Key Concept: Calendar-Based 2×2 Square Patterns, Verification of Patterns

c) 17
[Solution Description]
Given the 2×2 square pattern:
$\begin{matrix} \\ a & a+1 \\ \\ a+7 & a+8 \\ \end{matrix}$.
The diagonals are $(a, a+8)$ and $(a+1, a+7)$.
Product of first diagonal: $P_1 = a(a+8)$.
Product of second diagonal: $P_2 = (a+1)(a+7)$.
Given $P_1 = 72$, so $a(a+8) = 72$.
Solving $a^2 + 8a - 72 = 0$ gives $a = 4$ or $a = -12$ (invalid for calendar).
Thus, $a = 4$.
Now, $P_2 = (4+1)(4+7) = 5 \times 11 = 55$.
Difference $= P_1 - P_2 = 72 - 55 = 17$.

Key Concept: Interpreting calendar-based 2×2 patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- **Assertion (A)**: The difference between the diagonal products in a 2×2 calendar square is calculated as $(a + 1)(a + 7) - a(a + 8) = a^2 + 8a + 7 - a^2 - 8a = 7$. Therefore, the assertion is true.
- **Reason (R)**: A weekly calendar has numbers increasing by 1 in rows and by 7 in columns, making the labeling $\begin{array}{cc} a & (a + 1) \\ a + 7 & (a + 8) \end{array}$ valid. This spacing causes the diagonal product difference to be 7. So, the reason is true and correctly explains the assertion.
Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Verification of Patterns, Mathematical Exploration

a) No solution exists
[Solution Description]
Using the standard 2×2 pattern:
$\begin{matrix} \\ a & a+1 \\ \\ a+7 & a+8 \\ \end{matrix}$.
Diagonal products: $P_1 = a(a+8)$, $P_2 = (a+1)(a+7)$.
Given $|P_1 - P_2| = 9$.
First, compute $P_1 - P_2$:
$a(a+8) - (a+1)(a+7) = a^2 + 8a - (a^2 + 8a + 7) = -7$.
Thus, $|-7| = 7 \neq 9$.
Next, consider $P_2 - P_1 = 7$.
Since neither equals 9, the condition cannot be satisfied with integer $a$ in a calendar grid.
However, if we assume the question refers to $P_2 - P_1 = 9$, then:
$(a+1)(a+7) - a(a+8) = 9$.
Expanding: $a^2 + 8a + 7 - a^2 - 8a = 9$.
Simplifies to $7 = 9$, which is false.
Hence, no such $a$ exists.

Key Concept: Generalization of Diagonal Pattern

b) $n^2 + 8n + 7$
[Solution Description] Using the labeling:
$\begin{array}{cc} n & (n + 1) \\ (n + 7) & (n + 8) \end{array}$
The second diagonal product is:
$(n + 1) \times (n + 7) = n^2 + 8n + 7$.

Key Concept: Verification of Diagonal Pattern

b) $7$
[Solution Description]
The diagonal products are $9 \times 17 = 153$ and $10 \times 16 = 160$. The difference is:
$160 - 153 = 7$
Hence, the difference is $7$.

Key Concept: Generalization of Calendar Pattern

b) Due to the weekly structure of calendars (7-day rows)
[Solution Description]
The difference arises because the numbers in the rows are separated by $7$ days (days in a week). If we represent the numbers as $\begin{array}{cc} a & a+1 \\ a+7 & a+8 \end{array}$, then:
$(a + 1)(a + 7) - a(a + 8) = (a^2 + 8a + 7) - (a^2 + 8a) = 7$
This holds for any such square in a standard calendar.

Key Concept: Algebraic Pattern Investigation: Interpreting calendar-based 2×2 patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let the top-left number in the 2×2 square be $a$. The other numbers can be represented as $(a + 1)$, $(a + 7)$, and $(a + 8)$ due to the structure of the calendar.
The product of the first diagonal is $a(a + 8) = a^2 + 8a$.
The product of the second diagonal is $(a + 1)(a + 7) = a^2 + 8a + 7$.
The difference between the two products is $(a^2 + 8a + 7) - (a^2 + 8a) = 7$.
Hence, the Assertion is true. The Reason provides the correct algebraic explanation for this observation, so it is also true and correctly explains the Assertion.

Key Concept: Diagonal Products in 2×2 Calendar Grid

c) $7$
[Solution Description]
The diagonal products are $(a)(a + 8) = a^2 + 8a$ and $(a + 1)(a + 7) = a^2 + 8a + 7$. The difference between them is calculated as:
$(a + 1)(a + 7) - a(a + 8) = (a^2 + 8a + 7) - (a^2 + 8a) = 7$
Thus, the difference is always $7$.

Key Concept: Algebraic Pattern Investigation, Calendar-based 2×2 patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in a 2×2 calendar square, the difference between diagonal products is always 7. This is true because, for any 2×2 square labeled as $a$, $(a + 1)$, $(a + 7)$, and $(a + 8)$, the diagonal products are:
$a \times (a + 8) = a^2 + 8a$
and
$(a + 1) \times (a + 7) = a^2 + 8a + 7$
The difference between these two products is $(a^2 + 8a + 7) - (a^2 + 8a) = 7$. Therefore, the assertion is correct.
The reason explains why this happens: the numbers in a calendar row increase by 1 and the numbers in a column increase by 7. This explanation is also correct and directly justifies the assertion.
Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Mathematical Exploration, Application

a) No solution exists
[Solution Description]
Using the standard 2×2 pattern:
$\begin{matrix} \\ a & a+1 \\ \\ a+7 & a+8 \\ \end{matrix}$.
The diagonal products are $P_1 = a(a+8)$ and $P_2 = (a+1)(a+7)$.
Given $P_1 = P_2 + 6$:
$a(a+8) = (a+1)(a+7) + 6$.
Expanding both sides:
$a^2 + 8a = a^2 + 8a + 7 + 6$.
Simplifying: $0 = 13$, which implies no solution exists.
But this contradicts the question, so rechecking:
Alternatively, if $P_2 = P_1 + 6$:
$(a+1)(a+7) = a(a+8) + 6$.
Expanding: $a^2 + 8a + 7 = a^2 + 8a + 6$.
Simplifying: $7 = 6$, again no solution.
Therefore, the condition is impossible unless the difference is -6.
Let $P_1 = P_2 - 6$:
$a^2 + 8a = a^2 + 8a + 7 - 6$
Simplifies to $0 = 1$, still invalid.
Hence, the only possible valid case is when $a = 3$ (empirical check):
For $a = 3$, $P_1 = 33$, $P_2 = 40$, difference is 7 (not matching).
For $a = 2$, $P_1 = 20$, $P_2 = 24$, difference is 4.
For $a = 1$, $P_1 = 9$, $P_2 = 16$, difference is 7.
Thus, no integer $a$ satisfies the given condition.

Key Concept: Diagonal Product Difference

c) 7
[Solution Description] The diagonal products are calculated as follows:
First diagonal product: $a \times (a + 8) = a^2 + 8a$
Second diagonal product: $(a + 1) \times (a + 7) = a^2 + 8a + 7$
The difference between the two products is:
$(a^2 + 8a + 7) - (a^2 + 8a) = 7$
Therefore, the difference is always 7.

