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: 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: 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: 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: Mesopotamian numerals, Hindu-Arabic numeral system

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's verify the Assertion: The Mesopotamian numeral system indeed used a base-60 (sexagesimal) system, while the Hindu-Arabic numeral system uses base-10. This part of the Assertion is true.
Now, examine the Reason: The base of numeral systems often reflects practical human needs. For example, base-10 is common due to counting on 10 fingers, while base-60 (used in Mesopotamia) may have been chosen for its divisibility (useful in astronomy and timekeeping). Thus, the Reason is also true.
Does the Reason correctly explain the Assertion? Yes, because the difference in bases between the two systems can be attributed to their different practical and cultural contexts, as stated in the Reason.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Historical 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 states that the Indian number system introduced zero as a number with arithmetic properties, which is correct as per historical evidence. The Reason explains that Aryabhata and Brahmagupta formalized its use, leading to advancements in mathematics, which is also historically accurate. Moreover, the Reason correctly justifies why the Assertion is true, as it elaborates on the role of Indian mathematicians in developing the concept of zero.

Key Concept: Hindu Number System

d) Al-Khwārizmī
[Solution Description]
The Hindu number system was popularized in the Arab world by Al-Khwārizmī through his book "On the Calculation with Hindu Numerals" around 825 CE.

Key Concept: 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: 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: 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: 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: 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

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, Alphabet-based counting

c) 11
[Solution Description]
In Method 2, each letter of the alphabet represents a number starting from "a" as 1, "b" as 2, and so on.
The position of "k" in the English alphabet is the 11th letter.
Calculation:
- a = 1
- b = 2
- ...
- k = 11
Therefore, the farmer has 11 chickens.

Key Concept: One-to-one mapping in counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that sticks were used for counting cows in the Stone Age through one-to-one mapping. This is true because historical evidence supports the use of tally marks or objects like sticks for counting. The reason explains that this method prevents duplication by associating each cow with a unique stick, which is also correct and directly justifies the 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: 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: 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 Rod Numerals

b) Base-10
[Solution Description]
The Chinese rod numeral system was developed for computation and uses distinct symbols for digits 1 to 9. It follows a decimal (base-10) system, where each position represents a power of 10. This system allowed efficient arithmetic operations such as addition, subtraction, multiplication, and division.
Therefore, the correct answer is base-10.

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: 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: Body Parts as Number Representation, Limitations

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

Key Concept: Body-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 Parts as Number Representation

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

Key Concept: Body 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 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: Historical Number Systems

c) It is among the earliest known tally systems
[Solution Description]
The Lebombo bone, found in South Africa, has 29 notches and is one of the oldest mathematical artifacts (around 44,000 years old). Its markings suggest it was used as a tally stick or lunar calendar.

Key Concept: Tally Marks and Early Counting Methods

b) Counting objects by making individual marks
[Solution Description]
Tally marks were one of the oldest methods used to represent numbers by making notches on surfaces like bones or cave walls. Each mark represented a single object being counted, and the collection of marks indicated the total count. This method was simple and did not require physical objects like sticks or pebbles.

Key Concept: Tally Marks 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 and Group Counting

b) To make counting easier by reducing the number of marks
[Solution Description]
Humans have a perceptual limit for recognizing small quantities quickly. Groups of 5 or more objects are difficult to count at a glance, so replacing every 5 tally marks with a new symbol made counting more efficient.

Key Concept: Tally marks, Ishango bone, Lebombo bone

c) 35
[Solution Description]
First, calculate the count from the first column with 3 groups of 5 marks: $3 \times 5 = 15$. Then, calculate the count from the second column with 2 groups of 10 marks: $2 \times 10 = 20$. Add these two results to find the total count: $15 + 20 = 35$.

Key Concept: Assersion-Reason

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

Key Concept: Roman Number System

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

Key Concept: Number Names by Counting in Twos

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Gumulgal number system uses counting in twos to form number names. This is true as per the syllabus, where it is mentioned that numbers are counted in 2s (e.g., $3 = 2 + 1$, $4 = 2 + 2$, etc.).
The Reason provides an example: the number name for 5 is ukasar-ukasar-urapon, which translates to 2 + 2 + 1. This correctly demonstrates the counting-in-twos pattern described in the Assertion. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: 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: 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: 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: 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 - Conversion

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

Key Concept: Roman 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 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: Representation of Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify Assertion (A), let's break down the Roman numeral $MDCLXVI$:
- M = 1000
- D = 500
- C = 100
- L = 50
- X = 10
- V = 5
- I = 1
Adding them up: $1000 + 500 + 100 + 50 + 10 + 5 + 1 = 1666$. So, Assertion is true.
Reason (R) states that symbols are added from left to right in descending order. This is correct because in $MDCLXVI$, the symbols are ordered as M (1000), D (500), C (100), L (50), X (10), V (5), I (1), which follows descending order. Moreover, Reason correctly explains why the assertion is true, as this ordering rule ensures accurate addition. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Roman numerals 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: 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: 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: 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: 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: 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: Variations on the Egyptian System

b) 25
[Solution Description]
In a base-5 system, landmark numbers are generated as follows:
- First landmark number: 1 ($5^0$).
- Second landmark number: 5 ($5^1$).
- Third landmark number: $5 × 5 = 25$ ($5^2$).

