Mathematics Lecture Review: Exponents, Cubes, Divisibility, and Ratios

Laws of Exponents and Scientific Notation

Power notation is a mathematical representation used to express the repeated multiplication of a number by itself. It consists of a base number and an exponent. The base number identifies the value being multiplied, while the exponent specifies the number of times the base number is multiplied by itself.

For any two non-zero integers $a$ and $b$, and whole numbers $m$ and $n$, the fundamental laws of exponents are defined as follows:

$a^m \times a^n = a^{m+n}$

$a^m \times b^m = (ab)^m$

$a^m \frac{a^m}{a^n} = a^{m-n} \quad (\text{where } m > n)$

$a^m \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$

$(a^m)^n = a^{mn}$

$a^0 = 1$

$(-1)^{\text{even number}} = 1$

$(-1)^{\text{odd number}} = -1$

In scientific notation, a number is written as the product of a decimal term and an exponential term of base 10. The decimal term indicates the number of significant figures in the numerical value, whereas the exponential term specifies the precise location of the decimal point.

Evaluation problems based on exponential laws include the following calculations:

1. Prime factorisation of $1050$ is $2 \times 3 \times 5^2 \times 7$.

2. Comparing exponential values such as $3 \times 5^2 = 75$, $2 \times 5^3 = 250$, $5 \times 3^2 = 45$, and $3 \times 25 = 75$ identifies $2 \times 5^3$ as the greatest value.

3. Expression $m$ multiplied $n$ times by itself in exponential form is written as $m^n$.

4. Simplification of $(-1) \times (-1)^5 \times (-1) \times (-1)^{23} \times (-1)^2$ yields (-1)^{1 + 5 + 1 + 23 + 2} = (-1)^{32} = 1$.\n\n5. The worldwide population of sheep in 2024 is approximately 10^9$, and the population of goats is also approximately$10^9$. The combined total population is$10^9 + 10^9 = 2 \times 10^9.\n\n6. The distance between the Sun and Saturn is 1,433,500,000,000\,m$, which expressed in scientific notation is$1.4335 \times 10^{12}\,m.\n\n7. Comparison of 3^5 = 243$and$5^3 = 125$shows$3^5 > 5^3$.

5. Exponential identity equivalence: $432 = 2^4 \times 3^3$.

6. Unit conversion exponent: 1\text{ million} = 10^6$.\n\n10. Numerical evaluation: 2^3 \times 3^2 + 3^3 \times 2^3 = 8 \times 9 + 27 \times 8 = 72 + 216 = 288$.

7. Numerical evaluation: (2^{20} + 2^{16}) \times 2^6 = 2^{26} + 2^{22}$.\n\n12. Numerical evaluation: 2^4 \times 10^4 + (10^2 \times 2^2)^2 = 16 \times 10000 + (100 \times 4)^2 = 160000 + 160000 = 320000$.

8. Numerical evaluation: (8^0 + 3^0)(4^0 + 5^2)^2 = (1 + 1)(1 + 25)^2 = 2 \times 26^2 = 2 \times 676 = 1352$.\n\n14. Algebraic simplification: \frac{(-18)^4 \times 9^3 \times 4^8}{6^5 \times 8 \times 9^2} = \frac{(2 \times 3^2)^4 \times (3^2)^3 \times (2^2)^8}{(2 \times 3)^5 \times 2^3 \times (3^2)^2} = \frac{2^4 \times 3^8 \times 3^6 \times 2^{16}}{2^5 \times 3^5 \times 2^3 \times 3^4} = \frac{2^{20} \times 3^{14}}{2^8 \times 3^9} = 2^{12} \times 3^5 = 4096 \times 243 = 995328$.

9. Light year conversion: $1\text{ light year} = 9,460,000,000,000\,km = 9.46 \times 10^{12}\,km$. Comparing the average distance between the Sun and Earth ($1.496 \times 10^8\,km$) to 1 light year gives the ratio $\frac{1.496 \times 10^8}{9.46 \times 10^{12}} \approx 1.58 \times 10^{-5}$.

