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: Ancient Indian Numerals

c) A dot
[Solution Description]
In the Bakhshali manuscript, which dates back to the 3rd century CE, the digit zero was represented as a dot.

Key Concept: Origin of Modern Numbers, Transmission of Indian Numerals

c) Bakhshali manuscript
[Solution Description]
The Bakhshali manuscript, dating back to the 3rd century CE, is known for its use of a dot ($\cdot$) to represent zero. This was one of the earliest documented uses of zero in the Indian numeral system.

Key Concept: Representation of Zero

c) Bakhshali manuscript
[Solution Description]
The Bakhshali manuscript, dated around the 3rd century CE, contains the first known instance of zero represented as a dot. This was part of the Indian numeral system, which included digits 0 through 9.

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: Historical Context of Numbers

d) Indians
[Solution Description]
The Indian civilization is credited with developing the modern numerical system, including the digit 0 (initially represented as a dot), around 2000 years ago. This system was later transmitted to the Arab world and then to Europe. The key figures in this transmission were Al-Khwārizmī and Fibonacci.

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: Spread of numerals, historical timelines

d) 17th century
[Solution Description]
The Hindu-Arabic numeral system was introduced to Europe by Fibonacci around 1200 CE but faced resistance. It became widely adopted only by the 17th century after overcoming initial skepticism and proving its efficiency in calculations.

Key Concept: Origin of Zero

c) Aryabhata
[Solution Description]
Aryabhata, in his work Āryabhaṭīya (499 CE), was the first to use 0 as a number and perform arithmetic operations with it, such as $0 + a = a$ and $0 \times a = 0$.

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: Origin of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Indian number system introduced zero as a number with arithmetic properties, which is correct as per historical evidence. The Reason explains that Aryabhata and Brahmagupta formalized its use, leading to advancements in mathematics, which is also historically accurate. Moreover, the Reason correctly justifies why the Assertion is true, as it elaborates on the role of Indian mathematicians in developing the concept of zero.

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: 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, Counting with sticks

a) 18
[Solution Description]
In Method 1, each stick represents one goat. If the shepherd has 23 sticks but 5 are broken, those broken sticks do not correspond to any goats.
Calculation:
Total sticks = 23
Broken sticks = 5
Effective sticks (representing goats) = $23 - 5 = 18$
Therefore, the shepherd has 18 goats.

Key Concept: Number Representation

b) 26
[Solution Description]
The English alphabet consists of 26 letters ('a' to 'z'). In Method 2, each letter represents a unique number, starting from 'a' for 1 up to 'z' for 26. Therefore, the maximum number that can be represented using this method is 26.

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: 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: 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, 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: Mayan Number System

c) Three dots at the top and three bars and two dots at the bottom
[Solution Description]
The Mayan number system uses dots for 1 and bars for 5. For numbers up to 19, a dot represents 1 and a bar represents 5. For numbers above 19, they use a base-20 system with each higher position representing multiples of 20.
To represent 77 in the Mayan system:
1. Divide 77 by 20: $77 \div 20 = 3$ with remainder 17.
2. The remainder 17 is written as three bars (each bar is 5) and two dots (each dot is 1), i.e., ---$\cdot$$\cdot$.
3. The quotient 3 is written as three dots, i.e., $\cdot$$\cdot$$\cdot$, in the next higher position.
4. Combining them gives $\cdot$$\cdot$$\cdot$ below ---$\cdot$$\cdot$.
Thus, the correct representation is three dots at the top and three bars and two dots at the bottom.

Key Concept: Mayan Number System

b) Bar (――)
[Solution Description]
In the Mayan number system, a dot (⋅) represents 1, a bar (――) represents 5, and a shell symbol represents 0. Therefore, the symbol used to represent the number 5 is the bar (――).

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: Body-Part Number Representation

c) Shoulder of the right arm
[Solution Description]
From the given sequence, after the right ear ($12$), the next body part is the shoulder of the right arm ($13$).

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: Body-Part Number Representation

b) Elbow of the left arm
[Solution Description]
According to the sequence, the number $7$ is represented by the elbow of the left arm.

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 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: 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: Tally Marks and Early Number Systems

a) Representing lunar phases
[Solution Description]
The Ishango bone, discovered in the Democratic Republic of Congo, features notches arranged in columns. These markings are thought to represent numbers and may indicate a calendrical system or tally counting method.

Key Concept: Historical Number Systems

c) It is among the earliest known tally systems
[Solution Description]
The Lebombo bone, found in South Africa, has 29 notches and is one of the oldest mathematical artifacts (around 44,000 years old). Its markings suggest it was used as a tally stick or lunar calendar.

