Key Concept: Human Need for Counting, Origin of Modern Numbers

d) $āyuta$
[Solution Description]
The Yajurveda Samhita lists number names based on powers of 10. According to the text, $āyuta$ represents ten thousand ($10^4$).

Key Concept: 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: 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: 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: 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: Origin of Numbers, Hindu-Arabic Numerals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the digit 0 being introduced as a number revolutionized mathematics. This is true because, before its introduction, computations were limited. The Reason explains how zero was treated as a number with specific arithmetic properties, which is also true. Moreover, the Reason correctly explains the Assertion since these properties of zero enabled more advanced mathematical operations and theories to develop, such as algebra and analysis. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Origin of Numbers

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: 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: Origin of Zero

c) Aryabhata
[Solution Description]
Aryabhata, in his work Āryabhaṭīya (499 CE), was the first to use 0 as a number and perform arithmetic operations with it, such as $0 + a = a$ and $0 \times a = 0$.

Key Concept: 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: 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: 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: Counting Methods

b) It cannot count collections with more than 26 objects.
[Solution Description]
Using the English alphabet limits counting because there are only 26 distinct letters available. Therefore, collections larger than 26 objects cannot be counted using this method without repetition or modification of the sequence, making it impractical beyond small quantities.

Key Concept: One-to-one mapping, Alphabet-based counting

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

Key Concept: One-to-One Mapping, Roman 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: One-to-one mapping, Counting with sticks

a) 18
[Solution Description]
In Method 1, each stick represents one goat. If the shepherd has 23 sticks but 5 are broken, those broken sticks do not correspond to any goats.
Calculation:
Total sticks = 23
Broken sticks = 5
Effective sticks (representing goats) = $23 - 5 = 18$
Therefore, the shepherd has 18 goats.

Key Concept: 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: 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, Roman Number System

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

Key Concept: Mayan Number System, Hindu Number System

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

Key Concept: Chinese 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: 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: Name representation

b) ta ta
[Solution Description]
Given the additive nature of the system:
Step 1: We have representations for 1 ('la'), 2 ('ta') and 3 ('na')
Step 2: For 4, it would combine existing symbols through addition
Possible logical combinations:
'ta ta' (2+2=4)
'la na' (1+3=4)
'na la' (3+1=4)
The simplest form would be combining the largest available numbers

Key Concept: Body part counting

c) 2 complete counts
[Solution Description]
First, we need to understand that one complete count means using all 10 fingers (both hands). To find out how many complete counts are needed for 15:
Step 1: Divide 15 by 10 (fingers per count) = 1.5 counts
Step 2: Since they can't do half counts, they would need to round up to 2 complete counts (using fingers twice)

Key Concept: Body Parts as Number Representation, 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 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-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 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: 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 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: 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: 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: Number names by counting in twos

c) ahage ahage ahage
[Solution Description]
From the syllabus:
- Bushmen represent numbers as follows:
1: tokale
2: ahage
3: ahage tokale (or ahawao)
4: ahage ahage tokale
5: ahage ahage ahage
So, 5 is directly given as ahage ahage ahage.

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

a) ras
[Solution Description]
According to the Gumulgal system:
- Numbers greater than 6 are called "ras."
- The exact representation for numbers beyond 6 is not detailed, but since 9 = 6 + 3, and 6 is represented as ukasar-ukasar-ukasar, we can infer that 9 might be represented as "ras" with some additional notation or context.
However, based on the given information, any number >6 is simply called "ras."
Therefore, 9 would also be called "ras."

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 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 representation, subtraction principle

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

Key Concept: Roman 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 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 - 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: Representation in Base-5

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

Key Concept: Base-5 Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 143 can be represented as $125 + 5 + 5 + 5 + 1 + 1 + 1$ in a base-5 system. This is correct because $125 = 5^3$, $5 = 5^1$, and $1 = 5^0$, which are all valid landmark numbers in the base-5 system. The reason states that the landmark numbers in a base-5 system are powers of 5, which is also true. Moreover, the reason correctly explains the assertion since the representation of 143 uses powers of 5, which are the landmark numbers in this system.

