Key Concept: Sum of Consecutive Odd Numbers

b) 25
[Solution Description]
The sum of the first five consecutive odd numbers is calculated as follows: $1 + 3 + 5 + 7 + 9 = 25$. We know that $5^2 = 25$, so the result is the perfect square 25.

Key Concept: Sum of Successive Odd Numbers

c) 8
[Solution Description]
Perfect squares can be expressed as the sum of successive odd numbers starting from 1. Since $64 = 8^2$, we need to add the first 8 odd numbers: $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64$.

Key Concept: Units Digit of Perfect Squares

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that 144 is a perfect square because its units digit is 4. The reason states that a number is a perfect square only if its units digit is 0, 1, 4, 5, 6, or 9.
Checking the assertion: $12^2 = 144$, so 144 is indeed a perfect square.
Evaluating the reason: According to the syllabus, perfect squares can only have 0, 1, 4, 5, 6, or 9 as their units digit. However, having such a unit digit does not guarantee that the number is a perfect square — it is a necessary but not sufficient condition. For example, 24 has a units digit of 4 but is not a perfect square.
Thus, both the assertion and reason are true, but the reason does not fully explain why 144 is a perfect square (it just provides a necessary condition).

Key Concept: Perfect Squares, Sum of Odd Numbers

d) 50
[Solution Description]
To determine which number is not a perfect square, we check each option by verifying if it can be written as the sum of consecutive odd numbers starting from 1 or by confirming its non-membership in the list of perfect squares.
- $1 + 3 + 5 = 9$ (valid, so 9 is a perfect square).
- $1 + 3 + 5 + 7 = 16$ (valid, so 16 is a perfect square).
- The sum $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$ (valid, so 49 is a perfect square).
- The number 50 cannot be expressed as the sum of consecutive odd numbers starting from 1, nor is it listed among perfect squares.

Key Concept: Square Numbers, Properties of Perfect Squares, Units Digit Patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sum of the first 15 odd natural numbers is a perfect square. Let's verify this by calculating the sum of the first 15 odd numbers:
$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 = 225$
We know that $15^2 = 225$, which confirms that 225 is indeed a perfect square. Thus, the assertion is true.
The reason states that the sum of the first $n$ odd natural numbers is $n^2$. This is a well-known property of square numbers, where:
$\sum_{k=1}^{n} (2k - 1) = n^2$
For example, $1 + 3 = 4 = 2^2$, $1 + 3 + 5 = 9 = 3^2$, and so on. Therefore, the reason is also true and correctly explains the assertion.

Key Concept: Perfect Squares and Units Digit

c) Assertion is true, but Reason is false.
[Solution Description]
To verify the assertion, we check if 625 is a perfect square. Calculating $25 \times 25 = 625$, so 625 is indeed a perfect square. The reason states that all numbers ending with 5 are perfect squares, which is false. For example, 35 ends with 5 but is not a perfect square since $\sqrt{35}$ is not an integer. Therefore, the assertion is true, but the reason is false.

Key Concept: Prime Factorisation, Estimating Square Roots

c) 6
[Solution Description]
We first list the squares of numbers from 21 to 29:
- $21^2 = 441$ → Prime factors: $3^2 \times 7^2$ → Sum of exponents = $2 + 2 = 4$
- $22^2 = 484$ → Prime factors: $2^2 \times 11^2$ → Sum of exponents = $2 + 2 = 4$
- $23^2 = 529$ → Prime factors: $23^2$ → Sum of exponents = $2$
- $24^2 = 576$ → Prime factors: $2^6 \times 3^2$ → Sum of exponents = $6 + 2 = 8$
- $25^2 = 625$ → Prime factors: $5^4$ → Sum of exponents = $4$
- $26^2 = 676$ → Prime factors: $2^2 \times 13^2$ → Sum of exponents = $2 + 2 = 4$
- $27^2 = 729$ → Prime factors: $3^6$ → Sum of exponents = $6$
- $28^2 = 784$ → Prime factors: $2^4 \times 7^2$ → Sum of exponents = $4 + 2 = 6$
- $29^2 = 841$ → Prime factors: $29^2$ → Sum of exponents = $2$
From above, possible sums of exponents are: 2, 4, 6, 8. The correct option is 6.

Key Concept: Estimating Square Roots

b) 44
[Solution Description]
From the syllabus, we know that 1936 is a perfect square. To estimate its square root:
1. It lies between $1600$ ($40^2$) and $2500$ ($50^2$).
2. The last digit is 6, meaning the square root must end with 4 or 6 (i.e., 44 or 46).
3. Calculating $45^2 = 2025$, which is greater than 1936.
4. Now calculate $44^2 = 1936$.
Hence, the correct answer is 44.

Key Concept: Patterns in Perfect Squares, Odd Number Summation

b) 10
[Solution Description]
The smallest three-digit perfect square is $100$ ($10^2$).
According to the pattern:
$1 + 3 + 5 + \ldots + (2n-1) = n^2$.
Here, $n^2 = 100 \Rightarrow n = 10$.
So, $10$ odd numbers must be added:
$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100$.
The correct answer is 10.

