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A Level H1 Mathematics Numbers Ratio Proportion Quiz

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Questions

A-Level Maths H1 Quiz - Numbers Ratio Proportion

Name: __________________________
Class: __________________________
Date: __________________________
Score: _________ / 60

Duration: 60 Minutes
Total Marks: 60

Instructions to Candidates:

Answer all 20 questions.
Write your answers in the spaces provided.
All non-exact numerical answers must be correct to 3 significant figures, unless otherwise specified.
Give non-exact answers in terms of $\pi$ or surds where appropriate.
You are expected to use a graphing calculator.
Unsupported answers from a graphing calculator are allowed unless the question specifically states otherwise.

Section A: Algebraic Manipulation and Indices (15 Marks)

1. Simplify the expression $\frac{2^{n+2} - 2^{n+1}}{2^n}$ , giving your answer as an integer. [2]

2. Given that $x = 3^{2a}$ and $y = 3^{a+1}$ , express $\frac{x^2}{y}$ in the form $3^k$ , where $k$ is an integer in terms of $a$ . [2]

3. Solve the equation $4^{x} - 5(2^{x}) + 4 = 0$ . [3]

4. Without using a calculator, simplify $\sqrt{75} + \sqrt{12} - \sqrt{27}$ , giving your answer in the form $k\sqrt{3}$ where $k$ is an integer. [2]

5. Rationalize the denominator of $\frac{6}{\sqrt{5} - \sqrt{2}}$ and simplify your answer. [3]

6. Given that $\log_2 x + \log_2 (x-2) = 3$ , find the value of $x$ . [3]

Section B: Ratio, Proportion and Variation (20 Marks)

7. The variable $y$ is inversely proportional to the square of $x$ . Given that $y=12$ when $x=2$ , find the value of $y$ when $x=4$ . [3]

8. The resistance $R$ of a wire varies directly as its length $L$ and inversely as the square of its diameter $d$ . (a) Write down the formula connecting $R$ , $L$ , and $d$ , using $k$ as the constant of proportionality. [1] (b) If the length is doubled and the diameter is halved, find the factor by which the resistance changes. [3]

9. A sum of $5000 is divided among three people, A, B, and C, in the ratio $2:3:5$ . (a) Calculate the amount received by person B. [2] (b) Person C gives 20% of their share to person A. Calculate the new ratio of A's share to C's share, giving your answer in its simplest form. [3]

10. The cost of running a factory consists of a fixed cost and a variable cost that is proportional to the number of units produced. When 100 units are produced, the total cost is $1200. When 250 units are produced, the total cost is $2100. (a) Find the fixed cost. [3] (b) Find the total cost when 400 units are produced. [2]

11. The intensity of light $I$ from a source is inversely proportional to the square of the distance $d$ from the source. <image_placeholder> id: Q11-fig1 type: graph linked_question: Q11 description: A graph showing the relationship between Intensity I (y-axis) and 1/d^2 (x-axis). The line is straight and passes through the origin. labels: y-axis: Intensity I (units), x-axis: 1/d^2 (m^-2) values: The line passes through the point (0.25, 80). must_show: A straight line graph passing through (0,0) and (0.25, 80). </image_placeholder>

(a) Using the graph above, determine the constant of proportionality $k$ . [2] (b) Calculate the distance $d$ when the intensity is 20 units. [2]

Section C: Applications in Business and Social Sciences (25 Marks)

12. A company's profit $P$ is modeled by the function $P(x) = -2x^2 + 80x - 500$ , where $x$ is the number of items sold in hundreds. (a) Find the number of items (in hundreds) that must be sold to maximize profit. [2] (b) Calculate the maximum profit. [2]

13. The population of a city is growing exponentially. In 2010, the population was 1.2 million. In 2020, the population was 1.5 million. (a) Find the annual growth rate $r$ , assuming the model $P(t) = P_0 e^{rt}$ , where $t$ is the number of years since 2010. [3] (b) Estimate the population in 2030. [2]

14. A bank offers an interest rate of 4% per annum, compounded monthly. (a) Calculate the effective annual interest rate. [3] (b) How many years will it take for an investment to double in value? Give your answer to the nearest year. [3]