Number Theory, Divisibility Rules, and Remainder Properties

A divisibility test provides a method to determine whether a whole number is divisible by a given integer without performing long division. Standard divisibility criteria include:

• Divisibility by 2: The last digit of the number is $0$, $2$, $4$, $6$, or 8$.\n- Divisibility by 3: The sum of all individual digits in the number is divisible by 3$.
• Divisibility by 4: The number formed by the last two digits of the given number is divisible by 4$.\n- Divisibility by 5: The last digit of the number is 0$or$5$.
• Divisibility by 6: The number satisfies the divisibility criteria for both $2$ and 3$.\n- Divisibility by 8: The number formed by the last three digits is divisible by 8$.
• Divisibility by 9: The sum of all individual digits in the number is divisible by 9$.\n- Divisibility by 10: The last digit of the number is 0$.
• Divisibility by 11: The difference between the sum of the digits at odd places and the sum of the digits at even places is either $0$ or a multiple of 11$.\n- Divisibility by 24: A number is divisible by 24$if and only if it is divisible by both$3$and$8$, as$3$and$8$are co-prime factors of$24$($24 = 2^3 \times 3).\n\nAlgebraic proof for divisibility by 11 in a four-digit number abcd$: The number can be expanded as$1000a + 100b + 10c + d = (1001a - a) + (99b + b) + (11c - c) + d = (1001a + 99b + 11c) - (a - b + c - d) = 11(91a + 9b + c) - ((a + c) - (b + d))$. Since$11(91a + 9b + c)$is a multiple of$11$, the full number is divisible by$11$if and only if$(a + c) - (b + d)$is divisible by$11$.

General factor divisibility property: If a number is divisible by an integer $m$, then it is guaranteed to be divisible by every factor of $m$. For instance, if a number is divisible by $12$, it is necessarily divisible by $1$, $2$, $3$, $4$, and 6$.\n\nRemainder arithmetic properties under modulo operations:\n\n1. If x + 5 \equiv 4 \pmod k$and$x + 2 \equiv 1 \pmod k$, subtracting the two equations yields$(x + 5) - (x + 2) = 3 \equiv 3 \pmod k, which is consistent for base systems.\n\n2. Numbers between 80$and$105$that are divisible by$9$are$81$,$90$, and$99$.

3. Three-digit numbers completely divisible by $3$ formed using digits $0$, $4$, $5$, and $8$ without repetition must have a digit sum divisible by $3$. The sum of $4+5+0 = 9$ (divisible by 3) and $4+5+8 = 17$ (not divisible). Thus using $(0, 4, 5)$, the valid three-digit numbers are $405$, $450$, $504$, and 540$.\n\n4. Two-digit numbers divisible by 5$using digits$0$,$2$,$4$,$5$once must end in$0$or$5$. Valid numbers are$20$,$40$,$50$,$25$, and$45$.

4. Addition and subtraction of remainders: If $854 \equiv 2 \pmod 6$ and $1325 \equiv 5 \pmod 6$, then:

   • $1325 + 854 \equiv 5 + 2 = 7 \equiv 1 \pmod 6$.
   • 1325 - 854 \equiv 5 - 2 = 3 \pmod 6$.\n\n6. System of linear remainders: A number leaving remainder 1$when divided by$4$, remainder$2$when divided by$5$, and remainder$3$when divided by$6$can be analyzed using modular congruences:$N \equiv -3 \pmod 4$,$N \equiv -3 \pmod 5$, and$N \equiv -3 \pmod 6$. Therefore,$N + 3$must be a multiple of$\text{LCM}(4, 5, 6) = 60$. The general expression is$N = 60k - 3$. For$k = 1$, the smallest positive integer is$57$.

5. System of remainders: A number leaving remainder $2$ when divided by $3$, remainder $3$ when divided by $4$, and remainder $4$ when divided by $5$ satisfies $N \equiv -1 \pmod 3$, $N \equiv -1 \pmod 4$, and $N \equiv -1 \pmod 5$. Thus $N + 1$ is a multiple of $\text{LCM}(3, 4, 5) = 60$. The smallest positive integer is 59$.\n\n# Squares, Square Roots, and Special Numerical Patterns\n\nA square number is an integer produced by multiplying an integer by itself. Finding the square root of a number is the inverse operation of squaring.\n\nPractical application problems involving square roots:\n\n1. Smallest square number divisible by 5$,$10$, and$25$: The prime factorisations are$5 = 5$,$10 = 2 \times 5$, and$25 = 5^2$. The LCM is$2 \times 5^2 = 50$. To make all exponent powers even for a perfect square, multiply by$2$, yielding$2^2 \times 5^2 = 100$.