Key Concept: Prime Factorisation and Perfect Squares

a) 144
[Solution Description]
A perfect square has its prime factors grouped into two identical sets. Let's check each option:
1. 144: $12^2 = 144$ and its prime factors are $2 \times 2 \times 2 \times 2 \times 3 \times 3$, which can be grouped as $(2 \times 2 \times 3) \times (2 \times 2 \times 3)$, so it is a perfect square.
2. 99: Its prime factors are $3 \times 3 \times 11$, which cannot be grouped equally.
3. 50: Its prime factors are $2 \times 5 \times 5$, which cannot be grouped equally.
4. 72: Its prime factors are $2 \times 2 \times 2 \times 3 \times 3$, which cannot be grouped equally.
Therefore, only 144 is a perfect square.

Key Concept: Prime Factorisation and Perfect Squares

c) Yes, because its prime factors can be paired identically.
[Solution Description]
The prime factorisation of 1156 is $2 \times 2 \times 17 \times 17$. Since these factors can be grouped into two identical groups ($2 \times 17$ and $2 \times 17$), 1156 is a perfect square. Specifically, $(2 \times 17)^2 = 34^2 = 1156$.

Key Concept: Sum of Consecutive Odd Numbers

b) 25, representing $5^2$
[Solution Description]
The sum of the first $n$ consecutive odd numbers starting from $1$ is equal to $n^2$. For $n = 5$, the sum is $1 + 3 + 5 + 7 + 9 = 25$, which represents the perfect square $5^2$.

Key Concept: Difference Between Consecutive Squares

b) 49
[Solution Description]
The difference between consecutive squares forms a sequence of odd numbers: $3, 5, 7, 9, \dots$. Given the difference is $15$, we can find the smaller square by solving $2n + 1 = 15 \Rightarrow n = 7$. Thus, the smaller square is $7^2 = 49$.

Key Concept: Patterns in Units Place

c) 537
[Solution Description]
The units digit of a perfect square can only be $0, 1, 4, 5, 6,$ or $9$. Numbers ending with $2, 3, 7,$ or $8$ are not perfect squares. Among the options, only the number ending with $7$ does not fit this pattern.

Key Concept: Square Numbers, Patterns in Units Place, Consecutive Odd Numbers and Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate $45^2 = 2025$, which confirms that 2025 is indeed a perfect square. For the reason, the sum of the first $n$ odd numbers starting from 1 is given by $n^2$. Here, the sum of the first 45 odd numbers is $45^2 = 2025$. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Difference Between Consecutive Squares

c) 9
[Solution Description]
The next perfect square after $16$ ($4^2$) is $25$ ($5^2$). The difference is $25 - 16 = 9$, which matches one of the options.

Key Concept: Patterns in Units Place, Non-Integer Squares

d) 6
[Solution Description]
A number with 3 zeros at the end can be written as $N = k \times 1000$, where $k$ is an integer not divisible by 10.
Squaring $N$ gives $N^2 = k^2 \times 1000^2 = k^2 \times 1,000,000$.
Since $k^2$ does not contribute additional zeros, the square will have exactly $6$ zeros.

Key Concept: Sum of Odd Numbers

c) 25
[Solution Description]
Sum of first $n$ odd natural numbers equals $n^2$. Here, $n = 5$, so sum $= 5^2 = 25$.

Key Concept: Square Numbers Ending with 6, Sum of Odd Numbers

a) 36
[Solution Description]
To determine which number is a perfect square ending with 6:
1) Check each option to see if it ends with 6.
2) Verify if the number can be expressed as the sum of consecutive odd numbers starting from 1.
- For 36: $1 + 3 + 5 + 7 + 9 + 11 = 36$ (valid).
- For 26: Ends with 6 but cannot be expressed as the sum of consecutive odd numbers starting from 1.
- For 46: Ends with 6 but fails the sum test.
- For 56: Ends with 6 but fails the sum test.
Thus, only 36 satisfies both conditions.

Key Concept: Zeros in Perfect Squares

d) 6
[Solution Description]
According to the properties, if a number has 3 zeros at the end, its square will have 6 zeros. Since 5000 has 3 zeros ($5000 = 5 \times 10^3$), squaring it gives $(5 \times 10^3)^2 = 25 \times 10^6 = 25,000,000$. Thus, the square of 5000 has 6 zeros at the end.

Key Concept: Difference Between Consecutive Squares

b) 25
[Solution Description]
Difference between $(k+1)^2$ and $k^2$ is $2k + 1$. Given $2k + 1 = 11$, solving gives $k = 5$. Thus, smaller square is $5^2 = 25$.

Key Concept: Units Place Property

c) 198
[Solution Description]
Perfect squares can only end with digits 0, 1, 4, 5, 6, or 9. The number ending with 8 is not a perfect square.