Key Concept: Calendar Structure Impact

c) Because each row in the calendar is offset by 7 days
[Solution Description] The rows in a calendar are offset by exactly 7 days due to weekly structure. For the labeling $\begin{array}{cc} a & (a + 1) \\ (a + 7) & (a + 8) \end{array}$, the first diagonal product is $a(a + 8)$ and the second diagonal product is $(a + 1)(a + 7)$. Expanding both:
First diagonal: $a^2 + 8a$
Second diagonal: $a^2 + 8a + 7$
The difference $(a^2 + 8a + 7) - (a^2 + 8a) = 7$ arises because of the fixed 7-day row offset in calendars.

Key Concept: Error Spotting in Algebraic Simplification

c) $5x + 10$
[Solution Description]
The expression is given as $2(x - 1) + 3(x + 4)$. The mistake here is incorrect distribution of $2$ and $3$. To solve correctly, first distribute $2$ and $3$ to their respective terms:
$2(x - 1) = 2x - 2 \\ \\ 3(x + 4) = 3x + 12$
Now, combine like terms:
$2x - 2 + 3x + 12 = 5x + 10$
Thus, the simplified form is $5x + 10$.

Key Concept: Distributive Property and Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, we apply the distributive property:
$-3p(-5p + 2q) = (-3p)(-5p) + (-3p)(2q) = 15p^2 - 6pq$
This matches the assertion, so Assertion (A) is true.
The Reason (R) correctly states the distributive property in algebra and explains why the simplification in (A) is valid. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Error spotting in algebraic simplification, combining like terms

c) $4x^2 + 4x + 9$
[Solution Description]
Combine like terms: $7x^2 - 3x^2 = 4x^2$, and $4x$ and $9$ remain as they are.
Thus, $7x^2 + 4x - 3x^2 + 9 = 4x^2 + 4x + 9$.

Key Concept: Error spotting in algebraic simplification, distribution

b) $-12k^2 + 20km$
[Solution Description]
To simplify $-4k(3k - 5m)$, distribute $-4k$ to each term inside the parentheses:
$-4k \times 3k = -12k^2$ and $-4k \times (-5m) = 20km$.
Thus, $-4k(3k - 5m) = -12k^2 + 20km$.

Key Concept: Error in Expansion

c) $25m^2 + 60mn + 36n^2$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, we expand:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2$
Simplify each term:
$25m^2 + 60mn + 36n^2$
The correct expansion is $\boxed{25m^2 + 60mn + 36n^2}$.

Key Concept: Error Spotting in Algebraic Simplification

c) $15p^2 - 6pq$
[Solution Description]
The expression is given as $-3p(-5p + 2q)$. The mistake here is incorrect distribution of $-3p$. To solve correctly, distribute $-3p$ to each term inside the parentheses:
$-3p \times (-5p) = 15p^2 \\ \\ -3p \times (2q) = -6pq$
Thus, the simplified form is $15p^2 - 6pq$.

Key Concept: Error in Combining Like Terms

d) $5x + 10$
[Solution Description]
Expand and simplify step-by-step:
$2(x - 1) + 3(x + 4) = 2x - 2 + 3x + 12$
Combine like terms:
$2x + 3x - 2 + 12 = 5x + 10$
The correct simplified form is $\boxed{5x + 10}$.

Key Concept: Error in Distributive Property

d) $15p^2 - 6pq$
[Solution Description]
To simplify $-3p(-5p + 2q)$, we distribute $-3p$ to each term inside the brackets:
$-3p(-5p) = 15p^2 \quad \text{and} \quad -3p(2q) = -6pq$
So the simplified form is:
$\boxed{15p^2 - 6pq}$

Key Concept: Error Spotting in Algebraic Simplification

b) $3y + 4$
[Solution Description]
The expression is given as $y + 2(y + 2)$. The mistake here is incorrectly expanding it as $(y + 2)^2$. To solve correctly, first distribute $2$ to the terms inside the parentheses:
$2(y + 2) = 2y + 4$
Now, combine with the remaining term:
$y + 2y + 4 = 3y + 4$
Thus, the simplified form is $3y + 4$.

Key Concept: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, apply the distributive property correctly:
$-3p(-5p + 2q) = (-3p)(-5p) + (-3p)(2q) = 15p^2 - 6pq$.
The reason correctly states the distributive property formula.
Thus, both Assertion and Reason are true, and Reason explains the Assertion.

Key Concept: Error spotting in algebraic simplification, binomial expansion

c) $4y^2 - 28y + 49$
[Solution Description]
Use the identity $(a - b)^2 = a^2 - 2ab + b^2$:
$(2y)^2 - 2 \times 2y \times 7 + 7^2 = 4y^2 - 28y + 49$.

Key Concept: Distributive property, Like terms

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that $5w^2 + 6w$ simplifies to $11w^3$, which is false. The correct simplification is $5w^2 + 6w$ itself since $5w^2$ and $6w$ are not like terms ($w^2$ and $w$ have different exponents). The reason states that terms with the same variable raised to the same power are like terms, which is true. However, this does not explain the assertion because the assertion incorrectly combines unlike terms. Therefore, the assertion is false, but the reason is true.

Key Concept: Square of the Difference, Pattern Recognition

c) 9801
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 100$ and $b = 1$:
$(100 - 1)^2 = 100^2 - 2 \times 100 \times 1 + 1^2$
$= 10000 - 200 + 1$
$= 9801$.

Key Concept: Square of the Sum, Difference of Squares

c) 10816
[Solution Description]
We use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 100$ and $b = 4$. Substituting these values:
$(100 + 4)^2 = 100^2 + 2 \times 100 \times 4 + 4^2$
$= 10000 + 800 + 16$
$= 10816$.

Key Concept: Square of the Difference of Two Numbers

c) $25y^2 - 20y + 4$
[Solution Description]
The identity $(a - b)^2 = a^2 - 2ab + b^2$ is used here with $a = 5y$ and $b = 2$. Substituting:
$(5y - 2)^2 = (5y)^2 - 2(5y)(2) + 2^2 = 25y^2 - 20y + 4$.

Key Concept: Application of Identity 1A

c) 121
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = 10$ and $b = 1$.
Step 1: Square the first term: $10^2 = 100$.
Step 2: Multiply the two terms and double the result: $2 \times (10 \times 1) = 20$.
Step 3: Square the second term: $1^2 = 1$.
Final calculation: $100 + 20 + 1 = 121$.

Key Concept: Square of the Sum of Two Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, use the identity for the square of the sum of two numbers: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 3$ and $b = 4$.
Step 1: Calculate $a^2$ which is $3^2 = 9$.
Step 2: Calculate $2ab$ which is $2 \times 3 \times 4 = 24$.
Step 3: Calculate $b^2$ which is $4^2 = 16$.
Adding these together gives $9 + 24 + 16 = 49$, which matches $(3 + 4)^2 = 7^2 = 49$.
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Application of Square of Sum/Difference

c) $42km$
[Solution Description]
Expanding $(7k + 3m)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$, where $a = 7k$ and $b = 3m$:
$(7k + 3m)^2 = (7k)^2 + 2(7k)(3m) + (3m)^2 = 49k^2 + 42km + 9m^2$. The middle term is $42km$.

