Vedic Mathematics: Principles and Applications of the Sixteen Sutras
Executive Summary
This briefing document synthesizes the core principles, historical context, and mathematical applications of "Vedic Mathematics," as rediscovered and propounded by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja (1884–1960). Based on an intensive eight-year study of the Atharvaveda, Tirthaji reconstructed sixteen fundamental Sutras (aphorisms) and several sub-sutras that provide a comprehensive, "mental" system for solving complex mathematical problems.
The system is characterized by its "one-line" mental approach, which significantly reduces the time and steps required by conventional Western methods—often by a factor of ten or more. While the system appears "magical" due to its speed, it is rooted in a logical framework that Tirthaji argued is the "fountain-head" of all knowledge, both spiritual and secular. The document outlines the author's unique methodology, the philosophical underpinnings of Vedic science, and specific arithmetical and algebraic applications that range from basic multiplication to differential calculus and analytical conics.
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Historical and Philosophical Context
The Author: Jagadguru Swami Sri Bharati Krishna Tirthaji
The system’s architect was an exceptional scholar with a world-record academic pedigree, passing his M.A. in seven subjects (including Mathematics, Sanskrit, and Philosophy) simultaneously in 1904. After years of service in national education, he spent eight years (1911–1919) in concentrated contemplation and research in the forests of Sringeri.
During this period, he recovered the "long lost keys" to the mathematical mysteries of the Vedas. Although he originally authored sixteen volumes—one for each Sutra—the manuscripts were irretrievably lost in 1956. The current work was rewritten from memory in 1957, shortly before his death.
The Definition of "Veda"
The document establishes a specific definition of the term "Veda," derived from its root meaning as an "illimitable store-house of all knowledge."
- Total Knowledge: The Vedas are presented not merely as religious texts but as repositories of all knowledge needed by mankind, including "secular," "temporal," or "worldly" matters.
- Sub-Sciences (Upavedas): Mathematics is categorized under Sthapatyaveda (engineering and architecture), a subsidiary of the Atharvaveda.
- Intuitional Visualization: Tirthaji claimed the Sutras were not conceived pragmatically but were the result of "intuitional visualization of fundamental mathematical truths."
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Core Mathematical Principles and Sutras
The Vedic system relies on sixteen primary Sutras. These are described as easy to understand, apply, and remember, often requiring only mental arithmetic.
Arithmetical Computations
Method/Sutra | Mathematical Concept | Application/Example |
Ekadhikena Purvena ("By one more than the previous") | Conversion of vulgar fractions into recurring decimals. | Solving 1/19, 1/29, or 1/49 in a single line by using a multiplier (e.g., for 1/19, the multiplier is 2). |
Nikhilam Navatashcaramam Dashatah ("All from 9 and last from 10") | Multiplication and division for numbers near a base (power of 10). | Efficiently multiplying 9 \times 7 or 49 \times 49 using "working bases" like 50. |
Urdhva-Tiryagbhyam ("Vertically and Crosswise") | A general formula for all multiplication and certain divisions. | Multi-digit multiplication (e.g., 5-digit by 5-digit) solved in a single line of work. |
Paravartya Yojayet ("Transpose and Apply") | Division when divisor digits are small; also used for equations. | Used in algebraic division and solving linear equations via mental transposition. |
Dhvajanka ("Flag digit") | Known as the "Crowning Gem" of division. | A universal, at-sight mental division method (e.g., 38982 \div 73 in one line). |
Algebra and Advanced Calculus
The system extends beyond basic arithmetic into complex algebraic structures and calculus:
- Factorisation: Uses sub-sutras like Anurupyena ("Proportionately") and Lopana-Sthapana ("Elimination and Retention") to factorize quadratics and cubics.
- Simple and Simultaneous Equations: Solved via Sunyam Samyasamuccaye ("When the sum is the same, that sum is zero"), allowing for mental solutions to systems that typically require multiple steps of substitution or elimination.
- Quadratic Equations and Calculus: Employs Calam-Kalana (Differential Calculus) to reduce a quadratic equation to two simple first-degree equations. For example, x^2-5x+6=0 is reduced using its first differential (2x-5).
- Analytical Conics: Provides one-line mental solutions for determining equations of straight lines passing through given points or identifying asymptotes of hyperbolas.
Specialized Techniques
- Vinculum: An ingenious device to reduce digits larger than 5, facilitating mental calculation (e.g., writing 18 as 2\bar{2}).
