Key Concept: Indian Number System and Spread

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the Indian number system was revolutionary due to its inclusion of zero as both a placeholder and a digit, which is true historically. The reason claims that Fibonacci's introduction of this system to Europe led to its immediate universal adoption in the 12th century. While Fibonacci did play a key role in introducing the system to Europe via his book *Liber Abaci*, widespread adoption was delayed until the Renaissance due to resistance against replacing Roman numerals. Thus, the assertion is true, but the reason is false.

Key Concept: Transmission of Numerals

c) Fibonacci
[Solution Description]
The Italian mathematician Fibonacci introduced the Indian numeral system to Europe in his book "Liber Abaci" (1202). Despite initial resistance due to the entrenched Roman numeral system, it eventually became widely adopted during the Renaissance.

Key Concept: Transmission of Indian Numerals

d) Al-Khwārizmī
[Solution Description]
Al-Khwārizmī wrote the book *On the Calculation with Hindu Numerals* in 825 CE, which was instrumental in introducing Indian numerals to the Arab world.

Key Concept: Human Need for Counting, Origin of Modern Numbers

d) $āyuta$
[Solution Description]
The Yajurveda Samhita lists number names based on powers of 10. According to the text, $āyuta$ represents ten thousand ($10^4$).

Key Concept: Mesopotamian numerals, Hindu-Arabic numeral system

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's verify the Assertion: The Mesopotamian numeral system indeed used a base-60 (sexagesimal) system, while the Hindu-Arabic numeral system uses base-10. This part of the Assertion is true.
Now, examine the Reason: The base of numeral systems often reflects practical human needs. For example, base-10 is common due to counting on 10 fingers, while base-60 (used in Mesopotamia) may have been chosen for its divisibility (useful in astronomy and timekeeping). Thus, the Reason is also true.
Does the Reason correctly explain the Assertion? Yes, because the difference in bases between the two systems can be attributed to their different practical and cultural contexts, as stated in the Reason.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Purpose of Counting

c) To track food supplies
[Solution Description]
Ancient humans needed to count for various practical reasons, such as keeping track of food supplies, livestock, and trade goods. They also counted to track time and predict seasonal events.

Key Concept: Hindu Number System

d) Al-Khwārizmī
[Solution Description]
The Hindu number system was popularized in the Arab world by Al-Khwārizmī through his book "On the Calculation with Hindu Numerals" around 825 CE.

Key Concept: Mesopotamian numerals, Indian numeral system, spread of numbers

a) 45
[Solution Description]
Mesopotamians used a base-60 (sexagesimal) system. The number 45 in their system would be represented as $45_{60}$. To convert to decimal, we calculate:
$4 \times 60^1 + 5 \times 60^0 = 4 \times 60 + 5 \times 1 = 240 + 5 = 245.$
However, the question asks for representing 45 in their symbols and converting back to decimal, so the correct answer is 45 itself.

Key Concept: Place Value System

c) Place value system with base 10
[Solution Description]
The Hindu-Arabic numeral system is a positional or place value system where the position of a digit determines its value. It originated in India and includes the digit 0 as a placeholder and as a number.

Key Concept: Origin of Numbers, Hindu-Arabic Numerals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the digit 0 being introduced as a number revolutionized mathematics. This is true because, before its introduction, computations were limited. The Reason explains how zero was treated as a number with specific arithmetic properties, which is also true. Moreover, the Reason correctly explains the Assertion since these properties of zero enabled more advanced mathematical operations and theories to develop, such as algebra and analysis. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Origin of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true as the Hindu-Arabic numeral system did indeed originate in India and was later transmitted to the Arab world. The reason is also true because the introduction of 0 as both a digit and a number revolutionized mathematics and science. Furthermore, the reason correctly explains why the Hindu-Arabic numeral system was significant, as it included the concept of zero, which was a major mathematical advancement.

Key Concept: Origin of Numbers

c) India
[Solution Description]
The modern number system, which includes digits 0 to 9, originated in India around 2000 years ago. This system was later transmitted to the Arab world and then to Europe.

Key Concept: One-to-One Mapping

b) No pebble should be used more than once for mapping.
[Solution Description]
In the one-to-one mapping method, each object (stone in this case) is uniquely paired with another object (pebble). The total number of pebbles will exactly match the number of stones if the mapping is correctly done, ensuring no two stones are mapped to the same pebble and no pebble is left unmapped.

Key Concept: Tally Marks

b) As a lunar calendar or counting aid.
[Solution Description]
The Lebombo bone's notches suggest it might have been used as a lunar calendar, tracking phases of the moon over time. Alternatively, it could have served as a simple tally stick to count objects like days, animals, or resources, given the prehistoric context where such markers were common for record-keeping.

