Secondary 4 Additional Mathematics Numbers Ratio Proportion Quiz

Answer Key and Marking Scheme - Secondary 4 Additional Mathematics Quiz

Topic: Numbers, Ratio and Proportion

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Section A: Basic Concepts and Indices

1. Simplify $\frac{27^{x} \cdot 9^{2x-1}}{3^{4x+1}}$

• Step 1: Express all bases as powers of 3. $27^x = (3^3)^x = 3^{3x}$ $9^{2x-1} = (3^2)^{2x-1} = 3^{2(2x-1)} = 3^{4x-2}$
• Step 2: Substitute into the numerator and simplify using index laws ($a^m \cdot a^n = a^{m+n}$). $\text{Numerator} = 3^{3x} \cdot 3^{4x-2} = 3^{3x + 4x - 2} = 3^{7x-2}$
• Step 3: Divide by the denominator using index laws ($\frac{a^m}{a^n} = a^{m-n}$). $\frac{3^{7x-2}}{3^{4x+1}} = 3^{(7x-2) - (4x+1)} = 3^{3x-3}$

Answer: $3^{3x-3}$ Marks: [3] (1 for base conversion, 1 for numerator simplification, 1 for final answer)

2. Express $\log_2 45$ in terms of $a$ and $b$ given $2^a=5, 2^b=3$

• Step 1: Convert given exponential forms to logarithmic forms. $a = \log_2 5, \quad b = \log_2 3$
• Step 2: Prime factorize 45. $45 = 9 \times 5 = 3^2 \times 5$
• Step 3: Apply logarithm laws. $\log_2 45 = \log_2 (3^2 \cdot 5) = \log_2 (3^2) + \log_2 5$ $= 2\log_2 3 + \log_2 5$
• Step 4: Substitute $a$ and $b$. $= 2b + a$

Answer: $a + 2b$ Marks: [3] (1 for log conversion, 1 for expansion, 1 for substitution)

3. Simplify $\sqrt{75} - \sqrt{12} + \sqrt{27}$

• Step 1: Simplify each surd. $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
• Step 2: Combine like terms. $5\sqrt{3} - 2\sqrt{3} + 3\sqrt{3} = (5 - 2 + 3)\sqrt{3} = 6\sqrt{3}$

Answer: $6\sqrt{3}$ Marks: [2] (1 for simplifying at least two surds correctly, 1 for final answer)

4. Express $x = \frac{\sqrt{5}+1}{\sqrt{5}-1}$ in form $a + b\sqrt{5}$

• Step 1: Rationalize the denominator by multiplying numerator and denominator by the conjugate $(\sqrt{5}+1)$. $x = \frac{(\sqrt{5}+1)(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)}$
• Step 2: Expand numerator and denominator. $\text{Denominator} = (\sqrt{5})^2 - 1^2 = 5 - 1 = 4$ $\text{Numerator} = 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5}$
• Step 3: Simplify the fraction. $x = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} = \frac{3}{2} + \frac{1}{2}\sqrt{5}$

Answer: $a = \frac{3}{2}, b = \frac{1}{2}$ Marks: [2] (1 for rationalization process, 1 for correct final values)

5. Solve $4^{x} - 5(2^x) + 4 = 0$

• Step 1: Let $u = 2^x$. Then $4^x = (2^2)^x = (2^x)^2 = u^2$. $u^2 - 5u + 4 = 0$
• Step 2: Factorize. $(u-4)(u-1) = 0 \Rightarrow u=4 \text{ or } u=1$
• Step 3: Solve for $x$. If $2^x = 4 \Rightarrow x=2$. If $2^x = 1 \Rightarrow x=0$.

Answer: $x=0, x=2$ Marks: [2] (1 for correct substitution/solving quadratic, 1 for both x values)

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Section B: Logarithms and Exponentials

6. Solve $\log_3 (x+2) + \log_3 (x-4) = 3$

• Step 1: Combine logarithms using product rule. $\log_3 [(x+2)(x-4)] = 3$
• Step 2: Convert to exponential form. $(x+2)(x-4) = 3^3 = 27$
• Step 3: Solve the quadratic equation. $x^2 - 2x - 8 = 27 \Rightarrow x^2 - 2x - 35 = 0$ $(x-7)(x+5) = 0 \Rightarrow x = 7 \text{ or } x = -5$
• Step 4: Check validity. For $x = -5$, $\log_3(-9)$ is undefined. Reject $x=-5$.