Key Concept: Difference Between Consecutive Squares

b) 25
[Solution Description]
The difference between consecutive squares is an odd number. The general pattern shows that the difference between $n^2$ and $(n+1)^2$ is given by $(2n + 1)$.
Here, $n = 12$.
So, the difference is $2(12) + 1 = 25$.

Key Concept: Perfect Squares and Odd Numbers, Sum of Consecutive Odd Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify if 144 is a perfect square, we check if it can be expressed as the sum of consecutive odd numbers starting from 1. The sum of the first $n$ odd numbers is given by $n^2$. For 144:
$n^2 = 144 \implies n = 12$
Now, let’s verify by adding the first 12 odd numbers:
$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = 144$
Thus, the Assertion is true. The Reason states that the sum of the first $n$ odd numbers equals $n^2$, which is also true. Moreover, the Reason correctly explains why 144 is a perfect square, as it follows this property.

Key Concept: Perfect Squares and Odd Numbers, Differences Between Consecutive Squares

a) The difference between consecutive squares is always an odd number.
[Solution Description]
To solve this, we analyze each option based on the properties of perfect squares:
1) The difference between consecutive squares is always an odd number: This is true as shown by examples like $4 - 1 = 3$, $9 - 4 = 5$.
2) A perfect square can end with the digit 8: This is false since perfect squares can only end with $0, 1, 4, 5, 6,$ or $9$.
3) The sum of the first $n$ odd numbers is $(n+1)^2$: This is false; the correct formula is $n^2$.
4) A number with 5 zeros at the end cannot be a perfect square: This is true because squares must have an even number of zeros at the end.
Thus, options 1 and 4 are correct, but since only one option is allowed, the most encompassing truth is option 1.

Key Concept: Sum of Odd Numbers

d) 16
[Solution Description]
The sum of the first $n$ odd numbers is equal to $n^2$. Here, $n = 4$.
Step 1: Calculate $n^2 = 4^2 = 16$.
Therefore, the sum of the first 4 odd numbers is $16$.

Key Concept: Perfect Squares and Odd Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify if 36 can be expressed as the sum of successive odd natural numbers, let's calculate the sum of the first $n$ odd numbers:
$1 + 3 + 5 + 7 + 9 + 11 = 36$. Here, $n = 6$ since there are 6 terms.
According to the property, the sum of the first $n$ odd numbers is $n^2$. Hence, $6^2 = 36$, which matches the assertion.
Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Perfect Squares and Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The $5$-th triangular number is calculated using the formula for triangular numbers: $T_n = \frac{n(n+1)}{2}$. For $n=5$, $T_5 = \frac{5(5+1)}{2} = 15$. The $6$-th triangular number is $T_6 = \frac{6(6+1)}{2} = 21$. The sum of $T_5 + T_6 = 15 + 21 = 36$, which is indeed $6^2$.
The reason states a general property that the sum of the $n$-th and $(n+1)$-th triangular numbers is $(n+1)^2$. This is verified by the calculation above, as it holds for $n=5$. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Properties of Perfect Squares

d) 2
[Solution Description]
Perfect squares can only end with 0, 1, 4, 5, 6, or 9. Among the given options, 2 is not one of these digits. Hence, 2 cannot be the units digit of a perfect square.

Key Concept: Perfect Squares

d) 25
[Solution Description]
To find the square of 5, multiply 5 by itself: $5 \times 5 = 25$.

Key Concept: Triangular Numbers and Perfect Squares

c) 21
[Solution Description]
The sum of two consecutive triangular numbers gives a perfect square. Here, $36 = 6^2$. The two consecutive triangular numbers whose sum is 36 are 15 and 21 ($15 + 21 = 36$). Hence, the larger triangular number is 21.

Key Concept: Difference Between Perfect Squares

b) 23
[Solution Description]
Calculate $12^2 = 144$ and $11^2 = 121$.
Difference: $144 - 121 = 23$.
According to the pattern, differences between consecutive squares are odd numbers. Here, $23$ is indeed an odd number.

Key Concept: Sum of Consecutive Triangular Numbers

b) 49
[Solution Description]
The 6th triangular number is $T_6 = 1 + 2 + 3 + \cdots + 6 = 21$.
The 7th triangular number is $T_7 = 1 + 2 + 3 + \cdots + 7 = 28$.
Sum: $T_6 + T_7 = 21 + 28 = 49$.
Since $49 = 7^2$, it is a perfect square.

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 square root of 144 is 12, which is correct since $12^2 = 144$. The reason given is $12 \times 12 = 144$, which is also true and correctly explains why 12 is the square root of 144.

Key Concept: Square Numbers and Odd Number Summation

c) 15
[Solution Description]
A perfect square can be expressed as the sum of consecutive odd numbers starting from 1.
The sum of the first $n$ odd numbers is $n^2$.
Given the sum is 225, we need to find $n$ such that $n^2 = 225$.
Taking the square root of both sides: $n = \sqrt{225} = 15$.
Therefore, the first 15 odd numbers must be added to get 225.