Key Concept: One-to-one mapping, Alphabet-based counting

c) 11
[Solution Description]
In Method 2, each letter of the alphabet represents a number starting from "a" as 1, "b" as 2, and so on.
The position of "k" in the English alphabet is the 11th letter.
Calculation:
- a = 1
- b = 2
- ...
- k = 11
Therefore, the farmer has 11 chickens.

Key Concept: One-to-one mapping, Roman numerals

c) 17
[Solution Description]
To solve this question, we need to understand the Roman numeral system representation mentioned in Method 3.
The Roman numeral "XVII" can be broken down as follows:
- X represents 10.
- V represents 5.
- II represents 2 (I + I).
Adding these together: $10 + 5 + 2 = 17$.
Therefore, "XVII" represents 17 sheep.

Key Concept: One-to-One Mapping, Roman Number System

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Roman number system is an unending standard sequence for counting because it introduces new symbols for larger numbers. This is true as the Roman system uses combinations of symbols (like I, V, X, L, C, D, M) to represent larger numbers. However, the Reason states that the Roman number system does not require an infinite set of unique symbols, which is also true but does not fully explain why it is considered "unending." The system's unending nature comes from its ability to combine existing symbols in specific ways, not merely from avoiding an infinite set of symbols. Hence, both are true, but Reason does not correctly explain Assertion.

Key Concept: One-to-One Mapping

a) Using pebbles or sticks
[Solution Description]
The method described involves creating a one-to-one mapping between each cow and a stick. This ensures that each cow is uniquely represented by a stick, allowing for accurate counting. This technique is known as one-to-one mapping, where no two cows are associated with the same stick.

Key Concept: Mayan Number System, Hindu Number System

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that the Mayan system's use of a placeholder for zero was as advanced as the Hindu number system. This is false because while the Mayans did use a placeholder for zero, they did not treat it as a number with arithmetic properties like the Hindus did. The Reason claims that both systems recognized zero as both a digit and a number with arithmetic properties, which is also false because the Mayans only used zero as a placeholder, not as a number. Therefore, both Assertion and Reason are false.

Key Concept: Chinese Rod Numerals

b) Base-10
[Solution Description]
The Chinese rod numeral system was developed for computation and uses distinct symbols for digits 1 to 9. It follows a decimal (base-10) system, where each position represents a power of 10. This system allowed efficient arithmetic operations such as addition, subtraction, multiplication, and division.
Therefore, the correct answer is base-10.

Key Concept: Hindu Number System, Roman Number System

c) M
[Solution Description]
To solve this problem, we need to multiply the Roman numeral CXXV by 8, both in their respective forms.
Step 1: Convert the Roman numeral CXXV to its Hindu-Arabic equivalent.
- C = 100, X = 10, X = 10, V = 5.
- So, CXXV = 100 + 10 + 10 + 5 = 125.
Step 2: Multiply the Hindu-Arabic equivalent by 8.
- $125 \times 8 = 1000$.
Step 3: Convert the product back to Roman numerals.
- 1000 in Roman numerals is M.
Therefore, the correct answer is M.

Key Concept: Chinese Number System, Mayan Number System

d) Two dots over one bar and two dots
[Solution Description]
To solve this problem, we need to express the decimal number 47 in the Mayan number system.
Step 1: Break down the number 47 into Mayan place values.
- The Mayan system uses base-20 with some variations. For numbers less than 360, the first level is 1s and the second level is 20s.
- Divide 47 by 20 to find the value for the 20s place: $47 \div 20 = 2$ with a remainder of 7.
- So, the 20s place has a value of 2, and the 1s place has a value of 7.
Step 2: Represent the values using Mayan symbols (dots and bars).
- For the 20s place (value = 2):
- 2 is represented as two dots ($\cdot\cdot$).
- For the 1s place (value = 7):
- 7 is represented as one bar (5) and two dots (2), i.e., $\_\cdot\cdot$.
Step 3: Combine the representations vertically with the 20s place on top.
- The Mayan representation for 47 is:
- Two dots at the top (20s place).
- One bar and two dots at the bottom (1s place).
Therefore, the correct Mayan representation is two dots over one bar and two dots.

Key Concept: Mayan Number System

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion is correct because the Mayan number system indeed used a dot for 1 and a bar for 5 for representing numbers from 1 to 19. However, the Reason is false because their system was not a strict base-20 system; it had modifications where landmark numbers included 360 and 7200 instead of powers of 20 beyond $20^1$.

Key Concept: Body-Part Number Representation

d) Index finger of the left hand
[Solution Description]
According to the given sequence, the number $4$ is represented by the index finger of the left hand.

Key Concept: Name representation

b) ta ta
[Solution Description]
Given the additive nature of the system:
Step 1: We have representations for 1 ('la'), 2 ('ta') and 3 ('na')
Step 2: For 4, it would combine existing symbols through addition
Possible logical combinations:
'ta ta' (2+2=4)
'la na' (1+3=4)
'na la' (3+1=4)
The simplest form would be combining the largest available numbers

Key Concept: Body Parts Counting System, Limitations

b) Right elbow
[Solution Description]
To find the 50th count in a repeating sequence of 27 body parts:
Step 1: Divide 50 by 27 to find the complete cycles and remainder.
Step 2: $50 ÷ 27 = 1$ cycle with a remainder of $23$.
Step 3: The 50th count corresponds to the 23rd body part in the sequence.

