Evaluation G - 9 | 1.1 Solutions
1.1.1 Simplify $x^{\frac{2}{3}} \times x^{\frac{1}{6}}$. (A) $x^{\frac{1}{2}}$ (B) $x^{\frac{5}{6}}$ (C) $x^{\frac{1}{3}}$ (D) $x^{\frac{3}{4}}$ (E) $x^{\frac{2}{9}}$.
Add exponents: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$ → $x^{\frac{5}{6}}$ → Answer: B. $x^{\frac{5}{6}}$
1.1.2 Evaluate $16^{\frac{3}{4}}$. (A) 4 (B) 6 (C) 8 (D) 12 (E) 2.
$16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = (2)^3 = 8$ or $\sqrt[4]{16^3} = \sqrt[4]{4096} = 8$ → Answer: C. 8
1.1.3 If $27^{k} = 9$, what is $k$? (A) $\frac{2}{3}$ (B) $\frac{3}{2}$ (C) $\frac{1}{3}$ (D) $\frac{1}{2}$ (E) 3.
$27 = 3^3$, $9 = 3^2$ → $(3^3)^k = 3^{3k} = 3^2$ → $3k = 2$ → $k = \frac{2}{3}$ → Answer: A. $\frac{2}{3}$
1.1.4 Simplify $\left( a^{\frac{1}{3}} b^{\frac{2}{3}} \right)^{6}$. (A) $a^2 b^4$ (B) $a^3 b^2$ (C) $a^4 b^2$ (D) $a^2 b^3$ (E) $a^3 b^4$.
Multiply exponents: $a^{\frac{1}{3} \times 6} = a^2$, $b^{\frac{2}{3} \times 6} = b^4$ → $a^2 b^4$ → Answer: A. $a^2 b^4$
1.1.5 Solve for $x$: $x^{\frac{5}{2}} = 32$. (A) 2 (B) 4 (C) 6 (D) 8 (E) 16.
Raise both sides to $\frac{2}{5}$: $x = 32^{\frac{2}{5}} = (32^{\frac{1}{5}})^2 = 2^2 = 4$ → Answer: B. 4
1.1.6 Which is equivalent to $\frac{1}{\sqrt[3]{x^2}}$? (A) $x^{\frac{2}{3}}$ (B) $x^{-\frac{2}{3}}$ (C) $x^{\frac{3}{2}}$ (D) $x^{-\frac{3}{2}}$ (E) $x^{\frac{1}{3}}$.
$\sqrt[3]{x^2} = x^{\frac{2}{3}}$, so $\frac{1}{x^{\frac{2}{3}}} = x^{-\frac{2}{3}}$ → Answer: B. $x^{-\frac{2}{3}}$
1.1.7 The volume of a cube is $125 \text{ cm}^3$. What is the length of one side in cm? (A) 3 (B) 5 (C) 15 (D) 25 (E) $125^{\frac{1}{3}}$.
Side length = $\sqrt[3]{125} = 5$ → Answer: B. 5
1.1.8 Simplify $\sqrt[4]{81x^8 y^4}$. (A) $3x^2 y$ (B) $9x^2 y$ (C) $3x^4 y$ (D) $3x^2 y^2$ (E) $9x^4 y^2$.
$\sqrt[4]{81} = 3$, $\sqrt[4]{x^8} = x^{8/4}=x^2$, $\sqrt[4]{y^4}=y$ → $3x^2 y$ → Answer: A. $3x^2 y$
1.1.9 If $p = 8$ and $q = 27$, find $p^{\frac{2}{3}} + q^{\frac{1}{3}}$. (A) 5 (B) 7 (C) 9 (D) 11 (E) 13.
$8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4$, $27^{\frac{1}{3}} = 3$ → $4 + 3 = 7$ → Answer: B. 7
1.1.10 Simplify $\frac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}}$. (A) $x^{\frac{1}{4}}$ (B) $x^{\frac{5}{4}}$ (C) $x^{\frac{1}{2}}$ (D) $x^{\frac{3}{2}}$ (E) $x^{\frac{1}{3}}$.
Subtract exponents: $\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$ → $x^{\frac{1}{4}}$ → Answer: A. $x^{\frac{1}{4}}$
1.1.11 Which is larger: $16^{\frac{3}{2}}$ or $32^{\frac{4}{5}}$? (A) $16^{\frac{3}{2}}$ (B) $32^{\frac{4}{5}}$ (C) They are equal (D) Cannot be compared (E) Both are 64.
$16^{\frac{3}{2}} = (16^{\frac{1}{2}})^3 = 4^3 = 64$, $32^{\frac{4}{5}} = (32^{\frac{1}{5}})^4 = 2^4 = 16$ → $64 > 16$ → Answer: A. $16^{\frac{3}{2}}$
1.1.12 Simplify $\sqrt[3]{x} \times \sqrt[6]{x}$. (A) $\sqrt{x}$ (B) $\sqrt[9]{x}$ (C) $x$ (D) $x^{\frac{1}{2}}$ (E) $x^2$.
$x^{\frac{1}{3}} \times x^{\frac{1}{6}} = x^{\frac{1}{3}+\frac{1}{6}} = x^{\frac{2}{6}+\frac{1}{6}} = x^{\frac{3}{6}} = x^{\frac{1}{2}} = \sqrt{x}$ → Answer: A. $\sqrt{x}$
1.1.13 A bacteria culture doubles every hour. The population after $t$ hours is $P(t) = 100 \times 2^t$. After how many hours does the population reach 25600? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10.
$100 \times 2^t = 25600$ → $2^t = 256$ → $2^t = 2^8$ → $t = 8$ → Answer: C. 8
1.1.14 If $m^{\frac{1}{3}} = 5$, find $m^{\frac{2}{3}}$. (A) 5 (B) 10 (C) 15 (D) 20 (E) 25.
$m^{\frac{2}{3}} = (m^{\frac{1}{3}})^2 = 5^2 = 25$ → Answer: E. 25
1.1.15 A square has area $72 \text{ cm}^2$. Find its side length in simplest radical form. (A) $6\sqrt{2}$ (B) $3\sqrt{8}$ (C) $8\sqrt{3}$ (D) $2\sqrt{18}$ (E) $9\sqrt{2}$.
Side = $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ → Answer: A. $6\sqrt{2}$
1.1.16 Simplify $\left( \frac{8}{27} \right)^{-\frac{2}{3}}$. (A) $\frac{4}{9}$ (B) $\frac{9}{4}$ (C) $\frac{2}{3}$ (D) $\frac{3}{2}$ (E) $\frac{16}{81}$.
$\left( \frac{8}{27} \right)^{-\frac{2}{3}} = \left( \frac{27}{8} \right)^{\frac{2}{3}} = \frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}} = \frac{(27^{\frac{1}{3}})^2}{(8^{\frac{1}{3}})^2} = \frac{3^2}{2^2} = \frac{9}{4}$ → Answer: B. $\frac{9}{4}$
1.1.17 If $y = x^{\frac{3}{2}}$ and $y = 64$, what is $x$? (A) 4 (B) 8 (C) 12 (D) 16 (E) 24.
$x^{\frac{3}{2}} = 64$ → raise both sides to $\frac{2}{3}$: $x = 64^{\frac{2}{3}} = (64^{\frac{1}{3}})^2 = 4^2 = 16$ → Answer: D. 16
1.1.18 Simplify $\sqrt[3]{54} - \sqrt[3]{16}$. (A) $\sqrt[3]{38}$ (B) $2\sqrt[3]{2}$ (C) $\sqrt[3]{2}$ (D) $3\sqrt[3]{2}$ (E) 0.
$\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}$, $\sqrt[3]{16} = \sqrt[3]{8 \times 2} = 2\sqrt[3]{2}$ → $3\sqrt[3]{2} - 2\sqrt[3]{2} = \sqrt[3]{2}$ → Answer: C. $\sqrt[3]{2}$
1.1.19 An investment of ₦50,000 grows to ₦405,000 after 4 years. The growth follows $A = P \times r^t$ where $r$ is the yearly multiplier. Find $r$. (A) 3 (B) 9 (C) $\sqrt{3}$ (D) $\sqrt[4]{8.1}$ (E) $\sqrt[3]{5}$.
$405000 = 50000 \times r^4$ → $r^4 = \frac{405000}{50000} = 8.1$ → $r = \sqrt[4]{8.1}$ → Answer: D. $\sqrt[4]{8.1}$
1.1.20 Simplify fully: $\frac{\sqrt[4]{x^3}}{\sqrt{x}}$. (A) $x^{\frac{1}{4}}$ (B) $x^{\frac{5}{4}}$ (C) $x^{\frac{3}{2}}$ (D) $x^{-\frac{1}{4}}$ (E) $x^{\frac{1}{2}}$.
$\sqrt[4]{x^3} = x^{\frac{3}{4}}$, $\sqrt{x} = x^{\frac{1}{2}}$ → $\frac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}} = x^{\frac{3}{4}-\frac{2}{4}} = x^{\frac{1}{4}}$ → Answer: A. $x^{\frac{1}{4}}$
Evaluation G - 9 | 1.2 Solutions
1.2.1 Express 0.000075 in standard form. (A) $7.5 \times 10^{-5}$ (B) $7.5 \times 10^{-4}$ (C) $7.5 \times 10^{5}$ (D) $7.5 \times 10^{4}$ (E) $7.5 \times 10^{-6}$.