Key Concept: Square Root Properties

d) 9
[Solution Description]
The square root of 81 can be both $+9$ and $-9$ as $9^2 = 81$ and $(-9)^2 = 81$. However, in this context, the syllabus specifies considering only the positive square root. Therefore, $\sqrt{81} = 9$.

Key Concept: Square Roots and Perfect Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if 324 is a perfect square, we analyze its prime factorization: $324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$. These factors can be grouped as $(2 \times 3 \times 3) \times (2 \times 3 \times 3) = 18^2$, which shows that 324 is a perfect square. The Reason correctly explains the Assertion because it states the condition for a number to be a perfect square based on its prime factorization.

Key Concept: Square Root Calculation

c) 12
[Solution Description]
We need to find a number $x$ such that $x^2 = 144$. From the given syllabus, we know that $12 \times 12 = 144$. Therefore, the positive square root of 144 is 12.

Key Concept: Cube Roots

c) 7
[Solution Description]
The cube root of a number $y$ is a number $x$ such that $x^3 = y$. Here, we need to find $\sqrt[3]{343}$.
We know that $7 \times 7 \times 7 = 343$, so:
$\sqrt[3]{343} = 7.$
Therefore, the correct answer is 7.

Key Concept: Definition of Cubic Numbers

c) 64
[Solution Description]
To find the cube of 4, multiply 4 by itself three times: $4 \times 4 \times 4 = 64$. Thus, the cube of 4 is 64.

Key Concept: Cube Roots

b) 2
[Solution Description]
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, to find $\sqrt[3]{8}$, we look for a number such that $x \times x \times x = 8$. Since $2 \times 2 \times 2 = 8$, the cube root of 8 is 2.

Key Concept: Perfect Cubes

b) 216
[Solution Description]
To identify a perfect cube, we need to check if the number can be expressed as $n^3$ where $n$ is an integer. Let's evaluate each option:
- For 144: The prime factorization is $12 \times 12 = 144$, which cannot be split into three identical groups.
- For 216: $6^3 = 216$ and its prime factors are $(2 \times 3)^3 = 2^3 \times 3^3$, which forms three identical groups.
- For 99: The prime factors are $9 \times 11$, which do not form three identical groups.
- For 200: The prime factorization is $8 \times 25 = 200$, which does not split into three identical groups.
Thus, the correct answer is the perfect cube 216.

Key Concept: Identifying Perfect Cubes

b) 27
[Solution Description]
A perfect cube is a number obtained by multiplying an integer by itself three times. Checking the options:
- 9 is not a perfect cube ($2^3 = 8$ and $3^3 = 27$).
- 27 is a perfect cube ($3 \times 3 \times 3 = 27$).
- 36 is not a perfect cube.
- 49 is not a perfect cube.
Thus, 27 is the perfect cube among the given options.

Key Concept: Perfect Cubes and Prime Factorization

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion and reason are correct:
1. **Assertion**: Check if 216 is a perfect cube.
$6^3 = 6 \times 6 \times 6 = 216$, so 216 is indeed a perfect cube.
2. **Reason**: Analyze the prime factorization of 216.
$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3$. Here, both prime factors (2 and 3) appear three times, meaning they can be split into three identical groups. Thus, the reason correctly explains why 216 is a perfect cube.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Perfect Cubes and Prime Factorization

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if 216 is a perfect cube, we examine its prime factorization. Let's break it down step by step:
- Prime factorization of 216:
$216 = 6 \times 6 \times 6 = (2 \times 3)^3 = 2^3 \times 3^3$.
- Since the exponents of all prime factors (2 and 3) are multiples of 3, 216 is indeed a perfect cube.
- The assertion (A) states that 216 is a perfect cube, which is true.
- The reason (R) provides the correct prime factorization of 216, which also justifies why it is a perfect cube. Therefore, the reason correctly explains the assertion.

Key Concept: Perfect Cubes, Prime Factorisation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if 5832 is a perfect cube, we first find its prime factorisation:
$5832 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2^3 \times 3^6$
Since both prime factors ($2$ and $3$) appear in multiples of three, 5832 is indeed a perfect cube. Further, $(2 \times 3^2)^3 = 2^3 \times 3^6 = 5832$, so $\sqrt[3]{5832} = 18$.
Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains why 5832 is a perfect cube.

Key Concept: Perfect Cubes

c) 64
[Solution Description]
To find the cube of 4, multiply 4 by itself three times: $4 \times 4 \times 4 = 64$. Thus, the cube of 4 is 64.

Key Concept: Sum of consecutive odd numbers, Perfect cubes

c) 7
[Solution Description]
The sum of consecutive odd numbers forming a perfect cube follows the pattern where the sum of the first $n$ such numbers is $n^3$. Given that $343 = 7^3$, the number of terms being added must be $7$.