- Duplex (Dwandwa Yoga): A specific process used for the extraction of square roots in a single line.
- Vedic Numerical Code: A system for representing numbers through Sanskrit letters, facilitating the memorization of mathematical constants or results.
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Comparison with Conventional Methods
The text highlights a stark contrast between Vedic "mental" mathematics and the "cumbrous" Western methods taught in modern universities.
- Efficiency: Problems requiring 30 to 100 steps in Western mathematics (such as large recurring decimals or complex divisions) are reduced to a single "one-line" step.
- Speed: The time taken for Vedic methods is described as a "third, a fourth, a tenth, or even a much smaller fraction" of conventional time.
- Educational Timeline: Tirthaji estimated that the entire course of mathematical studies (from arithmetic to calculus) could be mastered in 8 to 12 months using Vedic lines, compared to the 16 to 20 years required by current university systems.
- Verification: The system is described as having "beatific beauty" because each digit often automatically yields its predecessor and successor, making the process self-verifying.
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Key Quotes and Philosophical Insights
On the Nature of the System
"It is magic until you understand it; and it is mathematics thereafter." — Jagadguru Swami Sri Bharati Krishna Tirthaji
On the Creation of Zero
"The importance of the creation of the ZERO mark can never be exaggerated... It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." — Prof. G.P. Halstead
On the Scope of the Vedas
"The very word 'Veda' has this derivational meaning i.e. the fountain-head and illimitable store-house of all knowledge... [it implies] that the Vedas contain within themselves all the knowledge needed by mankind relating not only to the so-called 'spiritual' matters but also to those usually described as purely 'secular'." — Jagadguru Swami Sri Bharati Krishna Tirthaji
On Ancient vs. Modern Intuition
"While all great and true knowledge is born of intuition... the modern method is to get the intuition by suggestion from an appearance in life or nature... the ancient Indian method of knowledge had for its business to disclose something of the Self, the Infinite or the Divine to the regard of the soul." — Dr. Prem Lata Sharma
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Conclusion
Vedic Mathematics is presented as a "monumental work" that bridges the gap between ancient intuitional wisdom and modern analytical needs. Its primary value lies in its radical simplification of mathematical processes, making high-level computation accessible even to young students. The system's utility has been demonstrated to university audiences in India and the United States, consistently striking experts with its "originality and simplicity." As an introductory volume, it serves as a proof of concept for a broader application of Sutras across all branches of pure and applied mathematics.
Based on the provided sources, here are 5 multiple-choice questions for each distinct chapter or chapter-group, followed by the answer key.
Chapter I: Actual Applications (Recurring Decimals)
1. Which Sutra is used to convert vulgar fractions into recurring decimals in a single line?
A. Nikhilam Navatashcaramam Dashatah B. Ekadhikena Purvena C. Urdhva-Tiryagbhyam D. Paravartya Yojayet
2. In the Vedic method for $1/19$, what is the multiplier (the "one more" portion)?
A. 1 B. 9 C. 2 D. 18
3. The multiplier for the decimal conversion of $1/29$ is:
A. 2 B. 3 C. 10 D. 28
4. How many digits are in the recurring decimal answer for $1/19$?
A. 9 B. 10 C. 18 D. 20
5. For $1/49$, the multiplier used to "reel off" the answer is:
A. 4 B. 5 C. 9 D. 50
Chapter II: Arithmetical Computations (Nikhilam Multiplication)
1. What does "Nikhilam Navatashcaramam Dashatah" literally mean?
A. Vertically and Crosswise B. All from 9 and the last from 10 C. Transpose and Apply D. Proportionately
2. The Nikhilam method is most effective for numbers near:
A. Zero B. A prime number C. A power of 10 (Base) D. A fraction
3. To multiply $9 \times 7$ using this method, what are the deficiencies from the base of 10?
A. 9 and 7 B. 1 and 3 C. 0 and 1 D. 3 and 1
4. When multiplying $41 \times 41$, what is the suggested "working base" to avoid cumbrous work?
A. 10 B. 40 C. 50 D. 100
5. For the multiplication $49 \times 49$, which working bases are suggested as options?
A. $10 \times 4$ or $10 \times 5$ B. $100/2$ or $10 \times 5$ C. 10 or 100 D. $50/2$ or $10 \times 2$
Chapter III: Multiplication (Urdhva-Tiryak)