Key Concept: Square of the Sum of Two Numbers

c) $9x^2 + 24x + 16$
[Solution Description]
We use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 3x$ and $b = 4$. Substituting these values into the identity:
$(3x + 4)^2 = (3x)^2 + 2(3x)(4) + 4^2 = 9x^2 + 24x + 16$.

Key Concept: Square of Sum/Difference, Distributive Property

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion claims that $(a + b)^2$ is always greater than $a^2 + b^2$. The reason given is that the term $2ab$ ensures this inequality holds for non-zero $a$ and $b$.
Step 1: Expand $(a + b)^2$ using Identity 1A:
$(a + b)^2 = a^2 + 2ab + b^2$
Step 2: Compare $(a + b)^2$ with $a^2 + b^2$. The difference is $2ab$.
Step 3: The term $2ab$ can be positive, negative, or zero depending on the signs of $a$ and $b$:
- If $a$ and $b$ are both positive or both negative, $2ab > 0$, so $(a + b)^2 > a^2 + b^2$.
- If one of them is positive and the other is negative, $2ab < 0$, which may make $(a + b)^2 < a^2 + b^2$ (e.g., $a = 1$, $b = -1$ gives $(1 - 1)^2 = 0 < 1 + 1 = 2$). - If either $a$ or $b$ is zero, $2ab = 0$, making $(a + b)^2 = a^2 + b^2$. Thus, the assertion is false because $(a + b)^2$ is not always greater than $a^2 + b^2$. The reason is true when $a$ and $b$ are both non-zero and have the same sign, but it fails to account for cases where $2ab \leq 0$. Therefore, the reason does not fully explain the assertion, but it is conditionally correct.

Key Concept: Square of the Difference of Two Numbers

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the Assertion, we apply Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 3x$ and $b = 4y$. Substituting these values into the identity, we get $(3x - 4y)^2 = (3x)^2 - 2 \times 3x \times 4y + (4y)^2 = 9x^2 - 24xy + 16y^2$. However, the Assertion states the result as $9x^2 - 12xy + 16y^2$, which is incorrect because the middle term should be $-24xy$ instead of $-12xy$. The Reason correctly states the identity for the square of the difference of two numbers. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Square of the Difference of Two Numbers

b) $25 - 10y + y^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, substitute $a = 5$ and $b = y$.
Step 1: Square the first term: $5^2 = 25$.
Step 2: Multiply the two terms and double the result: $-2 \times (5 \times y) = -10y$.
Step 3: Square the second term: $y^2$.
Final expansion: $25 - 10y + y^2$.

Key Concept: Difference of Squares, Multi-step Solution

c) 9996
[Solution Description]
We use the identity $(a + b)(a - b) = a^2 - b^2$ with $a = 100$ and $b = 2$:
$(100 - 2)(100 + 2) = 100^2 - 2^2$
$= 10{,}000 - 4$
$= 9996$.

Key Concept: Square of the Sum of Two Numbers

c) $x^2 + 8x + 16$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = x$ and $b = 4$.
Step 1: Square the first term: $x^2$.
Step 2: Multiply the two terms and double the result: $2 \times (x \times 4) = 8x$.
Step 3: Square the second term: $4^2 = 16$.
Final expansion: $x^2 + 8x + 16$.

Key Concept: Verifying patterns

a) $100 + 16 = 116$
[Solution Description]
The given pattern is $(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$. Substitute $a = 7$ and $b = 3$ into the equation.
First calculate $(a + b)^2$:
$(7 + 3)^2 = 10^2 = 100$
Next calculate $(a - b)^2$:
$(7 - 3)^2 = 4^2 = 16$
Add the two results:
$100 + 16 = 116$
Now compute $2(a^2 + b^2)$:
$a^2 = 7^2 = 49$
$b^2 = 3^2 = 9$
Sum them:
$49 + 9 = 58$
Multiply by 2:
$2 \times 58 = 116$
Both sides are equal, confirming the pattern holds.

Key Concept: Using the identity $(a + b

c) 25
[Solution Description]

We know that
$(a+b)^2 = a^2 + 2ab + b^2$

Here, $((x+5)^2)$ means $(a=x)$ and $(b=5).$ So,
$(x+5)^2 = x^2 + 2(x)(5) + 5^2$
$= x^2 + 10x + 25$

Comparing this with
$x^2 + 10x + k$

we get:
$k = 25$

So, the value of $k$ is $(\boxed{25}\).$

Key Concept: Algebraic Identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the identity $(a + b)(a - b) = a^2 - b^2$ simplifies multiplication, which is true as shown in examples like $98 \times 102 = 100^2 - 2^2 = 9996$. The reason explains that this identity comes from the distributive property, which is also correct because expanding $(a + b)(a - b)$ gives $a^2 - ab + ab - b^2 = a^2 - b^2$, confirming the identity. Thus, the reason correctly explains the assertion.

Key Concept: Using identities to explain numerical puzzles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(10 + 5)^2$ can be expanded using the identity $(a + b)^2 = a^2 + 2ab + b^2$. This is correct because the identity holds for any real numbers $a$ and $b$, including 10 and 5. Applying it, we get:
$(10 + 5)^2 = 10^2 + 2 \times 10 \times 5 + 5^2 = 100 + 100 + 25 = 225$
The reason correctly explains why the identity works: the square of a binomial indeed equals the sum of the squares of the terms and twice their product. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Sum of squares, Algebraic identities

a) 100
[Solution Description]
Plugging in $a = 8$ and $b = 6$ into the identity:
$2(8^2 + 6^2) = (8 + 6)^2 + (8 - 6)^2$
$2(64 + 36) = 14^2 + 2^2$
$200 = 196 + 4$
Now, solving for $a^2 + b^2$:
$2(a^2 + b^2) = 200 \implies a^2 + b^2 = 100$

Key Concept: Algebraic Pattern Investigation: Sum of Squares and Difference of Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any two natural numbers $a$ and $b$, $2(a^2 + b^2)$ can be written as $(a + b)^2 + (a - b)^2$. This is supported by the algebraic identity provided in the reason. By expanding $(a + b)^2$ and $(a - b)^2$:
$(a + b)^2 = a^2 + 2ab + b^2 \\ \\ (a - b)^2 = a^2 - 2ab + b^2$
Adding these two equations gives:
$(a + b)^2 + (a - b)^2 = 2a^2 + 2b^2 = 2(a^2 + b^2)$
Thus, the assertion is true, and the reason correctly explains it using the derived identity.

Key Concept: Applying $(a + b

a) 95
[Solution Description]

We know that
$(a+b)(a-b)=a^2-b^2$

Here,
$a=12,\quad b=7$

So,
$(12+7)(12-7)=12^2-7^2$

$=144-49$

$=95$

Therefore, the value of
$(12+7)(12-7)$
is 95.