0.000075 = $7.5 \times 10^{-5}$ (move decimal 5 places right) → Answer: A. $7.5 \times 10^{-5}$
1.2.2 Calculate $(4 \times 10^{3}) \times (2 \times 10^{5})$ and give your answer in standard form. (A) $8 \times 10^{8}$ (B) $8 \times 10^{15}$ (C) $8 \times 10^{7}$ (D) $8 \times 10^{2}$ (E) $8 \times 10^{9}$.
Multiply coefficients: $4 \times 2 = 8$; add exponents: $10^{3+5} = 10^{8}$ → $8 \times 10^{8}$ → Answer: A. $8 \times 10^{8}$
1.2.3 The mass of a dust particle is $7.3 \times 10^{-10}$ kg. How many particles are needed to make a total mass of $1.46 \times 10^{-4}$ kg? (A) $2 \times 10^{5}$ (B) $2 \times 10^{4}$ (C) $5 \times 10^{4}$ (D) $2 \times 10^{6}$ (E) $2 \times 10^{5}$.
Number = $\frac{1.46 \times 10^{-4}}{7.3 \times 10^{-10}} = \frac{1.46}{7.3} \times 10^{-4 - (-10)} = 0.2 \times 10^{6} = 2 \times 10^{5}$ → Answer: E. $2 \times 10^{5}$
1.2.4 Write $9.54 \times 10^{-3}$ as an ordinary number. (A) 0.00954 (B) 0.0954 (C) 9540 (D) 0.000954 (E) 954.
$10^{-3} = 0.001$, so $9.54 \times 0.001 = 0.00954$ → Answer: A. 0.00954
1.2.5 Light travels at approximately $3 \times 10^{8}$ m/s. How far does light travel in one year (365 days)? Give your answer in standard form. (A) $9.46 \times 10^{15}$ m (B) $9.46 \times 10^{12}$ m (C) $3.15 \times 10^{7}$ m (D) $9.46 \times 10^{8}$ m (E) $3.15 \times 10^{15}$ m.
Seconds in a year: $365 \times 24 \times 3600 = 31,536,000 = 3.1536 \times 10^{7}$ s. Distance = $(3 \times 10^{8}) \times (3.1536 \times 10^{7}) = 9.4608 \times 10^{15}$ m → Answer: A. $9.46 \times 10^{15}$ m
1.2.6 Simplify $\frac{6 \times 10^{8}}{2 \times 10^{-3}}$ and express in standard form. (A) $3 \times 10^{11}$ (B) $3 \times 10^{5}$ (C) $3 \times 10^{4}$ (D) $3 \times 10^{10}$ (E) $3 \times 10^{12}$.
Divide coefficients: $6 \div 2 = 3$; subtract exponents: $10^{8 - (-3)} = 10^{11}$ → $3 \times 10^{11}$ → Answer: A. $3 \times 10^{11}$
1.2.7 The population of Nigeria is approximately $2.2 \times 10^{8}$. If the land area is $9.24 \times 10^{5}$ km$^{2}$, what is the population density (people per km$^{2}$)? (A) $2.38 \times 10^{2}$ (B) $2.38 \times 10^{3}$ (C) $4.2 \times 10^{2}$ (D) $2.38 \times 10^{1}$ (E) $4.2 \times 10^{1}$.
Density = $\frac{2.2 \times 10^{8}}{9.24 \times 10^{5}} = \frac{2.2}{9.24} \times 10^{3} \approx 0.238 \times 10^{3} = 2.38 \times 10^{2}$ → Answer: A. $2.38 \times 10^{2}$
1.2.8 Which of the following is NOT equal to $5.6 \times 10^{4}$? (A) 56000 (B) $56 \times 10^{3}$ (C) $0.56 \times 10^{5}$ (D) $560 \times 10^{2}$ (E) $5600 \times 10^{0}$.
$5.6 \times 10^{4} = 56000$. Check E: $5600 \times 10^{0} = 5600$, not equal → Answer: E. $5600 \times 10^{0}$
1.2.9 Calculate $(5 \times 10^{-2})^{3}$ and give your answer in standard form. (A) $1.25 \times 10^{-4}$ (B) $125 \times 10^{-6}$ (C) $1.25 \times 10^{-5}$ (D) $1.25 \times 10^{-7}$ (E) $1.25 \times 10^{5}$.
$(5)^{3} = 125$, $(10^{-2})^{3} = 10^{-6}$ → $125 \times 10^{-6} = 1.25 \times 10^{-4}$ → Answer: A. $1.25 \times 10^{-4}$
1.2.10 The thickness of a human hair is about $1.7 \times 10^{-4}$ m. How many hairs stacked would reach a height of $3.4$ m? (A) $2 \times 10^{3}$ (B) $2 \times 10^{4}$ (C) $5 \times 10^{3}$ (D) $2 \times 10^{2}$ (E) $2 \times 10^{4}$.
Number = $\frac{3.4}{1.7 \times 10^{-4}} = \frac{3.4}{1.7} \times 10^{4} = 2 \times 10^{4}$ → Answer: B. $2 \times 10^{4}$
1.2.11 Add $3.2 \times 10^{5} + 4.8 \times 10^{4}$ and express in standard form. (A) $3.68 \times 10^{5}$ (B) $8.0 \times 10^{4}$ (C) $3.68 \times 10^{4}$ (D) $8.0 \times 10^{5}$ (E) $3.2 \times 10^{5}$.
$3.2 \times 10^{5} = 320000$, $4.8 \times 10^{4} = 48000$, sum = $368000 = 3.68 \times 10^{5}$ → Answer: A. $3.68 \times 10^{5}$
1.2.12 A microchip has $2.5 \times 10^{9}$ transistors. If each transistor uses $3 \times 10^{-8}$ joules per operation, find the total energy used per operation. (A) $7.5 \times 10^{1}$ J (B) $7.5 \times 10^{-17}$ J (C) $7.5 \times 10^{17}$ J (D) $7.5 \times 10^{1}$ J (E) $7.5 \times 10^{1}$ J.
Total energy = $(2.5 \times 10^{9}) \times (3 \times 10^{-8}) = 7.5 \times 10^{1} = 75$ J → Answer: A. $7.5 \times 10^{1}$ J
1.2.13 The distance from Earth to Mars is approximately $2.25 \times 10^{8}$ km. A spacecraft travels at $5 \times 10^{4}$ km/h. How many hours does the journey take? (A) $4.5 \times 10^{3}$ (B) $4.5 \times 10^{4}$ (C) $1.125 \times 10^{4}$ (D) $4.5 \times 10^{2}$ (E) $4.5 \times 10^{3}$.
Time = $\frac{2.25 \times 10^{8}}{5 \times 10^{4}} = \frac{2.25}{5} \times 10^{4} = 0.45 \times 10^{4} = 4.5 \times 10^{3}$ hours → Answer: A. $4.5 \times 10^{3}$
1.2.14 Which number is largest? (A) $8.2 \times 10^{-3}$ (B) $8.2 \times 10^{-4}$ (C) $8.2 \times 10^{-5}$ (D) $8.2 \times 10^{-6}$ (E) $8.2 \times 10^{-7}$.
Larger exponent (less negative) = larger number. $-3 > -4 > -5 > -6 > -7$ → $8.2 \times 10^{-3}$ is largest → Answer: A. $8.2 \times 10^{-3}$
1.2.15 If $x = 4 \times 10^{7}$ and $y = 2 \times 10^{-3}$, find $x \times y$. (A) $8 \times 10^{4}$ (B) $8 \times 10^{10}$ (C) $2 \times 10^{4}$ (D) $8 \times 10^{-4}$ (E) $8 \times 10^{4}$.
$x \times y = (4 \times 2) \times 10^{7 + (-3)} = 8 \times 10^{4}$ → Answer: A. $8 \times 10^{4}$
1.2.16 A factory produces $1.25 \times 10^{6}$ bottles per day. How many bottles are produced in 400 days? Give your answer in standard form. (A) $5.0 \times 10^{8}$ (B) $5.0 \times 10^{7}$ (C) $5.0 \times 10^{9}$ (D) $3.125 \times 10^{3}$ (E) $5.0 \times 10^{6}$.
$400 = 4 \times 10^{2}$; total = $(1.25 \times 10^{6}) \times (4 \times 10^{2}) = 5.0 \times 10^{8}$ → Answer: A. $5.0 \times 10^{8}$
1.2.17 Write $0.000000506$ in standard form. (A) $5.06 \times 10^{-7}$ (B) $5.06 \times 10^{-8}$ (C) $5.06 \times 10^{-6}$ (D) $5.06 \times 10^{-9}$ (E) $5.06 \times 10^{7}$.
Move decimal 7 places right: $0.000000506 = 5.06 \times 10^{-7}$ → Answer: A. $5.06 \times 10^{-7}$
1.2.18 A nanosecond is $10^{-9}$ seconds. How many nanoseconds are there in $3.6 \times 10^{3}$ seconds? (A) $3.6 \times 10^{12}$ (B) $3.6 \times 10^{6}$ (C) $3.6 \times 10^{-12}$ (D) $3.6 \times 10^{3}$ (E) $3.6 \times 10^{9}$.
Number = $\frac{3.6 \times 10^{3}}{10^{-9}} = 3.6 \times 10^{3 - (-9)} = 3.6 \times 10^{12}$ → Answer: A. $3.6 \times 10^{12}$
1.2.19 Which is equal to $2.5 \times 10^{6}$? (A) 2500000 (B) 250000 (C) 0.0000025 (D) 0.000025 (E) 25000000.
$2.5 \times 10^{6} = 2.5 \times 1000000 = 2,500,000$ → Answer: A. 2500000
1.2.20 The speed of sound is $3.4 \times 10^{2}$ m/s. How long (in seconds) does it take sound to travel $1.53 \times 10^{3}$ m? (A) $4.5 \times 10^{0}$ (B) $4.5 \times 10^{-1}$ (C) $4.5 \times 10^{1}$ (D) $4.5 \times 10^{-2}$ (E) $4.5 \times 10^{0}$.