Key Concept: Base of a Number System

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

Key Concept: 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 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: Base conversion, Grouping principle

b) 514 in base-7
[Solution Description]
To represent 256 in base-7:
Step 1: Find the largest power of 7 less than or equal to 256.
$7^3 = 343$ is too large. So, $7^2 = 49$ comes first.
Step 2: Divide 256 by 49. We get 5 with remainder $256 - (49 \times 5) = 11$.
Step 3: Divide the remainder 11 by $7^1 = 7$. We get 1 with remainder $11 - 7 = 4$.
Step 4: The remaining 4 is represented by $7^0 = 1$ units.
Thus, $256 = 5 \times 7^2 + 1 \times 7^1 + 4 \times 7^0$, which is written as 514 in base-7.

Key Concept: Egyptian Number System - Representation

a) A single symbol for $10$
[Solution Description]
The Egyptian number system uses landmark numbers based on powers of $10$. The first landmark number is $1$, and the second landmark number is obtained by grouping $10$ collections of $1$, which gives $10$.

Key Concept: Multiplication in Egyptian Numerals

c) 1000
[Solution Description]
In the Egyptian system, each multiplication of a landmark number by 10 results in the next landmark number:
- $10 × 1 = 10$ (second landmark number).
- $10 × 10 = 100$ (third landmark number).
- Therefore, $10 × 100 = 1000$ (fourth landmark number).

Key Concept: 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: 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: 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: Base-5 number system, Landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's convert $43_5$ to base-10: The rightmost digit (3) is multiplied by $5^0 = 1$, and the leftmost digit (4) is multiplied by $5^1 = 5$. Thus, $43_5 = 4 \times 5 + 3 \times 1 = 20 + 3 = 23_{10}$. The reason correctly explains that the positional value is determined by powers of 5, which aligns with the conversion process. Therefore, both statements are true, and the reason explains the 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: Egyptian System Multiplication

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

Key Concept: Base Conversion in Base-5 System

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

Key Concept: Base Systems, Multiplication

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

Key Concept: 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 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: 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 conversion, Landmark numbers

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

Key Concept: 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: Shortcomings of Egyptian System

b) It involved complex calculations for simple multiplications.
[Solution Description]
In the Egyptian system, multiplication was more complex compared to addition and subtraction. It required understanding that each landmark number is a power of 10 and applying this logic manually for each operation. For example, multiplying $10 \times 100$ required recognizing that the product is $1000$.

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: 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: 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: 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

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, 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: Zero in Number Systems

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

Key Concept: 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: Positional representation, conversion from decimal

c) Three symbols for 60 followed by three symbols for 1
[Solution Description]
To convert 183 to the Mesopotamian system:
1. Divide 183 by 60: $183 \div 60 = 3$ with remainder $3$.
2. The number consists of $3 \times 60^1 + 3 \times 60^0$ = 3 groups of 60 and 3 units.
3. The Mesopotamian representation would show 3 symbols for 60 (each being a symbol for 10 repeated three times) followed by 3 symbols for 1 (since they used base-60).

Key Concept: Conversion from Mesopotamian System

c) 35
[Solution Description]

In the Mesopotamian number system, one wedge symbol represented 10, and one vertical stroke symbol represented 1. Therefore, three wedge symbols represent:

$3×10=30$

and five vertical stroke symbols represent:

$5 × 1 = 5$

So, the total value is:

$30 + 5 = 35$

Hence, the representation stands for 35.

Key Concept: 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: 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 Representation, Mesopotamian Number System

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

Key Concept: 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: Place Value in Mayan System

c) One dot at the top and a seashell at the bottom ($\cdot$ over seashell)
[Solution Description]
In the Mayan place value system:
- The lower level represents units (1s), while the level above represents multiples of 20.
- For 20, we write one dot at the second level (representing $1 \times 20$) and a zero placeholder (seashell symbol) at the units level.
- The correct representation is a dot over a seashell symbol.

Key Concept: 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: 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: 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 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: 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: 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

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: 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: 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: 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, 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: 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.