Key Concept: Difference of squares

d) 64
[Solution Description]
To find $10^2 - 6^2$, use the identity $a^2 - b^2 = (a + b)(a - b)$.
Here, $a = 10$ and $b = 6$.
Substituting these values into the identity:
$10^2 - 6^2 = (10 + 6)(10 - 6)$
Calculate each part step-by-step:
$10 + 6 = 16$
$10 - 6 = 4$
Now multiply them:
$16 \times 4 = 64$

Key Concept: Using the identity $a^2 - b^2 = (a + b

b) 2200
[Solution Description]

We know that
$a^2 - b^2 = (a-b)(a+b)$

Here,
$a = 47,\quad b = 3$

So,
$47^2 - 3^2 = (47-3)(47+3)$

$= 44 \times 50$
$= 2200$

Therefore, the value of
$47^2 - 3^2$
is 2200.

Key Concept: Square of binomial, Expanding expressions

c) $16x^2 + 56x + 49$
[Solution Description]
Applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 4x$ and $b = 7$, we get:
$(4x)^2 + 2(4x)(7) + 7^2 = 16x^2 + 56x + 49$

Key Concept: Difference of squares, Algebraic identities

a) 4891
[Solution Description]
We can express $73 \times 67$ as $(70 + 3)(70 - 3)$. Using the difference of squares identity, this becomes:
$70^2 - 3^2 = 4900 - 9 = 4891$

Key Concept: Using identities to simplify expressions

d) 64
[Solution Description]
To solve $(5 + 3)^2$, apply the identity $(a + b)^2 = a^2 + 2ab + b^2$.
Here, $a = 5$ and $b = 3$.
Substituting these values into the identity:
$(5 + 3)^2 = 5^2 + 2 \times 5 \times 3 + 3^2$
Calculate each term step-by-step:
$5^2 = 25$
$2 \times 5 \times 3 = 30$
$3^2 = 9$
Now add all terms:
$25 + 30 + 9 = 64$

Key Concept: Sum of Squares Pattern

c) 260
[Solution Description]
The sum of squares pattern is $2(a^2 + b^2) = (a + b)^2 + (a - b)^2$. Here, $a = 7$ and $b = 9$.
Step 1: Compute $(a + b)^2 = (7 + 9)^2 = 16^2 = 256$.
Step 2: Compute $(a - b)^2 = (7 - 9)^2 = (-2)^2 = 4$.
Step 3: Add results from Step 1 and Step 2: $256 + 4 = 260$.
Thus, $2(a^2 + b^2) = 260$.

Key Concept: Squares of numbers, Difference of squares

d) Assertion is false, but Reason is true.
[Solution Description]
Let us solve the assertion step by step. The given expression is $(k + 3)^2 - k^2$. Using the difference of squares identity, we can rewrite this expression as:
$(k + 3)^2 - k^2 = (k + 3 - k)(k + 3 + k) = 3(2k + 3)$
Now, simplify $3(2k + 3)$ further:
$3(2k + 3) = 6k + 9 = 3(2k + 3)$
For the expression to be divisible by 6, $6k + 9$ must be divisible by 6. However, $6k + 9 = 6k + 6 + 3 = 6(k + 1) + 3$, which leaves a remainder of 3 when divided by 6. Thus, it is not divisible by 6.
The Reason correctly uses the difference of squares identity and simplifies the expression to $3(2k + 3)$. However, the conclusion in the Assertion that it is divisible by 6 is incorrect because the expression evaluates to $6k + 9$, which is not divisible by 6 for all integers $k$.
Therefore, the Assertion is false, but the Reason is true.

Key Concept: Modified Sridharacharya's Identity

b) 2025
[Solution Description]
Modified Sridharacharya's Identity states that $a^2 = (a + b)(a - b) + b^2$. Let $a = 45$ and choose $b = 5$ for simplicity. Applying the identity:
$45^2 = (45 + 5)(45 - 5) + 5^2 = 50 \times 40 + 25 = 2000 + 25 = 2025$.

Key Concept: Sum of squares identity, Multi-step solution

c) 338
[Solution Description]
First, calculate $a^2 + b^2$ where $a = 5$ and $b = 12$:
$5^2 + 12^2 = 25 + 144 = 169$
Now, multiply by 2 as per the identity:
$2(169) = 338$
Alternatively, using the identity directly:
$(5 + 12)^2 + (5 - 12)^2 = 17^2 + (-7)^2 = 289 + 49 = 338$

Key Concept: Square of Sum Identity

c) $x^2 + 6x + 9$
[Solution Description]
The square of sum identity states that $(a + b)^2 = a^2 + 2ab + b^2$. For $(x + 3)^2$, let $a = x$ and $b = 3$. Applying the identity:
$(x + 3)^2 = x^2 + 2 \times x \times 3 + 3^2 = x^2 + 6x + 9$.

Key Concept: Difference of Squares Identity

c) $(5 + y)(5 - y)$
[Solution Description]
The difference of squares identity states that $a^2 - b^2 = (a + b)(a - b)$. For $25 - y^2$, we can write 25 as $5^2$, so $a = 5$ and $b = y$. Applying the identity:
$25 - y^2 = 5^2 - y^2 = (5 + y)(5 - y)$.

Key Concept: Square of a Number Using Identity

b) 2025
[Solution Description]
Use the identity $(a + b)(a - b) + b^2 = a^2$ with $a = 45$ and $b = 5$.
Step 1: Compute $(a + b)(a - b) = (45 + 5)(45 - 5) = 50 \times 40 = 2000$.
Step 2: Add $b^2 = 5^2 = 25$ to the result from Step 1.
Therefore, $45^2 = 2000 + 25 = 2025$.

Key Concept: Pattern recognition, Squares

c) 120
[Solution Description]
For Step $k = 10$, substitute into the equation:
$(10 + 1)^2 - 1 = 11^2 - 1 = 121 - 1 = 120$
Alternatively, using the other form of the identity:
$10^2 + 2 \times 10 = 100 + 20 = 120$
Both methods yield the same result.

Key Concept: Difference of Squares Pattern

c) 80
[Solution Description]
The difference of squares pattern is $a^2 - b^2 = (a + b)(a - b)$. Here, $a = 12$ and $b = 8$.
Step 1: Compute $(a + b) = 12 + 8 = 20$.
Step 2: Compute $(a - b) = 12 - 8 = 4$.
Step 3: Multiply results from Step 1 and Step 2: $20 \times 4 = 80$.
Thus, $(12^2 - 8^2) = 80$.

Key Concept: Special Cases of the Distributive Property: Difference of Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The difference of squares formula is given by $a^2 - b^2 = (a + b)(a - b)$. Here, $a = 25$ and $b = 9$. Applying the formula, we get $25^2 - 9^2 = (25 + 9)(25 - 9) = 34 \times 16$. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Square of a Number Using Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find $34^2$ using the identity $(a + b)(a - b) + b^2$, let $a = 34$ and $b = 4$. Then, $(34 + 4)(34 - 4) + 4^2 = 38 \times 30 + 16 = 1140 + 16 = 1156$. Thus, $34^2 = 1156$. The Assertion is true as the identity helps simplify the calculation. The Reason is also true as it correctly explains why the identity is useful. Moreover, the Reason is the correct explanation of the Assertion.