Key Concept: Prime Factorisation

c) 8
[Solution Description]
A number is a perfect cube if its prime factors can be grouped into triplets. For example, 8 can be written as $2 \times 2 \times 2 = 2^3$, making it a perfect cube.

Key Concept: Last Digit Pattern in Cubes

c) 3
[Solution Description]
From the pattern observed in cubes:
- Numbers ending with 7 have cubes ending with 3.
For example:
$7^3 = 343 \quad (\text{last digit is } 3)$
$17^3 = 4913 \quad (\text{last digit is } 3)$
Therefore, if a number ends with 7, its cube ends with 3.

Key Concept: Taxicab Numbers

c) 1729
[Solution Description]
The smallest such number is called a taxicab number. From the syllabus, the first taxicab number is 1729, which can be written as $1^3 + 12^3$ and $9^3 + 10^3$.

Key Concept: Prime Factorisation of Cubes

c) $2^3 \times 3^3$
[Solution Description]
The cube of a number has each prime factor raised to three times its original power. For $6 = 2 \times 3$, its cube is:
$6^3 = (2 \times 3)^3 = 2^3 \times 3^3$

Key Concept: Cubic Numbers: Properties of Cube Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion and reason step by step:
1. The assertion states that the cube of a negative integer is always negative. For example, $(-2)^3 = -8$, $(-3)^3 = -27$, which are both negative.
2. The reason states that multiplying a number by itself three times preserves its sign. This is true because:
- Negative $\times$ Negative $\times$ Negative = Negative.
- Positive $\times$ Positive $\times$ Positive = Positive.
3. Therefore, the reason correctly explains why the cube of a negative integer is always negative.

Key Concept: Taxicab Number

c) 1729
[Solution Description]
A taxicab number is expressible as the sum of two cubes in two different ways. The smallest known example is 1729:
$1729 = 1^3 + 12^3 = 9^3 + 10^3$
Among the options, 1729 fits this property.

Key Concept: Taxicab Number Identification

b) $(2, 16)$
[Solution Description]
The number $4104$ is a taxicab number and satisfies:
$4104 = 2^3 + 16^3 = 9^3 + 15^3.$
Among the options, only $(2, 16)$ corresponds to one valid pair.

Key Concept: Cube Root Calculation

d) 12
[Solution Description]
To find the cube root of $1728$, we perform prime factorization:
$1728 = 12 \times 12 \times 12 = 12^3$.
Therefore, $\sqrt[3]{1728} = 12$.

Key Concept: Cube Root

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The cube root of a number $n$ is the value that, when multiplied by itself three times, gives $n$. Here, $3 \times 3 \times 3 = 27$, so the cube root of 27 is indeed 3. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Prime Factorization, Cube Roots

b) 18
[Solution Description]
To make the number a perfect cube after multiplying by 7, the exponents in its prime factorization must all be multiples of 3. The given prime factorization is $2^2 \times 3^1 \times 7^2$. Multiplying by 7 gives $2^2 \times 3^1 \times 7^3$. To make all exponents divisible by 3:
- For 2: $2^2$ needs another $2^1$ to become $2^3$.
- For 3: $3^1$ needs another $3^2$ to become $3^3$.
Thus, the smallest number to multiply is $2^1 \times 3^2 = 2 \times 9 = 18$.

Key Concept: Cube Root Calculation

c) 5
[Solution Description]
To find the cube root of 125, we look for a number that, when multiplied by itself three times, gives 125. We know that $5 \times 5 \times 5 = 125$. Therefore, $\sqrt[3]{125} = 5$.

Key Concept: Taxicab Numbers, Perfect Cubes

a) 1729
[Solution Description]
The smallest taxicab number is 1729, known as the Hardy-Ramanujan Number. It can be expressed as:
$1729 = 1^3 + 12^3 = 9^3 + 10^3$.
The next taxicab numbers are larger (e.g., 4104, 13832), so 1729 is the correct answer.

Key Concept: Taxicab Numbers, Sum of Two Cubes

a) 1729
[Solution Description]
The smallest such $k$ is 1729, which has two pairs: $(1, 12)$ and $(9, 10)$. This makes 1729 the smallest taxicab number.

Key Concept: Taxicab Numbers

b) 2
[Solution Description]
There are exactly two pairs of positive integers $(a, b)$ that satisfy $a^3 + b^3 = 1729$. These are $(1, 12)$ and $(9, 10)$, since $1^3 + 12^3 = 1729$ and $9^3 + 10^3 = 1729$.

Key Concept: Taxicab Numbers

c) 4104
[Solution Description]
After 1729, the next taxicab numbers are 4104 and 13832. Here, 4104 is one such taxicab number. It can be expressed as $2^3 + 16^3 = 4104$ and $9^3 + 15^3 = 4104$.

Key Concept: Taxicab Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 1729 is the smallest number expressible as the sum of two cubes in two different ways, which is true as proven by Ramanujan. The reason provides the two valid cube pairs $(1^3 + 12^3)$ and $(9^3 + 10^3)$ that sum to 1729, correctly explaining why the assertion is true.