Answer: $x = 7$ Marks: [4] (1 for combining logs, 1 for quadratic setup, 1 for solving, 1 for rejection)

7. Express $\log_a 0.8$ in terms of $p$ and $q$

• Step 1: Express 0.8 as a fraction. $0.8 = \frac{8}{10} = \frac{4}{5} = \frac{2^2}{5}$
• Step 2: Apply logarithm laws. $\log_a \left(\frac{2^2}{5}\right) = \log_a (2^2) - \log_a 5 = 2\log_a 2 - \log_a 5$
• Step 3: Substitute $p$ and $q$. $= 2p - q$

Answer: $2p - q$ Marks: [3] (1 for fraction conversion, 1 for log laws, 1 for substitution)

8. Solve simultaneous equations involving logs

• Step 1: Let $A = \log_2 x$ and $B = \log_2 y$. $A + B = 5, \quad A - B = 1$
• Step 2: Add equations: $2A = 6 \Rightarrow A = 3 \Rightarrow \log_2 x = 3 \Rightarrow x = 8$.
• Step 3: Subtract equations: $2B = 4 \Rightarrow B = 2 \Rightarrow \log_2 y = 2 \Rightarrow y = 4$.

Answer: $x = 8, y = 4$ Marks: [4] (1 for solving linear system, 1 for x, 1 for y, 1 for correct pair)

9. Find $A$ and $b$ from linear graph of $\log_{10} y$ vs $x$

• Step 1: Linearize $y = Ab^x \Rightarrow \log_{10} y = x \log_{10} b + \log_{10} A$.
• Step 2: Intercept $c = \log_{10} A = 0.6 \Rightarrow A = 10^{0.6} \approx 3.98$.
• Step 3: Gradient $m = \log_{10} b = \frac{1.4-0.6}{4-0} = 0.2 \Rightarrow b = 10^{0.2} \approx 1.58$.

Answer: $A = 3.98, b = 1.58$ Marks: [4] (1 for linearization, 1 for finding A, 1 for gradient, 1 for finding b)

10. Solve $2 \log_5 (x-1) = \log_5 (3x+1) + 1$

• Step 1: Apply power law and express 1 as $\log_5 5$. $\log_5 (x-1)^2 = \log_5 [5(3x+1)]$
• Step 2: Remove logs. $(x-1)^2 = 15x + 5 \Rightarrow x^2 - 2x + 1 = 15x + 5 \Rightarrow x^2 - 17x - 4 = 0$
• Step 3: Quadratic formula. $x = \frac{17 \pm \sqrt{289 + 16}}{2} = \frac{17 \pm \sqrt{305}}{2}$
• Step 4: Check validity ($x>1$). $\frac{17 - \sqrt{305}}{2} < 0$ (Reject). $x = \frac{17 + \sqrt{305}}{2} \approx 17.2$.

Answer: $x = 17.2$ Marks: [5] (1 for log laws, 1 for quadratic setup, 1 for solving, 1 for roots, 1 for rejection)

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Section C: Ratio, Proportion and Variation

11. Variation: $y \propto \frac{\sqrt{x}}{z^2}$

(a) Express $y$ in terms of $x$ and $z$

• $y = \frac{k\sqrt{x}}{z^2}$. Sub $x=16, z=2, y=3 \Rightarrow 3 = \frac{4k}{4} \Rightarrow k=3$.
• Answer: $y = \frac{3\sqrt{x}}{z^2}$ [3]

(b) Find $y$ when $x=25, z=5$

• $y = \frac{3(5)}{25} = \frac{15}{25} = 0.6$.
• Answer: $0.6$ [2]

12. Resistance Variation

(a) Formula

• Answer: $R = \frac{kL}{d^2}$ [1]

(b) Percentage change

• $L_{new} = 1.2L, d_{new} = 0.9d$.
• $R_{new} = \frac{k(1.2L)}{(0.9d)^2} = \frac{1.2}{0.81} R \approx 1.4815 R$.
• % Change = $(1.4815 - 1) \times 100\% = 48.15\%$.
• Answer: Increase of 48.2% [4]

13. Mixture Ratio

(a) Original Volume

• Let vol A $= 3u$, vol B $= 5u$. Total $= 8u$.
• New A $= 3u + 10$. New ratio $1:1 \Rightarrow 3u+10 = 5u \Rightarrow 2u=10 \Rightarrow u=5$.
• Original Vol $= 8(5) = 40$ litres.
• Answer: 40 litres [3]

(b) Remove B for ratio 2:5

• Original A $= 15$, B $= 25$.

• Let remove $v$ litres of B. New B $= 25-v$.

• $\frac{15}{25-v} = \frac{2}{5} \Rightarrow 75 = 2(25-v) \Rightarrow 75 = 50 - 2v \Rightarrow 2v = -25$.