Key Concept: Body Parts as Number Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in some cultures, the wrist represents the number 6 in their body-part counting system. This is true based on the provided syllabus example where counting starts with fingers (1-5) and then moves to wrist (6). The reason explains the standard sequence followed by such systems, starting with fingers and moving to other body parts, which is also correct. Moreover, the reason correctly explains why the wrist would represent 6, as it comes after counting all five fingers. Hence, both are true, and the reason provides the correct explanation for the assertion.

Key Concept: Body Parts as Number Representation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the little finger represents the number 1 in some early number systems, which is true based on the syllabus example. The reason states that counting starts with the smallest body part and progresses to larger ones. While this may be true in general, it does not specifically explain why the little finger represents 1. There could be other reasons for choosing the little finger first, such as cultural significance or ease of pointing. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: Body part counting

c) 2 complete counts
[Solution Description]
First, we need to understand that one complete count means using all 10 fingers (both hands). To find out how many complete counts are needed for 15:
Step 1: Divide 15 by 10 (fingers per count) = 1.5 counts
Step 2: Since they can't do half counts, they would need to round up to 2 complete counts (using fingers twice)

Key Concept: Assersion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion claims the Ishango bone was used for counting hunted animals, which is speculative and lacks direct archaeological evidence. However, the Reason correctly states that the grouping of notches aligns with lunar cycles, which is supported by researchers' interpretations of the Ishango bone as a possible lunar calendar. Therefore, while the Reason is true and well-supported, the Assertion is unverified and likely false.

Key Concept: Tally marks, Ishango bone, Lebombo bone

c) 35
[Solution Description]
First, calculate the count from the first column with 3 groups of 5 marks: $3 \times 5 = 15$. Then, calculate the count from the second column with 2 groups of 10 marks: $2 \times 10 = 20$. Add these two results to find the total count: $15 + 20 = 35$.

Key Concept: Tally Marks and Early Counting Methods

b) Counting objects by making individual marks
[Solution Description]
Tally marks were one of the oldest methods used to represent numbers by making notches on surfaces like bones or cave walls. Each mark represented a single object being counted, and the collection of marks indicated the total count. This method was simple and did not require physical objects like sticks or pebbles.

Key Concept: Tally Marks on Bones

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Ishango bone has notches arranged in columns, which is true as mentioned in the syllabus. The Reason suggests that these notches could represent a calendrical system, which is also mentioned in the syllabus as a possible interpretation. Furthermore, the Reason correctly explains why the notches are arranged in columns, indicating a potential calendrical system.

Key Concept: Lebombo bone, Tally marks

a) 5 groups, 4 marks
[Solution Description]
Divide 29 by 5 to find the number of complete groups and the remainder: $29 \div 5 = 5$ with a remainder of 4. So, there are 5 complete groups and 4 ungrouped marks.

Key Concept: Tally Marks and Group Counting

b) To make counting easier by reducing the number of marks
[Solution Description]
Humans have a perceptual limit for recognizing small quantities quickly. Groups of 5 or more objects are difficult to count at a glance, so replacing every 5 tally marks with a new symbol made counting more efficient.

Key Concept: Number Names in Gumulgal System

d) ukasar-ukasar-urapon
[Solution Description]
In the Gumulgal system, numbers are formed by counting in twos. For 5, the representation is $2 + 2 + 1$, which translates to ukasar-ukasar-urapon.

Key Concept: Gumulgal Number System, Counting in Twos

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Gumulgal number system uses counting in twos for numbers up to 6 and labels anything above $6 \text{ as } \textit{ras}$. From the syllabus, this is correct, as $3 = 2 + 1$, $4 = 2 + 2$, etc., and numbers greater than 6 are indeed called $\textit{ras}$.
The reason explains that counting in groups like twos emerged due to human limitations in counting larger quantities without grouping. This is also true as per the syllabus, which mentions that most humans struggle to count groups of 5+ objects at a glance.
Furthermore, the reason correctly explains why counting in groups like twos became necessary, which aligns with the assertion about the Gumulgal system. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Number names by counting in twos, Bakairi system

b) t'oa-t'oa-t'oa-t'oa
[Solution Description]
The Bakairi system counts in twos:
1: xa
2: t'oa
3: 'quo
4: t'oa-t'oa
5: t'oa-t'oa-t'a
6: t'oa-t'oa-t'oa
To represent 8, we note that 8 = 6 + 2. Since 6 is t'oa-t'oa-t'oa and 2 is t'oa, the logical extension would be t'oa-t'oa-t'oa-t'oa.