Key Concept: Taxicab Numbers

c) 1729
[Solution Description]
The smallest number expressible as the sum of two positive cubes in two distinct ways is 1729, also known as the Hardy-Ramanujan Number. The two expressions are $1^3 + 12^3 = 1729$ and $9^3 + 10^3 = 1729$.

Key Concept: Cube Roots

c) 8
[Solution Description]
To find the cube root of 512, we can use prime factorization.
$512 = 8 \times 64 = (2^3) \times (4^3) = (2 \times 4)^3 = 8^3$
Therefore, $\sqrt[3]{512} = 8$.

Key Concept: Cube Roots

c) 9
[Solution Description]
We know that $9 \times 9 \times 9 = 729$. Therefore, $\sqrt[3]{729} = 9$.

Key Concept: Perfect Cubes and Consecutive Odd Numbers

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's analyze the given pattern:
- For $n=1$, the sum is $1 = 1^3$.
- For $n=2$, the sum is $3 + 5 = 8 = 2^3$.
- For $n=3$, the sum is $7 + 9 + 11 = 27 = 3^3$.
Observing the pattern, the sum of $n$ consecutive odd numbers (starting from the correct odd number) equals $n^3$. Here, there are $10$ terms ($n=10$) in the sum $91 + 93 + \ldots + 109$.
Calculating the mid-point of the range: $\frac{91 + 109}{2} = 100$.
Total sum = Mid-point $\times$ number of terms = $100 \times 10 = 1000 = 10^3$.
Thus, the sum is indeed a perfect cube.
The reason states that the sum of the first $n$ consecutive odd numbers starting from an appropriate odd number results in $n^3$. This is true as demonstrated by the pattern. However, the reason does not directly explain why the specific sum $91 + \ldots + 109$ equals $10^3$ but rather provides a general rule.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Taxicab Numbers

b) $(2, 16)$ and $(9, 15)$
[Solution Description]
The taxicab number $4104$ can be expressed as the sum of two cubes in two different ways:
$4104 = 2^3 + 16^3 = 9^3 + 15^3$
Calculating:
$2^3 + 16^3 = 8 + 4096 = 4104$
$9^3 + 15^3 = 729 + 3375 = 4104$
Therefore, the correct pairs are $(2, 16)$ and $(9, 15)$.

Key Concept: Perfect Cubes and Consecutive Odd Numbers

b) $31 + 33 + 35 + 37 + 39 + 41$
[Solution Description]
To find the sequence of consecutive odd numbers whose sum is $216$, we refer to the pattern given in the syllabus. The sum of $n$ consecutive odd numbers starting from the appropriate term will be $n^3$. Here, $216 = 6^3$, so $n = 6$.
The first term of the sequence can be found using the formula for the first term of the $k^{th}$ group of consecutive odd numbers:
$\text{First term} = k^2 - k + 1$
For $k = 6$:
$\text{First term} = 6^2 - 6 + 1 = 36 - 6 + 1 = 31$
Thus, the sequence is $31, 33, 35, 37, 39, 41$.

Key Concept: Cubic Numbers: Successive Differences

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for perfect cubes, the third differences are always constant. This is true as demonstrated in the syllabus example where the third differences were all 6 for the sequence $1, 8, 27, 64, 125, 216, \ldots$.
The reason claims that this property is unique to cubic sequences and helps identify them. This is also true because, unlike squares or higher powers, only cubic numbers exhibit this specific pattern of constant third differences.
Since both the assertion and reason are true, and the reason correctly explains why perfect cubes have constant third differences, the correct option is the one where both are true and the reason is the correct explanation.

Key Concept: Cube Roots, Patterns in Squares

a) 6
[Solution Description]
We need $n^3 \equiv 216 \ (\text{mod} \ 1000)$.
First, solve $n^3 \equiv 216 \ (\text{mod} \ 8)$:
$216 \equiv 0 \ (\text{mod} \ 8)$, so $n$ must be even.
Next, solve $n^3 \equiv 216 \ (\text{mod} \ 125)$:
$216 = 6^3$, so $n \equiv 6 \ (\text{mod} \ 5)$ (using Fermat's Little Theorem).
Combining, the smallest $n$ satisfying both is $6$ (since $6^3 = 216$, which ends with $216$).
However, checking higher candidates:
$16^3 = 4096$ (does not end with $216$),
$26^3 = 17576$ (ends with $576$),
$36^3 = 46656$ (ends with $656$),
$46^3 = 97336$ (ends with $336$),
$56^3 = 175616$ (ends with $616$),
$66^3 = 287496$ (ends with $496$),
$76^3 = 438976$ (ends with $976$),
$86^3 = 636056$ (ends with $056$),
$96^3 = 884736$ (ends with $736$),
$106^3 = 1191016$ (ends with $016$),
$116^3 = 1560896$ (ends with $896$),
$126^3 = 2000376$ (ends with $376$),
$136^3 = 2515456$ (ends with $456$),
$146^3 = 3112136$ (ends with $136$),
$156^3 = 3796416$ (ends with $416$),
$166^3 = 4574296$ (ends with $296$),
$176^3 = 5451776$ (ends with $776$),
$186^3 = 6434856$ (ends with $856$),
$196^3 = 7529536$ (ends with $536$).
The pattern repeats every $100$ units. The smallest such $n$ is $6 + 100k$ where $k$ is an integer. Here, $6$ is the smallest.

