_{10} = 1010_2$ Fractional part 0.25: $0.25 \times 2 = 0.5$ (bit 0) $0.5 \times 2 = 1.0$ (bit 1) So fractional part = $0.01_2$ Therefore, $10.25_{10} = 1010.01_2$ 1.4 Convert binary $11111101_2$ to octal: Group bits in 3 from right: $111\111\101_2$ Convert each group: $111_2=7$, $111_2=7$, $101_2=5$ So, $11111101_2 = 775_8$ 1.5 Convert denary $(1988.75)_{10}$ to hexadecimal: Integer part 1988: $1988 \div 16 = 124$ remainder $4$ $124 \div 16 = 7$ remainder $12 (C)$ $7 \div 16 = 0$ remainder $7$ So integer part = $7C4_{16}$ Fractional part 0.75: $0.75 \times 16 = 12.0$ (C) So fractional part = $0.C_{16}$ Therefore, $1988.75_{10} = 7C4.C_{16}$ 2. Simplify equations: 2.1 $-\frac{1}{2}(10x-2)+3=7(1-2x)$ Expand: $-5x + 1 + 3 = 7 - 14x$ Simplify: $-5x + 4 = 7 - 14x$ Add $14x$ both sides: $9x + 4 = 7$ Subtract 4: $9x = 3$ Divide by 9: $x = \frac{1}{3}$ 2.2 $4 - 3q = 2q - 11$ Add $3q$ both sides: $4 = 5q - 11$ Add 11: $15 = 5q$ Divide by 5: $q = 3$ 3. Solve inequalities: 3.1 $4x + 5 < 6x + 9$ Subtract $4x$: $5 < 2x + 9$ Subtract 9: $-4 < 2x$ Divide by 2: $-2 < x$ 3.2 $\frac{2x - 5}{3} > \frac{3x + 3}{4}$ Cross multiply: $4(2x - 5) > 3(3x + 3)$ $8x - 20 > 9x + 9$ Subtract $8x$: $-20 > x + 9$ Subtract 9: $-29 > x$ Or $x < -29$ 3.3 $\frac{7 - 2t}{4} \geq 1$ Multiply both sides by 4: $7 - 2t \geq 4$ Subtract 7: $-2t \geq -3$ Divide by -2 (reverse inequality): $t \leq \frac{3}{2}$ 4. Sets and Cartesian product: 4.1 Equal sets have exactly the same elements. Example: $A = \{1,2,3\}$ and $B = \{3,2,1\}$ are equal. Equivalent sets have the same number of elements but not necessarily the same elements. Example: $A = \{1,2,3\}$ and $C = \{a,b,c\}$ are equivalent. 4.2 Cartesian product $P \times Q$: $P = \{2,3,6,7\}$, $Q = \{r,s,t\}$ $P \times Q = \{(2,r),(2,s),(2,t),(3,r),(3,s),(3,t),(6,r),(6,s),(6,t),(7,r),(7,s),(7,t)\}$ $Q \times P = \{(r,2),(r,3),(r,6),(r,7),(s,2),(s,3),(s,6),(s,7),(t,2),(t,3),(t,6),(t,7)\}$ $P \times Q \neq Q \times P$ 4.3 Survey problem: Let total surveyed = $N$ Use inclusion-exclusion: $$N = |M| + |P| + |R| - |M \cap P| - |P \cap R| - |M \cap R| + |M \cap P \cap R| + \text{none}$$ $$= 51 + 49 + 60 - 34 - 32 - 36 + 24 + 1 = 83$$ 4.4 Number who like Rice but not Posho nor Matooke: $$|R| - |P \cap R| - |M \cap R| + |M \cap P \cap R| = 60 - 32 - 36 + 24 = 16$$ 4.5 Number who like Posho or Rice: $$|P \cup R| = |P| + |R| - |P \cap R| = 49 + 60 - 32 = 77$$ 5. Function graph questions omitted (no graph provided). 6. Simplify expressions: 6.1 $24 \times 23 = 2^4 \times 2^3 = 2^{7} = 128$ 6.2 $(42)^3 = 4^{2 \times 3} = 4^6 = (2^2)^6 = 2^{12} = 4096$ 6.3 Solve for $x$: 6.3.1 $2x = 8 \Rightarrow x = 4$ 6.3.2 $3x + 1 = 27 \Rightarrow 3x = 26 \Rightarrow x = \frac{26}{3}$ 6.3.3 $10x = 0.01 \Rightarrow x = \log_{10}(0.01) = -2$ 7. Logarithms: 7.1 $\log_4(64x^2y^3)$ $$= \log_4(64) + \log_4(x^2) + \log_4(y^3)$$ $$= 3 + 2\log_4(x) + 3\log_4(y)$$ 7.2 $\log\left(\frac{100x^3}{y^5}\right)$ $$= \log(100) + \log(x^3) - \log(y^5)$$ $$= 2 + 3\log(x) - 5\log(y)$$ 7.3 Simplify $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$ $$= \frac{\log \frac{25}{125} + \log 625^{1/2}}{3 \log 5} = \frac{\log \frac{25}{125} + \log 25}{3 \log 5}$$ $$= \frac{\log \frac{25}{125} \times 25}{3 \log 5} = \frac{\log 5}{3 \log 5} = \frac{1}{3}$$ 7.4 Solve $\log(x-1) + \log(x+1) = 2(\log x + 2)$ $$\log((x-1)(x+1)) = 2\log x + 4$$ $$\log(x^2 - 1) = \log(x^2) + 4$$ $$\log(x^2 - 1) - \log(x^2) = 4$$ $$\log\left(\frac{x^2 - 1}{x^2}\right) = 4$$ $$\frac{x^2 - 1}{x^2} = 10^4 = 10000$$ $$x^2 - 1 = 10000 x^2$$ $$-1 = 9999 x^2$$ No real solution since left side negative and right side positive. 7.5 Simplify $\log_3(\sqrt{27} x^2)$ $$= \log_3(27^{1/2}) + \log_3(x^2) = \frac{1}{2} \log_3(27) + 2 \log_3(x)$$ $$= \frac{1}{2} \times 3 + 2 \log_3(x) = \frac{3}{2} + 2 \log_3(x)$$