15. The demand $D$ for a product is related to its price $p$ by the equation $D = \frac{1000}{p+10}$ . (a) Find the price $p$ when the demand is 50 units. [2] (b) If the price increases by 10%, calculate the percentage change in demand. [3]

16. A mixture of two chemicals, A and B, is prepared in the ratio $3:2$ by volume. Chemical A costs $12 per liter and Chemical B costs $18 per liter. (a) Find the cost per liter of the mixture. [3] (b) If the cost of Chemical A increases by 20% and the cost of Chemical B decreases by 10%, find the new cost per liter of the mixture. [3]

17. The index number for the price of a basket of goods in 2022, with 2020 as the base year, is 115. (a) If the cost of the basket in 2020 was $800, calculate the cost in 2022. [2] (b) If the inflation rate from 2022 to 2023 is 5%, calculate the index number for 2023 with 2020 as the base year. [3]

18. A car depreciates in value by 15% each year. (a) Write down an expression for the value of the car $V$ after $n$ years, if its initial value is $30,000. [2] (b) Find the number of years it takes for the car's value to drop below $10,000. [3]

19. The ratio of men to women in a company is $3:5$ . The average salary of men is $45,000 and the average salary of women is $55,000. (a) Find the average salary of all employees in the company. [3] (b) If 10 men and 10 women are hired, does the average salary increase, decrease, or remain the same? Justify your answer. [2]

20. A business project requires an initial investment of $10,000. The returns are expected to be $3,000 at the end of year 1, $4,000 at the end of year 2, and $5,000 at the end of year 3. Using a discount rate of 10% per annum, calculate the Net Present Value (NPV) of the project. Formula: $NPV = \sum \frac{C_t}{(1+r)^t} - C_0$ [4]

Answers

A-Level Maths H1 Quiz - Numbers Ratio Proportion - Answer Key

1. Simplify $\frac{2^{n+2} - 2^{n+1}}{2^n}$

Numerator: $2^{n+2} - 2^{n+1} = 2^n \cdot 2^2 - 2^n \cdot 2^1 = 2^n(4 - 2) = 2^n(2)$
Expression: $\frac{2^n \cdot 2}{2^n} = 2$
Answer: 2 [2]

2. Express $\frac{x^2}{y}$ in the form $3^k$

$x = 3^{2a} \implies x^2 = (3^{2a})^2 = 3^{4a}$
$y = 3^{a+1}$
$\frac{x^2}{y} = \frac{3^{4a}}{3^{a+1}} = 3^{4a - (a+1)} = 3^{3a-1}$
Answer: $3^{3a-1}$ so $k = 3a-1$ [2]

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

Let $u = 2^x$ . Then $4^x = (2^2)^x = (2^x)^2 = u^2$ .
Equation becomes $u^2 - 5u + 4 = 0$ .
Factorize: $(u-4)(u-1) = 0$ .
$u = 4$ or $u = 1$ .
Case 1: $2^x = 4 \implies 2^x = 2^2 \implies x = 2$ .
Case 2: $2^x = 1 \implies 2^x = 2^0 \implies x = 0$ .
Answer: $x = 0, 2$ [3]

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

$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$
$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
Sum: $5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (5+2-3)\sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$ [2]

5. Rationalize $\frac{6}{\sqrt{5} - \sqrt{2}}$

Multiply numerator and denominator by conjugate $\sqrt{5} + \sqrt{2}$ .
$\frac{6(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} = \frac{6(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{6(\sqrt{5} + \sqrt{2})}{3}$
Simplify: $2(\sqrt{5} + \sqrt{2}) = 2\sqrt{5} + 2\sqrt{2}$
Answer: $2\sqrt{5} + 2\sqrt{2}$ [3]

6. Solve $\log_2 x + \log_2 (x-2) = 3$

Combine logs: $\log_2 (x(x-2)) = 3$
Convert to index form: $x(x-2) = 2^3 = 8$
$x^2 - 2x - 8 = 0$
Factorize: $(x-4)(x+2) = 0$
$x = 4$ or $x = -2$ .
Check validity: For $\log_2 x$ , $x > 0$ . For $\log_2 (x-2)$ , $x > 2$ .
$x = -2$ is rejected. $x = 4$ is valid.
Answer: $x = 4$ [3]