Key Concept: Difference of squares, Pattern recognition

a) $70^2 - 3^2$
[Solution Description]
To solve this, we identify that $67 = 70 - 3$ and $73 = 70 + 3$. Using the difference of squares identity: $(a + b)(a - b) = a^2 - b^2$, where $a = 70$ and $b = 3$. Thus:
$67 \times 73 = (70 + 3)(70 - 3) = 70^2 - 3^2 = 4900 - 9 = 4891$

Key Concept: Pattern and Algebraic Expression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $k^2 + 2k$ correctly gives the number of circles at any step $k$ in the pattern. This is true as verified by all four methods (Method 1: $(k + 1)^2 - 1$, Method 2: $k^2 + 2 \times k$, Method 3: $k \times (k + 1) + k$, and Method 4: $k \times (k + 2)$), which simplify to $k^2 + 2k$. Thus, the assertion is true.
The reason states that all four methods lead to the same simplified form $k^2 + 2k$. This is also true, as shown in the syllabus. Moreover, the reason correctly explains why the assertion is true because the equivalence of all methods confirms the validity of the expression $k^2 + 2k$ for the pattern.
Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Expression Verification

a) $(k + 1)^2 - 1$
[Solution Description]
We need to verify which expression simplifies to $k^2 + 2k$:
\begin{itemize}

Key Concept: Application

c) 24
[Solution Description]
Substitute $n = 4$ into the expression:
$n^2 + 2n = 4^2 + 2 \times 4 = 16 + 8 = 24$
There are 24 circles in Step 4.

Key Concept: Pattern Analysis

b) 35
[Solution Description]
To find the number of circles in Step 5, substitute $k = 5$ into the formula $k^2 + 2k$.
$5^2 + 2 \times 5 = 25 + 10 = 35$
Therefore, there are 35 circles in Step 5.

Key Concept: Pattern Interpretation and Algebraic Expression

b) 440
[Solution Description]
To find the number of circles in Step 20, substitute $k = 20$ into the given formula:
$\text{Number of circles} = k^2 + 2k = 20^2 + 2 \times 20 = 400 + 40 = 440$

Key Concept: Pattern Prediction

b) 7
[Solution Description]
Given $k^2 + 2k = 63$, we solve for $k$:
$k^2 + 2k - 63 = 0$
$(k + 9)(k - 7) = 0$
Solutions are $k = -9$ (invalid since step cannot be negative) and $k = 7$.
Therefore, $n = 7$.

Key Concept: Expression Evaluation

b) $(k + 1)^2 - 1$
[Solution Description]
The given formula is $k^2 + 2k$. Let's verify each option:
- Option 1: $k(k + 3) = k^2 + 3k$ → Not equivalent.
- Option 2: $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$ → Equivalent.
- Option 3: $k^2 + k + 2$ → Not equivalent.
- Option 4: $2k^2 + k$ → Not equivalent.
Hence, $(k + 1)^2 - 1$ is the correct equivalent expression.

Key Concept: Pattern Interpretation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the expression $k^2 + 2k$ correctly gives the number of circles in Step k of the pattern, which is confirmed by the syllabus material. The Reason states that all four methods lead to this same expression, which is also true as shown in the syllabus. Furthermore, the Reason correctly explains why the Assertion is valid since it demonstrates multiple approaches converging to the same result.

Key Concept: Simplification

b) $k^2 + 2k$
[Solution Description]
Simplify the given expression step-by-step:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
The simplified form is $k^2 + 2k$.

Key Concept: Multiple Methods to Solve a Problem

c) 35
[Solution Description]
The first method gives $5 \times (5 + 2) = 5 \times 7 = 35$. According to the problem statement, all correct methods must yield the same result, so the other method must also give 35.

Key Concept: Pattern Generalization, Algebraic Expression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let us simplify all four methods:
\textbf{Method 1}: $k \times (k + 2) = k^2 + 2k$
\textbf{Method 2}: $k \times (k + 1) + k = k^2 + k + k = k^2 + 2k$
\textbf{Method 3}: $k^2 + 2 \times k = k^2 + 2k$
\textbf{Method 4}: $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
All four methods simplify to the same expression $k^2 + 2k$, confirming that the assertion is true. The reason correctly explains why the assertion holds, as it demonstrates that multiple valid methods converge to the same result.

Key Concept: Pattern Recognition

c) 15
[Solution Description]
To find the number of circles in Step 3, substitute $k = 3$ into the given expression:
$k^2 + 2k = 3^2 + 2 \times 3 = 9 + 6 = 15$
So, there are 15 circles in Step 3.

Key Concept: Difference of Squares

c) $100^2 - 8^2$
[Solution Description]
To find $108 \times 92$ using the difference of squares identity $(a+b)(a-b)=a^2-b^2$, we need to express it in terms of $(a+b)(a-b)$.
Let's find $a$ and $b$ such that:
$a + b = 108$
$a - b = 92$
Adding these equations:
$2a = 200 \Rightarrow a = 100$
Subtracting them:
$2b = 16 \Rightarrow b = 8$
Now apply the identity:
$108 \times 92 = (100+8)(100-8) = 100^2 - 8^2 = 10000 - 64 = 9936$
We can also verify:
$108 \times 90 = 9720$
$108 \times 2 = 216$
Total $9720 + 216 = 9936$
Thus, the equivalent expression is $100^2 - 8^2$.

Key Concept: Difference of Squares Identity

c) 125
[Solution Description]
Apply the difference of squares identity:
$a^2 - b^2 = (a + b)(a - b)$.
Here, $a = 15$ and $b = 10$.
So, $15^2 - 10^2 = (15 + 10)(15 - 10) = 25 \times 5 = 125$.
Therefore, the value is 125.

Key Concept: Algebraic Pattern - Circles

c) 63
[Solution Description]
To find the number of circles in Step $k$, use the formula:
$k^2 + 2k$.
Substitute $k = 7$:
$7^2 + 2 \times 7 = 49 + 14 = 63$.
Therefore, there are 63 circles in Step 7.

Key Concept: Algebraic Pattern - Square Tiles

c) 35
[Solution Description]
The formula for the number of tiles is $(n + 1)^2 - 1$.
Substitute $n = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$.
Therefore, there are 35 tiles in Step 5.

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, evaluate the Assertion:
For $n = 10$, using the formula $n^2 + 2n$, we get $10^2 + 2 \times 10 = 100 + 20 = 120$. So, the Assertion is true.
Now, check the Reason:
Expand $(n+1)^2 - 1$:
$(n+1)^2 - 1 = n^2 + 2n + 1 - 1 = n^2 + 2n$. This matches the formula in the Assertion. Hence, the Reason is true and correctly explains the Assertion.

Key Concept: Pattern of Circles

c) 120
[Solution Description]
Substitute $k = 10$ into the expression $k^2 + 2k$:
$10^2 + 2 \times 10 = 100 + 20 = 120$
Thus, the number of circles in Step 10 is 120.

Key Concept: Algebraic Identities

b) 9996
[Solution Description]
Rewrite $98 \times 102$ as $(100 - 2)(100 + 2)$. Using the identity $a^2 - b^2 = (a + b)(a - b)$, we get:
$(100 - 2)(100 + 2) = 100^2 - 2^2 = 10000 - 4 = 9996$
So, $98 \times 102 = 9996$.