1. What is the literal meaning of "Urdhva-Tiryagbhyam"?
A. Proportionately B. Transpose and Apply C. Vertically and Crosswise D. Elimination and Retention
2. The Urdhva-Tiryak formula is described as:
A. A special case only for base-10 numbers B. A general formula applicable to all multiplication cases C. A method specifically for decimals D. A formula for division only
3. Which of the following can be solved in a single line using this method?
A. $73 \times 37$ B. $87265 \times 32117$ C. $123 \times 89$ D. All of the above
4. Unlike the Nikhilam method, Urdhva-Tiryak does not require numbers to be near:
A. A base B. Zero C. Each other D. Whole numbers
5. The system is designed to allow calculations to be done:
A. Only on paper B. Using a calculator C. Mentally or in one line D. Only by advanced mathematicians
Chapter IV: Division by Nikhilam Method
1. The Nikhilam division method is specifically suited for cases where divisor digits are:
A. Small B. Large (e.g., 9, 98, 997) C. Prime D. Even
2. This division method eliminates the need for:
A. Subtraction B. Addition C. Large-number multiplication or subtraction D. Using a divisor
3. In successive divisions of two-digit numbers by 9, the method is used for:
A. Quick mental results B. Long division steps C. Squaring D. Finding the H.C.F.
4. Which example demonstrates the use of deficiencies in division?
A. $1/19$ B. $1234 \div 112$ C. $9 \times 7$ D. $x^2-5x+6$
5. The Nikhilam division procedure involves using:
A. Transposition B. Flag digits C. Deficiencies from the base D. Geometrical Progression
Chapter V: Division by Paravartya Method
1. What does "Paravartya Yojayet" mean?
A. Vertically and Crosswise B. All from 9 C. Transpose and Apply D. Proportionately
2. This method is preferred when divisor digits are:
A. Large B. Small C. Recurring D. Negative
3. In the division $(7x^2+5x+3) \div (x-1)$, what is the remainder?
A. 7 B. 12 C. 15 D. 0
4. If the divisor is 112, how are the digits transposed in this method?
A. 1 and 2 become -1 and -2 B. 1 and 1 become -1 and -1 C. They remain 1 and 2 D. They are multiplied by 9
5. The Paravartya method is highly effective for:
A. Multiplication of large numbers B. Finding square roots C. Algebraic division of polynomials D. Determining divisibility by 9
Chapter VI: Argumental Division
1. Argumental division is essentially the reverse process of which formula?
A. Ekadhikena B. Urdhva-Tiryak C. Nikhilam D. Paravartya
2. This method relies on:
A. Complex long division B. Simple argumentation C. The Flag method D. Logarithms
3. In the example $(3x^2-x-5) \div (x-7)$, the method is used to find:
A. The H.C.F. B. The square root C. The quotient and remainder D. The differential
4. For the division $(x^4-4x^2+12x-9) \div (x^2+2x-3)$, the method requires:
A. Identifying coefficients mentally B. Transposing all terms to zero C. Squaring the divisor D. Using a base of 10
5. Which operation does Chapter VI focus on?
A. Factorisation B. Division C. Conics D. Recurring Decimals
Chapters VII - IX: Factorisation and H.C.F.
1. "Anurupyena" means:
A. Elimination B. Proportionately C. Vertically D. First by first
2. Which sub-sutra is translated as "Elimination and Retention"?
A. Adyamadyena B. Anurupyena C. Lopana-Sthapana D. Sunyam Samyasamuccaye
3. The "Adyamadyena" Sutra for H.C.F. involves comparing:
A. The first terms and the last terms B. The middle terms C. Only the highest powers D. Only the absolute terms
4. To find the H.C.F. of expressions $P$ and $Q$, one can use the principle that it is also the H.C.F. of:
A. $P \times Q$ B. $P + Q$ only C. $MP \pm NQ$ D. $P^2 - Q^2$
5. Factorising the quadratic $2x^2+5x+2$ is done by applying the principle of:
A. Flag digits B. The first by the first and the last by the last C. Transpose and Apply D. Geometrical Progression
Chapters X - XVI: Simple Equations
1. "Vilokanam" refers to solving equations by:
A. Cross-multiplication B. Mental observation C. Long division D. Integration
2. "Sunyam Samyasamuccaye" implies that when a certain sum is the same on both sides:
A. That sum is zero B. The answer is 1 C. The equation is impossible D. Multiply by the base
3. In the equation $\frac{1}{x-7} + \frac{1}{x-9} = \frac{1}{x-6} + \frac{1}{x-10}$, what is the value of $x$?
A. 7 B. 8 C. 16 D. 0
4. The "Antyayoreva" Sutra is introduced in Chapter XVI to solve:
A. Quadratic equations B. Fractional additions and series summations C. Simultaneous equations D. Cubic roots
5. If the product of absolute terms is identical on both sides of a simple equation, $x$ is often:
A. 1 B. 10 C. 0 D. $\infty$
Chapters XV, XX, XXI: Simultaneous Equations
1. If the $y$-coefficients and constants are in the same ratio ($a:b = c:d$), what is the value of $x$?
A. 1 B. 0 C. The ratio itself D. Unsolvable
2. In $6x+7y=8$ and $19x+14y=16$, $x$ is 0 because $7:14$ is equal to:
A. $6:19$ B. $8:16$ C. $1:2$ D. $14:16$
3. Equations with large coefficients like $45x-23y=113$ and $23x-45y=91$ are solved by:
A. Substitution B. Addition and subtraction devices C. Flag division D. Calculus
4. The "Sunyam Anyat" Sutra states that:
A. Everything is zero B. If one variable is in ratio, the other is zero C. All from 9 D. Transpose and apply
5. Solving $x+y=4$ and $x^2+xy+4x=24$ mentally involves spotting:
A. Prime numbers B. Common factors C. The discriminant D. Recurring decimals
Chapters XVII, XXII: Quadratic Equations & Calculus
1. The "Chalan-Kalana" Sutra refers to:
A. Multiplication B. Differential Calculus C. Square roots D. Geometry
2. How does Calculus help solve a quadratic equation?
A. By increasing its degree B. By reducing it to two simple first-degree equations C. By finding the recurring decimal D. By eliminating the $x^2$ term only
3. In the Vedic system, the first differential of a quadratic is equal to:
A. The sum of its binomial factors B. The square root of the discriminant C. Both A and B D. Neither A nor B
4. For $x^2-5x+6=0$, the first differential is:
A. $x-5$ B. $2x-5$ C. $2x+6$ D. $5x-6$
5. "Gunaka-Samuccaya" explains that the first differential is:
A. Zero B. The sum of binomial factors C. The H.C.F. D. The product of digits
Chapters XVIII - XIX: Cubic and Bi-quadratic Equations
1. The "Purana" method is used to:
A. Multiply large numbers B. Reduce degree complexity of equations C. Find the value of $\pi$ D. Solve simple linear equations
2. The first step in solving a cubic equation in this chapter is identifying the first root through:
A. Calculus B. Long division C. Inspection D. Flag division
3. Argumentation-cum-factorisation aims to break high-degree equations into:
A. Complex numbers B. Simpler linear components C. Recurring decimals D. Partial fractions
4. Bi-quadratic equations are solved using logic identical to:
A. Simple equations B. Cubic equations C. Simultaneous equations D. Conics
5. These chapters focus on equations of which degrees?
A. 1st and 2nd B. 3rd and 4th C. 5th and 6th D. 10th and 20th
Chapter XXIII: Partial Fractions
1. Which Sutra is used to resolve complex fractions into partial fractions in one line?
A. Nikhilam B. Urdhva-Tiryak C. Paravartya D. Adyamadyena
2. The Vedic method for partial fractions avoids:
A. Addition B. Lengthy conventional substitution C. Mental arithmetic D. Using the denominator
3. For the fraction $\frac{3x^2+12x+11}{(x+1)(x+2)(x+3)}$, the coefficients are found:
A. By solving three simultaneous equations B. Instantly through mental arithmetic C. Using logarithms D. Using the flag method