7. Inverse proportion $y \propto \frac{1}{x^2}$

Formula: $y = \frac{k}{x^2}$
Find $k$ : $12 = \frac{k}{2^2} \implies 12 = \frac{k}{4} \implies k = 48$ .
Equation: $y = \frac{48}{x^2}$
Find $y$ when $x=4$ : $y = \frac{48}{4^2} = \frac{48}{16} = 3$ .
Answer: 3 [3]

8. Resistance variation (a) $R = \frac{kL}{d^2}$ [1] (b) New $L' = 2L$ , New $d' = \frac{1}{2}d$ .

$R' = \frac{k(2L)}{(\frac{1}{2}d)^2} = \frac{2kL}{\frac{1}{4}d^2} = 8 \frac{kL}{d^2} = 8R$ .
The resistance increases by a factor of 8.
Answer: 8 times [3]

9. Ratio division (a) Total parts = $2+3+5 = 10$ .

Value of one part = $\frac{5000}{10} = 500$ .
B's share = $3 \times 500 = 1500$ .
Answer: $1500 [2] (b) C's initial share = $5 \times 500 = 2500$ .
C gives 20% to A: $0.20 \times 2500 = 500$ .
New C = $2500 - 500 = 2000$ .
A's initial share = $2 \times 500 = 1000$ .
New A = $1000 + 500 = 1500$ .
New Ratio A:C = $1500 : 2000 = 15 : 20 = 3 : 4$ .
Answer: $3:4$ [3]

10. Linear Cost Model $C = a + bN$

Eq 1: $1200 = a + 100b$
Eq 2: $2100 = a + 250b$
Subtract Eq 1 from Eq 2: $900 = 150b \implies b = 6$ .
Substitute $b=6$ into Eq 1: $1200 = a + 600 \implies a = 600$ . (a) Fixed cost $a = 600$ . Answer: $600 [3] (b) Cost for 400 units: $C = 600 + 6(400) = 600 + 2400 = 3000$ .
Answer: $3000 [2]

11. Light Intensity Graph (a) Graph is $I$ vs $\frac{1}{d^2}$ . Equation $I = k(\frac{1}{d^2})$ .

Gradient $k = \frac{\Delta I}{\Delta (1/d^2)} = \frac{80 - 0}{0.25 - 0} = \frac{80}{0.25} = 320$ .
Answer: $k = 320$ [2] (b) $I = 20$ . $20 = \frac{320}{d^2} \implies d^2 = \frac{320}{20} = 16$ .
$d = \sqrt{16} = 4$ meters.
Answer: 4 m [2]

12. Profit Maximization (a) $P(x) = -2x^2 + 80x - 500$ . This is a downward parabola.

Vertex at $x = -\frac{b}{2a} = -\frac{80}{2(-2)} = \frac{80}{4} = 20$ .
Answer: 20 (hundred items) [2] (b) Max Profit $P(20) = -2(20)^2 + 80(20) - 500$ .
$P(20) = -2(400) + 1600 - 500 = -800 + 1600 - 500 = 300$ .
Answer: $300 [2]

13. Exponential Growth (a) $P(t) = 1.2 e^{rt}$ . At $t=10$ (2020), $P=1.5$ .

$1.5 = 1.2 e^{10r} \implies e^{10r} = \frac{1.5}{1.2} = 1.25$ .
$10r = \ln(1.25) \implies r = \frac{\ln(1.25)}{10} \approx 0.02231$ .
Answer: $r \approx 0.0223$ [3] (b) 2030 is $t=20$ .
$P(20) = 1.2 e^{20(0.02231)} = 1.2 (e^{10r})^2 = 1.2 (1.25)^2$ .
$P(20) = 1.2 (1.5625) = 1.875$ .
Answer: 1.875 million [2]