Key Concept: Number Names by Counting in Twos

d) $\text{azag\^{a} azag\^{a}}$​
[Solution Description]

In the Bakairi number system, $\text{tokalȃ}$ represents 1 and $\text{azagȃ}$ represents 2. Larger numbers are formed by combining these basic number words. Therefore, the number 4 is represented as:

2 + 2 = 4

So, 4 is written as:

$\text{azagȃ azagȃ}$​

Some sources also write it as $\text{azagȃ azagȃ}$, using a simplified spelling.

Key Concept: Number names by counting in twos

c) ahage ahage ahage
[Solution Description]
From the syllabus:
- Bushmen represent numbers as follows:
1: tokale
2: ahage
3: ahage tokale (or ahawao)
4: ahage ahage tokale
5: ahage ahage ahage
So, 5 is directly given as ahage ahage ahage.

Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the Gumulgal number system constructs the name for 5 as ukasar-ukasar-urapon, which is correct because $5 = 2 + 2 + 1$ and hence the name ukasar (2) repeated twice followed by urapon (1). The reason states that numbers greater than 6 are called ras because they rely on counting by twos. This is also true, but it does not explain why the name for 5 is constructed as given in the assertion. It only explains another aspect of the Gumulgal number system. Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Roman Numeral Addition

b) XIX
[Solution Description]
First, convert both Roman numerals to Hindu-Arabic numbers:
Step 1: XII = 10 + 1 + 1 = 12.
Step 2: VII = 5 + 1 + 1 = 7.
Step 3: Add them together: 12 + 7 = 19.
Step 4: Convert 19 back to Roman numerals: 19 = 10 + 9 = XIX.
Therefore, the sum is XIX.

Key Concept: Roman Numerals - Landmark Numbers

d) M
[Solution Description]
Convert each option to Hindu-Arabic numerals:
a) D = 500
b) CM = 900 (1000 - 100)
c) DC = 600 (500 + 100)
d) M = 1000.
Therefore, M (1000) is the largest.

Key Concept: Representation of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify Assertion (A), let's break down the Roman numeral $MDCLXVI$:
- M = 1000
- D = 500
- C = 100
- L = 50
- X = 10
- V = 5
- I = 1
Adding them up: $1000 + 500 + 100 + 50 + 10 + 5 + 1 = 1666$. So, Assertion is true.
Reason (R) states that symbols are added from left to right in descending order. This is correct because in $MDCLXVI$, the symbols are ordered as M (1000), D (500), C (100), L (50), X (10), V (5), I (1), which follows descending order. Moreover, Reason correctly explains why the assertion is true, as this ordering rule ensures accurate addition. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Roman numerals arithmetic, addition

a) MCCXI
[Solution Description]
To add $DCCLXVII$ ($767$) and $CDXLIV$ ($444$):
1. Convert to Hindu-Arabic numerals: $DCCLXVII = 500 + 200 + 50 + 10 + 5 + 1 + 1 = 767$; $CDXLIV = (500 - 100) + (50 - 10) + (5 - 1) = 444$.
2. Add the numbers: $767 + 444 = 1211$.
3. Convert back to Roman numerals: $1211 = 1000 + 100 + 100 + 10 + 1 = MCCXI$.

Key Concept: Roman Numeral Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To represent 45 in Roman numerals:
1. Start with the largest landmark number less than or equal to 45, which is L (50).
2. Since 40 is 10 less than 50, we write XL (where X is 10 and L is 50).
3. Add V (5) to make it 45, resulting in XLV.
The Reason correctly explains the subtraction rule used in the Assertion (X before L means 50 - 10 = 40).

Key Concept: Roman numerals representation, complex cases

a) MMMCMXCIX
[Solution Description]
To represent 3999 in Roman numerals:
1. Break down 3999 into its largest possible components: $1000 + 1000 + 1000 + (1000 - 100) + (100 - 10) + (10 - 1)$.
2. Substitute the Roman symbols: $M$ for $1000$, $CM$ for $900$ ($1000 - 100$), $XC$ for $90$ ($100 - 10$), and $IX$ for $9$ ($10 - 1$).
3. Combine them: $M + M + M + CM + XC + IX = MMMCMXCIX$.

Key Concept: Base of a Number System

b) A system with landmark numbers 1, 5, 25, 125
[Solution Description]
A base-5 number system has landmark numbers as powers of 5, such as $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, etc.

Key Concept: Base-n System, Landmark Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that in a base-5 number system, the landmark number $5^4$ is 625. This is true because $5^4 = 5 \times 5 \times 5 \times 5 = 625$. The Reason explains that landmark numbers in a base-n system are powers of the base, which is also true and directly explains why the Assertion holds. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Landmark Numbers

c) 1, 7, 49, 343
[Solution Description]
In a base-n system, the landmark numbers are the powers of n starting from $n^0 = 1$. For a base-7 system, the landmark numbers are:
$7^0 = 1, \quad 7^1 = 7, \quad 7^2 = 49, \quad 7^3 = 343, \quad \dots$
Therefore, the landmark numbers are 1, 7, 49, 343, and so on.