Key Concept: First Differences in Cubic Numbers

c) 30
[Solution Description]
First, calculate the first differences for consecutive cubes:
$8 - 1 = 7, \quad 27 - 8 = 19, \quad 64 - 27 = 37, \quad 125 - 64 = 61, \quad 216 - 125 = 91.$
The question asks for the difference between the first differences of $216$ and $125$, which is $91 - 61 = 30$.

Key Concept: Second Differences in Cubic Numbers

b) 18
[Solution Description]
Calculate the first differences:
$8 - 1 = 7, \quad 27 - 8 = 19, \quad 64 - 27 = 37.$
Now, compute the second difference:
$37 - 19 = 18.$

Key Concept: Stabilized Differences for Cubic Numbers

c) 3
[Solution Description]
From the syllabus, we observe that for perfect cubes, the differences stabilize at the third level. At this level, all differences become $6$.

Key Concept: Sanskrit Terminology

b) Both the square figure and the square power
[Solution Description]
As per Aryabhata's description, *varga* refers to both the square figure (or its area) and the square power (product of two equal quantities).

Key Concept: Origin of the Term 'Root'

c) Because it represented the basis or origin of a square/cube
[Solution Description]
The term *mula* was used because it signified the basis or origin of a square or cube, similar to how roots are the basis of plants. This terminology influenced Arabic (*jidhr*) and Latin (*radix*).

Key Concept: Historical Origin of Mathematical Terms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

The term $\textit{varga-mula}$ indeed refers to the square root in Sanskrit, as it combines $\textit{varga}$ (square) and $\textit{mula}$ (root). The reason correctly explains that $\textit{mula}$ was used to denote the origin or basis of a square, and this concept was later adopted in Arabic $(\textit{jidhr})$ and Latin $(\textit{radix}),$ eventually leading to the modern term "root." Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Babylonian Contributions

a) Quickly finding square roots and cube roots for geometric problems
[Solution Description]
The Babylonian clay tablets contained lists of perfect squares and cubes, which were used for quick calculations in land measurement and architectural design.

Key Concept: Ancient Mathematical Terminology

b) Cube root
[Solution Description]
In ancient Sanskrit mathematics, the term *ghana* was used for the cube of a number or the solid cube figure, and *mula* means root (origin). Hence, *ghana-mula* refers to the cube root.

Key Concept: Cube root calculation

c) 30
[Solution Description]
To find the cube root of 27000, we can factorize it:
$27000 = 27 \times 1000 = 3^3 \times 10^3 = (3 \times 10)^3 = 30^3$
Therefore, the cube root of 27000 is 30.

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 Babylonians compiled lists of perfect squares and cubes, which is historically accurate as evidenced by ancient clay tablets dating back to 1700 BCE. The reason provided explains the practical application of these lists in solving problems related to land measurement and architectural design, which aligns with historical records. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Properties of cubes, Historical context

a) Varga-varga of any number has an even number of factors
[Solution Description]
We need to identify which statement contradicts known properties of cubes from historical context.
Analyzing each concept:
1) Cubes of odd numbers: Always odd (3³=27, 5³=125)
2) Perfect cubes ending with 8: Exist (2³=8, 12³=1728)
3) Two-digit cubes: Yes (4³=64, 5³=125)
4) Two-digit cubes becoming 7+ digits: Impossible (max 99³=970299 is 6 digits)
5) Factors of cubes: All have odd number of factors (as perfect squares)
Ancient texts would recognize all except the false claim about seven digit cubes from two-digit numbers.

Key Concept: Perfect cubes, Multiplication factors

c) 36, ghana
[Solution Description]
To find the smallest multiplier to make 8232 a perfect cube and identify its correct Sanskrit terminology.
Step 1: Factorize 8232:
$8232 = 2^3 \times 3 \times 7^3 \times 2 = 2^4 \times 3^1 \times 7^3$
Step 2: For perfect cube, all exponents must be multiples of 3:
- Need 2 more 2's (to make exponent 6)
- Need 2 more 3's (to make exponent 3)
- 7's exponent is already 3
Step 3: Multiplier needed:
$2^2 \times 3^2 = 4 \times 9 = 36$
Step 4: Brahmagupta would call the perfect cube "krti" (square) or "ghana" (cube), but since it's a perfect cube, "ghana" is correct.