6. Smallest square number divisible by $8$, $12$, and $16$: The prime factorisations are $8 = 2^3$, $12 = 2^2 \times 3$, and $16 = 2^4$. The LCM is $2^4 \times 3 = 48$. To make exponents even, multiply by $3$, giving 2^4 \times 3^2 = 144$.\n\n3. Equal donation word problem: Three sections of class VIII donated a total of \text{₹}5929$. Each student donated an amount equal to the total number of students. If$x$is the number of students, then$x^2 = 5929$. Taking the square root gives$x = \sqrt{5929} = 77 students.\n\n4. Library arrangement word problem: A librarian places 1225$books in rows such that the number of rows equals the number of books per row. The equation is$x^2 = 1225$, which yields$x = \sqrt{1225} = 35 books per row.\n\n5. Military square formation word problem: A General has 7774$soldiers. When forming a perfect square,$30$soldiers are left out. The number of soldiers in the square is$7774 - 30 = 7744$. The number of soldiers in each row is$\sqrt{7744} = 88$.

7. Square plot fencing cost word problem: The area of a square plot is $60025\,m^2$. The side length is $s = \sqrt{60025} = 245\,m$. The perimeter of the plot is $4 \times 245 = 980\,m$. At a rate of $\text{₹}50$ per metre, the cost of fencing is 980 \times 50 = \text{₹}49000$.\n\n7. Farm planting word problem: Out of 6250$plants,$9$plants are left over when forming equal rows and columns. The plants in the square formation equal$6250 - 9 = 6241$. The number of plants per row is$\sqrt{6241} = 79$.

Patterns and critical thinking puzzles in squares:

1. Middle number pattern rules: In geometric pattern grids, if the middle number is the square root of the sum of extreme numbers, or the sum of square roots of extreme numbers, individual values can be computed by isolating the unknown term.

2. Square root sign error analysis: A student stated that if the square roots of a number are $0.9$ and $-0.9$, the number is their product $-0.81$. The error is that the original number is the square of either square root, meaning $(0.9)^2 = (-0.9)^2 = 0.81$. Square roots cannot yield a negative product for a real square number.

3. Number identities and strange squares:

   • An odd square number that is a multiple of $7$ and less than $50$ is $49$ ($7^2$).
   • The square root of the number of squares on a chessboard ($64$ total 1x1 squares) is \sqrt{64} = 8$.\n - Numbers whose square equals themselves are 0$and$1.\n - Mirror image squares: 13^2 = 169$and$31^2 = 961$are mirror images. Other strange pairs include$12^2 = 144$and$21^2 = 441$, as well as$102^2 = 10404$and$201^2 = 40401$.
   • The square of $567$ is 567^2 = 321489$.\n - A three-digit perfect square whose digit reversal is also a perfect square includes 144$(reverse$441 = 21^2$) and$169$(reverse$961 = 31^2).\n\n4. Pythagorean Triplets: A set of three positive integers (a, b, c)$satisfying$a^2 + b^2 = c^2$. For a triplet with greatest member$37$($c = 37$), using$c = m^2 + 1 = 37$gives$m^2 = 36 \implies m = 6$. The remaining members are$2m = 12$and$m^2 - 1 = 35$. The triplet is$(12, 35, 37).\n\n5. Multiples and Long Division Square Roots:\n - Prime factorisation of 2592 = 2^5 \times 3^4$. To make it a perfect square, multiply by$2$, giving$2592 \times 2 = 5184$. The square root is$\sqrt{5184} = 72$.
   • The greatest four-digit number is $9999$. Long division shows $99^2 = 9801$ with remainder $198$. Thus, the greatest four-digit perfect square is 9801$.\n - Long division of decimal 0.041616$yields$\sqrt{0.041616} = 0.204$.

4. Assertion-Reason Statement: Assertion (A): The square of an integer is always a non-negative number. This statement is True because the product of two positive numbers or two negative numbers is always positive, and 0^2 = 0$.\n\n# Cubes, Cube Roots, and Cube Sum Identities\n\nThe cube of a number is defined as the product obtained by multiplying a number by itself three times. If x$is a non-zero number, then$x^3 = x \times x \times x$. If$m = n^3$for natural numbers$m$and$n$, then$m is called a perfect cube or cubic number.\n\nKey properties of cubes and cube roots:\n\n1. The cube of every even number is even, and the cube of every odd number is odd.\n\n2. The cubes of all negative numbers are negative.\n\n3. Ending digits of cubes:\n - Numbers ending in 0, 1, 4, 5, 6, 9$have cubes ending in$0, 1, 4, 5, 6, 9 respectively.\n - Numbers ending in 2$have cubes ending in$8$, and numbers ending in$8$have cubes ending in$2$.

   • Numbers ending in $3$ have cubes ending in $7$, and numbers ending in $7$ have cubes ending in 3$.\n\n4. Cube root radical notation: The cube root of a number y$is denoted by$\sqrt[3]{y}$. If$x^3 = y$, then$\sqrt[3]{y} = x$.