Key Concept: Concept merging, Base-n arithmetic

c) 97 in base-10
[Solution Description]
Step 1: Convert both numbers to base-10.
For 124 (base-5):
$1 \times 5^2 + 2 \times 5^1 + 4 \times 5^0 = 25 + 10 + 4 = 39$
For 213 (base-5):
$2 \times 5^2 + 1 \times 5^1 + 3 \times 5^0 = 50 + 5 + 3 = 58$
Step 2: Add them:
$39 + 58 = 97$ in base-10.

Key Concept: Base-5 Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 143 can be represented as $125 + 5 + 5 + 5 + 1 + 1 + 1$ in a base-5 system. This is correct because $125 = 5^3$, $5 = 5^1$, and $1 = 5^0$, which are all valid landmark numbers in the base-5 system. The reason states that the landmark numbers in a base-5 system are powers of 5, which is also true. Moreover, the reason correctly explains the assertion since the representation of 143 uses powers of 5, which are the landmark numbers in this system.

Key Concept: Advantage of Base-n Systems

b) Because multiplication in base-n involves only addition of exponents
[Solution Description]
In a base-n system, the landmark numbers are powers of n, so multiplying two landmark numbers results in another landmark number (e.g., $10^a \times 10^b = 10^{a+b}$). This simplifies multiplication because it reduces to adding exponents. In contrast, the Roman numeral system does not have landmark numbers as powers of a fixed base, making multiplication complex and less systematic.

Key Concept: Multiplication in Egyptian Numerals

c) 1000
[Solution Description]
In the Egyptian system, each multiplication of a landmark number by 10 results in the next landmark number:
- $10 × 1 = 10$ (second landmark number).
- $10 × 10 = 100$ (third landmark number).
- Therefore, $10 × 100 = 1000$ (fourth landmark number).

Key Concept: Grouping by Different Base

d) 25
[Solution Description]
In this system:
$1^{st} \text{ landmark } = 1 \\ \\ 2^{nd} \text{ landmark } = 5 \times 1 = 5 \\ \\ 3^{rd} \text{ landmark } = 5 \times 5 = 25$
Thus, the third landmark number is 25.

Key Concept: Variations on the Egyptian System

b) 25
[Solution Description]
In a base-5 system, landmark numbers are generated as follows:
- First landmark number: 1 ($5^0$).
- Second landmark number: 5 ($5^1$).
- Third landmark number: $5 × 5 = 25$ ($5^2$).

Key Concept: Variations on the Egyptian System - Grouping

d) $25$
[Solution Description]
When grouping by $5$, the first landmark number is $1$. The second landmark number is $5$ (grouping $5$ collections of $1$). The third landmark number is obtained by grouping $5$ collections of $5$, which equals $25$.

Key Concept: Landmark Numbers in Egyptian System

b) 3
[Solution Description]
The Egyptian system uses powers of 10 as landmark numbers: 1, 10, 100, 1000, etc. To represent 7,250, we break it down as:
$7,250 = 7 \times 1000 + 2 \times 100 + 5 \times 10$
Here, the landmark numbers used are 1000, 100, and 10. Since each power of 10 is a distinct symbol, the total landmark numbers needed are 3.

Key Concept: Shortcomings of the Egyptian System

c) It required an unending sequence of symbols for higher powers of $10$
[Solution Description]
The Egyptian number system required an unending sequence of symbols for higher powers of $10$ as numbers grew larger, making it impractical for very large representations.

Key Concept: Base-5 number system, Landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's convert $43_5$ to base-10: The rightmost digit (3) is multiplied by $5^0 = 1$, and the leftmost digit (4) is multiplied by $5^1 = 5$. Thus, $43_5 = 4 \times 5 + 3 \times 1 = 20 + 3 = 23_{10}$. The reason correctly explains that the positional value is determined by powers of 5, which aligns with the conversion process. Therefore, both statements are true, and the reason explains the assertion.

Key Concept: Base Conversion in Base-5 System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's convert 24$_5$ to decimal. The rightmost digit is 4 (position 0), so it contributes $4 \times 5^0 = 4$. The next digit is 2 (position 1), contributing $2 \times 5^1 = 10$. Summing them gives $10 + 4 = 14$ in decimal, which matches the assertion. The reason explains the correct method for converting base-5 numbers to decimal, which justifies why the assertion is true. Hence, both are true, and Reason correctly explains Assertion.

Key Concept: Base Systems, Multiplication

b) $432_5$
[Solution Description]
Convert $14_5$ and $23_5$ to decimal:
$14_5 = 1 \times 5^1 + 4 \times 5^0 = 9$.
$23_5 = 2 \times 5^1 + 3 \times 5^0 = 13$.
Product in decimal: $9 \times 13 = 117$.
Convert 117 to base-5:
$117 \div 5 = 23$ with remainder $2$,
$23 \div 5 = 4$ with remainder $3$,
$4 \div 5 = 0$ with remainder $4$.
Reading remainders from bottom to top: $432_5$.