Key Concept: Counting Methods

b) To make counting easier and more efficient
[Solution Description]
Early humans found it difficult to count more than 4 or 5 objects at a glance due to perceptual limits. Grouping tally marks in sets (e.g., 5) made counting and representation more efficient, leading to the development of more advanced number systems.

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 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: 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: 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: 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: 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

c) They all represent numbers by counting in twos.
[Solution Description]
All three groups use a counting system based on grouping numbers in twos. Their number names are formed by combining words for smaller numbers (especially 2 and 1) to represent larger numbers.

Key Concept: Bushmen Number System

c) t'oa-t'oa
[Solution Description]
The Bushmen system also uses counting in twos. The number 4 is represented as two occurrences of the word for 2, which is t'oa-t'oa.

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: Roman Numerals - Subtraction Rule

c) XLIX
[Solution Description]
The number 49 is represented using the subtraction rule where 40 is written as XL (50 - 10) and 9 as IX (10 - 1). Thus, 49 is written as XLIX.

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: 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: Basic Roman Numeral Conversion

c) XV
[Solution Description]
To convert the number 15 to Roman numerals, we use the landmark numbers: V (5) and X (10).
Step 1: Since 15 = 10 + 5.
Step 2: The Roman numeral for 10 is X, and for 5 is V.
Therefore, 15 in Roman numerals is XV.

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 and landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert 999 to Roman numerals:
900 is represented as CM (1000 - 100),
90 as XC (100 - 10),
and 9 as IX (10 - 1).
Thus, 999 = CMXCIX, confirming the Assertion.
The Reason correctly explains that the subtractive principle (like IV for 4 or IX for 9) was used but not always consistently, as mentioned in the syllabus regarding the representation of 40 as XL or XXXX.

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: Representation in Base-5

b) 5 + 1 + 1
[Solution Description]
In base-5, landmark numbers are 1, 5, 25, etc. To represent 7, we use one 5 and two 1's: $5 + 1 + 1 = 7$.

Key Concept: Base conversion, Grouping principle

b) 514 in base-7
[Solution Description]
To represent 256 in base-7:
Step 1: Find the largest power of 7 less than or equal to 256.
$7^3 = 343$ is too large. So, $7^2 = 49$ comes first.
Step 2: Divide 256 by 49. We get 5 with remainder $256 - (49 \times 5) = 11$.
Step 3: Divide the remainder 11 by $7^1 = 7$. We get 1 with remainder $11 - 7 = 4$.
Step 4: The remaining 4 is represented by $7^0 = 1$ units.
Thus, $256 = 5 \times 7^2 + 1 \times 7^1 + 4 \times 7^0$, which is written as 514 in base-7.

Key Concept: Base-5 Number System

c) 200
[Solution Description]
To convert 50 to base-5, we divide the number by successive powers of 5. The largest power of 5 less than 50 is $5^3 = 125$, which is too large, so we start with $5^2 = 25$. Divide 50 by 25: $50 \div 25 = 2$ with remainder 0. Next, divide the remaining part (0) by $5^1 = 5$: $0 \div 5 = 0$ with remainder 0. Finally, divide by $5^0 = 1$: $0 \div 1 = 0$ with remainder 0. Thus, 50 in base-5 is written as 200.

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: Landmark Numbers

c) 27
[Solution Description]
The landmark numbers in a base-3 system are powers of 3: $3^0 = 1$, $3^1 = 3$, $3^2 = 9$, $3^3 = 27$. The fourth landmark number is $3^3 = 27$.

Key Concept: Assertion-Reason

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 uses base 10 for grouping numbers, which is true as the system groups numbers in powers of 10 (e.g., 1, 10, 100, etc.). The Reason explains that each new landmark number is formed by grouping 10 collections of the previous landmark number, which is also true and correctly justifies the Assertion. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

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: 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: 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: Multiplication in Egyptian Numerals

b) 1000
[Solution Description]
In the Egyptian system, multiplying a landmark number by 10 shifts it to the next higher landmark number. Since:
$100 = 10^2 \\ \\ 10 = 10^1 \\ \\ 100 \times 10 = 10^2 \times 10^1 = 10^3 = 1000$
The result is 1000.

Key Concept: Landmark Numbers, Representation of Numbers

a) Five 100s, six 10s, seven 1s
[Solution Description]
To represent 567 in the Egyptian system:
- Start with the largest landmark number less than 567, which is 100.
- Divide 567 by 100: $567 ÷ 100 = 5$ with a remainder of 67. So, we use five 100s.
- Next, for the remainder 67, use the next largest landmark number, 10.
- Divide 67 by 10: $67 ÷ 10 = 6$ with a remainder of 7. So, we use six 10s.
- The remaining 7 is represented by seven 1s.
- Total representation: five 100s + six 10s + seven 1s.