5. Product and quotient rules for cube roots:

$\sqrt[3]{xy} = \sqrt[3]{x} \times \sqrt[3]{y}$

$\sqrt[3]{\frac{x}{y}} = \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \quad (\text{where } y \neq 0)$

Evaluation problems and algebraic puzzles on cubes:

1. Number of zeros at the end: If a perfect cube ends with 9 zeros, its cube root contains $\frac{9}{3} = 3$ zeros.

2. Maximum digits in cube root: For a six-digit number, grouping digits in triplets from right to left gives 2 groups, so the cube root contains $2$ digits.

3. Cube root calculation: $\sqrt[3]{13824} = 24$ (tens digit 2, units digit 4).

4. Cube of $35$: 35^3 = 42875$.\n\n5. Cube root of decimal: \sqrt[3]{0.010648} = 0.22$.

5. Units place of $124^3$: Since the last digit is $4$, $4^3 = 64$, so the units digit is 4$.\n\n7. Functional equivalence: If p$is the cube root of$q$($p = \sqrt[3]{q}$), then$p^3 = q$.

6. Non-perfect cube identification: $100000$ ($10^5$) is not a perfect cube because the exponent of 10 is not a multiple of 3.

7. Smallest multiple for perfect cube: Prime factorisation of $36 = 2^2 \times 3^2$. To form triplets, multiply by $2 \times 3 = 6$. The smallest multiple is 36 \times 6 = 216$.\n\n10. Special number characteristics:\n - A two-digit even number that is a square and a cube is 64$($8^2 = 4^3).\n - The largest negative perfect cube integer is -1.\n - The LCM of \sqrt[3]{125} = 5$and$\sqrt{64} = 8$is$\text{LCM}(5, 8) = 40$.

8. Armstrong numbers (sum of cubes of digits): Three-digit numbers equal to the sum of the cubes of their digits include $153$, $370$, $407$, and the fourth number is $371$ ($3^3 + 7^3 + 1^3 = 27 + 343 + 1 = 371$).

9. Division to form a perfect cube: Prime factorisation of $106480 = 2^4 \times 5 \times 11^3 = (2^3 \times 11^3) \times (2 \times 5)$. The un-grouped factor is $2 \times 5 = 10$. Dividing $106480$ by $10$ yields the perfect cube $10648$ ($22^3$).

10. Cube root evaluation using factors:

    • \sqrt[3]{209584584} = \sqrt[3]{5832 \times 35937} = \sqrt[3]{5832} \times \sqrt[3]{35937} = 18 \times 33 = 594$.\n - \sqrt[3]{29554216} = \sqrt[3]{10648 \times 12167} = \sqrt[3]{10648} \times \sqrt[3]{12167} = 22 \times 23 = 506$.

11. Ratio of cubes problem: Three numbers are in ratio $2:3:5$. Let the numbers be $2x, 3x, 5x$. The sum of their cubes is $(2x)^3 + (3x)^3 + (5x)^3 = 8x^3 + 27x^3 + 125x^3 = 160x^3 = 10240$. Solving for $x^3$ gives x^3 = \frac{10240}{160} = 64 \implies x = 4$. The numbers are 8$,$12$, and$20$.

12. Triangular sum of cubes formula: The sum of the first $n$ cubes is equal to the square of the $n$-th triangular number:

$1^3 + 2^3 + 3^3 + \dots + n^3 = \left[\frac{n(n+1)}{2}\right]^2 = (1 + 2 + 3 + \dots + n)^2$

For $n = 5$: 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = (1 + 2 + 3 + 4 + 5)^2 = 15^2 = 225$.\n\nFor n = 15$:$1^3 + 2^3 + 3^3 + \dots + 15^3 = \left[\frac{15 \times 16}{2}\right]^2 = (120)^2 = 14400$.

Assertion-Reason Questions on Cubes:

• Assertion (A): The volume of a cube can be calculated by raising the length of one edge to the power of three. Reason (R): A cube has three equal dimensions and volume is length times width times height. Both (A) and (R) are true, and (R) is the correct explanation of (A).
• Assertion (A): The cube root of a number is always an integer if the original number is a perfect cube. Reason (R): A perfect cube is defined as the cube of an integer. Both (A) and (R) are true, and (R) is the correct explanation of (A).
• Assertion (A): To find the cube root of 27, you determine which number multiplied by itself three times equals 27. Reason (R): The cube root of 27 is 3 because $3 \times 3 \times 3 = 27$. Both (A) and (R) are true, and (R) is the correct explanation of (A).

Algebraic Expressions, Polynomial Operations, and Historical Multiplication Techniques

An algebraic expression is formed from variables and constants using algebraic operations. A monomial has one term, a binomial has two terms, a trinomial has three terms, and a polynomial is a general expression with one or more terms.

