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: 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: 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 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: 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: Transmission of Numerals

c) Fibonacci
[Solution Description]
The Italian mathematician Fibonacci introduced the Indian numeral system to Europe in his book "Liber Abaci" (1202). Despite initial resistance due to the entrenched Roman numeral system, it eventually became widely adopted during the Renaissance.

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

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 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: 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: Mesopotamian numerals, Indian numeral system, spread of numbers

a) 45
[Solution Description]
Mesopotamians used a base-60 (sexagesimal) system. The number 45 in their system would be represented as $45_{60}$. To convert to decimal, we calculate:
$4 \times 60^1 + 5 \times 60^0 = 4 \times 60 + 5 \times 1 = 240 + 5 = 245.$
However, the question asks for representing 45 in their symbols and converting back to decimal, so the correct answer is 45 itself.

Key Concept: 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: 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, 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: Tally Marks

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

Key Concept: 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: 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

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: Hindu Number System, Roman Number System

c) M
[Solution Description]
To solve this problem, we need to multiply the Roman numeral CXXV by 8, both in their respective forms.
Step 1: Convert the Roman numeral CXXV to its Hindu-Arabic equivalent.
- C = 100, X = 10, X = 10, V = 5.
- So, CXXV = 100 + 10 + 10 + 5 = 125.
Step 2: Multiply the Hindu-Arabic equivalent by 8.
- $125 \times 8 = 1000$.
Step 3: Convert the product back to Roman numerals.
- 1000 in Roman numerals is M.
Therefore, the correct answer is M.

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: 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: 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: 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: 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

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

Key Concept: Body-Part Number Representation

d) Index finger of the left hand
[Solution Description]
According to the given sequence, the number $4$ is represented by the index finger of the left hand.

Key Concept: Assersion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion claims the Ishango bone was used for counting hunted animals, which is speculative and lacks direct archaeological evidence. However, the Reason correctly states that the grouping of notches aligns with lunar cycles, which is supported by researchers' interpretations of the Ishango bone as a possible lunar calendar. Therefore, while the Reason is true and well-supported, the Assertion is unverified and likely false.

Key Concept: Tally Marks on Bones

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Ishango bone has notches arranged in columns, which is true as mentioned in the syllabus. The Reason suggests that these notches could represent a calendrical system, which is also mentioned in the syllabus as a possible interpretation. Furthermore, the Reason correctly explains why the notches are arranged in columns, indicating a potential calendrical system.

Key Concept: 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: 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: 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: 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 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 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

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 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: 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 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: Roman Numerals - Landmark Numbers

d) M
[Solution Description]
Convert each option to Hindu-Arabic numerals:
a) D = 500
b) CM = 900 (1000 - 100)
c) DC = 600 (500 + 100)
d) M = 1000.
Therefore, M (1000) is the largest.

Key Concept: Roman numerals representation, complex cases

a) MMMCMXCIX
[Solution Description]
To represent 3999 in Roman numerals:
1. Break down 3999 into its largest possible components: $1000 + 1000 + 1000 + (1000 - 100) + (100 - 10) + (10 - 1)$.
2. Substitute the Roman symbols: $M$ for $1000$, $CM$ for $900$ ($1000 - 100$), $XC$ for $90$ ($100 - 10$), and $IX$ for $9$ ($10 - 1$).
3. Combine them: $M + M + M + CM + XC + IX = MMMCMXCIX$.

Key Concept: Roman 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: 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: Advantage of Base-n Systems

b) Because multiplication in base-n involves only addition of exponents
[Solution Description]
In a base-n system, the landmark numbers are powers of n, so multiplying two landmark numbers results in another landmark number (e.g., $10^a \times 10^b = 10^{a+b}$). This simplifies multiplication because it reduces to adding exponents. In contrast, the Roman numeral system does not have landmark numbers as powers of a fixed base, making multiplication complex and less systematic.

