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: Indian Number System and Spread

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

Key Concept: Early human need for counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that early humans needed to count livestock for trade and survival, which is true as per the syllabus. The reason given is that counting helped them track the number of animals they owned, which is also true and directly explains why they needed to count livestock.

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: 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: 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: 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: 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: Origin of Numbers

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

Key Concept: Origin of 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

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: 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 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: 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, 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: 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: 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: 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: Mayan Number System, Hindu Number System

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

Key Concept: Chinese 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: 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

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: Scalability of Body Parts Counting System

c) 16
[Solution Description]
Given each joint counts as 5 units:
Step 1: Wrist (6) to elbow (11) covers $11 - 6 = 5$ units (as expected).
Step 2: Similarly, elbow (11) to shoulder will also cover 5 units.
Step 3: Thus, right shoulder = $11 + 5 = 16$.

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 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 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: 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: Historical Mathematical Artefacts

c) Lebombo bone
[Solution Description]
The Lebombo bone, discovered in South Africa, is one of the oldest known mathematical artefacts with tally marks. It has 29 notches and is estimated to be around 44,000 years old. It may have been used as a tally stick or lunar calendar.

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 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: 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: 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: 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: 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, 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) 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: 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 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: Landmark Numbers Identification

c) C
[Solution Description]
According to the landmark numbers listed in the syllabus, C represents 100 in Roman numerals.
Step 1: Refer to the table: C = 100.
Therefore, the correct option is C.

Key Concept: Roman numerals representation, subtraction principle

a) MCMXLIX
[Solution Description]
To represent 1949 in Roman numerals:
1. Break down 1949 into its largest landmark numbers: $1000 + (1000 - 100) + (50 - 10) + (10 - 1)$
2. Substitute the Roman symbols: $M$ for $1000$, $CM$ for $900$ ($1000 - 100$), $XL$ for $40$ ($50 - 10$), and $IX$ for $9$ ($10 - 1$).
3. Combine them: $M + CM + XL + IX = MCMXLIX$.

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 Numeral Addition

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

Key Concept: Roman numerals 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: 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: 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-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 of a Number System

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

Key Concept: Base-n System, Landmark Numbers

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

Key Concept: 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: 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: 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: 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: 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: 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: 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: Egyptian System Multiplication

b) 10
[Solution Description]
The first landmark number in the Egyptian system is 1, and the second landmark number is 10. The product of these two numbers is $1 \times 10 = 10$.

Key Concept: Base-5 Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 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 Conversion in Base-5 System

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

Key Concept: Base Systems, Multiplication

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

Key Concept: 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: Base-n representation, Arithmetic operations

d) $101_7$
[Solution Description]
First, convert both numbers from base-7 to base-10:
$43_7 = 4 \times 7^1 + 3 \times 7^0 = 28 + 3 = 31_{10}$
$25_7 = 2 \times 7^1 + 5 \times 7^0 = 14 + 5 = 19_{10}$
Add them in base-10: $31 + 19 = 50_{10}$
Now, convert $50_{10}$ back to base-7:
Find the largest power of 7 less than or equal to 50, which is $7^2 = 49$.
$50 \div 49 = 1$ with remainder $1$.
Next, $1 \div 7^0 = 1$ with remainder $0$.
So, $50_{10} = 1 \times 7^2 + 0 \times 7^1 + 1 \times 7^0 = 101_7$.

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: 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: Advantages of Base-n System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that base-5 allows efficient number representation using powers of 5, which is true because in base-n systems, numbers are represented using combinations of landmark numbers (powers of the base). The reason correctly explains that these representations use unique combinations of $5^0, 5^1, 5^2,$ etc., demonstrating why the base-5 system is efficient. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Shortcomings of Egyptian System

c) Large numbers needed too many symbols due to lack of positional notation
[Solution Description]
The Egyptian number system used repetitive symbols for landmark numbers (powers of 10). As numbers grew larger, they required an impractical number of symbols because the system lacked positional notation and higher powers were not compactly represented.

Key Concept: Shortcomings of Egyptian System

c) It uses positional notation to represent values.
[Solution Description]
The Egyptian number system had several limitations, including the need for new symbols for larger numbers, lack of positional value, and absence of a zero symbol. However, it did not have the limitation of requiring complex calculations for addition and subtraction, as these operations were relatively straightforward in the system.

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: Arithmetic in Egyptian System

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

Key Concept: Shortcomings of the Egyptian System, Base Representation

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

Key Concept: 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: Base-60 System Conversion

a) $2 \times 60 + 5$
[Solution Description]
To convert 125 to base-60, divide it by 60:
125 ÷ 60 = 2 with a remainder of 5.
Therefore, 125 = 2 × 60 + 5.
In the Mesopotamian base-60 system, this is represented as two 60s and five 1s.

Key Concept: Place Value Representation

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

Key Concept: Place Value Representation

b) 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: Zero in Number Systems

d) Zero was treated as a number with defined arithmetic properties.
[Solution Description]
The Hindu number system introduced zero as both a placeholder and an independent number. This allowed unambiguous representation and arithmetic operations on zero, revolutionizing mathematics. Aryabhata and Brahmagupta formalized its properties.

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: 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: Place Value Representation, Mesopotamian Number System

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the Mesopotamian number system is a fully developed place value system, which is false because it lacked a symbol for zero and had inconsistent spacing, leading to confusion. The reason claims that the Mesopotamian system used consistent spacing between symbols, which is also false as historical evidence shows spacing was not standardized. Therefore, both Assertion and Reason are false. However, since the question requires choosing from the given options, the correct answer is that Assertion is false, but Reason is true (though in reality, both are false). For the purpose of this MCQ, we select the closest matching option.

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: 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 addition

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

Key Concept: 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 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: 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: 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) 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: 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, 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: 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, 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: $5000 + 0 + 80 + 3 = 5083$.

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."

Key Concept: Place Value Representation

a) $4 \times 100 + 2 \times 10 + 6 \times 1$
[Solution Description]
The number 426 can be expanded using place values as follows:
$4$ is in the hundreds place: $4 \times 10^2 = 400$
$2$ is in the tens place: $2 \times 10 = 20$
$6$ is in the units place: $6 \times 1 = 6$
Adding them together: $400 + 20 + 6 = 426$.
So, the correct expanded form is $4 \times 10^2 + 2 \times 10 + 6 \times 1$.