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: 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: 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: 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: 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: 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: Landmark Numbers

c) Easily recognizable reference numbers
[Solution Description]
Landmark numbers are easily recognizable numbers used as reference points for understanding other numbers. They help people make sense of quantities, especially larger ones.

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 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: 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: 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: 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: 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, 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: 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

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: Use of Body Parts

c) People in Papua New Guinea
[Solution Description]
The syllabus mentions that people in Papua New Guinea used their body parts as the standard sequence/number system for counting.

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: Mayan Number System, Roman Number System

b) CCLXI
[Solution Description]
To solve this problem, we first need to convert the given Mayan number into its decimal equivalent. The Mayan number has two bars and three dots in the 20s place and one dot in the 1s place.
Step 1: Calculate the value of the 20s place.
- Each bar represents 5, so two bars represent $2 \times 5 = 10$.
- Each dot represents 1, so three dots represent $3 \times 1 = 3$.
- Total value in the 20s place is $(10 + 3) \times 20 = 13 \times 20 = 260$.
Step 2: Calculate the value of the 1s place.
- One dot in the 1s place represents $1 \times 1 = 1$.
Step 3: Add the values from both places to get the total decimal value.
- Total decimal value is $260 + 1 = 261$.
Step 4: Convert the decimal value 261 to Roman numerals.
- 261 = 200 (CC) + 60 (LX) + 1 (I) = CCLXI.
Therefore, the correct Roman numeral representation is CCLXI.

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: 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: Extension of Body Parts Counting System

c) 27
[Solution Description]
Since the system uses both sides symmetrically:
Step 1: Total unique parts per side = 15.
Step 2: Right knee is 12th part, so left knee will be $12 + 15 = 27$.
Step 3: Hence, left knee corresponds to the 27th count.

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: 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-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 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 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 on Bones and Other Surfaces

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion states that the Ishango bone's tally marks are arranged in columns, possibly indicating a calendrical system. Archaeological evidence supports this claim, as the notches on the Ishango bone are grouped in patterns that may represent lunar cycles or other time-keeping methods.
The Reason claims that tally marks on ancient bones were solely used for simple counting. However, this is false because archaeological findings like the Ishango and Lebombo bones suggest these tally marks had more complex uses beyond just counting, such as tracking lunar months or calendrical systems.
Thus, the Assertion is true, but the Reason is false.

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: Counting in groups, Human perception limit

c) 33
[Solution Description]
Each new symbol represents 7 marks, so 4 symbols represent $4 \times 7 = 28$ marks. Adding the 5 separate marks gives the total count: $28 + 5 = 33$.

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: 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: 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: Number Names by 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 to form number names. This is true as per the syllabus, where it is mentioned that numbers are counted in 2s (e.g., $3 = 2 + 1$, $4 = 2 + 2$, etc.).
The Reason provides an example: the number name for 5 is ukasar-ukasar-urapon, which translates to 2 + 2 + 1. This correctly demonstrates the counting-in-twos pattern described in the Assertion. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Number Names by Counting in Twos

c) ukasar-ukasar-urapon
[Solution Description]
The Gumulgal count numbers in twos. For 5, they combine the name for 2 ("ukasar") and 1 ("urapon"). Since $5 = 2 + 2 + 1$, the representation is "ukasar-ukasar-urapon".

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 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: 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: 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: 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: 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: Concept merging, Base comparison

c) 1210 in base-4
[Solution Description]
Step 1: Find the largest power of 4 less than or equal to 100.
$4^3 = 64$ is the largest.
Step 2: Divide 100 by 64. Quotient is 1, remainder is $100 - 64 = 36$.
Step 3: Divide 36 by $4^2 = 16$. Quotient is 2, remainder is $36 - 32 = 4$.
Step 4: Divide 4 by $4^1 = 4$. Quotient is 1, remainder is 0.
Step 5: Final representation is $1 \times 4^3 + 2 \times 4^2 + 1 \times 4^1 + 0 \times 4^0 = 1210$ (base-4).