Historical Note on Multiplication: Methods for quick multiplication using the distributive property were developed in ancient Indian mathematics. These techniques appear in the foundational texts of Brahmagupta (628 CE), Sridharacharya (750 CE), and Bhaskaracharya (Lilavati, 1150 CE). Brahmagupta formally termed these methods ishta-gunana in his treatise Brahmasphutasiddhanta.

Monomial and Polynomial Operations:

1. Monomial Products:

   • $3m^2 \times \left(-3m - \frac{1}{2}n + mn\right) = -9m^3 - \frac{3}{2}m^2n + 3m^3n$
   • $-\frac{3}{4}ab \times (2a - 8b + 4ac) = -\frac{3}{2}a^2b + 6ab^2 - 3a^2bc$
   • $-xy \times (-x - y + xy - 1) = x^2y + xy^2 - x^2y^2 + xy$

2. Algebraic Simplification and Evaluation:

   • Simplify 3a(a - b) + b(a + b) = 3a^2 - 3ab + ab + b^2 = 3a^2 - 2ab + b^2$.\n - Evaluating for a = 1, b = -1$gives$3(1)^2 - 2(1)(-1) + (-1)^2 = 3 + 2 + 1 = 6$.

3. Addition of Expressions:

   • Add $l(l - m)$, $m(m - n)$, and $n(n - l)$: (l^2 - lm) + (m^2 - mn) + (n^2 - nl) = l^2 + m^2 + n^2 - lm - mn - nl$.\n - Add 2a(c - a - b)$and$2b(c - b - a)$:$(2ac - 2a^2 - 2ab) + (2bc - 2b^2 - 2ab) = -2a^2 - 2b^2 - 4ab + 2ac + 2bc$.

4. Subtraction of Polynomials:

   • Subtract $2p(p - q + 3r) = 2p^2 - 2pq + 6pr$ from $4p(5r - 2q + 3p) = 20pr - 8pq + 12p^2$:      (12p^2 - 8pq + 20pr) - (2p^2 - 2pq + 6pr) = 10p^2 - 6pq + 14pr$.\n - Subtract the sum of a(a + 2b + c) = a^2 + 2ab + ac$and$-b(a - b + 2c) = -ab + b^2 - 2bc$(Sum =$a^2 + b^2 + ab + ac - 2bc$) from$c(-a - b + c) = -ac - bc + c^2:\n     (-ac - bc + c^2) - (a^2 + b^2 + ab + ac - 2bc) = -a^2 - b^2 + c^2 - ab - 2ac + bc$.

5. Multi-term Simplifications:

   • 2x^2(x^3 - x) - 3x(x^3 + 2x) - 2(x^4 - 3x^2) = (2x^5 - 2x^3) - (3x^4 + 6x^2) - (2x^4 - 6x^2) = 2x^5 - 5x^4 - 2x^3$.\n - 4ab(ab - b) - 6a^2(b^2 - b) - 3b^2(2a - a^2) + 2ab(a - b) = (4a^2b^2 - 4ab^2) - (6a^2b^2 - 6a^2b) - (6ab^2 - 3a^2b^2) + (2a^2b - 2ab^2) = a^2b^2 + 8a^2b - 12ab^2$.

6. Binomial Multiplication:

   • Degree of product $(3x^2y^2 - 5)(x - 1)$ is $4 + 1 = 5$.
   • Expansion of (x - 4)(x + 5) = x^2 + 5x - 4x - 20 = x^2 + x - 20$.\n - Evaluating [2a + (-b)] \times [-3a + (-5)]$for$a = 0, b = -1$gives$[0 - (-1)] \times [0 - 5] = (1)(-5) = -5$.

Standard Algebraic Identities, Applications, and Higher-Order Thinking Problems

Standard algebraic identities serve as foundational equivalence laws for simplifying polynomials and computing numerical products:

Identity 1: $(a + b)^2 = a^2 + 2ab + b^2$

Identity 2: $(a - b)^2 = a^2 - 2ab + b^2$

Identity 3: $(a + b)(a - b) = a^2 - b^2$

Identity 4: $(x + a)(x + b) = x^2 + (a + b)x + ab$

Evaluation and Proof Applications:

1. Numerical identity computation:

   • 536^2 - 136^2 = (536 - 136)(536 + 136) = (400)(672) = 268800$.\n - 62^2 = (60 + 2)^2 = 3600 + 240 + 4 = 3844$.
   • 108^2 = (100 + 8)^2 = 10000 + 1600 + 64 = 11664$.\n - 403^2 = (400 + 3)^2 = 160000 + 2400 + 9 = 162409$.
   • 10.2^2 = (10 + 0.2)^2 = 100 + 4 + 0.04 = 104.04$.\n - 58 \times 62 = (60 - 2)(60 + 2) = 60^2 - 2^2 = 3600 - 4 = 3596$.
   • 103 \times 97 = (100 + 3)(100 - 3) = 10000 - 9 = 9991$.\n\n2. Reciprocal transformations:\n - If x + \frac{1}{x} = 11$, squaring both sides gives$x^2 + 2 + \frac{1}{x^2} = 121 \implies x^2 + \frac{1}{x^2} = 119$.
   • Squaring again gives x^4 + 2 + \frac{1}{x^4} = 119^2 = 14161 \implies x^4 + \frac{1}{x^4} = 14159$.\n - If x - \frac{1}{x} = 5$, squaring both sides gives$x^2 - 2 + \frac{1}{x^2} = 25 \implies x^2 + \frac{1}{x^2} = 27$.

2. Expression evaluation: Find $36x^2 + 49y^2 - 84xy = (6x - 7y)^2$. For $x = 2, y = -2$, the value is [6(2) - 7(-2)]^2 = (12 + 14)^2 = 26^2 = 676$.\n\n4. Higher-Order Thinking Skills (HOTS):\n - Given a + b = 5$and$ab = 2:\n     (a) (a + b)^2 = 5^2 = 25$.      (b) a^2 + b^2 = (a + b)^2 - 2ab = 25 - 4 = 21$.\n     (c) (a - b)^2 = a^2 + b^2 - 2ab = 21 - 4 = 17$.

   • Solve for $p$ in $7p = 76^2 - 71^2$: 7p = (76 - 71)(76 + 71) = 5 \times 147 = 735 \implies p = 105$.\n - Solve for p$in$15p = 25^2 - 10^2$:$15p = (25 - 10)(25 + 10) = 15 \times 35 \implies p = 35$.
   • Solve for $x$ in $576^2 - 224^2 = 704x$: (576 - 224)(576 + 224) = (352)(800) = 281600 = 704x \implies x = 400$.\n - Solve for m$in$\frac{1.75 \times 1.75 - 0.25 \times 0.25}{1.75 + 0.25} = 3m$:$\frac{(1.75 - 0.25)(1.75 + 0.25)}{1.75 + 0.25} = 1.50 = 3m \implies m = 0.5$.
   • Solve for $m$ in $\frac{599^2 - 401^2}{660} = m$: \frac{(599 - 401)(599 + 401)}{660} = \frac{198 \times 1000}{660} = 300 \implies m = 300$.\n\n5. Algebraic Identity Verification Statements:\n - (3a - 2b)(3a + 2b) + (2a - 3b)(2a + 3b) = (9a^2 - 4b^2) + (4a^2 - 9b^2) = 13a^2 - 13b^2 = 13(a + b)(a - b).\n - (k + 1)(k + 2) - k(k + 3) = (k^2 + 3k + 2) - (k^2 + 3k) = 2 (always constant equal to 2).\n\n# Polynomial Division and Factorization Theorems\n\nDivision of algebraic expressions involves dividing polynomial terms by monomials, binomials, or trinomials using factorisation or long division algorithms.\n\nDivision Algorithm: \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}\n\nPolynomial Long Division Examples:\n\n1. Divide 15x^3 + 37x^2 - 53x + 55$by$3x + 5:\n - First term of quotient: \frac{15x^3}{3x} = 5x^2.\n - Subtract (15x^3 + 25x^2)$to get remainder$12x^2 - 53x + 55$.
   • Second term of quotient: \frac{12x^2}{3x} = 4x$.\n - Subtract (12x^2 + 20x)$to get remainder$-73x + 55$.
   • Third term of quotient: \frac{-73x}{3x} = -\frac{73}{3}$.\n\n2. Divide 6x^2 - 10x - 4$by$x - 1:\n - Quotient is 6x - 4$with a remainder of$-8$.
   • Verification: (x - 1)(6x - 4) + (-8) = 6x^2 - 10x + 4 - 8 = 6x^2 - 10x - 4$.\n\n3. Factor Determination using Remainder Theorem: To determine if 2x - 3$is a factor of$6x^3 - x^2 - 10x + m$, substitute$x = \frac{3}{2} and set the expression to zero:\n   6\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - 10\left(\frac{3}{2}\right) + m = 0\n   6\left(\frac{27}{8}\right) - \frac{9}{4} - 15 + m = 0\n   \frac{81}{4} - \frac{9}{4} - 15 + m = 0\n   18 - 15 + m = 0 \implies 3 + m = 0 \implies m = -3\n\n4. Addition required for exact divisibility: To make 6x^3 - x^2 - 10x - 3$divisible by$2x - 3$, evaluate the remainder at$x = \frac{3}{2}$. The value is$3 - 3 = 0$, so the remainder is already$0$. If a non-zero remainder$R$were obtained,$-R must be added.\n\n5. Calendar Matrix Diagonal Product Property: In any 2 by 2 square matrix on a calendar of the form:\n\n\begin{pmatrix} a & a+1 \ a+7 & a+8 \end{pmatrix}\n\nThe product of the diagonal entries gives:\n- Primary diagonal product: a(a + 8) = a^2 + 8a\n- Secondary diagonal product: (a + 1)(a + 7) = a^2 + 8a + 7\n\nThe difference between the two diagonal products is always (a^2 + 8a + 7) - (a^2 + 8a) = 7.\n\n# Geometry of Quadrilaterals, Polygons, and Diagonal Properties\n\nA quadrilateral is a closed two-dimensional polygon with four edges and four vertices.\n\nProperties of Quadrilaterals and Parallelograms:\n\n1. Square: Possesses all individual geometric properties of both a rhombus and a rectangle (all sides equal, all interior angles 90^\circ, diagonals equal and perpendicular bisectors).\n\n2. Trapezium: A quadrilateral having exactly one pair of parallel sides.\n\n3. Isosceles Trapezium: A trapezium in which the non-parallel sides are equal in length.\n\n4. Rhombus: A parallelogram having all four sides of equal length. Diagonals bisect each other at right angles (90^\circ).\n\n5. Parallelogram Rules:\n - Adjacent interior angles are supplementary (sum to 180^\circ).\n - Opposite sides and opposite angles are equal.\n - Diagonals bisect each other.\n - If the diagonals are equal and bisect each other at right angles, the parallelogram is a square.\n\nRegular Polygon Angle Formula:\nEach interior angle of a regular polygon with n sides is given by:\n\n\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}\n\nIf each interior angle measures 165^\circ:\n165 = \frac{(n - 2) \times 180}{n} \implies 165n = 180n - 360 \implies 15n = 360 \implies n = 24\text{ sides}\n\nGeometric Rhombus Problem:\nIn a rhombus RICE$, diagonals$IE$and$RC$intersect at point$O$. Given$OC = 12\,cm$and$OE = 5\,cm:\n- Since diagonals bisect each other, OI = OE = 5\,cm$, and$OR = OC = 12\,cm$.

