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: 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: Historical Number Systems

c) India
[Solution Description]
The Hindu-Arabic numeral system, which includes the digits 0 through 9, originated in India around 2000 years ago. It was later transmitted to the Arab world and then to Europe.

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: 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: 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: 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: 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: Landmark Numbers

c) 100
[Solution Description]
Landmark numbers are reference points in a number system, often powers of the base. In the base-10 system, landmark numbers include 1, 10, 100, 1000, etc., which are powers of 10.

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: 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: Indian numeral system, evolution of zero

c) Defining arithmetic properties of zero
[Solution Description]
Brahmagupta formalized the arithmetic rules for zero as both a digit and a number. He defined operations like addition ($a + 0 = a$) and subtraction ($a - 0 = a$), which were crucial for modern mathematics. The Bakhshali manuscript introduced zero as a dot, but Brahmagupta expanded its use.

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: Tally Marks

c) The Lebombo bone
[Solution Description]
Archaeologists have discovered ancient tally sticks like the Ishango bone and the Lebombo bone. The Lebombo bone, estimated to be around 44,000 years old, is considered one of the oldest mathematical artefacts. It features 29 notches and may have been used as a tally stick or lunar calendar.

Key Concept: Counting Methods

b) It cannot count collections with more than 26 objects.
[Solution Description]
Using the English alphabet limits counting because there are only 26 distinct letters available. Therefore, collections larger than 26 objects cannot be counted using this method without repetition or modification of the sequence, making it impractical beyond small quantities.

Key Concept: Counting Mechanisms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that the Ishango bone is one of the oldest mathematical artifacts with tally marks, which is true as it dates back 20,000 to 35,000 years and was used for counting or calendrical purposes. The Reason (R) explains that tally marks are a simple method for counting by creating a one-to-one correspondence between objects and marks, which is also true. Additionally, Reason (R) correctly explains why tally marks were used in ancient artifacts like the Ishango bone—because they provide an easy way to count objects without complex symbols or systems.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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

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: Chinese Number System

b) 10
[Solution Description]
According to the syllabus, the Chinese rod numerals were a decimal (base-10) system with symbols for 1 to 9.

Key Concept: Roman Number System

c) MCMLXXXVII
[Solution Description]
The Roman number system uses the following symbols: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). Numbers are formed by combining these symbols, where smaller numerals placed before larger ones are subtracted (e.g., IV = 4 or IX = 9).
To convert 1987 to Roman numerals:
1. Break down 1987: 1000 + 900 + 80 + 7.
2. 1000 is represented as M.
3. 900 is CM (1000 - 100).
4. 80 is LXXX (50 + 30, where 30 is XXX).
5. 7 is VII (5 + 2).
6. Combine all parts: MCMLXXXVII.
Thus, the correct representation is MCMLXXXVII.

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) 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: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
The Mayan number system primarily used a base-20 system but introduced a third landmark number at 360, likely due to its calendar system. This made it not a pure base-20 system and complicated computations. Therefore, the Assertion is false as the system was not purely base-20, while the Reason correctly explains why the system was not purely base-20 due to the introduction of the 360 landmark number.

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, Limitations

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the body part counting system can represent any large number without limitations, which is false because the syllabus explicitly mentions that the system is limited by the number of distinguishable body parts. The reason correctly identifies this limitation and also mentions the requirement of memorizing the sequence, which is true. Therefore, Assertion is false, but Reason is true.

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 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-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: 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 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: 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: 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: 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: 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: 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

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: Roman Number System

c) XIV
[Solution Description]
In the Roman number system, 14 is represented by combining the symbols for 10 (X) and 4 (IV), giving XIV.

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: 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: 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 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: 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 - Conversion

b) MDCCLXXXIX
[Solution Description]
To convert 1789 to Roman numerals, we break it down into landmark numbers:
1789 = 1000 (M) + 500 (D) + 200 (CC) + 50 (L) + 30 (XXX) + 9 (IX).
Combining these, we get MDCCLXXXIX.

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 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: 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: 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: 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: 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: 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 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: Egyptian Number System - Representation

a) A single symbol for $10$
[Solution Description]
The Egyptian number system uses landmark numbers based on powers of $10$. The first landmark number is $1$, and the second landmark number is obtained by grouping $10$ collections of $1$, which gives $10$.

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: Landmark Numbers, Representation of Numbers

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that 3426 can be represented using exactly 15 symbols in the Egyptian number system. According to the rules of the Egyptian system, numbers are represented by grouping them into sums of landmark numbers (1, 10, 100, 1000, etc.), with each group requiring a separate symbol. For 3426, the breakdown is:
$3426 = 1000 + 1000 + 1000 + 100 + 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1$
This uses 3 symbols for 1000, 4 for 100, 2 for 10, and 6 for 1, totaling 15 symbols.
The Reason explains that each landmark number requires a unique symbol and is grouped in multiples up to 9, which is true as per the Egyptian system's rules. However, the Reason does not directly explain why 3426 specifically uses 15 symbols; it only states a general rule. Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

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

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: 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: Base System Representation

b) 12
[Solution Description]
To represent 7 in base-5, we find the largest power of 5 less than or equal to 7, which is $5^1 = 5$. We can subtract 5 from 7 to get 2. So, 7 in base-5 is represented as $12_5$ (1 group of 5 and 2 units).

