Secondary 3 Additional Mathematics Numbers Ratio Proportion Quiz

Secondary 3 Additional Mathematics Quiz - Numbers Ratio Proportion (Answer Key)

1. Simplify $\sqrt{75} - 2\sqrt{12} + \sqrt{27}$
$\sqrt{75} = 5\sqrt{3}$
$2\sqrt{12} = 4\sqrt{3}$
$\sqrt{27} = 3\sqrt{3}$
$5\sqrt{3} - 4\sqrt{3} + 3\sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$
[M1 for simplifying surds, A1 for final answer]

2. Rationalise $\frac{6}{\sqrt{5} - \sqrt{2}}$
Multiply by conjugate $\sqrt{5} + \sqrt{2}$:
$\frac{6(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{6(\sqrt{5} + \sqrt{2})}{3} = 2(\sqrt{5} + \sqrt{2})$
Answer: $2\sqrt{5} + 2\sqrt{2}$
[M1 for conjugate method, A1 for final answer]

3. Find $x^2 + y^2$ given $x = \sqrt{3} + 1, y = \sqrt{3} - 1$
$x^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}$
$y^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$
Sum $= 8$
Answer: $8$
[M1 for expansion, A1 for final answer]

4. Solve $\sqrt{2x + 5} = x - 1$
Square both sides: $2x + 5 = x^2 - 2x + 1 \Rightarrow x^2 - 4x - 4 = 0$
$x = \frac{4 \pm \sqrt{16 + 16}}{2} = 2 \pm 2\sqrt{2}$
Check: $x = 2 - 2\sqrt{2}$ gives negative RHS, reject.
Answer: $x = 2 + 2\sqrt{2}$
[M1 for quadratic, A1 for valid root]

5. Express $\frac{3\sqrt{2}}{\sqrt{6} - \sqrt{2}}$ in form $a + b\sqrt{3}$
Multiply by $\sqrt{6} + \sqrt{2}$:
Numerator: $3\sqrt{12} + 3(2) = 6\sqrt{3} + 6$
Denominator: $6 - 2 = 4$
Result: $\frac{6 + 6\sqrt{3}}{4} = \frac{3}{2} + \frac{3}{2}\sqrt{3}$
Answer: $\frac{3}{2} + \frac{3}{2}\sqrt{3}$
[M1 for rationalisation, A1 for final form]

6. Variation $y \propto \frac{x^2}{z}$
$y = \frac{kx^2}{z}$. Sub $x=2, z=3, y=8$:
$8 = \frac{4k}{3} \Rightarrow k = 6$
Answer: $y = \frac{6x^2}{z}$
[M1 for finding k, A1 for equation]

7. Ratio Problem
Let boys $= 5u$, girls $= 4u$.
$\frac{5u - 5}{4u + 5} = \frac{4}{5} \Rightarrow 25u - 25 = 16u + 20 \Rightarrow 9u = 45 \Rightarrow u = 5$
Original Boys $= 5(5) = 25$
Answer: $25$
[M1 for equation, A1 for answer]

8. Joint Variation $A = kB\sqrt{C}$
$24 = k(3)\sqrt{16} = k(3)(4) = 12k \Rightarrow k = 2$
Answer: $2$
[M1 for substitution, A1 for k]

9. Partial Variation $C = k_1 + k_2 A$
$120 = k_1 + 20k_2$
$180 = k_1 + 35k_2$
Subtract: $60 = 15k_2 \Rightarrow k_2 = 4$
Answer: $4$
[M1 for solving simultaneous eq, A1 for variable part]

10. Profit Sharing
Total parts $= 3 + 4 + 5 = 12$
Bob's share $= \frac{4}{12} \times 24000 = \frac{1}{3} \times 24000 = 8000$
Answer: \8,000$
[M1 for fraction, A1 for amount]

11. Calculate $y$ when $x=3, z=4$ using $y = \frac{6x^2}{z}$
$y = \frac{6(3^2)}{4} = \frac{54}{4} = 13.5$
Answer: $13.5$
[M1 for substitution, A1 for answer]

12. Find $C$ when $A=10, B=5$ using $A = 2B\sqrt{C}$
$10 = 2(5)\sqrt{C} \Rightarrow 10 = 10\sqrt{C} \Rightarrow \sqrt{C} = 1 \Rightarrow C = 1$
Answer: $1$
[M1 for substitution, M1 for solving, A1 for answer]

13. Cost for Area $50 \text{ m}^2$
From Q9, $k_2 = 4$. Find $k_1$: $120 = k_1 + 20(4) \Rightarrow k_1 = 40$.
Formula: $C = 40 + 4A$
$C = 40 + 4(50) = 40 + 200 = 240$
Answer: \240$
[M1 for finding constant, M1 for final calc, A1 for answer]

14. Percentage Increase in Period
$T = k\sqrt{l}$. New $l' = 1.44l$.
$T' = k\sqrt{1.44l} = 1.2 k\sqrt{l} = 1.2 T$
Increase $= 20\%$
Answer: $20\%$
[M1 for relationship, M1 for percentage, A1 for answer]

15. Express $z$ in terms of $x, y$
$\frac{1}{z} = \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}$
$z = \frac{xy}{x + y}$
Answer: $z = \frac{xy}{x + y}$
[M1 for common denominator, M1 for inversion, A1 for answer]

16. Value of $z$ given $x = 2 + \sqrt{3}, y = 2 - \sqrt{3}$
$x + y = 4$
$xy = (2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1$
$z = \frac{1}{4}$
Answer: $\frac{1}{4}$
[M1 for sum/product, A1 for final value]

17. Map Area Calculation
Scale $1 : 50,000$. Area scale $1^2 : 50,000^2 = 1 : 2.5 \times 10^9$.
Actual Area $= 12 \text{ km}^2 = 12 \times (10^5 \text{ cm})^2 = 12 \times 10^{10} \text{ cm}^2$.
Map Area $= \frac{12 \times 10^{10}}{2.5 \times 10^9} = \frac{120}{2.5} = 48 \text{ cm}^2$.
Answer: $48 \text{ cm}^2$
[M1 for area scale, M1 for conversion, A1 for answer]

18. Actual Perimeter
Map dims: $4 \text{ cm}, 6 \text{ cm}$. Map Perim $= 2(4+6) = 20 \text{ cm}$.
Actual Perim $= 20 \times 50,000 \text{ cm} = 1,000,000 \text{ cm}$.
In km: $1,000,000 / 100,000 = 10 \text{ km}$.
Answer: $10 \text{ km}$
[M1 for map perim, M1 for scale conversion, A1 for answer]

19. Simplify $\frac{\sqrt{x^2 - 4x + 4}}{x - 2}$ for $x > 2$
$\sqrt{(x-2)^2} = |x-2|$. Since $x > 2$, $|x-2| = x - 2$.
$\frac{x - 2}{x - 2} = 1$
Answer: $1$
[M1 for factorisation, M1 for modulus logic, A1 for answer]

20. Bob's Cash Take-home
Bob's Share = \8,000$(from Q10). Reinvest$20%$:$0.20 \times 8000 = 1600$. Take home:$8000 - 1600 = 6400$. **Answer:**$$6,400$
[M1 for percentage, M1 for subtraction, A1 for answer]