Time = $\frac{1.53 \times 10^{3}}{3.4 \times 10^{2}} = \frac{1.53}{3.4} \times 10^{1} = 0.45 \times 10^{1} = 4.5$ seconds = $4.5 \times 10^{0}$ → Answer: A. $4.5 \times 10^{0}$
Evaluation G - 9 | 1.3 Solutions (Recurring Decimals → Fractions)
1.3.1 Convert $0.\dot{7}$ (0.7777...) to a fraction. (A) $\frac{7}{9}$ (B) $\frac{7}{10}$ (C) $\frac{7}{99}$ (D) $\frac{7}{11}$ (E) $\frac{7}{8}$.
Let $x = 0.\dot{7} = 0.777...$; $10x = 7.777...$; subtract: $10x - x = 7$ → $9x = 7$ → $x = \frac{7}{9}$ → Answer: A. $\frac{7}{9}$
1.3.2 Express $0.\dot{4}2\dot{3}$ (0.423423423...) as a fraction. (A) $\frac{423}{999}$ (B) $\frac{42}{99}$ (C) $\frac{423}{1000}$ (D) $\frac{423}{990}$ (E) $\frac{47}{111}$.
Repeating block "423" of length 3: $x = 0.\overline{423}$; $1000x = 423.\overline{423}$; subtract: $999x = 423$ → $x = \frac{423}{999} = \frac{47}{111}$ → Answer: E. $\frac{47}{111}$
1.3.3 Convert $0.1\dot{6}$ (0.166666...) to a fraction in simplest form. (A) $\frac{1}{6}$ (B) $\frac{16}{99}$ (C) $\frac{5}{30}$ (D) $\frac{1}{5}$ (E) $\frac{4}{25}$.
Let $x = 0.1\dot{6} = 0.1666...$; $10x = 1.666...$, $100x = 16.666...$; subtract: $100x - 10x = 15$ → $90x = 15$ → $x = \frac{15}{90} = \frac{1}{6}$ → Answer: A. $\frac{1}{6}$
1.3.4 Write $0.\dot{2}5\dot{7}$ (0.257257257...) as a fraction. (A) $\frac{257}{999}$ (B) $\frac{257}{1000}$ (C) $\frac{25}{99}$ (D) $\frac{257}{990}$ (E) $\frac{257}{900}$.
Repeating block "257" length 3: $x = 0.\overline{257}$; $1000x = 257.\overline{257}$; $999x = 257$ → $x = \frac{257}{999}$ → Answer: A. $\frac{257}{999}$
1.3.5 Express $0.8\dot{3}$ (0.83333...) as a fraction. (A) $\frac{5}{6}$ (B) $\frac{83}{99}$ (C) $\frac{25}{30}$ (D) $\frac{4}{5}$ (E) $\frac{7}{8}$.
Let $x = 0.8\dot{3} = 0.8333...$; $10x = 8.333...$, $100x = 83.333...$; subtract: $100x - 10x = 75$ → $90x = 75$ → $x = \frac{75}{90} = \frac{5}{6}$ → Answer: A. $\frac{5}{6}$
1.3.6 Convert $0.\dot{0}9\dot{0}$ (0.090090090...) to a fraction. (A) $\frac{90}{999}$ (B) $\frac{9}{100}$ (C) $\frac{10}{111}$ (D) $\frac{1}{11}$ (E) $\frac{90}{1000}$.
Repeating block "090" length 3: $x = 0.\overline{090}$; $1000x = 90.\overline{090}$; $999x = 90$ → $x = \frac{90}{999} = \frac{10}{111}$ → Answer: C. $\frac{10}{111}$
1.3.7 Express $2.\dot{3}$ (2.3333...) as an improper fraction. (A) $\frac{7}{3}$ (B) $\frac{23}{10}$ (C) $\frac{21}{9}$ (D) $\frac{5}{2}$ (E) $\frac{23}{9}$.
$2.\dot{3} = 2 + 0.\dot{3} = 2 + \frac{3}{9} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}$ → Answer: A. $\frac{7}{3}$
1.3.8 Convert $0.2\dot{3}\dot{4}$ (0.234343434...) to a fraction. (A) $\frac{232}{990}$ (B) $\frac{234}{999}$ (C) $\frac{23}{99}$ (D) $\frac{116}{495}$ (E) $\frac{234}{1000}$.
Let $x = 0.2\overline{34}$; $10x = 2.\overline{34}$, $1000x = 234.\overline{34}$; subtract: $1000x - 10x = 232$ → $990x = 232$ → $x = \frac{232}{990} = \frac{116}{495}$ → Answer: D. $\frac{116}{495}$
1.3.9 Write $\frac{5}{11}$ as a recurring decimal. (A) $0.\dot{4}5\dot{4}$ (B) $0.\dot{4}$ (C) $0.45$ (D) $0.\dot{4}5$ (E) $0.4\dot{5}$.
$5 \div 11 = 0.454545... = 0.\overline{45}$ → Answer: D. $0.\dot{4}5$
1.3.10 Convert $0.2\dot{0}9\dot{0}$ (0.20909090...) to a fraction in simplest form. (A) $\frac{209}{999}$ (B) $\frac{207}{990}$ (C) $\frac{23}{110}$ (D) $\frac{209}{1000}$ (E) $\frac{23}{99}$.
Let $x = 0.2\overline{90}$; $10x = 2.\overline{90}$, $1000x = 209.\overline{90}$; subtract: $1000x - 10x = 207$ → $990x = 207$ → $x = \frac{207}{990} = \frac{23}{110}$ → Answer: C. $\frac{23}{110}$
1.3.11 Express $\frac{7}{15}$ as a recurring decimal. (A) $0.4\dot{6}$ (B) $0.\dot{4}6$ (C) $0.46$ (D) $0.4\dot{6}\dot{6}$ (E) $0.\dot{4}6\dot{6}$.
$7 \div 15 = 0.46666... = 0.4\dot{6}$ → Answer: A. $0.4\dot{6}$
1.3.12 Convert $1.\dot{2}5\dot{7}$ (1.257257257...) to a mixed number. (A) $1\frac{257}{999}$ (B) $1\frac{257}{1000}$ (C) $1\frac{25}{99}$ (D) $1\frac{257}{990}$ (E) $1\frac{257}{900}$.
$1.\overline{257} = 1 + \frac{257}{999}$ → Answer: A. $1\frac{257}{999}$
1.3.13 If $x = 0.\dot{1}8$ (0.181818...), what is $100x - x$? (A) 18 (B) 18.18 (C) 18.1818 (D) 18.18... (E) 18.1.
$x = 0.\overline{18}$; $100x = 18.\overline{18}$; $100x - x = 18.\overline{18} - 0.\overline{18} = 18$ → Answer: A. 18
1.3.14 Convert $2.\dot{4}5\dot{6}$ (2.456456456...) into a fraction. (A) $\frac{2456}{999}$ (B) $\frac{2454}{999}$ (C) $\frac{818}{333}$ (D) $\frac{2456}{1000}$ (E) $\frac{2456}{990}$.
$2.\overline{456} = 2 + \frac{456}{999} = \frac{2 \times 999 + 456}{999} = \frac{1998 + 456}{999} = \frac{2454}{999}$ → Answer: B. $\frac{2454}{999}$
1.3.15 Which fraction is equal to $0.\dot{1}2\dot{3}$? (A) $\frac{41}{333}$ (B) $\frac{123}{999}$ (C) $\frac{123}{1000}$ (D) $\frac{41}{333}$ (E) $\frac{123}{990}$.
$0.\overline{123} = \frac{123}{999} = \frac{41}{333}$ → Answer: D. $\frac{41}{333}$
1.3.16 Convert $0.\dot{3}5$ (0.353535...) to a fraction. (A) $\frac{35}{99}$ (B) $\frac{35}{100}$ (C) $\frac{7}{20}$ (D) $\frac{35}{999}$ (E) $\frac{7}{200}$.
$0.\overline{35} = \frac{35}{99}$ → Answer: A. $\frac{35}{99}$
1.3.17 Express $0.5\dot{3}6$ (0.5363636...) as a fraction in simplest form. (A) $\frac{531}{990}$ (B) $\frac{536}{999}$ (C) $\frac{59}{110}$ (D) $\frac{536}{990}$ (E) $\frac{59}{100}$.
Let $x = 0.5\overline{36}$; $10x = 5.\overline{36}$, $1000x = 536.\overline{36}$; subtract: $1000x - 10x = 531$ → $990x = 531$ → $x = \frac{531}{990} = \frac{59}{110}$ → Answer: C. $\frac{59}{110}$
1.3.18 Write $\frac{2}{7}$ as a recurring decimal. (A) $0.\dot{2}8571\dot{4}$ (B) $0.285714$ (C) $0.2857\dot{1}4$ (D) $0.2\dot{8}5714$ (E) $0.\dot{2}85714$.
$\frac{2}{7} = 0.\overline{285714}$ → The repeating block is 285714 → Answer: A. $0.\dot{2}8571\dot{4}$
1.3.19 Convert $0.6\dot{1}2\dot{8}$ (0.6128128128...) to a fraction. (A) $\frac{6122}{9990}$ (B) $\frac{3061}{4995}$ (C) $\frac{6128}{9990}$ (D) $\frac{6128}{10000}$ (E) $\frac{6128}{9900}$.
