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: 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: Transmission of Indian Numerals

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

Key Concept: 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: 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: Ancient Indian Numerals

c) A dot
[Solution Description]
In the Bakhshali manuscript, which dates back to the 3rd century CE, the digit zero was represented as a dot.

Key Concept: Origin of Numbers

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

Key Concept: Origin of Numbers, 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: 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) 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: Spread of numerals, historical timelines

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

Key Concept: 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: Counting Mechanisms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that the Ishango bone is one of the oldest mathematical artifacts with tally marks, which is true as it dates back 20,000 to 35,000 years and was used for counting or calendrical purposes. The Reason (R) explains that tally marks are a simple method for counting by creating a one-to-one correspondence between objects and marks, which is also true. Additionally, Reason (R) correctly explains why tally marks were used in ancient artifacts like the Ishango bone—because they provide an easy way to count objects without complex symbols or systems.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Number Representation

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

Key Concept: One-to-one mapping 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: 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: 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: Mayan Number System

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion is correct because the Mayan number system indeed used a dot for 1 and a bar for 5 for representing numbers from 1 to 19. However, the Reason is false because their system was not a strict base-20 system; it had modifications where landmark numbers included 360 and 7200 instead of powers of 20 beyond $20^1$.

Key Concept: Use of Body Parts

c) People in Papua New Guinea
[Solution Description]
The syllabus mentions that people in Papua New Guinea used their body parts as the standard sequence/number system for counting.

Key Concept: Mayan Number System, 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: Scalability of Body Parts Counting System

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

Key Concept: Body-Part Number Representation

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

Key Concept: Body-Part Number Representation

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: Extension of Body Parts Counting System

c) 27
[Solution Description]
Since the system uses both sides symmetrically:
Step 1: Total unique parts per side = 15.
Step 2: Right knee is 12th part, so left knee will be $12 + 15 = 27$.
Step 3: Hence, left knee corresponds to the 27th count.

Key Concept: 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: Stick representation

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

Key Concept: 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 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 on Bones

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

Key Concept: Counting in groups, Human perception limit

c) 33
[Solution Description]
Each new symbol represents 7 marks, so 4 symbols represent $4 \times 7 = 28$ marks. Adding the 5 separate marks gives the total count: $28 + 5 = 33$.

Key Concept: Tally Marks and Early Number Systems

a) Representing lunar phases
[Solution Description]
The Ishango bone, discovered in the Democratic Republic of Congo, features notches arranged in columns. These markings are thought to represent numbers and may indicate a calendrical system or tally counting method.

Key Concept: Tally Marks 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: 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: Number Names by Counting in Twos

c) They all represent numbers by counting in twos.
[Solution Description]
All three groups use a counting system based on grouping numbers in twos. Their number names are formed by combining words for smaller numbers (especially 2 and 1) to represent larger numbers.

Key Concept: Number names by counting in twos, 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: Bushmen Number System

c) t'oa-t'oa
[Solution Description]
The Bushmen system also uses counting in twos. The number 4 is represented as two occurrences of the word for 2, which is t'oa-t'oa.

Key Concept: 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

d) $\text{azag\^{a} azag\^{a}}$​
[Solution Description]

In the Bakairi number system, $\text{tokalȃ}$ represents 1 and $\text{azagȃ}$ represents 2. Larger numbers are formed by combining these basic number words. Therefore, the number 4 is represented as:

2 + 2 = 4

So, 4 is written as:

$\text{azagȃ azagȃ}$​

Some sources also write it as $\text{azagȃ azagȃ}$, using a simplified spelling.

Key Concept: Roman numerals representation, subtraction principle

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

Key Concept: Roman numerals and landmark numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert 999 to Roman numerals:
900 is represented as CM (1000 - 100),
90 as XC (100 - 10),
and 9 as IX (10 - 1).
Thus, 999 = CMXCIX, confirming the Assertion.
The Reason correctly explains that the subtractive principle (like IV for 4 or IX for 9) was used but not always consistently, as mentioned in the syllabus regarding the representation of 40 as XL or XXXX.

Key Concept: Landmark Numbers 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 Numeral Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To represent 45 in Roman numerals:
1. Start with the largest landmark number less than or equal to 45, which is L (50).
2. Since 40 is 10 less than 50, we write XL (where X is 10 and L is 50).
3. Add V (5) to make it 45, resulting in XLV.
The Reason correctly explains the subtraction rule used in the Assertion (X before L means 50 - 10 = 40).