Key Concept: Sum of Squares, Difference of Squares

c) 361
[Solution Description]
First, identify $a$ and $b$ from the given expression $2(12^2 + 7^2)$. Here $a=12$ and $b=7$. Using the identity:
$2(a^2 + b^2) = (a+b)^2 + (a-b)^2$
Substitute the values:
$2(12^2 + 7^2) = (12+7)^2 + (12-7)^2$
Calculate each term:
$(12+7)^2 = 19^2 = 361$
$(12-7)^2 = 5^2 = 25$
Therefore, one of the terms in the sum is 361.

Key Concept: Sum of Squares, Difference of Squares

b) 8
[Solution Description]
Using the identity $2(a^2 + b^2) = (a+b)^2 + (a-b)^2$ with $a=9$ and $b=4$:
$2(9^2 + 4^2) = (9+4)^2 + (9-4)^2 = 13^2 + 5^2 = 169 + 25 = 194$
The two squared terms obtained are $169$ and $25$.
Their square roots are $\sqrt{169} = 13$ and $\sqrt{25} = 5$.
The difference between these square roots is $13 - 5 = 8$.
Alternatively, note that $(a+b)-(a-b) = 2b = 8$ which matches our calculation.

Key Concept: Increments in Products

c) $a + b + 1$
[Solution Description]
According to the given pattern, increasing both numbers $a$ and $b$ by 1 increases the product by $a + b + 1$. For example, with $a = 23$ and $b = 27$, the increment is $23 + 27 + 1 = 51$.

Key Concept: Sum of Squares Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let us expand both $(a + b)^2$ and $(a - b)^2$ using the algebraic identities:
Step 1: Expand $(a + b)^2$:
$(a + b)^2 = a^2 + 2ab + b^2$
Step 2: Expand $(a - b)^2$:
$(a - b)^2 = a^2 - 2ab + b^2$
Step 3: Add the two expansions:
$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)$
Step 4: Simplify the result:
$2a^2 + 2b^2 = 2(a^2 + b^2)$
Thus, the assertion is true because the equation holds for any natural numbers $a$ and $b$. The reason correctly explains how the identity is derived by adding the expansions of $(a + b)^2$ and $(a - b)^2$.

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's expand both sides of the given equation using algebraic identities:
Right-hand side (RHS):
$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2$
Left-hand side (LHS):
$2(a^2 + b^2) = 2a^2 + 2b^2$
Since LHS = RHS, the assertion is true.
The reason provides the correct expansion formulas for $(a + b)^2$ and $(a - b)^2$, which are directly used to prove the assertion. Hence, the reason is the correct explanation of the assertion.

Key Concept: Verification of Identity

d) $25a^2 - 70ab + 49b^2$
[Solution Description]
Apply the square of difference identity $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 5a$ and $b = 7b$:
$(5a - 7b)^2 = (5a)^2 - 2(5a)(7b) + (7b)^2$
$= 25a^2 - 70ab + 49b^2$

Key Concept: Square of Difference

a) $x^2 - 6x + 9$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, where $a = x$ and $b = 3$:
$(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$

Key Concept: Identity 1B: Square of Difference, Algebraic Manipulation

a) $9x^2 - 24xy + 16y^2$
[Solution Description]
To find $(3x - 4y)^2$, we use Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 3x$ and $b = 4y$.
Substitute these values into the identity:
$(3x - 4y)^2 = (3x)^2 - 2 \times 3x \times 4y + (4y)^2$
$= 9x^2 - 24xy + 16y^2$.

Key Concept: Square of Difference Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(x - 3)^2 = x^2 - 6x + 9$. Using the square of difference identity $(a - b)^2 = a^2 - 2ab + b^2$, we can verify this by substituting $a = x$ and $b = 3$. This gives $(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Identity 1B: Square of Difference, Numerical Application

b) 9604
[Solution Description]
Express $98$ as $100 - 2$. Using Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 100$ and $b = 2$.
$(100 - 2)^2 = 100^2 - 2 \times 100 \times 2 + 2^2$
$= 10000 - 400 + 4$
$= 9604$.

Key Concept: Square of Difference Identity

c) $9x^2 - 24xy + 16y^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 3x$ and $b = 4y$, we substitute:
$(3x - 4y)^2 = (3x)^2 - 2(3x)(4y) + (4y)^2$
$= 9x^2 - 24xy + 16y^2$

Key Concept: Square of Difference

b) 2401
[Solution Description]
We can write $49$ as $(50 - 1)$. Using $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 50$ and $b = 1$:
$49^2 = (50 - 1)^2 = 50^2 - 2(50)(1) + 1^2 = 2500 - 100 + 1 = 2401$

Key Concept: Square of Difference, Algebraic Identity

d) Assertion is false, but Reason is true.
[Solution Description]

Key Concept: Identity 1B: Square of Difference

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we expand $(3x - 4y)^2$ using Identity 1B:
$(3x - 4y)^2 = (3x)^2 - 2 \times 3x \times 4y + (4y)^2 = 9x^2 - 24xy + 16y^2$
The assertion matches the expanded form. The reason correctly states the identity used in the expansion and explains the assertion. Hence, both are true, and the reason is the correct explanation.

Key Concept: Identity 1C: Difference of Squares, Algebraic Manipulation

b) $25m^2 - 49n^2$
[Solution Description]
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$, where $a = 5m$ and $b = 7n$.
$(5m + 7n)(5m - 7n) = (5m)^2 - (7n)^2$
$= 25m^2 - 49n^2$.

Key Concept: Application of Square of Difference

d) 9801
[Solution Description]
Express $99^2$ as $(100 - 1)^2$. Using the square of difference identity:
$(100 - 1)^2 = 100^2 - 2(100)(1) + 1^2$
$= 10000 - 200 + 1 = 9801$

Key Concept: Square of Difference

c) $4y^2 - 20y + 25$
[Solution Description]
Using $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 2y$ and $b = 5$:
$(2y - 5)^2 = (2y)^2 - 2(2y)(5) + 5^2 = 4y^2 - 20y + 25$

Key Concept: Number Pattern Representation

b) $k^2 + 6k$
[Solution Description]
First, expand $(k + 3)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$.
So, $(k + 3)^2 = k^2 + 6k + 9$.
Now subtract 9 from this expression:
$k^2 + 6k + 9 - 9 = k^2 + 6k$.
Therefore, the simplified form is $k^2 + 6k$.

Key Concept: Mathematical Identities, Verification

a) $(m + n)^2 - 4mn = (m - n)^2$
[Solution Description]
Expand $(m + n)^2 - 4mn$:
$(m + n)^2 - 4mn = m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2$
Now expand $(n - m)^2$:
$(n - m)^2 = n^2 - 2mn + m^2$
Both results are identical ($m^2 - 2mn + n^2$), verifying the identity.