Let $x = 0.6\overline{128}$; $10x = 6.\overline{128}$, $10000x = 6128.\overline{128}$; subtract: $10000x - 10x = 6122$ → $9990x = 6122$ → $x = \frac{6122}{9990} = \frac{3061}{4995}$ → Answer: B. $\frac{3061}{4995}$
1.3.20 If $y = 0.\dot{3}6\dot{3}$ (0.363363363...), what is $999y$? (A) 363 (B) 36.3 (C) 363.363 (D) 36 (E) 363.363... .
$y = 0.\overline{363}$; $999y = 363$ → Answer: A. 363
.
Evaluation G - 9 | 1.4 Solutions (Surds)
1.4.1 Which is irrational? (A) $\sqrt{16}=4$ (B) $\sqrt{25}=5$ (C) $\sqrt{20}=2\sqrt{5}$ (D) $\sqrt{9}=3$ (E) $\sqrt{36}=6$.
$\sqrt{20}$ cannot be simplified to an integer or fraction → irrational → Answer: C
1.4.2 Simplify $\sqrt{72}$. (A) $6\sqrt{2}$ (B) $6\sqrt{3}$ (C) $8\sqrt{2}$ (D) $6\sqrt{6}$ (E) $8\sqrt{3}$.
$72 = 36 \times 2$, so $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ → Answer: A
1.4.3 True statement about $\sqrt{2}$? (A) rational (B) fraction (C) recurring decimal (D) irrational (E) integer.
$\sqrt{2}$ cannot be expressed as a fraction → irrational → Answer: D
1.4.4 Simplify $\sqrt{18} + \sqrt{8}$. (A) $5\sqrt{2}$ (B) $5\sqrt{3}$ (C) $5\sqrt{26}$ (D) $\sqrt{26}$ (E) $3\sqrt{2}+2\sqrt{2}$.
$\sqrt{18} = 3\sqrt{2}$, $\sqrt{8} = 2\sqrt{2}$, sum = $5\sqrt{2}$ → Answer: A
1.4.5 Which is a surd? (A) $\sqrt{64}=8$ (B) $\sqrt{49}=7$ (C) $\sqrt{100}=10$ (D) $\sqrt{50}=5\sqrt{2}$ (E) $\sqrt{81}=9$.
A surd is an irrational root. $\sqrt{50}$ simplifies to $5\sqrt{2}$ which is irrational → Answer: D
1.4.6 Simplify $3\sqrt{5} \times 2\sqrt{10}$. (A) $6\sqrt{50}$ (B) $30\sqrt{2}$ (C) $60\sqrt{2}$ (D) $30\sqrt{5}$ (E) $6\sqrt{15}$.
$3 \times 2 = 6$, $\sqrt{5} \times \sqrt{10} = \sqrt{50} = 5\sqrt{2}$, so $6 \times 5\sqrt{2} = 30\sqrt{2}$ → Answer: B
1.4.7 Rationalise $\frac{5}{\sqrt{3}}$. (A) $\frac{5\sqrt{3}}{3}$ (B) $\frac{5}{3}\sqrt{3}$ (C) $\frac{5}{\sqrt{3}}$ (D) $5\sqrt{3}$ (E) $\frac{5}{3}$.
Multiply numerator and denominator by $\sqrt{3}$: $\frac{5\sqrt{3}}{3}$ → Answer: A
1.4.8 Simplify $\frac{\sqrt{75}}{\sqrt{3}}$. (A) 3 (B) 5 (C) 15 (D) $\sqrt{25}$ (E) $\sqrt{5}$.
$\sqrt{75} = 5\sqrt{3}$, then $\frac{5\sqrt{3}}{\sqrt{3}} = 5$ → Answer: B
1.4.9 Which is NOT a surd? (A) $\sqrt{7}$ (B) $\sqrt{11}$ (C) $\sqrt{13}$ (D) $\sqrt{21}$ (E) $\sqrt{25}=5$.
$\sqrt{25}=5$ is rational, not a surd → Answer: E
1.4.10 Simplify $\sqrt{2} \times \sqrt{8}$. (A) 4 (B) $\sqrt{10}$ (C) $2\sqrt{2}$ (D) $\sqrt{16}$ (E) 8.
$\sqrt{2 \times 8} = \sqrt{16} = 4$ → Answer: A
1.4.11 Simplify $\sqrt{48} - \sqrt{27}$. (A) $\sqrt{21}$ (B) $\sqrt{3}$ (C) $\sqrt{75}$ (D) $2\sqrt{3}$ (E) $3\sqrt{3}$.
$\sqrt{48} = 4\sqrt{3}$, $\sqrt{27} = 3\sqrt{3}$, difference = $\sqrt{3}$ → Answer: B
1.4.12 Rationalise $\frac{2}{\sqrt{5} - 1}$. (A) $\frac{\sqrt{5}+1}{4}$ (B) $\frac{\sqrt{5}+1}{2}$ (C) $\frac{2\sqrt{5}+2}{4}$ (D) $\frac{2\sqrt{5}-2}{4}$ (E) $\frac{\sqrt{5}-1}{2}$.
Multiply by $\frac{\sqrt{5}+1}{\sqrt{5}+1}$: $\frac{2(\sqrt{5}+1)}{5-1} = \frac{2(\sqrt{5}+1)}{4} = \frac{\sqrt{5}+1}{2}$ → Answer: B
1.4.13 Which is larger: $2\sqrt{3}$ or $3\sqrt{2}$? (A) $2\sqrt{3}$ (B) $3\sqrt{2}$ (C) equal (D) cannot compare (E) $2\sqrt{3}=3\sqrt{2}$.
Square both: $(2\sqrt{3})^2 = 12$, $(3\sqrt{2})^2 = 18$, so $3\sqrt{2} > 2\sqrt{3}$ → Answer: B
1.4.14 Simplify $(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})$. (A) 2 (B) 8 (C) $5 - \sqrt{15}$ (D) $5 + 2\sqrt{15} + 3$ (E) $5 - 3$.
Difference of squares: $(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2$ → Answer: A
1.4.15 Which is irrational? (A) $0.\dot{3}$ (B) $\frac{22}{7}$ (C) $\pi$ (D) $\sqrt{9}=3$ (E) $3.14$.
$\pi$ is a well-known irrational number → Answer: C
1.4.16 Simplify $\sqrt[3]{54} - \sqrt[3]{16}$. (A) $\sqrt[3]{2}$ (B) $2\sqrt[3]{2}$ (C) $\sqrt[3]{38}$ (D) $3\sqrt[3]{2}$ (E) $0$.
$\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}$, $\sqrt[3]{16} = \sqrt[3]{8 \times 2} = 2\sqrt[3]{2}$, difference = $\sqrt[3]{2}$ → Answer: A
1.4.17 Write $\frac{\sqrt{32}}{\sqrt{2}}$ in simplest form. (A) 4 (B) $\sqrt{16}$ (C) $2\sqrt{2}$ (D) $\sqrt{30}$ (E) 8.
$\sqrt{32} = 4\sqrt{2}$, then $\frac{4\sqrt{2}}{\sqrt{2}} = 4$ → Answer: A
1.4.18 Simplify $3\sqrt{2} + 5\sqrt{2} - 4\sqrt{2}$. (A) $4\sqrt{2}$ (B) $4\sqrt{6}$ (C) $12\sqrt{2}$ (D) $2\sqrt{2}$ (E) $6\sqrt{2}$.
$3 + 5 - 4 = 4$, so $4\sqrt{2}$ → Answer: A
1.4.19 If $x = 3 + \sqrt{2}$, find $x + \frac{1}{x}$ in simplest surd form. (A) 6 (B) $\frac{24 + 6\sqrt{2}}{7}$ (C) $6 - 2\sqrt{2}$ (D) $3 + \sqrt{2}$ (E) $\frac{11}{7}$.
$\frac{1}{x} = \frac{1}{3+\sqrt{2}} = \frac{3-\sqrt{2}}{9-2} = \frac{3-\sqrt{2}}{7}$. Then $x + \frac{1}{x} = (3+\sqrt{2}) + \frac{3-\sqrt{2}}{7} = \frac{21+7\sqrt{2} + 3 - \sqrt{2}}{7} = \frac{24 + 6\sqrt{2}}{7}$ → Answer: B
1.4.20 Which is equivalent to $\sqrt{50} + 2\sqrt{18}$? (A) $5\sqrt{2} + 6\sqrt{2}$ (B) $11\sqrt{2}$ (C) $10\sqrt{2}$ (D) $12\sqrt{2}$ (E) $9\sqrt{2}$.
$\sqrt{50} = 5\sqrt{2}$, $2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}$, sum = $11\sqrt{2}$ → Answer: B
Evaluation G - 9 | 1.5 Solutions (Percentages)
1.5.1 A shirt costs ₦8,000. Increased by 15%. New price? (A) ₦9,000 (B) ₦9,200 (C) ₦9,500 (D) ₦9,800 (E) ₦10,000.
Increase = 15% of 8000 = 0.15 × 8000 = 1200. New price = 8000 + 1200 = 9200 → Answer: B
1.5.2 Laptop sold for ₦450,000 after 10% discount. Original price? (A) ₦495,000 (B) ₦500,000 (C) ₦405,000 (D) ₦540,000 (E) ₦480,000.