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: 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 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-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: 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: 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: 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, 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: 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: Egyptian Number System and Base Concept

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 a base of 10 because it groups numbers in powers of 10. This is true as the system's landmark numbers are defined as powers of 10 (1, 10, 100, etc.). The Reason explains that each landmark number is 10 times the previous one, which is also true and directly supports why the system is base-10. Thus, the Reason correctly explains the Assertion.

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: 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: 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 Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the third landmark number in a base-5 system is 25. This is correct because:
- First landmark number: 1
- Second landmark number: $5 \times 1 = 5$
- Third landmark number: $5 \times 5 = 25$
The reason explains that each new landmark number is obtained by multiplying the previous landmark number by 5, which is true for any base-n system. Here, the reason correctly explains why the assertion is true since it demonstrates how the third landmark number (25) is derived from the second (5).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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-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-n System: Grouping Efficiency

b) $3 \times 25 + 0 \times 5 + 3 \times 1$
[Solution Description]
To express 78 in base-5, we use the largest landmark numbers (powers of 5) less than or equal to 78. The landmark numbers are $5^0 = 1$, $5^1 = 5$, $5^2 = 25$, and $5^3 = 125$ (which is greater than 78). Divide 78 by 25 ($5^2$) to get 3 with a remainder of 3. Next, divide the remainder (3) by 5 ($5^1$) to get 0 with a remainder of 3. Finally, divide the remaining 3 by 1 ($5^0$) to get 3. Thus, 78 in base-5 is represented as $3 \times 25 + 0 \times 5 + 3 \times 1$.

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: 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: Base conversion, Landmark numbers

b) $2000_4$
[Solution Description]
To convert $128_{10}$ to base-4, find the largest power of 4 less than or equal to 128:
$4^3 = 64$
$128 \div 64 = 2$ with remainder $0$.
Next, $0 \div 4^2 = 0$ with remainder $0$.
Then, $0 \div 4^1 = 0$ with remainder $0$.
Finally, $0 \div 4^0 = 0$ with remainder $0$.
So, $128_{10} = 2 \times 4^3 + 0 \times 4^2 + 0 \times 4^1 + 0 \times 4^0 = 2000_4$.

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: 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: 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 needed new symbols for higher powers of 10, which is true because it used landmark numbers without a base concept. The Reason correctly identifies the absence of positional notation and zero in the system, which are key shortcomings. The reason directly explains why the assertion is true, as the lack of these features necessitated distinct symbols for each power of 10.

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: 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: 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

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, 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: 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

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: 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: 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: 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: 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: Conversion to decimal, multi-step interpretation

b) 24
[Solution Description]
To decode this:
1. Two symbols for 10 represent $2 \times 10 = 20$.
2. Four symbols for 1 represent $4 \times 1 = 4$.
3. Since there is no higher power of 60, this is simply $20 + 4 = 24$.
4. However, if this were in the "60s" place, it would be $(20 + 4) \times 60 = 1440$.
5. The question implies the simplest case (units place only) due to lack of positional context.

Key Concept: Mayan Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Mayan number system indeed uses a placeholder symbol for zero, which resembles a seashell. This assertion is true. The reason states that this placeholder symbol helps indicate an empty place value in the positional notation, which is also true and correctly explains why the Mayans used a placeholder symbol. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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 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: 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: 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: 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: 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 rod numeral system used a blank space as a placeholder for zero, which is true according to the syllabus. The reason explains that this method made identifying skipped place values easier compared to the Mesopotamian system, which is also true and correctly explains why the blank space was significant in the Chinese system. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: 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: 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 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.

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: 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: Digits and Zero

b) Zero multiplied by any number gives zero.
[Solution Description]
According to the Hindu number system, zero has arithmetic properties:
1. $0 + \text{any number} = \text{the same number}$
2. $0 \times \text{any number} = 0$
Thus, the correct statement is: "Zero multiplied by any number gives zero."