• In right triangle $ROC$ (or $ROE$), side length RE = \sqrt{OR^2 + OE^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\,cm$.\n\nMidpoint Theorem in Right Triangles:\nIn a right-angled triangle ABC$with the right angle at$B$, the midpoint$O$of the hypotenuse$AC$is equidistant from vertices$A$,$B$, and$C$. Constructing a rectangle$ABCD$shows that diagonals$AC$and$BD$are equal and bisect each other at$O$. Thus,$OA = OB = OC = \frac{1}{2}AC$.

Ratio, Proportion, and Practical Unit Conversions

A ratio compares two quantities of the same unit in terms of multiplication or division. In the ratio $a:b$, $a$ is the antecedent and $b$ is the consequent.

A proportion expresses the equivalence of two ratios: $a:b :: c:d \implies \frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$, where $a, d$ are extremes and $b, c$ are means.

Three quantities $a, b, c$ are in continued proportion if $a:b :: b:c \implies b^2 = ac$, where $b = \sqrt{ac}$ is the mean proportional.

Standard Unit Conversion Reference Guide:

• Length Conversion: $1\text{ metre} = 3.281\text{ feet}$
• Area Conversion: $1\text{ square metre} = 10.764\text{ square feet}$
• Acre Conversion: $1\text{ acre} = 43,560\text{ square feet}$
• Hectare Conversion: $1\text{ hectare} = 10,000\text{ square metres}$
• Volume Conversion: $1\text{ Litre} = 1,000\text{ mL} = 1,000\text{ cc}$
• Temperature Formulas:   $\text{Fahrenheit} = \left(\frac{9}{5} \times \text{Celsius}\right) + 32$$\text{Celsius} = \frac{5}{9} \times (\text{Fahrenheit} - 32)$   Example: $25^\circ\text{C} = \left(\frac{9}{5} \times 25\right) + 32 = 45 + 32 = 77^\circ\text{F}$.