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 System, Conversion to Base-10

c) 59
[Solution Description]
To convert $214_5$ to decimal, we use the place values in base-5:
$2 \times 5^2 + 1 \times 5^1 + 4 \times 5^0$
$= 2 \times 25 + 1 \times 5 + 4 \times 1$
$= 50 + 5 + 4 = 59$.
The decimal equivalent is 59.

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: Arithmetic operations in different bases

b) $41_5$
[Solution Description]
First, convert both numbers to base-10:
$12_5 = 1 \times 5^1 + 2 \times 5^0 = 5 + 2 = 7_{10}$
$3_5 = 3 \times 5^0 = 3_{10}$
Multiply them: $7 \times 3 = 21_{10}$.
Convert $21_{10}$ back to base-5:
Largest power of 5 less than or equal to 21 is $5^2 = 25$, which is too large, so use $5^1 = 5$.
$21 \div 5 = 4$ with remainder $1$.
Next, $1 \div 5^0 = 1$ with remainder $0$.
So, $21_{10} = 4 \times 5^1 + 1 \times 5^0 = 41_5$.

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: 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 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: Representation Comparison

b) Roman numerals lack fixed landmark numbers, while base-n systems use powers of n.
[Solution Description]
In the Roman numeral system, landmark numbers (like I, V, X, L, etc.) do not follow a fixed multiplicative pattern. For example, grouping is not uniform (e.g., V is 5, but X is 10; there is no consistent multiplier). In contrast, a base-n system groups numbers strictly in powers of n, making arithmetic operations systematic.

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 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: 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 Egyptian System

b) It required new symbols for every power of 10.
[Solution Description]
The Egyptian system required a unique symbol for each power of 10. As numbers grew larger, new symbols had to be invented continuously, making the system cumbersome and impractical for very large numbers. For example, representing $10^7$ required a new symbol, and even larger numbers would demand more symbols.

Key Concept: Place Value Representation

c) 60
[Solution Description]
The Mesopotamian number system is a base-60 system, also known as the sexagesimal system.

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

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

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 is superior due to its base-10 place value system with zero, which is true. The reason explains why this is the case by highlighting zero's role in preventing ambiguity and enabling arithmetic operations, which correctly explains the assertion. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: Mesopotamian Place Value System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 3600 is represented by a single symbol in the 3600s place in the Mesopotamian system. This is correct because 3600 = 60$^2$, and in their base-60 system, the third position from the right represents multiples of 3600.
The reason explains that the Mesopotamian system was a base-60 positional system. This is also true as the system used powers of 60 for different positions, with the rightmost being 60$^0$ (1s), next 60$^1$ (60s), then 60$^2$ (3600s), etc.
Furthermore, the reason correctly explains why the assertion is true: because it's a base-60 positional system, 3600 corresponds to the third position. Hence, both are true, and the reason is the correct explanation.

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: 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: 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 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: Conversion from Hindu-Arabic to Mayan

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

Key Concept: Place Value Representation - Mayan Number System

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion is true because scholars suggest that the Mayans chose 360 as their third landmark number due to its connection with their calendar rather than using 400, which would align with a pure base-20 system. The reason is also true as the Mayan calendar indeed included a 360-day cycle called the "Tun." However, while the reason provides factual information about the calendar, it does not directly explain why 360 was chosen over 400 in their number system—it merely states a coinciding fact. Thus, the reason is not the correct explanation for the assertion.

Key Concept: Mayan Landmark Numbers

d) It aligned with their 360-day calendar system
[Solution Description]
The Mayan civilization used 360 as their third landmark number (after 1 and 20) instead of 400 because it was likely related to their calendar system. Their calendar had 360 days in a year (tun), which influenced their numerical system.

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 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 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 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: Zero as a Digit, Historical Context

b) It allowed unambiguous representation of numbers and enabled arithmetic operations like addition and multiplication with zero.
[Solution Description]
Zero was introduced as a digit in the Hindu number system to serve as both a placeholder and a number. This allowed unambiguous representation of numbers in a place value system and enabled advanced arithmetic operations such as addition, subtraction, and multiplication involving zero. It also laid the foundation for modern mathematics, including algebra and analysis.

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, 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 both a digit and a number eliminated ambiguity and enabled efficient computation. This is true because the positional value system relies on 0 to represent absence in a place without altering the value of other digits. The Reason states that Brahmagupta codified the arithmetic properties of zero, treating it as a number, which is also true and directly explains why the system could handle computations efficiently. The Reason provides the conceptual basis for the Assertion, making it the correct explanation.

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.

Key Concept: Place Value, Landmark Numbers

a) $5 \times 1000 + 0 \times 100 + 8 \times 10 + 3$
[Solution Description]
The number 5083 can be broken down using place values of the Hindu number system. Starting from the rightmost digit:
$5$ is in the thousands place: $5 \times 10^3 = 5000$
$0$ is in the hundreds place: $0 \times 10^2 = 0$
$8$ is in the tens place: $8 \times 10^1 = 80$
$3$ is in the ones place: $3 \times 10^0 = 3$
Adding them together: $5000 + 0 + 80 + 3 = 5083$.