Key Concept: Base-5 Representation

d) $111_5$
[Solution Description]
To convert $31_{10}$ to base-5, divide 31 by powers of 5:
$31 \div 25 = 1 \text{ with remainder } 6 \\ \\ 6 \div 5 = 1 \text{ with remainder } 1 \\ \\ 1 \div 1 = 1 \text{ with remainder } 0$
Combining the coefficients gives $111_5$.

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: Multiplication in Base-7

a) $130_7$
[Solution Description]
First, convert $20_7$ and $5_7$ to base-10:
$20_7 = 2 \times 7^1 + 0 \times 7^0 = 14_{10} \\ \\ 5_7 = 5 \times 7^0 = 5_{10}$
Multiply them in base-10:
$14 \times 5 = 70_{10}$
Convert back to base-7 by dividing by powers of 7:
$70 \div 49 = 1 \text{ with remainder } 21 \\ \\ 21 \div 7 = 3 \text{ with remainder } 0 \\ \\ 0 \div 1 = 0 \text{ with remainder } 0$
Combining the coefficients gives $130_7$.

Key Concept: Base-4 System, Addition

b) $1020_4$
[Solution Description]
First, convert both numbers to decimal:
$123_4 = 1 \times 4^2 + 2 \times 4^1 + 3 \times 4^0 = 16 + 8 + 3 = 27$.
$231_4 = 2 \times 4^2 + 3 \times 4^1 + 1 \times 4^0 = 32 + 12 + 1 = 45$.
Sum in decimal: $27 + 45 = 72$.
Now, convert 72 to base-4:
$72 \div 4 = 18$ with remainder $0$,
$18 \div 4 = 4$ with remainder $2$,
$4 \div 4 = 1$ with remainder $0$,
$1 \div 4 = 0$ with remainder $1$.
Reading remainders from bottom to top: $1020_4$.

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, 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: 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: Simplified Arithmetic

c) Because of the positional value of digits.
[Solution Description] The positional nature of the base-n system makes operations like addition and subtraction systematic, as they involve digit-wise calculations with carry-over.

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: 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: 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, 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: 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: Symbol repetition in Egyptian numerals

a) No, because each symbol can only be repeated up to 9 times before moving to the next higher power of 10.
[Solution Description]
The Egyptian numeral system uses distinct symbols for powers of 10. Each symbol is used up to 9 times before moving to the next higher power of 10. Hence, no symbol can be repeated 10 or more times in a valid representation.

Key Concept: Shortcomings of Egyptian System

b) It involved complex calculations for simple multiplications.
[Solution Description]
In the Egyptian system, multiplication was more complex compared to addition and subtraction. It required understanding that each landmark number is a power of 10 and applying this logic manually for each operation. For example, multiplying $10 \times 100$ required recognizing that the product is $1000$.

Key Concept: Place Value Representation, Zero in Number Systems

b) It allowed unambiguous place value representation and advanced arithmetic
[Solution Description]
The introduction of zero as both a digit and a number was significant because:
1. It allowed unambiguous representation of numbers in a place value system (e.g., distinguishing 205 from 25).
2. It enabled arithmetic operations involving zero (e.g., addition and multiplication properties).
3. It laid the foundation for advanced mathematical concepts like negative numbers and algebra.

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, 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

b) A sideways/corner wedge
[Solution Description]

In the Mesopotamian number system, numbers were written using cuneiform wedge-shaped symbols. A vertical wedge was used to represent 1, while a sideways or corner wedge was used to represent 10. Therefore, the symbol for 10 in the Mesopotamian number system was the sideways/corner wedge.

Sideways or corner wedge = 10​

Key Concept: Place Value Representation, Babylonian Number System

b) $(1,12,1)_{60}$​
[Solution Description]

The Babylonian number system used base 60, also called the sexagesimal system. To convert 4321 into base 60, we divide it according to powers of 60.

$60^2 = 3600$

Now,

$4321 = 1 × 3600 + 721$

Next,

$721 = 12 × 60 + 1$

So,

$4321 = 1 \times 60^2 + 12 \times 60^1 + 1 \times 60^0$

$4321 = 3600 + 720 + 1$

$4321 = 4321$

Therefore, the Babylonian base-60 representation of 4321 is:

$(1,12,1)_{60}$​​

So, the correct answer is Option A.

Key Concept: Conversion from Mesopotamian System

c) 35
[Solution Description]

In the Mesopotamian number system, one wedge symbol represented 10, and one vertical stroke symbol represented 1. Therefore, three wedge symbols represent:

$3×10=30$

and five vertical stroke symbols represent:

$5 × 1 = 5$

So, the total value is:

$30 + 5 = 35$

Hence, the representation stands for 35.