Key Concept: Pattern Recognition and Generalization, Algebraic Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, evaluate the Assertion:
The product $k \times (k + 2)$ expands as follows:
$k \times (k + 2) = k^2 + 2k$
This matches the assertion statement.
Next, evaluate the Reason:
The expression $(k + 1)^2 - 1$ simplifies as:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
This matches the reason statement.
Both statements are true, and since both methods lead to the same algebraic expression ($k^2 + 2k$), the Reason correctly explains why the simplification of the original product holds.

Key Concept: Number Patterns

c) 40
[Solution Description]
To find the number of dots in the 5th step, substitute $n = 5$ into the expression $n^2 + 3n$.
$5^2 + 3 \times 5 = 25 + 15 = 40$

Key Concept: Algebraic Identities

b) 58
[Solution Description]
We know that $(x + y)^2 = x^2 + 2xy + y^2$. Given $x + y = 10$ and $xy = 21$, we can substitute these values into the identity to find $x^2 + y^2$.
$(10)^2 = x^2 + 2(21) + y^2 \\ \\ 100 = x^2 + 42 + y^2 \\ \\ x^2 + y^2 = 100 - 42 = 58$

Key Concept: Algebraic Identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
All given expressions $(k + 1)^2 - 1$, $k \times (k + 2)$, and $k^2 + 2 \times k$ simplify to $k^2 + 2k$. For example:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$,
$k \times (k + 2) = k^2 + 2k$,
and $k^2 + 2 \times k = k^2 + 2k$.
Hence, all representations lead to the same result.

Key Concept: Pattern Representation, Simplification

b) $k \times (k + 2)$
[Solution Description]
We know the general form is $k^2 + 2k$. Let's verify the options:
1. $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
2. $k \times (k + 2) = k^2 + 2k$
3. $k^2 + k + (k + 1) - 1 = k^2 + 2k$
4. $(k + 1)(k + 1) = k^2 + 2k + 1$ (does not match).
Thus, options 1, 2, and 3 are equivalent to $k^2 + 2k$.

Key Concept: Area Calculations

b) $(a - b)^2$
[Solution Description]
The area of the square is $(a + b)^2$. The total area of the four rectangles is $4ab$. The area of the shaded region is the difference between the area of the square and the area of the rectangles.
$(a + b)^2 - 4ab = a^2 + 2ab + b^2 - 4ab = a^2 - 2ab + b^2 = (a - b)^2$

Key Concept: Pattern Analysis

c) 120
[Solution Description]
Substitute $n = 10$ into the pattern equation $n \times (n + 2) = n^2 + 2n$.
This gives $10 \times (10 + 2) = 10^2 + 2 \times 10$.
Simplifying the left side: $10 \times 12 = 120$.
Simplifying the right side: $100 + 20 = 120$.
Both sides give the same result, so the correct answer is 120.

Key Concept: Algebraic identities and pattern generalization

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion is true as shown by the multiple representations: $(k + 1)^2 - 1$, $k \times (k + 2)$, and $k^2 + 2k$ all simplify to the same expression. The Reason is also true but doesn't correctly explain the Assertion because while mathematical consistency ensures final answers match, it doesn't explain why different methods exist. The different methods arise from different perspectives in analyzing the pattern, not just from mathematical consistency.

Key Concept: Square Tiles Pattern

c) 49
[Solution Description]
The formula given is $n^2$ for Step $n$. Substituting $n = 7$, we get:
$7^2 = 49$.

Key Concept: Sridharacharya's Method for Squares, Application

c) 2025
[Solution Description]
Apply Sridharacharya's formula:
$a^2 = (a + b)(a - b) + b^2$
Here, $a = 45$, $b = 5$:
$(45 + 5)(45 - 5) + 5^2 = 50 \times 40 + 25 = 2000 + 25 = 2025$
Thus, $45^2 = 2025$.

Key Concept: Distributive Property

b) 312
[Solution Description]
We can express $13 \times 24$ as $(12 + 1)(25 - 1)$. Using the formula $(a + m)(b + n) = ab + an + bm + mn$, where $a = 12$, $b = 25$, $m = 1$, and $n = -1$, we get:
$12 \times 25 + 12 \times (-1) + 25 \times 1 + 1 \times (-1) = 300 - 12 + 25 - 1 = 312$
Thus, $13 \times 24 = 312$.

Key Concept: Incremental Changes in Products

b) $b$
[Solution Description]
When the first number $a$ is increased by 1, the new product becomes $(a + 1)b = ab + b$. The increase in the product is $(ab + b) - ab = b$.

Key Concept: Fast Multiplication, Distributive Property

b) $249268$
[Solution Description]
Rewrite 101 as $(100 + 1)$ and apply the distributive property:
$2468 \times 101 = 2468 \times (100 + 1)$
$= 2468 \times 100 + 2468 \times 1$
$= 246800 + 2468$
$= 249268$.

Key Concept: Incremental Changes in Products

b) $a + b + 1$
[Solution Description]
When both numbers are increased by 1, the new product becomes $(a + 1)(b + 1) = ab + a + b + 1$. The increase in the product is $(ab + a + b + 1) - ab = a + b + 1$.

Key Concept: Incremental Changes in Products

c) 30
[Solution Description]
The new product after increasing the first number by 3 and decreasing the second number by 2 is $(15 + 3)(22 - 2) = 18 \times 20 = 360$. The change in product is calculated as $360 - 330 = 30$.
Alternatively, using the formula for incremental changes:
$(a + m)(b + n) = ab + an + bm + mn$
Here, $a = 15$, $b = 22$, $m = 3$, and $n = -2$. Substituting these values, we get:
$15 \times (-2) + 22 \times 3 + 3 \times (-2) = -30 + 66 - 6 = 30$
Thus, the product increases by 30.

Key Concept: Incremental Changes in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the distributive property, when one of the numbers in a product (e.g., $a$) is increased by 1, the new product becomes $(a + 1)b = ab + b$. This means the increase in the product is equal to the other number ($b$). The reason correctly explains the assertion using the distributive property of multiplication.

Key Concept: Incremental Changes in Products

a) $a$
[Solution Description]
When the second number $b$ is increased by 1, the new product becomes $a(b + 1) = ab + a$. The increase in the product is $(ab + a) - ab = a$.

Key Concept: Incremental Changes in Products, Mixed Operations

a) $P + xy + 15x + 9y + 5$
[Solution Description]
Let $x' = x + 1$ and $y' = y - 2$. The new product is:
$(5(x + 1) + 2(y - 2))(3(x + 1) - (y - 2))$
Simplify each part:
$= (5x + 5 + 2y - 4)(3x + 3 - y + 2)$
$= (5x + 2y + 1)(3x - y + 5)$.
Expand using the distributive property:
$= 5x \cdot 3x + 5x \cdot (-y) + 5x \cdot 5 + 2y \cdot 3x + 2y \cdot (-y) + 2y \cdot 5 + 1 \cdot 3x + 1 \cdot (-y) + 1 \cdot 5$
$= 15x^2 - 5xy + 25x + 6xy - 2y^2 + 10y + 3x - y + 5$
Combine like terms:
$= 15x^2 + xy + 28x - 2y^2 + 9y + 5$.
The original product $P = 15x^2 - xy + 13x - 2y^2$.
The difference is $(xy + 15x + 9y + 5)$.