14. Compound Interest (a) Effective Annual Rate (EAR) for 4% compounded monthly.

$EAR = (1 + \frac{0.04}{12})^{12} - 1$ .
$EAR = (1.00333...)^{12} - 1 \approx 1.04074 - 1 = 0.04074$ .
Answer: 4.07% [3] (b) Double value: $2 = (1 + \frac{0.04}{12})^{12t}$ .
$\ln 2 = 12t \ln(1 + \frac{0.04}{12})$ .
$t = \frac{\ln 2}{12 \ln(1.00333...)} \approx \frac{0.6931}{12(0.003327)} \approx \frac{0.6931}{0.0399} \approx 17.36$ .
Nearest year: 17 years.
Answer: 17 years [3]

15. Demand Function (a) $D = 50$ . $50 = \frac{1000}{p+10}$ .

$p+10 = \frac{1000}{50} = 20$ .
$p = 10$ .
Answer: $10 [2] (b) Price increases by 10%: New $p = 10(1.10) = 11$ .
New Demand $D' = \frac{1000}{11+10} = \frac{1000}{21} \approx 47.619$ .
% Change = $\frac{47.619 - 50}{50} \times 100\% = \frac{-2.381}{50} \times 100\% \approx -4.76\%$ .
Answer: Decrease of 4.76% [3]

16. Mixture Cost (a) Ratio 3:2. Total parts 5.

Cost = $\frac{3(12) + 2(18)}{5} = \frac{36 + 36}{5} = \frac{72}{5} = 14.4$ .
Answer: $14.40 per liter [3] (b) New Cost A = $12(1.20) = 14.40$ . New Cost B = $18(0.90) = 16.20$ .
New Mixture Cost = $\frac{3(14.40) + 2(16.20)}{5} = \frac{43.2 + 32.4}{5} = \frac{75.6}{5} = 15.12$ .
Answer: $15.12 per liter [3]

17. Index Numbers (a) Index 115 means 115% of base.

Cost 2022 = $800 \times \frac{115}{100} = 8 \times 115 = 920$ .
Answer: $920 [2] (b) Inflation 5% from 2022 to 2023.
Cost 2023 = $920 \times 1.05 = 966$ .
Index 2023 (Base 2020) = $\frac{966}{800} \times 100 = 1.2075 \times 100 = 120.75$ .
Answer: 120.75 [3]

18. Depreciation (a) $V = 30000(1 - 0.15)^n = 30000(0.85)^n$ .

Answer: $V = 30000(0.85)^n$ [2] (b) $10000 > 30000(0.85)^n$ .
$\frac{1}{3} > 0.85^n$ .
$\ln(1/3) > n \ln(0.85)$ .
$n > \frac{\ln(1/3)}{\ln(0.85)} = \frac{-1.0986}{-0.1625} \approx 6.76$ .
Since $n$ must be an integer year for "drop below", at $n=6$ , $V \approx 11296$ . At $n=7$ , $V \approx 9601$ .
It takes 7 years.
Answer: 7 years [3]

19. Weighted Average Salary (a) Let number of men = $3x$ , women = $5x$ .

Total Salary = $3x(45000) + 5x(55000) = 135000x + 275000x = 410000x$ .
Total Employees = $8x$ .
Average = $\frac{410000x}{8x} = 51250$ .
Answer: $51,250 [3] (b) New hires: 10 men ($45k) and 10 women ($55k).
Average of new hires = $\frac{10(45000)+10(55000)}{20} = 50000$ .
Since the average of the new group ($50,000) is less than the current average ($51,250), the overall average will decrease.
Answer: Decrease [2]

20. Net Present Value (NPV)

$r = 0.10$ .
$PV_1 = \frac{3000}{1.1^1} = 2727.27$
$PV_2 = \frac{4000}{1.1^2} = \frac{4000}{1.21} = 3305.79$
$PV_3 = \frac{5000}{1.1^3} = \frac{5000}{1.331} = 3756.57$
Total PV Inflows = $2727.27 + 3305.79 + 3756.57 = 9789.63$
$NPV = 9789.63 - 10000 = -210.37$
Answer: -$210.37 [4]