Key Concept: Landmark Numbers in Base-4

d) $16$
[Solution Description]
In any base-$n$ system, landmark numbers are powers of the base. The first three landmark numbers for base-4 are:
$1 = 4^0 \\ \\ 4 = 4^1 \\ \\ 16 = 4^2$
Thus, the third landmark number is 16.

Key Concept: Base-5 Landmark Numbers

c) 25
[Solution Description]
In a base-5 system, the first landmark number is 1. The second landmark number is obtained by grouping 5 collections of size 1, which gives $5 \times 1 = 5$. The third landmark number is obtained by grouping 5 collections of size 5, which gives $5 \times 5 = 25$.

Key Concept: Efficient Representation

c) It allows unique representation of numbers using a fixed set of digits.
[Solution Description] The base-n system allows every number to be uniquely represented using digits from $0$ to $n-1$, eliminating the need for multiple symbols for different landmark numbers.

Key Concept: Multiplication in Base-n

b) $3125$
[Solution Description]
In a base-5 system, multiplication of landmark numbers follows the rule of exponents: $5^a \times 5^b = 5^{a+b}$. Here, $5^2 \times 5^3 = 5^{2+3} = 5^5$. Calculating $5^5$ gives $3125$.

Key Concept: Efficient Representation in Base-n System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the base-5 number system can represent the number 143 using a combination of its landmark numbers. This is true because any number can be expressed as a sum of multiples of landmark numbers (powers of the base). The Reason explains that the landmark numbers in base-5 are powers of 5, which helps in compact representation by grouping large quantities. This correctly justifies the Assertion since the systematic grouping allows efficient representation. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Scalability

c) Extending the powers of the base.
[Solution Description] The base-n system extends the sequence of powers of the base (e.g., $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, etc.), allowing efficient representation of large or small numbers.

Key Concept: Efficient Representation, Simplified Arithmetic

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, we need to verify the assertion (A). Converting 143 to base-5:
Step 1: Find the largest power of 5 less than or equal to 143, which is $5^3 = 125$.
Step 2: Divide 143 by 125 to get quotient 1 and remainder 18.
Step 3: Next, divide 18 by $5^2 = 25$, but since 25 > 18, the coefficient is 0.
Step 4: Now divide 18 by $5^1 = 5$ to get quotient 3 and remainder 3.
Step 5: Finally, divide the remainder 3 by $5^0 = 1$ to get coefficient 3.
Thus, $143_{10} = 1 \times 5^3 + 0 \times 5^2 + 3 \times 5^1 + 3 \times 5^0 = 1033_5$. This representation is indeed more efficient than non-positional systems.
Now, check the reason (R): The base-n system simplifies arithmetic by using powers of the base for grouping, which is true. However, this does not directly explain why $143_{10}$ can be represented as $1033_5$, as stated in the assertion. Hence, both are true, but the reason does not correctly explain the assertion.

Key Concept: Base concept in number systems

a) It requires fewer symbols for large numbers.
[Solution Description]
A base-n number system simplifies number representation and computations by using a fixed set of symbols (digits) for all numbers, avoiding the need for an unending sequence of unique symbols for higher powers like in the Egyptian system.

Key Concept: Arithmetic in Egyptian System

c) Manual symbol grouping made operations tedious
[Solution Description]
The Egyptian system lacked positional notation, forcing manual grouping and regrouping of symbols during operations like addition or multiplication. For example, adding $2907 + 43$ involved counting individual symbols and converting groups (e.g., 10 ones to 1 ten), which was time-consuming and error-prone.

Key Concept: Shortcomings of the Egyptian System, Base Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian numeral system requires an infinite sequence of unique symbols for higher powers of 10, which is true because each power of 10 has its own distinct symbol. The Reason claims that this inefficiency arises because the system lacks positional notation, which is also true since positional notation allows reuse of symbols with place value. Moreover, the Reason correctly explains the Assertion as the absence of positional notation directly leads to the need for more symbols in the Egyptian system.

Key Concept: Symbol Representation

a) 5 lotus flowers, 2 coils of rope, 6 heel-bone symbols, and 3 strokes
[Solution Description]

Key Concept: Shortcomings of the Egyptian System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian number system is inefficient for large numbers due to the lack of positional value. This is true as the syllabus mentions "Inefficiency in Representation" and "Lack of Positional Value" as shortcomings. The Reason explains that the system uses repeated symbols, which is also true and directly supports why the system is inefficient, as stated in the syllabus under "Inefficiency in Representation."