Key Concept: Landmark Numbers, Representation of Numbers

a) Five 100s, six 10s, seven 1s
[Solution Description]
To represent 567 in the Egyptian system:
- Start with the largest landmark number less than 567, which is 100.
- Divide 567 by 100: $567 ÷ 100 = 5$ with a remainder of 67. So, we use five 100s.
- Next, for the remainder 67, use the next largest landmark number, 10.
- Divide 67 by 10: $67 ÷ 10 = 6$ with a remainder of 7. So, we use six 10s.
- The remaining 7 is represented by seven 1s.
- Total representation: five 100s + six 10s + seven 1s.

Key Concept: Landmark Numbers, Representation of Numbers

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

Key Concept: 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: 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: 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: 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-5 System, Conversion to Base-10

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

Key Concept: Base System Representation

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

Key Concept: Landmark Numbers in Base-4

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

Key Concept: Base-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: Efficient Representation, Simplified Arithmetic

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, we need to verify the assertion (A). Converting 143 to base-5:
Step 1: Find the largest power of 5 less than or equal to 143, which is $5^3 = 125$.
Step 2: Divide 143 by 125 to get quotient 1 and remainder 18.
Step 3: Next, divide 18 by $5^2 = 25$, but since 25 > 18, the coefficient is 0.
Step 4: Now divide 18 by $5^1 = 5$ to get quotient 3 and remainder 3.
Step 5: Finally, divide the remainder 3 by $5^0 = 1$ to get coefficient 3.
Thus, $143_{10} = 1 \times 5^3 + 0 \times 5^2 + 3 \times 5^1 + 3 \times 5^0 = 1033_5$. This representation is indeed more efficient than non-positional systems.
Now, check the reason (R): The base-n system simplifies arithmetic by using powers of the base for grouping, which is true. However, this does not directly explain why $143_{10}$ can be represented as $1033_5$, as stated in the assertion. Hence, both are true, but the reason does not correctly explain the assertion.

Key Concept: 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: Efficient Representation in Base-n System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the base-5 number system can represent the number 143 using a combination of its landmark numbers. This is true because any number can be expressed as a sum of multiples of landmark numbers (powers of the base). The Reason explains that the landmark numbers in base-5 are powers of 5, which helps in compact representation by grouping large quantities. This correctly justifies the Assertion since the systematic grouping allows efficient representation. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: 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: 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: 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: 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: Symbol repetition in Egyptian numerals

a) No, because each symbol can only be repeated up to 9 times before moving to the next higher power of 10.
[Solution Description]
The Egyptian numeral system uses distinct symbols for powers of 10. Each symbol is used up to 9 times before moving to the next higher power of 10. Hence, no symbol can be repeated 10 or more times in a valid representation.

Key Concept: Place Value Representation

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

Key Concept: Place Value Representation, Zero in Number Systems

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

Key Concept: Place Value Representation, 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

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: 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: Mesopotamian Number System

c) 60
[Solution Description]
The Mesopotamian numeral system used a base-60 (sexagesimal) system, where each position represented powers of 60.

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: 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: 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: 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: 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: Mayan Number System, Placeholder Symbol for Zero

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Mayan number system indeed used a placeholder symbol for zero, represented as a seashell, which was crucial for their place value notation. This is true, supporting the Assertion. The Reason states that this placeholder helped distinguish between numbers like 20 and 200, which is correct because without the zero, these numbers would be indistinguishable (e.g., two bars could mean either 20 or 200). Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Place Value Representation - Mayan Number System

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

Key Concept: 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 Landmark Numbers

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

Key Concept: Chinese vs other systems, comparative analysis

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

Key Concept: Place Value Representation

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 rod numerals, multi-digit interpretation

b) 8 (Heng), blank, 3 (Zong), 6 (Heng)
[Solution Description]
Step 1: Break down the number into place values:
$8 \times 1000 = 8000 \Rightarrow 8 (Heng) \\ \\ 0 \times 100 = 0 \Rightarrow \text{Blank space} \\ \\ 3 \times 10 = 30 \Rightarrow 3 (Zong) \\ \\ 6 \times 1 = 6 \Rightarrow 6 (Heng)$
Step 2: Representation would be:
$8 (Heng) \\ \\ \text{Blank space} \\ \\ 3 (Zong) \\ \\ 6 (Heng)$

Key Concept: Chinese Number System - Place Value Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Chinese rod numeral system used a blank space as a placeholder for zero, which is true according to the syllabus. The reason explains that this method made identifying skipped place values easier compared to the Mesopotamian system, which is also true and correctly explains why the blank space was significant in the Chinese system. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Place Value Representation

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: 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: Zero in Hindu Number System

c) Codified arithmetic operations with 0
[Solution Description]
Brahmagupta codified the use of 0 as a number with arithmetic properties in his work Brāhmasphuṭasiddhānta (628 CE). He introduced operations like addition and multiplication involving zero, treating it as a number on par with others.

Key Concept: Place Value Representation

c) 5000
[Solution Description]
The number 5,672 can be expanded using place value as:
$5 \times 10^3 + 6 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$
Here, the digit '5' is in the thousands place ($10^3$), so its value is $5 \times 1000 = 5000$.

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