Applied Ratio and Proportion Word Problems:

1. Fourth Proportional Calculations:

   • For $8, 12, 16, x$: \frac{8}{12} = \frac{16}{x} \implies 8x = 192 \implies x = 24$.\n - For 4, 7, 8, x$:$\frac{4}{7} = \frac{8}{x} \implies 4x = 56 \implies x = 14$.

2. Mean Proportional Calculations:

   • Mean proportional between $9$ and $4$ is \sqrt{9 \times 4} = \sqrt{36} = 6$.\n - Mean proportional between 2$and$8$is$\sqrt{2 \times 8} = \sqrt{16} = 4$.

3. Number addition to form proportion: To make $1, 3, 10, 18$ proportional by adding $x$:    $\frac{1 + x}{3 + x} = \frac{10 + x}{18 + x} \implies (1 + x)(18 + x) = (3 + x)(10 + x)$$18 + 19x + x^2 = 30 + 13x + x^2 \implies 6x = 12 \implies x = 2$

4. Defective Bulbs Ratio: If 3 out of 12 bulbs are defective, the fraction is $\frac{3}{12} = \frac{1}{4}$. In $100$ bulbs, defective bulbs equal $\frac{1}{4} \times 100 = 25$.

5. Age Ratio Problem: Present ages of two girls are in ratio $3:5$ ($3x$ and $5x$). Five years ago, ratio was $\frac{3x - 5}{5x - 5} = \frac{1}{2}$. Cross-multiplying gives 2(3x - 5) = 1(5x - 5) \implies 6x - 10 = 5x - 5 \implies x = 5$. Present ages are 15$and$25 years.\n\n6. Map Scale Calculations:\n - Scale 1:5,000,000$: Distance of$2\,cm$on map represents$2 \times 5,000,000\,cm = 10,000,000\,cm = 100,000\,m = 100\,km$.

   • Scale $1:20$: Actual length of $20\,m = 2000\,cm$ drawn on map equals \frac{2000}{20} = 100\,cm = 1\,m$.\n\n7. Temperature Shift Conversion: Temperature increases from morning 68^\circ\text{F}$by$18^\circ\text{F}$. An increase of$18^\circ\text{F}$converts to Celsius as$\Delta C = \frac{5}{9} \times 18 = 10^\circ\text{C}.\n\n8. Compost Field Calculation: Recommended compost application is 8\text{ tonnes}$per acre ($43,560\text{ sq ft}$). Plot dimensions are$300\text{ ft} \times 400\text{ ft} = 120,000\text{ sq ft}$. Plot area in acres is$\frac{120,000}{43,560} \approx 2.7548\text{ acres}$. Compost required is$2.7548 \times 8 \approx 22.04\text{ tonnes}$.

6. Tap Filling Rate: A tap fills $400\,mL$ in $12\text{ seconds}$. Flow rate is $\frac{400}{12} = \frac{100}{3}\,mL/s$. To fill a bucket of $12\text{ litres} = 12,000\,mL$, time required is \frac{12000}{100/3} = 360\text{ seconds} = 6\text{ minutes}$.\n\n10. Population Density Crowdedness Comparison:\n - City X: Area 1,600\text{ sq. km}$, Population$32\text{ million} = 32,000,000$. Density is$\frac{32,000,000}{1600} = 20,000\text{ persons/sq. km}$.

   • City Y: Area $800\text{ sq. km}$, Population $24\text{ million} = 24,000,000$. Density is \frac{24,000,000}{800} = 30,000\text{ persons/sq. km}$.\n - City Y is more crowded because it has a higher population density per square kilometre.\n\n11. Medical Dosage Proportion:\n - Saline containing 25\text{ mg}$medication per$200\text{ mL}$. To receive$750\text{ mg}$medication:$\text{Volume} = 750 \times \frac{200}{25} = 6000\text{ mL} = 6\text{ Litres}$.
   • Medication in $9\text{ Litres} = 9000\text{ mL}$ saline: \text{Medication} = 9000 \times \frac{25}{200} = 1125\text{ mg}$.\n\nAssertion-Reason Questions on Ratio:\n- Assertion (A): If the ratio of sand to cement in a mixture is 3:2$, sand forms$\frac{3}{5}$of the mixture. Reason (R): The total number of parts in the ratio is$3 + 2 = 5$$. Both (A) and (R) are true, and (R) is the correct explanation of (A).

• Assertion (A): Two ratios are proportional if they have the same simplest form. Reason (R): Ratios are proportional when the difference between their terms is the same. Assertion (A) is true, but Reason (R) is false because proportionality depends on equal ratios (equal quotient/product of extremes and means), not equal differences.