Key Concept: Place Value Representation, Hindu Number System

b) 4056
[Solution Description]
The given expression represents the number in expanded form. To find the decimal equivalent, compute each term:
$4 \times 10^3 = 4000$
$5 \times 10^1 = 50$
$6 \times 10^0 = 6$
Adding them together: $4000 + 50 + 6 = 4056$.

Key Concept: Place Value Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Babylonian number system used base-60 to simplify fraction representation, which is true as 60 has multiple factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), allowing for more straightforward division. The Reason provides a correct explanation for why base-60 simplifies fraction representation, as having numerous divisors facilitates easy division into smaller parts. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Place Value Representation

b) 497
[Solution Description]
Calculate step-by-step:
$4 \times 10^2 = 400 \\ \\ 9 \times 10 = 90 \\ \\ 7 \times 1 = 7 \\ \\ \text{Sum} = 400 + 90 + 7 = 497$

Key Concept: Place Value Representation, Hindu Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Hindu number system's use of 0 as a placeholder and a number was important for developing algebraic structures like rings. This is true because the concept of zero allowed for more advanced mathematical operations and structures. The reason given is that Brahmagupta's work defined arithmetic operations with zero, which indeed provided the foundation for rings by ensuring closure under these operations. Since the reason correctly explains why the assertion is true, the correct choice is the option where both are true and the reason explains the assertion.

Key Concept: Place Value Representation

a) $(2,5,30)_{60}$
[Solution Description]

The Mesopotamian number system used base 60, also called the sexagesimal system. In this system, each place has a value based on powers of 60. Since

$3600 = 60^2$

the given number can be written as:

$7530 = 2 \times 60^2 + 5 \times 60^1 + 30 \times 60^0$

So, the digits in base 60 are 2, 5, and 30.

$(2,5,30)_{60}$

Therefore, the correct answer is Option A.

Key Concept: Ambiguity in representation, positional interpretation

b) Because the system lacked a clear way to denote empty positions
[Solution Description]
The ambiguity arises because:
1. In the Mesopotamian system, positions represent powers of 60 ($60^0$, $60^1$, $60^2$, etc.).
2. A single wedge $\mathrm{\text{ }}$ (representing 1) in the second position means $1 \times 60^1 = 60$.
3. If spacing is unclear, the same wedge could be interpreted as being in the third position: $1 \times 60^2 = 3600$.
4. The lack of a zero symbol or clear spacing made it impossible to distinguish between positions.

Key Concept: Positional representation, conversion from decimal

c) Three symbols for 60 followed by three symbols for 1
[Solution Description]
To convert 183 to the Mesopotamian system:
1. Divide 183 by 60: $183 \div 60 = 3$ with remainder $3$.
2. The number consists of $3 \times 60^1 + 3 \times 60^0$ = 3 groups of 60 and 3 units.
3. The Mesopotamian representation would show 3 symbols for 60 (each being a symbol for 10 repeated three times) followed by 3 symbols for 1 (since they used base-60).

Key Concept: Symbol Representation

b) A corner wedge or sideways wedge
[Solution Description]
The Mesopotamians used a sexagesimal numeral system, which was based on the number 60. In their cuneiform writing, the number 1 was represented by a vertical wedge, while the number 10 was represented by a corner wedge or sideways wedge. These symbols were combined to write larger numbers. Therefore, the correct answer is a corner wedge or sideways wedge.

Key Concept: Number Representation

b) $(2 \times 60) + 12$
[Solution Description]
To represent 132 in the Mesopotamian system, break it down into multiples of 60: $132 = (2 \times 60) + 12$. Thus, it is represented as two 60s and twelve 1s.

Key Concept: Mesopotamian Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion (A) is true as the Mesopotamian number system indeed used a base-60 (sexagesimal) system. The Reason (R) is also true because base-60 was chosen due to its divisibility properties, which simplified calculations for trade and astronomical observations. Moreover, the Reason correctly explains why the Mesopotamians adopted the base-60 system. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Mayan Symbol Interpretation

c) 3 bars and 4 dots
[Solution Description]

In the Mayan numeral system, a dot represents 1 and a bar represents 5. To represent the number 19, we break it as:

19 = 5 + 5 + 5 + 4

So, 19 is represented by 3 bars and 4 dots.

3 bars and 4 dots​

Therefore, the correct answer is Option A.

Key Concept: Mayan Number Representation

d) Three bars ($\overline{\ }\overline{\ }\overline{\ }$)
[Solution Description]
In the Mayan number system, a dot $\cdot$ represents 1 and a bar $\overline{\ }$ represents 5. To represent 15:
- Three bars ($5 + 5 + 5 = 15$) are used.
- The correct representation is three horizontal bars: $\overline{\ }\overline{\ }\overline{\ }$.