Let original = P. Selling price = P × (1 - 0.10) = 0.9P = 450,000 → P = 450,000 ÷ 0.9 = 500,000 → Answer: B
1.5.3 Population increases from 25,000 to 28,000. Percentage increase? (A) 10% (B) 12% (C) 15% (D) 18% (E) 20%.
Increase = 28,000 - 25,000 = 3,000. % increase = (3000/25000) × 100% = 12% → Answer: B
1.5.4 Score 42 out of 60. Percentage? (A) 60% (B) 65% (C) 70% (D) 72% (E) 75%.
Percentage = (42/60) × 100% = 70% → Answer: C
1.5.5 Compound interest on ₦200,000 for 2 years at 5% per annum compounded annually. (A) ₦20,000 (B) ₦20,500 (C) ₦21,000 (D) ₦21,500 (E) ₦22,000.
Amount = 200,000 × (1.05)² = 200,000 × 1.1025 = 220,500. Interest = 220,500 - 200,000 = 20,500 → Answer: B
1.5.6 Car depreciates 15% per year. Current value ₦8,500,000. Value one year ago? (A) ₦9,775,000 (B) ₦10,000,000 (C) ₦9,500,000 (D) ₦9,250,000 (E) ₦10,500,000.
Current = Previous × (1 - 0.15) = Previous × 0.85. Previous = 8,500,000 ÷ 0.85 = 10,000,000 → Answer: B
1.5.7 Phone price from ₦150,000 to ₦180,000. Percentage increase? (A) 15% (B) 20% (C) 25% (D) 30% (E) 35%.
Increase = 30,000. % = (30,000/150,000) × 100% = 20% → Answer: B
1.5.8 After 12% pay rise, earns ₦560,000. Salary before rise? (A) ₦492,800 (B) ₦500,000 (C) ₦490,000 (D) ₦498,000 (E) ₦510,000.
Let original = S. S × 1.12 = 560,000 → S = 560,000 ÷ 1.12 = 500,000 → Answer: B
1.5.9 Increase by 20% then discount 10%. Overall change? (A) 8% increase (B) 10% increase (C) 8% decrease (D) 10% decrease (E) 12% increase.
Let original = 100. After 20% increase = 120. After 10% discount = 120 × 0.9 = 108. Overall increase = 8% → Answer: A
1.5.10 Invest ₦500,000 at 4% compound for 3 years. Final amount? (A) ₦562,432 (B) ₦560,000 (C) ₦564,500 (D) ₦562,000 (E) ₦561,500.
Amount = 500,000 × (1.04)³ = 500,000 × 1.124864 = 562,432 → Answer: A
1.5.11 A number increased by 25% becomes 200. Original? (A) 150 (B) 160 (C) 170 (D) 180 (E) 140.
Let original = N. N × 1.25 = 200 → N = 200 ÷ 1.25 = 160 → Answer: B
1.5.12 Sugar price from ₦800 to ₦680 per kg. Percentage decrease? (A) 12% (B) 15% (C) 18% (D) 20% (E) 25%.
Decrease = 800 - 680 = 120. % = (120/800) × 100% = 15% → Answer: B
1.5.13 House bought ₦25,000,000 sold ₦30,000,000. Percentage profit? (A) 15% (B) 20% (C) 25% (D) 30% (E) 35%.
Profit = 5,000,000. % = (5,000,000/25,000,000) × 100% = 20% → Answer: B
1.5.14 Candidate received 1,200 votes = 60% of total. Total votes? (A) 1,800 (B) 2,000 (C) 2,200 (D) 2,400 (E) 2,600.
0.60 × Total = 1,200 → Total = 1,200 ÷ 0.60 = 2,000 → Answer: B
1.5.15 Simple interest on ₦150,000 for 3 years = ₦27,000. Rate per annum? (A) 4% (B) 5% (C) 6% (D) 7% (E) 8%.
SI = P × R × T / 100 → 27,000 = 150,000 × R × 3 / 100 → 27,000 = 4,500 × R → R = 27,000 ÷ 4,500 = 6% → Answer: C
1.5.16 Population from 50,000 to 45,000. Percentage decrease? (A) 8% (B) 9% (C) 10% (D) 11% (E) 12%.
Decrease = 5,000. % = (5,000/50,000) × 100% = 10% → Answer: C
1.5.17 Trader marks ₦40,000 up by 25% then gives 10% discount. Final price? (A) ₦45,000 (B) ₦46,000 (C) ₦47,000 (D) ₦48,000 (E) ₦49,000.
Marked price = 40,000 × 1.25 = 50,000. After 10% discount = 50,000 × 0.9 = 45,000 → Answer: A
1.5.18 Compound interest on ₦250,000 for 2 years at 6% per annum. (A) ₦30,000 (B) ₦30,400 (C) ₦30,900 (D) ₦31,000 (E) ₦31,500.
Amount = 250,000 × (1.06)² = 250,000 × 1.1236 = 280,900. Interest = 280,900 - 250,000 = 30,900 → Answer: C
1.5.19 After 30% discount, TV costs ₦175,000. Original price? (A) ₦225,000 (B) ₦230,000 (C) ₦240,000 (D) ₦250,000 (E) ₦260,000.
Let original = P. P × 0.7 = 175,000 → P = 175,000 ÷ 0.7 = 250,000 → Answer: D
1.5.20 Car depreciates 20% per year. Bought for ₦12,000,000. Value after 2 years? (A) ₦7,200,000 (B) ₦7,680,000 (C) ₦8,000,000 (D) ₦8,400,000 (E) ₦8,800,000.
After 1 year: 12,000,000 × 0.8 = 9,600,000. After 2 years: 9,600,000 × 0.8 = 7,680,000 → Answer: B
Evaluation G - 9 | 1.6 Solutions (Ratio and Proportion)
1.6.1 Divide ₦500,000 in the ratio 3:5:2. Largest share? (A) ₦150,000 (B) ₦200,000 (C) ₦250,000 (D) ₦300,000 (E) ₦350,000.
Total parts = 3+5+2 = 10. Largest share = (5/10) × 500,000 = 250,000 → Answer: C
1.6.2 If x : y = 3 : 4 and y : z = 2 : 5, find x : z. (A) 3:5 (B) 3:10 (C) 6:20 (D) 3:20 (E) 6:10.
Make y same: x:y = 3:4 = 6:8, y:z = 2:5 = 8:20. Thus x:z = 6:20 = 3:10 → Answer: B
1.6.3 y is directly proportional to x. When x=6, y=24. Find y when x=15. (A) 40 (B) 50 (C) 60 (D) 70 (E) 80.
y = kx → 24 = k×6 → k=4. So y = 4×15 = 60 → Answer: C
1.6.4 P is inversely proportional to Q. When Q=8, P=3. Find P when Q=12. (A) 2 (B) 2.5 (C) 3 (D) 4 (E) 4.5.
P × Q = k → 3×8 = 24 = k. So P = 24 ÷ 12 = 2 → Answer: A
1.6.5 Three numbers in ratio 2:3:5. Sum of largest and smallest is 56. Find middle number. (A) 12 (B) 18 (C) 20 (D) 24 (E) 30.
Let numbers be 2k, 3k, 5k. Largest + smallest = 5k + 2k = 7k = 56 → k=8. Middle = 3×8 = 24 → Answer: D
1.6.6 5 men dig a trench in 12 days. How many days for 15 men? (A) 3 (B) 4 (C) 5 (D) 6 (E) 8.
Inverse proportion: men × days = constant = 5×12 = 60. Days = 60 ÷ 15 = 4 → Answer: B
1.6.7 Boys : girls = 7:5. 240 more boys than girls. Total students? (A) 960 (B) 1000 (C) 1200 (D) 1440 (E) 1680.
Difference in ratio parts = 7-5=2 parts = 240 → 1 part = 120. Total parts = 12 → Total students = 12×120 = 1440 → Answer: D
1.6.8 x varies directly as y². When y=4, x=32. Find x when y=6. (A) 48 (B) 64 (C) 72 (D) 96 (E) 108.
x = k y² → 32 = k×16 → k=2. So x = 2×36 = 72 → Answer: C
1.6.9 Divide ₦720,000 among A, B, C: A gets twice B, B gets three times C. A's share? (A) ₦360,000 (B) ₦384,000 (C) ₦400,000 (D) ₦420,000 (E) ₦432,000.
Let C = k, B = 3k, A = 2×3k = 6k. Total = 6k+3k+k = 10k = 720,000 → k=72,000. A = 6×72,000 = 432,000 → Answer: E
1.6.10 10 workers do a job in 6 days. Workers needed for 4 days? (A) 12 (B) 14 (C) 15 (D) 16 (E) 18.
Workers × days = constant = 10×6 = 60. Workers = 60 ÷ 4 = 15 → Answer: C
1.6.11 Triangle angles ratio 2:3:4. Largest angle? (A) 40° (B) 60° (C) 80° (D) 100° (E) 120°.
Total parts = 2+3+4=9. Largest angle = (4/9)×180° = 80° → Answer: C
1.6.12 y directly proportional to x. y=15 when x=3. Find y when x=7. (A) 25 (B) 30 (C) 35 (D) 40 (E) 45.
y = kx → 15 = 3k → k=5. y = 5×7 = 35 → Answer: C
1.6.13 Two numbers ratio 7:9, sum = 320. Find their difference. (A) 30 (B) 40 (C) 50 (D) 60 (E) 70.
Let numbers be 7k, 9k. Sum = 16k = 320 → k=20. Difference = 9k-7k = 2k = 40 → Answer: B
1.6.14 A varies inversely as r². A=10 when r=2. Find A when r=5. (A) 0.8 (B) 1.6 (C) 2.5 (D) 4.0 (E) 6.25.
A × r² = k → 10×4 = 40 = k. A = 40 ÷ 25 = 1.6 → Answer: B
1.6.15 Recipe: 200g flour for 5 people. Flour for 8 people? (A) 250g (B) 280g (C) 300g (D) 320g (E) 350g.
Flour per person = 200÷5 = 40g. For 8 people = 8×40 = 320g → Answer: D
1.6.16 a:b = 4:5, b:c = 3:7, find a:c. (A) 12:35 (B) 12:28 (C) 15:28 (D) 20:21 (E) 21:20.