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: 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: 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: 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-5 Number System

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

Key Concept: Landmark Numbers 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: 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: 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: 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: 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: Shortcomings of the Egyptian System

c) It required an unending sequence of symbols for higher powers of $10$
[Solution Description]
The Egyptian number system required an unending sequence of symbols for higher powers of $10$ as numbers grew larger, making it impractical for very large representations.

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 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 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-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 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: Efficient Representation

c) It allows unique representation of numbers using a fixed set of digits.
[Solution Description] The base-n system allows every number to be uniquely represented using digits from $0$ to $n-1$, eliminating the need for multiple symbols for different landmark numbers.

Key Concept: Simplified Arithmetic

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

Key Concept: 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: 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: 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: 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

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 is inefficient for large numbers due to the lack of positional value. This is true as the syllabus mentions "Inefficiency in Representation" and "Lack of Positional Value" as shortcomings. The Reason explains that the system uses repeated symbols, which is also true and directly supports why the system is inefficient, as stated in the syllabus under "Inefficiency in Representation."

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: 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: 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: 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: Place Value Representation

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

Key Concept: Place Value Representation, Hindu Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Hindu number system's use of 0 as a placeholder and a number was important for developing algebraic structures like rings. This is true because the concept of zero allowed for more advanced mathematical operations and structures. The reason given is that Brahmagupta's work defined arithmetic operations with zero, which indeed provided the foundation for rings by ensuring closure under these operations. Since the reason correctly explains why the assertion is true, the correct choice is the option where both are true and the reason explains the assertion.

Key Concept: 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) $(2,5,30)_{60}$
[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: 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: Ambiguity in representation, positional interpretation

b) Because the system lacked a clear way to denote empty positions
[Solution Description]
The ambiguity arises because:
1. In the Mesopotamian system, positions represent powers of 60 ($60^0$, $60^1$, $60^2$, etc.).
2. A single wedge $\mathrm{\text{ }}$ (representing 1) in the second position means $1 \times 60^1 = 60$.
3. If spacing is unclear, the same wedge could be interpreted as being in the third position: $1 \times 60^2 = 3600$.
4. The lack of a zero symbol or clear spacing made it impossible to distinguish between positions.

Key Concept: 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: 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 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: 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 Number Representation

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

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

To represent 77:

77 = 3 × 20 + 17

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

17 = 5 + 5 + 5 + 2

So, the Mayan numeral representation of 77 is:

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

Therefore, the correct answer is Option A.

Key Concept: Place Value Concepts

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

Key Concept: 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 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 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: Chinese vs other systems, comparative analysis

d) Fully developed place value system
[Solution Description]
Key advancement shared by both:
1) Used distinct symbols for digits 1-9 (unlike cumulative systems)
2) Had clear positional notation where each position represented a power of 10
3) Both could represent zero (though Chinese used blank spaces initially)
4) This combination made calculations more efficient than non-positional or partial-positional systems

Key Concept: Place Value Representation

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, Multi-Concept

b) $100A + 10B + C$
[Solution Description]
In the Hindu number system, a three-digit number $\overline{ABC}$ can be expanded using place values as follows:
$A$ is in the hundreds place: $A \times 10^2 = 100A$
$B$ is in the tens place: $B \times 10^1 = 10B$
$C$ is in the ones place: $C \times 10^0 = C$
Therefore, the expanded form is $100A + 10B + C$.

Key Concept: Landmark Numbers

c) $10^2$
[Solution Description]
In the Hindu number system, landmark numbers are powers of 10. The hundreds place corresponds to $10^2$, which equals 100.

Key Concept: Place Value System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Hindu number system indeed uses 10 symbols (0-9), making the Assertion true. The Reason is also true because the inclusion of 0 eliminates ambiguity in number representation by acting as a placeholder. Moreover, the Reason correctly explains why 0 is essential in the Hindu number system's place value structure. Thus, both statements are true, and the Reason justifies the Assertion.

Key Concept: Landmark Numbers

b) Hundreds
[Solution Description]
In the Hindu number system, landmark numbers are powers of 10. Here, $10^2$ represents $10$ multiplied by itself twice, which equals $100$. This is known as the "hundreds" place.

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