Key Concept: Mesopotamian Number System

c) 60
[Solution Description]
The Mesopotamian numeral system used a base-60 (sexagesimal) system, where each position represented powers of 60.

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: 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: Mesopotamian Number Representation

a) $(2,12)_{60}$
[Solution Description]

The Mesopotamian or Babylonian number system used base 60. This means numbers were grouped according to powers of 60.

To convert 132 into base 60:

132 = 2 × 60 + 12

So, 132 has 2 groups of 60 and 12 units.

Therefore, the Mesopotamian representation is:

$(2,12)_{60}$​​

So, the correct answer is Option A.

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: Place Value in Mayan System

c) One dot at the top and a seashell at the bottom ($\cdot$ over seashell)
[Solution Description]
In the Mayan place value system:
- The lower level represents units (1s), while the level above represents multiples of 20.
- For 20, we write one dot at the second level (representing $1 \times 20$) and a zero placeholder (seashell symbol) at the units level.
- The correct representation is a dot over a seashell symbol.

Key Concept: Mayan Number Representation

a) Top level: 3 dots; Bottom level: 3 bars and 2 dots
[Solution Description]

The Mayan numeral system is based on 20. In this system, a dot represents 1, a bar represents 5, and numbers are written vertically. The bottom level represents ones, and the level above it represents twenties.

To represent 77:

77 = 3 × 20 + 17

Here, 3 is written in the upper level as 3 dots. The remaining 17 is written in the lower level as 3 bars and 2 dots, because:

17 = 5 + 5 + 5 + 2

So, the Mayan numeral representation of 77 is:

Top level: 3 dots; Bottom level: 3 bars and 2 dots​

Therefore, the correct answer is Option A.

Key Concept: Place Value Concepts

b) Due to their calendar system having 360 days in a cycle
[Solution Description]
The Mayan civilization used a modified vigesimal (base-20) system, but their third place value was 360 ($20 \times 18$) instead of 400 ($20 \times 20$). This deviation is likely due to the importance of their calendar system, which had 360 days in their "Tun" (a calendar cycle). The choice of 360 aligns with their astronomical and calendrical calculations rather than a strict base-20 progression.

Key Concept: Mayan Number System, Placeholder Symbol for Zero

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Mayan number system indeed used a placeholder symbol for zero, represented as a seashell, which was crucial for their place value notation. This is true, supporting the Assertion. The Reason states that this placeholder helped distinguish between numbers like 20 and 200, which is correct because without the zero, these numbers would be indistinguishable (e.g., two bars could mean either 20 or 200). Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: Chinese Number System - 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 Chinese number system uses blank spaces to indicate skipped place values, which is true as per the syllabus. The Reason explains that this was possible due to the uniformity in symbol sizes, making blank spaces easier to locate compared to the Mesopotamian system. Both statements are true, and the Reason correctly explains the Assertion.

Key Concept: Place Value Representation

a) 7 (Heng), 5 (Zong), blank, 8 (Zong)
[Solution Description]
In the Chinese number system, heng represents tens, thousands, etc., while zong represents units, hundreds, etc. The number 7,508 can be broken down as:
$7 \times 1000 + 5 \times 100 + 0 \times 10 + 8 \times 1$.
Therefore, it would be written as:
7 (Heng for thousands),
5 (Zong for hundreds),
0 (Heng for tens - blank space or zero symbol if available),
8 (Zong for units).

Key Concept: Place Value System Basics

b) blank space
[Solution Description]
The Chinese rod numeral system used a blank space to indicate the skipping of a place value, acting as a placeholder. This is explicitly stated in the syllabus under the description of the system.

Key Concept: Place Value Identification

d) horizontal rod
[Solution Description]
The Chinese rod numeral system uses a horizontal rod to represent the number 5. This is clearly mentioned in the syllabus as: $5 = \text{(horizontal rod)}$.

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: Place Value Representation: The 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 numeral 405 can be expanded as $4 \times 10^2 + 0 \times 10 + 5 \times 1$, which is true for the Hindu number system, a base-10 place value system. The reason states that '0' serves as a placeholder to avoid ambiguity in number representation, which is also true. Moreover, the reason correctly explains why the assertion holds true, as the presence of '0' ensures the correct placement and value of each digit in the numeral.

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: 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: Zero in Hindu Number System

c) Codified arithmetic operations with 0
[Solution Description]
Brahmagupta codified the use of 0 as a number with arithmetic properties in his work Brāhmasphuṭasiddhānta (628 CE). He introduced operations like addition and multiplication involving zero, treating it as a number on par with others.