Make b same: a:b = 4:5 = 12:15, b:c = 3:7 = 15:35. Thus a:c = 12:35 → Answer: A
1.6.17 Map scale 1:200,000. What distance in km does 6 cm represent? (A) 6 km (B) 12 km (C) 15 km (D) 18 km (E) 20 km.
6 cm on map = 6 × 200,000 cm = 1,200,000 cm = 12 km → Answer: B
1.6.18 Share money in ratio 2:3:5. Middle person gets ₦45,000. Total amount? (A) ₦120,000 (B) ₦135,000 (C) ₦150,000 (D) ₦165,000 (E) ₦180,000.
Middle share = 3 parts = 45,000 → 1 part = 15,000. Total = 10 parts = 150,000 → Answer: C
1.6.19 16 cows graze a field in 15 days. How many days for 24 cows? (A) 8 (B) 10 (C) 12 (D) 14 (E) 18.
Cows × days = constant = 16×15 = 240. Days = 240 ÷ 24 = 10 → Answer: B
1.6.20 Copper : zinc = 7:3. How much zinc in 500g alloy? (A) 100g (B) 150g (C) 200g (D) 250g (E) 300g.
Total parts = 7+3=10. Zinc = (3/10)×500 = 150g → Answer: B
Evaluation G - 9 | 1.7 Solutions (Rounding, Bounds, Significant Figures, Percentage Error)
1.7.1 Round $47.829$ to one decimal place. (A) 48.0 (B) 47.9 (C) 47.8 (D) 47.83 (E) 47.0.
Look at the second decimal digit: $47.8\mathbf{2}9$ → digit is $2 < 5$, so round down → $47.8$ → Answer: C
1.7.2 Write $0.005678$ to two significant figures. (A) 0.0056 (B) 0.0057 (C) 0.00568 (D) 0.0058 (E) 0.0060.
First non-zero digit is 5 (first significant figure). Second significant figure is 6 (the 7). Look at the third significant figure (8) → $8 \ge 5$, so round up the 6 to 7 → $0.0057$ → Answer: B
1.7.3 A length is measured as $15.6$ cm to the nearest millimetre. Find the upper bound. (A) 15.55 cm (B) 15.65 cm (C) 15.60 cm (D) 15.70 cm (E) 15.50 cm.
To the nearest millimetre (0.1 cm) means half of 0.1 = 0.05 cm above or below. Upper bound = $15.6 + 0.05 = 15.65$ cm → Answer: B
1.7.4 A bag of sugar weighs $2.0$ kg to two significant figures. What is the lower bound? (A) 1.99 kg (B) 2.00 kg (C) 1.95 kg (D) 1.90 kg (E) 2.05 kg.
Two significant figures: $2.0$ means the measurement is between $1.95$ and $2.05$. Lower bound = $1.95$ kg → Answer: C
1.7.5 Round $12,499$ to the nearest thousand. (A) 13,000 (B) 12,000 (C) 12,500 (D) 12,400 (E) 10,000.
Look at the hundreds digit: $12,\mathbf{4}99$ → digit is $4 < 5$, so round down to $12,000$ → Answer: B
1.7.6 A rectangle has length $8.3$ cm and width $4.7$ cm, both measured to one decimal place. Find the maximum possible area. (A) 39.00 cm² (B) 39.20 cm² (C) 38.92 cm² (D) 39.66 cm² (E) 40.00 cm².
To one decimal place: length $8.3$ means $8.25 \le L < 8.35$, width $4.7$ means $4.65 \le W < 4.75$. Maximum area uses upper bounds: $L_{\text{max}} = 8.35$, $W_{\text{max}} = 4.75$, area $= 8.35 \times 4.75 = 39.6625$ cm² ≈ $39.66$ cm² → Answer: D
1.7.7 Write $0.003045$ to three significant figures. (A) 0.00304 (B) 0.0030 (C) 0.00305 (D) 0.003045 (E) 0.0031.
First non-zero digit is 3 (first sig fig). Second sig fig is 0, third sig fig is 4. Next digit is 5 → $5 \ge 5$, so round up the 4 to 5 → $0.00305$ → Answer: C
1.7.8 A car travels $248$ km in $4.5$ hours, both given to three significant figures. Find the lower bound of the speed. (A) 55.0 km/h (B) 54.4 km/h (C) 54.8 km/h (D) 55.1 km/h (E) 54.7 km/h.
Lower bound of speed = lower bound distance ÷ upper bound time. $248$ to 3 sig figs: $247.5 \le d < 248.5$; $4.5$ to 3 sig figs: $4.45 \le t < 4.55$. Lower speed = $247.5 \div 4.55 \approx 54.3956 \approx 54.4$ km/h → Answer: B
1.7.9 Round $5.9995$ to three decimal places. (A) 5.999 (B) 6.001 (C) 6.000 (D) 5.990 (E) 6.010.
Three decimal places: look at the fourth decimal digit. $5.999\mathbf{5}$ → fourth digit is $5$, round up the third decimal from 9 to 10, causing carry → $6.000$ → Answer: C
1.7.10 A number is given as $350$ to two significant figures. What is the greatest possible value? (A) 355 (B) 349.9 (C) 354.9 (D) 354.99 (E) 355.0.
Two sig figs: $350$ means between $345$ and $355$. The greatest possible value is just below $355$, i.e., $354.99...$ → $354.99$ is the greatest among the options → Answer: D
1.7.11 Express $0.0006709$ to two significant figures. (A) 0.00068 (B) 0.00067 (C) 0.000671 (D) 0.0007 (E) 0.00066.
First non-zero digit is 6 (first sig fig). Second sig fig is 7. Next digit is 0 → $0 < 5$, so round down → $0.00067$ → Answer: B
1.7.12 The mass of a letter is $85$ g to the nearest gram. Find the upper bound of the mass. (A) 85.4 g (B) 85.5 g (C) 85.49 g (D) 85.0 g (E) 86.0 g.
To nearest gram means half of 1 = 0.5. Upper bound = $85 + 0.5 = 85.5$ g → Answer: B
1.7.13 Round $67,849$ to three significant figures. (A) 67,900 (B) 67,800 (C) 67,850 (D) 68,000 (E) 67,700.
$67,849$: three sig figs means look at the fourth digit. $67,\mathbf{8}49$ → third digit is 8, next digit is 4 < 5 → round down to $67,800$ → Answer: B
1.7.14 A square has side length $5.6$ cm measured to one decimal place. Find the minimum possible perimeter. (A) 22.4 cm (B) 22.0 cm (C) 21.6 cm (D) 22.2 cm (E) 22.8 cm.
Side $5.6$ cm to one decimal place means $5.55 \le s < 5.65$. Minimum side = $5.55$ cm. Perimeter = $4 \times 5.55 = 22.2$ cm → Answer: D
1.7.15 Write $0.0000456$ in standard form to two significant figures. (A) $4.5 \times 10^{-6}$ (B) $4.5 \times 10^{-5}$ (C) $4.6 \times 10^{-6}$ (D) $4.6 \times 10^{-5}$ (E) $4.56 \times 10^{-5}$.
$0.0000456 = 4.56 \times 10^{-5}$. To two sig figs: $4.6 \times 10^{-5}$ → Answer: D
1.7.16 A field has length $120$ m and width $80$ m, both to the nearest 10 m. Find the upper bound of the area. (A) 9,600 m² (B) 10,000 m² (C) 9,025 m² (D) 10,125 m² (E) 10,625 m².
To nearest 10 m: half of 10 = 5. Length upper bound = $120 + 5 = 125$ m. Width upper bound = $80 + 5 = 85$ m. Upper area = $125 \times 85 = 10,625$ m² → Answer: E
1.7.17 Round $9.999$ to three significant figures. (A) 9.99 (B) 10.00 (C) 10.0 (D) 9.999 (E) 9.90.
$9.999$ has four sig figs. To three sig figs, look at the fourth digit (9) → $9 \ge 5$, round up the third digit (9) causing carry → $10.0$ (which has three sig figs) → Answer: C
1.7.18 A temperature is recorded as $-3.5^\circ C$ to one decimal place. What is the lower bound? (A) $-3.45^\circ C$ (B) $-3.55^\circ C$ (C) $-3.50^\circ C$ (D) $-3.6^\circ C$ (E) $-3.4^\circ C$.
For negative numbers, lower bound is more negative. Half of 0.1 = 0.05. Lower bound = $-3.5 - 0.05 = -3.55^\circ C$ → Answer: B
1.7.19 Calculate $32.6 \times 4.8$ and give your answer to two significant figures. (A) 156 (B) 150 (C) 160 (D) 156.48 (E) 155.
$32.6 \times 4.8 = 156.48$. To two sig figs: first two digits are 1 and 5, third digit is 6 → $6 \ge 5$, round up → $160$ → Answer: C
1.7.20 A distance of $250$ km is measured to the nearest 10 km. Find the percentage error in the measurement. (A) $1\%$ (B) $4\%$ (C) $5\%$ (D) $2\%$ (E) $0.5\%$.
