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: 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: 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 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: Transmission of Indian Numerals

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

Key Concept: Historical Context of Numbers

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

Key Concept: Origin of Numbers

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: 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: Spread of numerals, historical timelines

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

Key Concept: 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: 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

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: Number Representation

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

Key Concept: One-to-One Mapping

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

Key Concept: One-to-One Mapping

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: 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: 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, 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: 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: 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: 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: Stick representation

c) Remove one full bundle and two single sticks
[Solution Description]
In this system, 12 would be represented as 2 bundles of 5 sticks plus 2 single sticks.
7 would be 1 bundle of 5 sticks plus 2 single sticks.
Subtraction steps:
Step 1: Start with 2 bundles + 2 singles
Step 2: Remove 1 bundle first → leaves 1 bundle + 2 singles
Step 3: Now remove remaining 2 singles → leaves 1 bundle (which equals 5)

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 Counting System, Limitations

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

Key Concept: Body Parts as Number Representation

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

Key Concept: 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: 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 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: Lebombo bone, Tally marks

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

Key Concept: Tally Marks 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: 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, 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

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) 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: 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 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: 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: 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 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 - 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: 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: 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 of a Number System

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

Key Concept: Representation in Base-5

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

Key Concept: Base Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In a base-$n$ number system, any non-negative integer can be uniquely represented as a sum of distinct powers of $n$ with coefficients ranging from 0 to $(n-1)$. For example, in base-5, the number 143 is expressed as $125 + 3 \times 5 + 3 \times 1$, i.e., $1 \times 5^3 + 0 \times 5^2 + 3 \times 5^1 + 3 \times 5^0$. This unique representation is possible because the base-$n$ system ensures that each number has a distinct combination of coefficients for the powers of $n$. Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains Assertion.

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-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: 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: Multiplication in Egyptian Numerals

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

Key Concept: Landmark Numbers, Representation of Numbers

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: 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: Variations on the Egyptian System - Grouping

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

Key Concept: 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 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: 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: Landmark Numbers in Base-4

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

Key Concept: Base-5 Landmark Numbers

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

Key Concept: 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: 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: 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: Scalability

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

Key Concept: 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: 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: Landmark number multiplication in Egyptian system

a) It becomes the next higher landmark number.
[Solution Description]
In the Egyptian numeral system, landmark numbers are powers of 10. Multiplying a landmark number (e.g., $10^n$) by 10 results in the next higher landmark number ($10^{n+1}$) because $10^n \times 10 = 10^{n+1}$.

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: Symbol Representation

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

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, Zero in Number Systems

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

Key Concept: Place Value Representation

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

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

$3600 = 60^2$

the given number can be written as:

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

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

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

Therefore, the correct answer is Option A.

Key Concept: 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

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

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: 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: Ambiguity in Representation

b) Because the same symbol could represent different values depending on its place
[Solution Description]
The Mesopotamian number system was based on 60, called the sexagesimal system. In this system, the same symbol used for 1 could also represent 60 or 3600, depending on its position. Since early Mesopotamian numerals did not clearly use a zero or place-holder, it could sometimes be confusing to know whether a symbol meant 1, 60, or 3600. Therefore, the correct answer is B.

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: 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: Place Value Concepts

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

Key Concept: Mayan Symbol Interpretation

c) 3 bars and 4 dots
[Solution Description]

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

19 = 5 + 5 + 5 + 4

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

3 bars and 4 dots​

Therefore, the correct answer is Option A.

Key Concept: Mayan Number Representation

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

Key Concept: Mayan addition

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

Key Concept: Conversion from Mayan to Hindu-Arabic

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

Key Concept: 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 representation

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the Chinese rod numeral system used blank spaces to represent zero in computations. This is true as the system employed blank spaces as placeholders for missing place values, similar to modern zero. However, the reason claims that blank spaces could cause ambiguity in interpreting numbers with skipped place values. This is false because the Chinese system was designed with uniform symbol sizes and clear spacing rules, making it unambiguous to identify skipped places. Therefore, the assertion is true while the reason is false.

Key Concept: Place Value Identification

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

Key Concept: Place Value Representation

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

Key Concept: 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) $4 \times 100 + 2 \times 10 + 6 \times 1$
[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 $4 \times 10^2 + 2 \times 10 + 6 \times 1$.

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: 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."

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.