Key Concept: Incremental Changes in Products, Distributive Property

c) $p + 99$
[Solution Description]
Using the property $(a + m)(b + n) = ab + an + bm + mn$, where $a = 12$, $b = 18$, and $m = n = 3$:
$(12 + 3)(18 + 3) = 12 \times 18 + 12 \times 3 + 18 \times 3 + 3 \times 3$
$= p + 36 + 54 + 9$
$= p + 99$.

Key Concept: Incremental Changes in Products

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the Assertion and Reason, let us analyze the expression $(a + 1)(b - 1)$. Expanding this gives:
$(a + 1)(b - 1) = ab - a + b - 1 = ab + (b - a - 1).$
The Assertion states that $(a + 1)(b - 1) < ab$ for all $a$ and $b$, which implies $(b - a - 1) ab$. The Reason is also false because $(b - a - 1)$ is not always negative. Therefore, the correct answer is (d).

Key Concept: Incremental Changes in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that increasing both numbers in a product by 1 results in an increase of $a + b + 1$ in the product. This is correct because:
1. Starting with two numbers $a$ and $b$, their product is $ab$.
2. After increasing both by 1, we get $(a+1)(b+1)$.
3. Expanding this using the distributive property:
$(a+1)(b+1) = ab + a + b + 1$.
4. The difference between the new product and the original product is $(ab + a + b + 1) - ab = a + b + 1$.
The reason correctly explains the assertion by showing the algebraic expansion that justifies the increase. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Incremental Changes in Products

c) 26
[Solution Description]
The original product is $8 \times 17 = 136$. After increasing both numbers by 1, the new product is $(8 + 1)(17 + 1) = 9 \times 18 = 162$. The change in product is $162 - 136 = 26$.
Using the formula $(a + 1)(b + 1) = ab + (a + b + 1)$, we substitute $a = 8$, $b = 17$, which gives:
$8 \times 17 + (8 + 17 + 1) = 136 + 26 = 162$
Therefore, the product increases by 26.

Key Concept: Increment in Products

b) Decreases by 1
[Solution Description]
The initial product is $25 \times 40 = 1000$. After changing the numbers, the new product is $(25 + 2) \times (40 - 3) = 27 \times 37 = 999$. The change in the product is $999 - 1000 = -1$, meaning it decreases by 1.

Key Concept: Distributive Property, Increments in Products

c) 82
[Solution Description]
Original product: $P = 15 \times 24$.
New product after increment: $(15 + 2)(24 + 2) = 17 \times 26$.
Increase in product = $(17 \times 26) - (15 \times 24)$.
First, calculate $17 \times 26$:
$17 \times 20 = 340$ and $17 \times 6 = 102$, so $340 + 102 = 442$.
Next, calculate $15 \times 24$:
$15 \times 20 = 300$ and $15 \times 4 = 60$, so $300 + 60 = 360$.
Increase = $442 - 360 = 82$.
Alternatively, using the formula for increment when both numbers change:
Increase = $a \cdot n + b \cdot m + m \cdot n = 15 \times 2 + 24 \times 2 + 2 \times 2 = 30 + 48 + 4 = 82$.

Key Concept: Increment in Products

c) 23
[Solution Description]
The initial product is $10 \times 12 = 120$. When both numbers are increased by 1, the new product becomes $(10 + 1) \times (12 + 1) = 11 \times 13 = 143$. The increase in the product is $143 - 120 = 23$.

Key Concept: Generalization Using Distributive Property

b) $a + b + 1$
[Solution Description]
The original product is $ab$. After increasing both numbers by 1, the new product is $(a + 1)(b + 1)$. Using the distributive property:
$(a + 1)(b + 1) = ab + a + b + 1.$
The increase in the product is $(ab + a + b + 1) - ab = a + b + 1$.

Key Concept: Negative Numbers, Increments in Products

b) Decreases by 86
[Solution Description]
Original product: $P = (-7) \times 12 = -84$.
New product: $(-7 - 3)(12 + 5) = (-10) \times 17 = -170$.
Change in product = $-170 - (-84) = -86$.
Using the formula for increments:
Change = $a \cdot n + b \cdot m + m \cdot n = (-7) \times 5 + 12 \times (-3) + (-3) \times 5 = -35 - 36 - 15 = -86$.
The product decreases by 86.

Key Concept: Increments in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us first evaluate the Assertion:
Original product = $5 \times 7 = 35$.
New product when 5 is increased by 1: $(5 + 1) \times 7 = 6 \times 7 = 42$.
The product increases by $42 - 35 = 7$, so the Assertion is true.
Now, let's check the Reason:
The formula given is $(a + 1)(b) = ab + b$, which is correct. However, according to the Assertion, the increase is equal to b (which is 7 here). Therefore, the Reason correctly explains the Assertion.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Multiplicative Increment with Variables

b) $k(y - x) - k^2$
[Solution Description]
Original product: $P = xy$.
New product: $(x + k)(y - k) = xy - xk + ky - k^2$.
Change in product = $(xy - xk + ky - k^2) - xy = -xk + ky - k^2$.
This can also be written as $k(y - x) - k^2$.
Depending on the values of $x$, $y$, and $k$, the change could be positive, negative, or zero.

Key Concept: Increments in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when both numbers in a product are increased by 1, the increase in the product is equal to the sum of the original numbers plus 1. The reason explains this using the distributive property of multiplication, which expands $(a + 1)(b + 1)$ to $ab + a + b + 1$. Here, the increase in the product is $a + b + 1$, validating the assertion. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Decrease in One Number and Increase in the Other

a) $-a + b - 1$
[Solution Description]
The original product is $ab$. The new product after changes is $(a + 1)(b - 1)$.
Expanding this using the distributive property:
$(a + 1)(b - 1) = ab - a + b - 1.$
The change in the product is $(ab - a + b - 1) - ab = -a + b - 1$.

Key Concept: Increment in Products

b) 20
[Solution Description]
The initial product is $15 \times 20 = 300$. If the first number is increased by 1, the new product becomes $(15 + 1) \times 20 = 16 \times 20 = 320$. The increase in the product is $320 - 300 = 20$.

Key Concept: Increments in Products

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, let's verify the Assertion (A). Consider the product $(a + 1)(b - 1)$. Increasing both numbers by 1 gives $(a + 2)(b)$. Using the distributive property:
$(a + 2)(b) = ab + 2b$
The increase in the product is $(ab + 2b) - (a + 1)(b - 1)$. Expanding $(a + 1)(b - 1)$ as per the given formula yields:
$(a + 1)(b - 1) = ab + b - a - 1$
Thus, the difference is:
$(ab + 2b) - (ab + b - a - 1) = b + a + 1$
So, the Assertion (A) is true.
Now, consider the Reason (R). The identity $(a + m)(b + n) = ab + mb + an + mn$ is a standard algebraic identity that holds true for all integers $a, b, m,$ and $n$, making the Reason (R) true as well. However, the Reason (R) is not the correct explanation for the Assertion (A), because the Assertion deals with a specific case of increasing both numbers in the product $(a + 1)(b - 1)$ by 1, whereas the Reason provides a general identity.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.