Maximum absolute error = half of 10 = 5 km. Percentage error = $\frac{5}{250} \times 100\% = 2\%$ → Answer: D
Evaluation G - 9 | 2.1 Solutions (Algebraic Manipulation)
2.1.1 Simplify $\frac{3x^2 y^3}{6xy^5}$. (A) $\frac{x}{2y^2}$ (B) $\frac{x}{2y}$ (C) $\frac{2x}{y^2}$ (D) $\frac{x}{2y^3}$ (E) $\frac{3x}{y^2}$.
Cancel common factors: $\frac{3}{6} = \frac{1}{2}$, $x^2/x = x$, $y^3/y^5 = 1/y^2$ → $\frac{x}{2y^2}$ → Answer: A
2.1.2 Make $x$ the subject of $y = 3x - 7$. (A) $x = 3y + 7$ (B) $x = \frac{y + 7}{3}$ (C) $x = \frac{y - 7}{3}$ (D) $x = 3(y + 7)$ (E) $x = 3(y - 7)$.
$y = 3x - 7$ → $y + 7 = 3x$ → $x = \frac{y + 7}{3}$ → Answer: B
2.1.3 Simplify $\frac{x^2 - 9}{x + 3}$. (A) $x - 3$ (B) $x + 3$ (C) $x - 9$ (D) $x + 9$ (E) $x - 1$.
$x^2 - 9 = (x-3)(x+3)$, so $\frac{(x-3)(x+3)}{x+3} = x - 3$ → Answer: A
2.1.4 Rearrange $V = \frac{4}{3}\pi r^3$ to make $r$ the subject. (A) $r = \sqrt[3]{\frac{3V}{4\pi}}$ (B) $r = \sqrt[3]{\frac{4V}{3\pi}}$ (C) $r = \frac{3V}{4\pi}$ (D) $r = \sqrt[3]{\frac{3V}{\pi}}$ (E) $r = \sqrt[3]{\frac{3V}{4}}$.
Multiply both sides by 3: $3V = 4\pi r^3$ → $r^3 = \frac{3V}{4\pi}$ → $r = \sqrt[3]{\frac{3V}{4\pi}}$ → Answer: A
2.1.5 Simplify $\frac{2}{x} + \frac{3}{x+1}$. (A) $\frac{5x + 2}{x(x+1)}$ (B) $\frac{5x + 3}{x(x+1)}$ (C) $\frac{5x + 2}{2x+1}$ (D) $\frac{5x + 2}{x+1}$ (E) $\frac{5x + 2}{x^2 + 2x}$.
Common denominator $x(x+1)$: $\frac{2(x+1) + 3x}{x(x+1)} = \frac{2x+2+3x}{x(x+1)} = \frac{5x+2}{x(x+1)}$ → Answer: A
2.1.6 Make $t$ the subject of $s = ut + \frac{1}{2}at^2$. (A) $t = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$ (B) $t = \frac{u \pm \sqrt{u^2 + 2as}}{a}$ (C) $t = \frac{-u \pm \sqrt{u^2 + 4as}}{a}$ (D) $t = \frac{-u \pm \sqrt{u^2 - 2as}}{a}$ (E) $t = \frac{-2u \pm \sqrt{u^2 + 2as}}{a}$.
Rewrite as $\frac{1}{2}at^2 + ut - s = 0$ → multiply by 2: $at^2 + 2ut - 2s = 0$. Using quadratic formula: $t = \frac{-2u \pm \sqrt{4u^2 + 8as}}{2a} = \frac{-2u \pm 2\sqrt{u^2 + 2as}}{2a} = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$ → Answer: A
2.1.7 Simplify $\frac{3}{x-2} - \frac{2}{x+1}$. (A) $\frac{x + 7}{(x-2)(x+1)}$ (B) $\frac{x - 7}{(x-2)(x+1)}$ (C) $\frac{5x - 7}{(x-2)(x+1)}$ (D) $\frac{x + 5}{(x-2)(x+1)}$ (E) $\frac{x + 7}{x^2 - 1}$.
Common denominator $(x-2)(x+1)$: $\frac{3(x+1) - 2(x-2)}{(x-2)(x+1)} = \frac{3x+3 - 2x + 4}{(x-2)(x+1)} = \frac{x + 7}{(x-2)(x+1)}$ → Answer: A
2.1.8 Make $R$ the subject of $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$. (A) $R = \frac{R_1 R_2}{R_1 + R_2}$ (B) $R = \frac{R_1 + R_2}{R_1 R_2}$ (C) $R = R_1 + R_2$ (D) $R = R_1 R_2$ (E) $R = \frac{1}{R_1} + \frac{1}{R_2}$.
RHS = $\frac{R_2 + R_1}{R_1 R_2}$, so $\frac{1}{R} = \frac{R_1 + R_2}{R_1 R_2}$ → $R = \frac{R_1 R_2}{R_1 + R_2}$ → Answer: A
2.1.9 Simplify $\frac{4x^2 - 25}{2x - 5}$. (A) $2x - 5$ (B) $2x + 5$ (C) $4x + 10$ (D) $4x - 10$ (E) $2x - 25$.
$4x^2 - 25 = (2x-5)(2x+5)$, so $\frac{(2x-5)(2x+5)}{2x-5} = 2x + 5$ → Answer: B
2.1.10 Make $h$ the subject of $A = 2\pi r(r + h)$. (A) $h = \frac{A}{2\pi r} - r$ (B) $h = \frac{A}{2\pi r} + r$ (C) $h = \frac{A}{2\pi} - r$ (D) $h = A - 2\pi r - r$ (E) $h = A - 2\pi r^2$.
$A = 2\pi r^2 + 2\pi r h$ → $A - 2\pi r^2 = 2\pi r h$ → $h = \frac{A - 2\pi r^2}{2\pi r} = \frac{A}{2\pi r} - r$ → Answer: A
2.1.11 Simplify $\frac{x^2 - 4x + 3}{x - 1}$. (A) $x - 3$ (B) $x + 3$ (C) $x - 1$ (D) $x + 1$ (E) $x - 4$.
$x^2 - 4x + 3 = (x-1)(x-3)$, so $\frac{(x-1)(x-3)}{x-1} = x - 3$ → Answer: A
2.1.12 Make $k$ the subject of $T = 2\pi\sqrt{\frac{k}{m}}$. (A) $k = \frac{T^2 m}{4\pi^2}$ (B) $k = \frac{4\pi^2 m}{T^2}$ (C) $k = \frac{T m}{2\pi}$ (D) $k = \frac{T^2}{4\pi^2 m}$ (E) $k = 4\pi^2 m T^2$.
Divide both sides by $2\pi$: $\frac{T}{2\pi} = \sqrt{\frac{k}{m}}$ → square: $\frac{T^2}{4\pi^2} = \frac{k}{m}$ → $k = \frac{T^2 m}{4\pi^2}$ → Answer: A
2.1.13 Simplify $\frac{2x}{x^2 - 9} - \frac{1}{x - 3}$. (A) $\frac{1}{x + 3}$ (B) $\frac{1}{x - 3}$ (C) $\frac{3x - 3}{x^2 - 9}$ (D) $\frac{x - 3}{x^2 - 9}$ (E) $\frac{x + 3}{x^2 - 9}$.
Note $x^2-9 = (x-3)(x+3)$. Write over common denominator: $\frac{2x - (x+3)}{(x-3)(x+3)} = \frac{2x - x - 3}{(x-3)(x+3)} = \frac{x-3}{(x-3)(x+3)} = \frac{1}{x+3}$ → Answer: A
2.1.14 Make $u$ the subject of $v^2 = u^2 + 2as$. (A) $u = \sqrt{v^2 - 2as}$ (B) $u = v - 2as$ (C) $u = \sqrt{v^2 + 2as}$ (D) $u = v - \sqrt{2as}$ (E) $u = v^2 - 2as$.
$u^2 = v^2 - 2as$ → $u = \sqrt{v^2 - 2as}$ → Answer: A
2.1.15 Simplify $\frac{x^2 - 5x + 6}{x^2 - 9}$. (A) $\frac{x - 2}{x + 3}$ (B) $\frac{x - 2}{x - 3}$ (C) $\frac{x + 2}{x + 3}$ (D) $\frac{x + 2}{x - 3}$ (E) $\frac{x - 2}{x + 1}$.
$x^2 - 5x + 6 = (x-2)(x-3)$, $x^2 - 9 = (x-3)(x+3)$, so $\frac{(x-2)(x-3)}{(x-3)(x+3)} = \frac{x-2}{x+3}$ → Answer: A
2.1.16 Make $c$ the subject of $E = mc^2$. (A) $c = \sqrt{\frac{E}{m}}$ (B) $c = \sqrt{Em}$ (C) $c = \frac{E}{m}$ (D) $c = \sqrt{E} - m$ (E) $c = E - m$.
$c^2 = \frac{E}{m}$ → $c = \sqrt{\frac{E}{m}}$ → Answer: A
2.1.17 Simplify $\frac{1}{x} + \frac{1}{y} - \frac{2}{x+y}$. (A) $\frac{x^2 + y^2}{xy(x+y)}$ (B) $\frac{x^2 - y^2}{xy(x+y)}$ (C) $\frac{2x + 2y}{xy(x+y)}$ (D) $\frac{x^2 + 2xy + y^2}{xy(x+y)}$ (E) $\frac{x^2 - 2xy + y^2}{xy(x+y)}$.
Common denominator $xy(x+y)$: $\frac{y(x+y) + x(x+y) - 2xy}{xy(x+y)} = \frac{xy + y^2 + x^2 + xy - 2xy}{xy(x+y)} = \frac{x^2 + y^2}{xy(x+y)}$ → Answer: A
2.1.18 Make $f$ the subject of $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$. (A) $f = \frac{uv}{u+v}$ (B) $f = \frac{u+v}{uv}$ (C) $f = uv(u+v)$ (D) $f = u+v$ (E) $f = uv$.