Key Concept: Mayan addition

c) 274
[Solution Description]
Convert both numbers to Hindu-Arabic first:
1. First Number:
- Top level ($\overline{\ } \cdot$) = 5 + 1 = 6 units of 20.
- Bottom level ($\cdot \cdot \cdot$) = 1 + 1 + 1 = 3 units of 1.
- Total = $(6 \times 20) + 3 = 120 + 3 = 123$.
2. Second Number:
- Top level ($\overline{\ } \cdot \cdot$) = 5 + 1 + 1 = 7 units of 20.
- Bottom level ($\overline{\ } \overline{\ } \cdot$) = 5 + 5 + 1 = 11 units of 1.
- Total = $(7 \times 20) + 11 = 140 + 11 = 151$.
Sum = $123 + 151 = 274$.

Key Concept: Conversion from Mayan to Hindu-Arabic

c) 155
[Solution Description]
The Mayan number has two levels:
1. The top level represents 20s, where $\overline{\ } \cdot \cdot$ means $5 + 1 + 1 = 7$ units of 20.
2. The bottom level represents 1s, where $\overline{\ } \overline{\ } \overline{\ }$ means $5 + 5 + 5 = 15$ units of 1.
To convert to Hindu-Arabic, multiply the top level by 20 and add the bottom level:
Total = $(7 \times 20) + 15 = 140 + 15 = 155$.

Key Concept: Conversion from Hindu-Arabic to Mayan

a) Top level: 1 dot; Middle level: 2 dots; Bottom level: 2 dots
[Solution Description]

Key Concept: Chinese vs other systems, comparative analysis

d) Fully developed place value system
[Solution Description]
Key advancement shared by both:
1) Used distinct symbols for digits 1-9 (unlike cumulative systems)
2) Had clear positional notation where each position represented a power of 10
3) Both could represent zero (though Chinese used blank spaces initially)
4) This combination made calculations more efficient than non-positional or partial-positional systems

Key Concept: Place Value Representation

b) 4132
[Solution Description]
Here, 3 (Heng) represents $3 \times 10 = 30$,
2 (Zong) represents $2 \times 1 = 2$,
4 (Heng) represents $4 \times 1000 = 4000$,
1 (Zong) represents $1 \times 100 = 100$.
Adding them up: $4000 + 100 + 30 + 2 = 4132$.

Key Concept: Place Value Representation

a) 6 (Heng), blank, 9 (Heng), blank
[Solution Description]
The number 6,090 can be expanded as:
$6 \times 1000 + 0 \times 100 + 9 \times 10 + 0 \times 1$.
Hence, the representation is:
6 (Heng for thousands),
0 (Zong for hundreds - blank space or zero symbol),
9 (Heng for tens),
0 (Zong for units - blank space or zero symbol).

Key Concept: Chinese rod numerals, place value conversion

c) 5079
[Solution Description]
Step 1: Identify the positions - The blank space indicates a missing place value. Following the pattern:
$5 (Heng) = 5 \times 10^3 = 5000 \\ \\ \text{Blank} = 0 \times 10^2 = 0 \\ \\ 7 (Zong) = 7 \times 10^1 = 70 \\ \\ 9 (Heng) = 9 \times 10^0 = 9$
Step 2: Add all values together:
$5000 + 0 + 70 + 9 = 5079$

Key Concept: Alternating Place Values

c) Zongs represent units, hundreds, tens of thousands; hengs represent tens, thousands, hundreds of thousands.
[Solution Description]
Zongs represent units, hundreds, tens of thousands, etc., while hengs represent tens, thousands, hundreds of thousands, etc. These alternate place values are fundamental to understanding the Chinese rod numeral system.

Key Concept: Place Value, Multi-Concept

b) $100A + 10B + C$
[Solution Description]
In the Hindu number system, a three-digit number $\overline{ABC}$ can be expanded using place values as follows:
$A$ is in the hundreds place: $A \times 10^2 = 100A$
$B$ is in the tens place: $B \times 10^1 = 10B$
$C$ is in the ones place: $C \times 10^0 = C$
Therefore, the expanded form is $100A + 10B + C$.

Key Concept: Place Value Representation

c) 5000
[Solution Description]
The number 5,672 can be expanded using place value as:
$5 \times 10^3 + 6 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$
Here, the digit '5' is in the thousands place ($10^3$), so its value is $5 \times 1000 = 5000$.

Key Concept: Landmark Numbers

c) $10^2$
[Solution Description]
In the Hindu number system, landmark numbers are powers of 10. The hundreds place corresponds to $10^2$, which equals 100.

Key Concept: Landmark Numbers

b) Hundreds
[Solution Description]
In the Hindu number system, landmark numbers are powers of 10. Here, $10^2$ represents $10$ multiplied by itself twice, which equals $100$. This is known as the "hundreds" place.

Key Concept: Place Value System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Hindu number system indeed uses 10 symbols (0-9), making the Assertion true. The Reason is also true because the inclusion of 0 eliminates ambiguity in number representation by acting as a placeholder. Moreover, the Reason correctly explains why 0 is essential in the Hindu number system's place value structure. Thus, both statements are true, and the Reason justifies the Assertion.