Key Concept: Cube Roots and Perfect Cubes

c) 30
[Solution Description] To find the cube root of 27000, we factorize it:
$27000 = 27 \times 1000 = 3^3 \times 10^3 = (3 \times 10)^3 = 30^3$
Thus, the cube root of 27000 is 30.

Key Concept: Mathematical etymology, Root operations

b) Sanskrit through Arabic and Latin
[Solution Description]
The concept originated from Sanskrit 'mula' meaning plant root, which was adopted into Arabic as 'jidhr' and then Latin as 'radix'. These terms all referred both to plant roots and mathematical roots. The English term 'root' comes from this lineage via Latin influence on European mathematics during the Middle Ages.

Key Concept: Historical terms in mathematics, Square roots, Varga-mula

b) Basis of the square power
[Solution Description]
The term 'varga-mula' was used for square root in ancient India. It literally means "the basis/origin of the square" ('varga' = square, 'mula' = root/basis). This relates to finding the original number that was squared to obtain a given value. Aryabhata and Brahmagupta explained this concept where mula represents the root operation.

Key Concept: Ancient Mathematical Terminology

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the term $varga$ was used in ancient India to refer to both the square figure and its area as well as the square power. This is true as per the given syllabus text which mentions, "In ancient Sanskrit works the term $varga$ was used both for the square figure or its area, as well as the square power."
The reason states that this usage originates from the graphical representation of a square figure. This is also true as Aryabhata's statement explains, "Thus, the term $varga$ for square power has its origin in the graphical representation of a square figure."
Moreover, the reason correctly explains why $varga$ was used for square power, linking it to the visual basis of the concept. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Conceptual Understanding

c) Babylonians
[Solution Description]
The Babylonians were the first to compile lists of perfect squares and cubes on clay tablets, which were used for geometric calculations in land measurement and architecture.

Key Concept: Indian Contributions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that *varga-varga* was used for the fourth power, which is correct as per the syllabus. The reason explains that *varga* means square, and *varga-varga* is derived by applying the square operation twice (square of square), which correctly explains why it represents the fourth power. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Cube Roots

c) 17
[Solution Description]
Observing the pattern of perfect cubes:
- $17^3 = 4913$.
This can be deduced by checking nearby integers since $16^3 = 4096$ and $18^3 = 5832$.

Key Concept: Perfect Cubes and Their Properties

c) 64
[Solution Description]
A perfect cube is a number that can be expressed as $n^3$ where $n$ is an integer. Checking each option:
$64 = 4^3$
Hence, 64 is a perfect cube.

Key Concept: Famous Mathematicians

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 1729 is the smallest number expressible as the sum of two cubes in two different ways, which is true ($1729 = 1^3 + 12^3 = 9^3 + 10^3$). The reason explains Ramanujan's talent for recognizing numerical patterns, which is also true and directly explains why he identified this property of 1729. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Taxicab Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that 1729 is the smallest number expressible as the sum of two cubes in two different ways. This is true, as verified by the equations $1729 = 1^3 + 12^3$ and $1729 = 9^3 + 10^3$. The Reason correctly explains why Ramanujan considered 1729 special, linking it directly to the Assertion. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Cube Roots

c) 5
[Solution Description]
The cube root of a number $x$ is a number $y$ such that $y^3 = x$. Here,
$5^3 = 125$
So, the cube root of 125 is 5.

Key Concept: Concept: Origin of mathematical terms varga and mula

d) Assertion is false, but Reason is true.
[Solution Description]
Assertion (A) is false because the term 'varga' was not exclusively used for the square power of a number. It was also used for the square figure or its area, as mentioned in the syllabus. Reason (R) is true because Aryabhata indeed stated that the term 'varga' for square power has its origin in the graphical representation of a square figure. Therefore, Assertion is false, but Reason is true.

Key Concept: Square Roots

b) 21 meters
[Solution Description]
The area of a square is given by $\text{Side}^2$. To find the side length, take the square root of the area:
$\sqrt{441} = 21$ meters.

Key Concept: Basic Concept: Cube Root Application

b) 11
[Solution Description]
Since 1331 is known to be a perfect cube, and observing that $11^3 = 1331$, the cube root must be 11.

Key Concept: Cube Roots

c) 30
[Solution Description]
To find the cube root of 27000, we can perform prime factorization:
$27000 = 27 \times 1000 = 3^3 \times 10^3 = (3 \times 10)^3$. Therefore, $\sqrt[3]{27000} = 30$.

Key Concept: Perfect Cubes, Historical Terminology

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion states that $\textit{varga-mula}$ was used to denote the square root due to its connection with the origin of squares. This is correct as per historical records where $\textit{mula}$ (root) was used for the mathematical operation of taking roots. The Reason states that Aryabhata introduced $\textit{varga}$ for square numbers and the squaring operation, which is also true based on his work. However, the Reason does not explain why $\textit{varga-mula}$ denotes the square root; it only provides information about $\textit{varga}.$ Thus, both statements are true, but Reason is not the correct explanation of Assertion.