RHS = $\frac{v + u}{uv} = \frac{1}{f}$ → $f = \frac{uv}{u+v}$ → Answer: A
2.1.19 Simplify $\frac{2x^2 + 5x + 3}{x^2 + 2x + 1}$. (A) $\frac{2x + 3}{x + 1}$ (B) $\frac{2x + 1}{x + 1}$ (C) $\frac{2x - 3}{x + 1}$ (D) $\frac{2x - 1}{x + 1}$ (E) $\frac{2x + 3}{x - 1}$.
$2x^2 + 5x + 3 = (2x+3)(x+1)$, $x^2 + 2x + 1 = (x+1)^2$, so $\frac{(2x+3)(x+1)}{(x+1)^2} = \frac{2x+3}{x+1}$ → Answer: A
2.1.20 Make $r$ the subject of $S = 2\pi r(r + h)$. (A) $r = \frac{-h \pm \sqrt{h^2 + \frac{2S}{\pi}}}{2}$ (B) $r = \frac{-h \pm \sqrt{h^2 + \frac{S}{2\pi}}}{2}$ (C) $r = h \pm \sqrt{h^2 + \frac{S}{\pi}}$ (D) $r = \frac{-h \pm \sqrt{h^2 + \frac{4S}{\pi}}}{2}$ (E) $r = \frac{-h \pm \sqrt{h^2 - \frac{2S}{\pi}}}{2}$.
Rewrite: $2\pi r^2 + 2\pi h r - S = 0$ → divide by $2\pi$: $r^2 + h r - \frac{S}{2\pi} = 0$. Quadratic formula: $r = \frac{-h \pm \sqrt{h^2 + \frac{2S}{\pi}}}{2}$ → Answer: A
Evaluation G - 9 | 2.2 Solutions (Algebraic Manipulation)
2.2.1 Expand $(x + 5)(x - 3)$. (A) $x^2 - 2x - 15$ (B) $x^2 + 2x - 15$ (C) $x^2 + 8x - 15$ (D) $x^2 - 8x - 15$ (E) $x^2 + 2x + 15$.
$(x+5)(x-3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15$ → Answer: B
2.2.2 Factorise $x^2 + 7x + 12$. (A) $(x-3)(x-4)$ (B) $(x+6)(x+2)$ (C) $(x+3)(x+4)$ (D) $(x-6)(x-2)$ (E) $(x+12)(x+1)$.
Factors of 12 that add to 7 are 3 and 4 → $(x+3)(x+4)$ → Answer: C
2.2.3 Expand and simplify $(2x - 3)^2$. (A) $4x^2 - 12x + 9$ (B) $4x^2 - 6x + 9$ (C) $4x^2 - 12x - 9$ (D) $4x^2 + 12x + 9$ (E) $2x^2 - 12x + 9$.
$(2x-3)^2 = 4x^2 - 12x + 9$ → Answer: A
2.2.4 Factorise $x^2 - 25$. (A) $(x-5)^2$ (B) $(x+5)^2$ (C) $(x-25)(x+1)$ (D) $(x-5)(x+5)$ (E) $(x-1)(x+25)$.
Difference of squares: $x^2 - 25 = (x-5)(x+5)$ → Answer: D
2.2.5 Complete the square for $x^2 + 6x - 2$. (A) $(x+3)^2 - 7$ (B) $(x-3)^2 - 11$ (C) $(x+6)^2 - 38$ (D) $(x+3)^2 - 11$ (E) $(x-6)^2 - 38$.
$(x+3)^2 = x^2 + 6x + 9$, so $x^2+6x-2 = (x+3)^2 - 11$ → Answer: D
2.2.6 Factorise $2x^2 + 7x + 3$. (A) $(2x+3)(x+1)$ (B) $(2x-1)(x-3)$ (C) $(x+3)(2x-1)$ (D) $(2x+1)(x+3)$ (E) $(2x+3)(x-1)$.
$(2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ → Answer: D
2.2.7 Expand $(3x - 2)(x + 4)$. (A) $3x^2 - 10x - 8$ (B) $3x^2 + 14x - 8$ (C) $3x^2 + 10x + 8$ (D) $3x^2 - 14x + 8$ (E) $3x^2 + 10x - 8$.
$(3x-2)(x+4) = 3x^2 + 12x - 2x - 8 = 3x^2 + 10x - 8$ → Answer: E
2.2.8 Factorise $x^2 - 8x + 16$. (A) $(x+4)^2$ (B) $(x-8)(x-2)$ (C) $(x+8)(x+2)$ (D) $(x-2)^2$ (E) $(x-4)^2$.
Perfect square: $x^2 - 8x + 16 = (x-4)^2$ → Answer: E
2.2.9 Complete the square for $x^2 - 10x + 3$. (A) $(x-5)^2 - 25$ (B) $(x-10)^2 - 97$ (C) $(x+5)^2 - 22$ (D) $(x-5)^2 - 22$ (E) $(x+5)^2 - 25$.
$(x-5)^2 = x^2 - 10x + 25$, so $x^2-10x+3 = (x-5)^2 - 22$ → Answer: D
2.2.10 Factorise $6x^2 - 13x + 6$. (A) $(2x+3)(3x+2)$ (B) $(2x-3)(3x+2)$ (C) $(2x+3)(3x-2)$ (D) $(2x-3)(3x-2)$ (E) $(6x-1)(x-6)$.
$(2x-3)(3x-2) = 6x^2 - 4x - 9x + 6 = 6x^2 - 13x + 6$ → Answer: D
2.2.11 Expand $(4x + 3)(2x - 5)$. (A) $8x^2 + 14x - 15$ (B) $8x^2 - 14x + 15$ (C) $8x^2 - 22x - 15$ (D) $8x^2 + 22x - 15$ (E) $8x^2 - 14x - 15$.
$(4x+3)(2x-5) = 8x^2 - 20x + 6x - 15 = 8x^2 - 14x - 15$ → Answer: E
2.2.12 Factorise $4x^2 - 9$. (A) $(2x-3)^2$ (B) $(2x+3)^2$ (C) $(4x-3)(x+3)$ (D) $(4x+3)(x-3)$ (E) $(2x-3)(2x+3)$.
Difference of squares: $4x^2 - 9 = (2x-3)(2x+3)$ → Answer: E
2.2.13 Complete the square for $2x^2 - 8x + 5$. (A) $2(x-2)^2 + 5$ (B) $2(x-4)^2 - 27$ (C) $2(x+2)^2 - 3$ (D) $(x-2)^2 - 3$ (E) $2(x-2)^2 - 3$.
Factor 2: $2(x^2 - 4x) + 5 = 2[(x-2)^2 - 4] + 5 = 2(x-2)^2 - 8 + 5 = 2(x-2)^2 - 3$ → Answer: E
2.2.14 Factorise $x^2 - 11x + 24$. (A) $(x-4)(x-6)$ (B) $(x+3)(x+8)$ (C) $(x+4)(x+6)$ (D) $(x-2)(x-12)$ (E) $(x-3)(x-8)$.
Factors of 24 that add to 11 are 3 and 8 → $(x-3)(x-8)$ → Answer: E
2.2.15 Expand $(x + 6)^2$. (A) $x^2 + 6x + 36$ (B) $x^2 + 36$ (C) $x^2 - 12x + 36$ (D) $x^2 + 12x - 36$ (E) $x^2 + 12x + 36$.
$(x+6)^2 = x^2 + 12x + 36$ → Answer: E
2.2.16 Factorise $3x^2 - 10x + 8$. (A) $(3x-2)(x-4)$ (B) $(3x+4)(x+2)$ (C) $(3x+2)(x+4)$ (D) $(3x-8)(x-1)$ (E) $(3x-4)(x-2)$.
$(3x-4)(x-2) = 3x^2 - 6x - 4x + 8 = 3x^2 - 10x + 8$ → Answer: E
2.2.17 Complete the square for $x^2 + 4x - 7$. (A) $(x+2)^2 - 3$ (B) $(x-2)^2 - 11$ (C) $(x+4)^2 - 23$ (D) $(x-4)^2 - 23$ (E) $(x+2)^2 - 11$.
$(x+2)^2 = x^2 + 4x + 4$, so $x^2+4x-7 = (x+2)^2 - 11$ → Answer: E
2.2.18 Factorise $x^2 - 2x - 35$. (A) $(x+7)(x-5)$ (B) $(x-35)(x+1)$ (C) $(x+35)(x-1)$ (D) $(x-5)(x+7)$ (E) $(x-7)(x+5)$.
Factors of -35 that add to -2 are -7 and +5 → $(x-7)(x+5)$ → Answer: E
2.2.19 Expand $(5x - 2)(5x + 2)$. (A) $25x^2 + 4$ (B) $25x^2 - 20x + 4$ (C) $25x^2 + 20x + 4$ (D) $25x^2 - 20x - 4$ (E) $25x^2 - 4$.
Difference of squares: $(5x-2)(5x+2) = 25x^2 - 4$ → Answer: E
2.2.20 Complete the square for $3x^2 + 12x + 7$. (A) $3(x+2)^2 + 7$ (B) $3(x+4)^2 - 41$ (C) $3(x+2)^2 - 5$ (D) $3(x-2)^2 - 5$ (E) $3(x+2)^2 + 5$.
Factor 3: $3(x^2 + 4x) + 7 = 3[(x+2)^2 - 4] + 7 = 3(x+2)^2 - 12 + 7 = 3(x+2)^2 - 5$ → Answer: C