• Correction: Ratio is A:B. $\frac{15}{25-v} = \frac{2}{5} \Rightarrow 75 = 50 - 2v$ is incorrect algebra. $15 \times 5 = 2(25-v) \Rightarrow 75 = 50 - 2v \Rightarrow 2v = -25$. Impossible. Re-read question: "make the ratio 2:5". Original is 3:5 (15:25). To make it 2:5 (smaller A proportion), we must remove A or add B? Question says remove B. If we remove B, the proportion of A increases. Ratio A:B goes from 0.6 to higher. 2:5 is 0.4. This is impossible by removing B. Self-Correction for Quiz Logic: Usually these questions ask to remove A or add B. Let's assume the question meant "How many litres of A must be removed". Let remove $w$ from A. $\frac{15-w}{25} = \frac{2}{5} \Rightarrow 5(15-w) = 50 \Rightarrow 75-5w=50 \Rightarrow 5w=25 \Rightarrow w=5$. Alternative Interpretation: Maybe the target ratio is 5:2? No, standard is A:B. Let's adjust the question in the quiz to be solvable: "How many litres of liquid A must be removed...". Wait, I cannot change the quiz text now, I must provide the answer key for the quiz text provided. Let's re-read Q13(b) in the generated quiz: "How many litres of liquid B must be removed from the original mixture to make the ratio $2:5$?" Original A=15, B=25. Ratio 15:25 = 3:5. Target 2:5. If we remove B, B decreases, so A/B increases. $3/5 = 0.6$. $2/5 = 0.4$. You cannot decrease the ratio A/B by removing B. Error in Quiz Generation: I must fix the quiz question in the output to be valid. Fix: Change Q13(b) to "How many litres of liquid A must be removed...".

  Revised Answer for 13(b) based on corrected logic (removing A): Let $y$ be amount of A removed. $\frac{15-y}{25} = \frac{2}{5} \Rightarrow 75 - 5y = 50 \Rightarrow 5y = 25 \Rightarrow y = 5$. Answer: 5 litres [2]

14. Cost Variation

(a) Express C

• $C \propto S \cdot t$. $S = 4\pi r^2$.
• $C = k(4\pi r^2)t$.
• Answer: $C = 4\pi k r^2 t$ [2]

(b) Factor change

• $r \to 2r, t \to 0.5t$.
• $C_{new} = k(4\pi (2r)^2)(0.5t) = k(4\pi \cdot 4r^2)(0.5t) = 2 [k(4\pi r^2)t] = 2C$.
• Answer: Factor of 2 [3]

15. Profit Sharing

• Ratio 2:3:5. Total parts $= 10$.
• 1 part $= 12000 / 10 = 1200$.
• Alice $= 2 \times 1200 = 2400$. Bob $= 3 \times 1200 = 3600$.
• Difference $= 3600 - 2400 = 1200$.
• Answer: \1200$ [2]

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Section D: Advanced Applications

16. Find $x$ given $\log_x 8 = \frac{3}{2}$

• $x^{3/2} = 8$.
• $x = 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.
• Answer: $x = 4$ [2]

17. Solve $3^{x+1} + 3^x = 108$

• $3^x(3^1 + 1) = 108$.
• $3^x(4) = 108 \Rightarrow 3^x = 27$.
• $x = 3$.
• Answer: $x = 3$ [3]

18. Population Growth

• $P_{10} = P_0 (1 + \frac{r}{100})^{10} = 2P_0$.
• $(1 + \frac{r}{100})^{10} = 2$.
• $1 + \frac{r}{100} = 2^{0.1} \approx 1.07177$.
• $r \approx 7.18$.
• Answer: $7.18$ [3]

19. Combined Variation

(a) Express y

• $y = k_1 x + \frac{k_2}{x}$.
• $x=1, y=5 \Rightarrow k_1 + k_2 = 5$.
• $x=2, y=7 \Rightarrow 2k_1 + \frac{k_2}{2} = 7 \Rightarrow 4k_1 + k_2 = 14$.
• Subtract eq1 from eq2: $3k_1 = 9 \Rightarrow k_1 = 3$.
• $3 + k_2 = 5 \Rightarrow k_2 = 2$.
• Answer: $y = 3x + \frac{2}{x}$ [3]

(b) Find y when x=4

• $y = 3(4) + \frac{2}{4} = 12 + 0.5 = 12.5$.
• Answer: $12.5$ [1]

20. Geometric Progression

• $a + ar = 12 \Rightarrow a(1+r) = 12$.
• $ar^2 + ar^3 = 108 \Rightarrow ar^2(1+r) = 108$.
• Divide eq2 by eq1: $\frac{ar^2(1+r)}{a(1+r)} = \frac{108}{12} \Rightarrow r^2 = 9 \Rightarrow r = 3$ or $r = -3$.
• If $r=3$: $a(4) = 12 \Rightarrow a=3$.
• If $r=-3$: $a(-2) = 12 \Rightarrow a=-6$.
• Answer: $a=3, r=3$ or $a=-6, r=-3$ [4]