Key Concept: Roman Numerals - Landmark Numbers

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

Key Concept: Roman Numerals - 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: 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: 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-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: Landmark Numbers

c) 27
[Solution Description]
The landmark numbers in a base-3 system are powers of 3: $3^0 = 1$, $3^1 = 3$, $3^2 = 9$, $3^3 = 27$. The fourth landmark number is $3^3 = 27$.

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 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: Multiplication in Egyptian Numerals

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

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian number system uses base 10 for grouping numbers, which is true as the system groups numbers in powers of 10 (e.g., 1, 10, 100, etc.). The Reason explains that each new landmark number is formed by grouping 10 collections of the previous landmark number, which is also true and correctly justifies the Assertion. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Landmark Numbers, 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: Variations on the Egyptian System - Grouping

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

Key Concept: Egyptian Number System - Representation

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

Key Concept: Egyptian Number System and Base Concept

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Egyptian number system uses a base of 10 because it groups numbers in powers of 10. This is true as the system's landmark numbers are defined as powers of 10 (1, 10, 100, etc.). The Reason explains that each landmark number is 10 times the previous one, which is also true and directly supports why the system is base-10. Thus, the Reason correctly explains the Assertion.

Key Concept: Base-5 Representation

d) $111_5$
[Solution Description]
To convert $31_{10}$ to base-5, divide 31 by powers of 5:
$31 \div 25 = 1 \text{ with remainder } 6 \\ \\ 6 \div 5 = 1 \text{ with remainder } 1 \\ \\ 1 \div 1 = 1 \text{ with remainder } 0$
Combining the coefficients gives $111_5$.

Key Concept: 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 Systems, Multiplication

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

Key Concept: Base-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: Multiplication in Base-7

a) $130_7$
[Solution Description]
First, convert $20_7$ and $5_7$ to base-10:
$20_7 = 2 \times 7^1 + 0 \times 7^0 = 14_{10} \\ \\ 5_7 = 5 \times 7^0 = 5_{10}$
Multiply them in base-10:
$14 \times 5 = 70_{10}$
Convert back to base-7 by dividing by powers of 7:
$70 \div 49 = 1 \text{ with remainder } 21 \\ \\ 21 \div 7 = 3 \text{ with remainder } 0 \\ \\ 0 \div 1 = 0 \text{ with remainder } 0$
Combining the coefficients gives $130_7$.

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: 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: Simplified Arithmetic

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

Key Concept: Efficient Representation

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

Key Concept: 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: 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 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: 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: 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: 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, 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

b) 497
[Solution Description]
Calculate step-by-step:
$4 \times 10^2 = 400 \\ \\ 9 \times 10 = 90 \\ \\ 7 \times 1 = 7 \\ \\ \text{Sum} = 400 + 90 + 7 = 497$

Key Concept: Place Value Representation

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

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

$3600 = 60^2$

the given number can be written as:

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

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

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

Therefore, the correct answer is Option A.

Key Concept: Place Value Representation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the Hindu number system is superior due to its base-10 place value system with zero, which is true. The reason explains why this is the case by highlighting zero's role in preventing ambiguity and enabling arithmetic operations, which correctly explains the assertion. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: 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: 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: 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: 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: Mesopotamian Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion (A) is true as the Mesopotamian number system indeed used a base-60 (sexagesimal) system. The Reason (R) is also true because base-60 was chosen due to its divisibility properties, which simplified calculations for trade and astronomical observations. Moreover, the Reason correctly explains why the Mesopotamians adopted the base-60 system. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: 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: Conversion from Hindu-Arabic to Mayan

a) Top level: 1 dot; Middle level: 2 dots; Bottom level: 2 dots
[Solution Description]

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: 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 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: Chinese rod numerals, place value representation

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

Key Concept: Chinese vs other systems, comparative analysis

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

Key Concept: 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

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

Key Concept: Place Value System, Hindu Number System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Hindu number system's use of 0 as both a digit and a number eliminated ambiguity and enabled efficient computation. This is true because the positional value system relies on 0 to represent absence in a place without altering the value of other digits. The Reason states that Brahmagupta codified the arithmetic properties of zero, treating it as a number, which is also true and directly explains why the system could handle computations efficiently. The Reason provides the conceptual basis for the Assertion, making it the correct explanation.

Key Concept: Place Value Representation: The 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 numeral 405 can be expanded as $4 \times 10^2 + 0 \times 10 + 5 \times 1$, which is true for the Hindu number system, a base-10 place value system. The reason states that '0' serves as a placeholder to avoid ambiguity in number representation, which is also true. Moreover, the reason correctly explains why the assertion holds true, as the presence of '0' ensures the correct placement and value of each digit in the numeral.

Key Concept: Place Value, Multi-Concept

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

Key Concept: Landmark Numbers

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.