4. The method can handle denominators with:
A. Only prime factors B. Repeated factors (squares or cubes) C. No factors D. Only one term
5. The coefficients $A, B, C$ are determined by making specific factors:
A. Zero (transpose and apply logic) B. Negative C. Infinite D. Equal to 10
Chapter XXV: The Vedic Numerical Code
1. In the ancient Sanskrit code, numbers are represented by:
A. Dots and dashes B. Letters of the alphabet C. Roman numerals D. Geometric shapes
2. Which letters all represent the number 1?
A. $ka, ta, pa, ya$ B. $kha, tha, pha, ra$ C. $ks$ D. $cha, tra$
3. The letter $ks$ represents:
A. 1 B. 5 C. 7 D. Zero
4. Consonants like $cha$ and $tra$ represent the number:
A. 3 B. 7 C. 9 D. 0
5. The purpose of this code was to embed mathematical data into:
A. Stone tablets B. Hymns and verses for memorisation C. Private letters D. Bank ledgers
Chapters XXVI: Recurring Decimals (Advanced)
1. The remainders in a recurring decimal conversion often follow a:
A. Arithmetic Progression B. Geometrical Progression C. Random sequence D. Prime sequence
2. For $1/7 = .142857$, what is the common ratio between the remainders $3, 2, 6, 4, 5, 1$?
A. 2 B. 3 C. 7 D. 10
3. How many digits are in the recurring decimal of $1/17$?
A. 8 B. 16 C. 17 D. 32
4. Geometrical ratio for $1/13$ is:
A. 1 B. 2 C. 3 D. 4
5. Which Sutra is used for these advanced decimal conversions?
A. Paravartya B. Ekadhika C. Urdhva D. Nikhilam
Chapter XXVII: Straight Division
1. Chapter XXVII refers to "Straight Division" as the:
A. Base method B. Crowning Gem C. Starting point D. Last resort
2. What is the key digit used in this method called?
A. Deficiency digit B. Flag (Dhvajanka) digit C. Base digit D. Osculator
3. Straight division allows for division by any number in:
A. 10 steps B. A single line C. Two lines D. A calculator only
4. In the example $38982 \div 73$, 7 is the primary divisor and 3 is the:
A. Quotient B. Remainder C. Flag D. Osculator
5. The process involves subtracting "flag-products" from:
A. The divisor B. The prefixed remainders (Gross Dividends) C. The quotient D. The base
Chapter XXVIII: Auxiliary Fractions
1. Auxiliary fractions are used when the denominator is:
A. A perfect square B. Near a power of ten but not exactly a base C. A prime number D. Zero
2. The method provides results to how many decimal places?
A. 2 B. 5 C. 20 or more D. Exactly 10
3. Fractions can be processed as:
A. One above normal or one below normal B. Only positive C. Only negative D. Infinite
4. To solve $29/15001$, the method suggests using:
A. Flag division B. Groups of three digits C. Simultaneous equations D. Calculus
5. Auxiliary fractions simplify division by:
A. Changing the numerator B. Adjusting denominators to be near a "normal" base C. Removing all digits D. Multiplying by $\pi$
Chapter XXX: Divisibility & Complex Osculators
1. "Multiplex Osculation" is used to determine divisibility for:
A. Small numbers B. Large divisors C. Fractions D. Negative numbers
2. Instead of processing individual digits, this method processes:
A. Only the last digit B. Groups of digits C. Only prime digits D. Every other digit
3. $P$ represents a positive osculator and $Q$ represents a:
A. Quotient B. Negative osculator C. Quadratic D. Quality factor
4. What is the positive osculator ($P_2$) for 157?
A. 7 B. 11 C. 13 D. 157
5. Testing divisibility by 1001 uses a negative osculator ($Q_3$) of:
A. 1 B. 7 C. 11 D. 1001
Chapter XXXI: Sum and Difference of Squares
1. Any number can be expressed as the difference of:
A. Two primes B. Two squares C. Two cubes D. Two fractions
2. Expressing $9$ as the difference of squares can be written as:
A. $5^2 - 4^2$ B. $3^2 - 0^2$ C. Both A and B D. Neither A nor B
3. Which formula is used to find Pythagorean triplets?
A. $D^2+N^2=(N+1)^2$ B. $a+b=c$ C. $x^2+y=z$ D. $MP \pm NQ$
4. Which is described as more difficult than expressing a number as a difference of squares?
A. Multiplication B. Division C. Expressing as a sum of squares D. Square roots
5. $141^2 + 9940^2 = 9941^2$ is an example of:
A. A simple equation B. A Pythagorean triplet C. A recurring decimal D. A differential
Chapters XXXII - XXXIV: Squaring and Square Roots
1. "Dwandwa Yoga" is also known as the:
A. Vertical method B. Duplex Combination Process C. Flag method D. Transpose and Apply
2. Which Sutra is used to square numbers near a power of 10?
A. Adyamadyena B. Yavadunam C. Anurupyena D. Nikhilam
3. Straight square root extraction is similar to the process of:
A. Straight division B. Multiplication C. H.C.F. D. Conics
4. The square root of 529 is:
A. 13 B. 23 C. 33 D. 43
5. The Duplex process for square roots works for numbers of:
A. Only 2 digits B. Only 4 digits C. Any length D. Only prime digits
Chapters XXXV - XXXVI: Cube Roots
1. Cube roots of exact cubes can be found via:
A. Logarithms B. Inspection and argumentation C. Addition only D. Conics
2. The general method for cube roots uses the algebraic expansion of:
A. $(a+b)^2$ B. $(a+b+c+d)^3$ C. $x^2-y^2$ D. $\pi r^2$
3. Digits in the cube root process are found:
A. Simultaneously B. Sequentially C. Randomly D. Backwards only
4. The sub-multiple method is specifically mentioned for cube roots of:
A. Odd numbers B. Even numbers C. Negative numbers D. Prime numbers
5. Finding the cube root of $355,045,312,441$ is described as:
A. Impossible mentally B. Easy via the general Vedic method C. Requiring 100 steps D. Only for small numbers
Chapters XXXVII - XXXIX: Geometry and Conics
1. Simple Vedic proofs are provided for which famous theorem?
A. Fermat's Last Theorem B. Pythagoras' Theorem C. Goldbach's Conjecture D. Chaos Theory
2. Chapter XXXIX offers mental one-line solutions for:
A. Coordinate geometry (Conics) B. Biology C. Chemistry D. Literature
3. The equation of a straight line through $(9, 17)$ and $(7, -2)$ is:
A. $x+y=10$ B. $19x-2y=137$ C. $2x-19y=0$ D. $7x-2y=9$
4. Which Sutra is used to find the two straight lines represented by a quadratic?
A. Adyamadyena B. Ekadhika C. Nikhilam D. Yavadunam
5. Vedic methods can find equations for which conic sections?
A. Hyperbolas B. Asymptotes C. Conjugate hyperbolas D. All of the above
Chapter XL: Miscellaneous Matters
1. Which mathematical value is encoded to 32 decimal places in a Sanskrit hymn?
A. $e$ B. $\sqrt{2}$ C. $\pi/10$ D. The Golden Ratio
2. The hymn used for encoding $\pi/10$ is dedicated to:
A. King Kaqaa B. Lord Krishna C. Pythagoras D. Apollonius
3. The source proves that there can be only how many regular polyhedrons?
A. Three B. Five C. Seven D. Infinite
4. Vedic Mathematics includes calculations regarding:
A. Solar and lunar eclipses B. The Earth's rotation C. Determinants D. All of the above
5. Determinants are used in the Vedic system for:
A. Theory of Equations and Conics B. Only simple addition C. Finding square roots only D. Music theory
Answer Key
Chapter I
- B | 2. C | 3. B | 4. C | 5. B
Chapter II
- B | 2. C | 3. B | 4. C | 5. B
Chapter III
- C | 2. B | 3. D | 4. A | 5. C
Chapter IV
- B | 2. C | 3. A | 4. B | 5. C
Chapter V
- C | 2. B | 3. C | 4. A | 5. C
Chapter VI
- B | 2. B | 3. C | 4. A | 5. B
Chapters VII - IX
- B | 2. C | 3. A | 4. C | 5. B
Chapters X - XVI
- B | 2. A | 3. B | 4. B | 5. C
Chapters XV, XX, XXI
- B | 2. B | 3. B | 4. B | 5. B
Chapters XVII, XXII
- B | 2. B | 3. C | 4. B | 5. B
Chapters XVIII - XIX
- B | 2. C | 3. B | 4. B | 5. B
Chapter XXIII
- C | 2. B | 3. B | 4. B | 5. A
Chapter XXV
- B | 2. A | 3. D | 4. B | 5. B
Chapter XXVI
- B | 2. D | 3. B | 4. C | 5. B
Chapter XXVII
- B | 2. B | 3. B | 4. C | 5. B
Chapter XXVIII
- B | 2. C | 3. A | 4. B | 5. B
Chapter XXX
- B | 2. B | 3. B | 4. B | 5. A
Chapter XXXI
- B | 2. C | 3. A | 4. C | 5. B
Chapters XXXII - XXXIV
- B | 2. B | 3. A | 4. B | 5. C
Chapters XXXV - XXXVI
- B | 2. B | 3. B | 4. B | 5. B
Chapters XXXVII - XXXIX
- B | 2. A | 3. B | 4. A | 5. D
Chapter XL
- C | 2. B | 3. B | 4. D | 5. A
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