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 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: 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-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: 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: 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: 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: 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: 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: 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: 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: 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: Algebraic Identities

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that $(a + b)(a - b) = a^2 - b^2$, which is true as it represents the difference of squares identity. The reason states that the product of two binomials can be simplified using the distributive property, which is also true. However, the reason does not specifically explain why $(a + b)(a - b)$ equals $a^2 - b^2$; it merely states a general property. Therefore, both are true, but the reason does not correctly explain the assertion.

Key Concept: Verification of Identity

c) $4p^2 + 4pq + q^2$
[Solution Description]
Use the identity $(a + b)^2 = a^2 + 2ab + b^2$:
$(2p + q)^2 = (2p)^2 + 2 \times 2p \times q + q^2 = 4p^2 + 4pq + q^2$
The correct expansion is $4p^2 + 4pq + q^2$.

Key Concept: Difference of squares, Remainder theorem

c) 5
[Solution Description]
Using remainder theorem:
Let $A = 7k + 3$, $B = 7m + 5$
Using difference of squares:
$A^2 - B^2 = (A+B)(A-B) = (7k+3+7m+5)(7k+3-7m-5) = (7(k+m)+8)(7(k-m)-2)$
When divided by 7:
$(A+B) \equiv 1 \mod 7$ (since 8 mod 7 = 1)
$(A-B) \equiv 5 \mod 7$ (since -2 mod 7 = 5)
Final remainder: $1 \times 5 = 5 \mod 7$

Key Concept: Simplifying Expressions

b) $5s$
[Solution Description]
First, distribute $\frac{1}{2}$ inside the brackets:
$\frac{1}{2}(10s - 6) = 5s - 3$
Then add 3 to the result:
$5s - 3 + 3 = 5s$
The simplified form is $5s$.

Key Concept: Number patterns, Algebraic identities

b) $n^2 - (n^2 - 1) = 1$
[Solution Description]
Let the numbers be $(n-1)$, $n$, and $(n+1)$.
Square of middle number: $n^2$
Product of other two: $(n-1)(n+1) = n^2 - 1$
Required expression: $n^2 - (n^2 - 1) = 1$
This shows the result is always 1 regardless of $n$.

Key Concept: Simplification of Expressions

c) $3a + 17$
[Solution Description]
First, expand both terms:
$5(a + 3) = 5a + 15$
$2(a - 1) = 2a - 2$
Now subtract the second expansion from the first:
$(5a + 15) - (2a - 2) = 3a + 17$
The simplified form is $3a + 17$.

Key Concept: Correcting misconceptions in algebraic expressions

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the expansion of $(5m + 6n)^2$ is $25m^2 + 36n^2$. However, this is incorrect because the correct expansion using the identity $(a + b)^2 = a^2 + 2ab + b^2$ gives $25m^2 + 60mn + 36n^2$. The reason states the incorrect identity $(a + b)^2 = a^2 + b^2$, which is false. Therefore, the assertion is false, and the reason is also false.

Key Concept: Algebraic Expansion

c) $-12x^2 + 8xy$
[Solution Description]
To simplify $-4x(3x - 2y)$, distribute $-4x$ to each term inside the parentheses:
$-4x \times 3x = -12x^2$
$-4x \times (-2y) = 8xy$
The simplified form is $-12x^2 + 8xy$.

Key Concept: Algebraic identities, Error correction

b) $9x^2 + 24xy + 16y^2$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, we expand $(3x + 4y)^2$ step by step:
$(3x)^2 + 2 \times 3x \times 4y + (4y)^2 = 9x^2 + 24xy + 16y^2$
The mistake was missing the middle term $24xy$.

Key Concept: Algebraic Identities

b) $a^2 - 2ab + b^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, we get:
$(a - b)^2 = a^2 - 2ab + b^2$
This matches the algebraic identity perfectly.

Key Concept: Correcting algebraic misconceptions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(4x + 3)^2 = 16x^2 + 24x + 9$, which is correct as it follows the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 4x$ and $b = 3$. Applying the identity: $(4x)^2 + 2(4x)(3) + 3^2 = 16x^2 + 24x + 9$. The reason correctly states the formula for expanding $(a + b)^2$ and explains why the assertion is true.

Key Concept: Expanding and Simplifying Expressions

b) $5x + 10$
[Solution Description]
First, distribute the coefficients inside the brackets:
$2(x - 1) = 2x - 2$
$3(x + 4) = 3x + 12$
Now, combine like terms:
$2x - 2 + 3x + 12 = (2x + 3x) + (-2 + 12) = 5x + 10$
The correct simplified form is $5x + 10$.

Key Concept: Squaring numbers using identities

c) 27225
[Solution Description]
To compute $165^2$ using the given identity, let us choose $b = 5$:
Step 1: Apply the identity:
$165^2 = (165 + 5)(165 - 5) + 5^2$
Step 2: Compute $(165 + 5) = 170$ and $(165 - 5) = 160$
Step 3: Multiply them: $170 \times 160 = 27200$
Step 4: Add $5^2 = 25$: $27200 + 25 = 27225$
Therefore, $165^2 = 27225$.

Key Concept: Quick multiplication using identities

c) 88804
[Solution Description]
To find $298^2$ using Sridharacharya's identity $a^2 = (a + b)(a - b) + b^2$, we choose $b = 2$ since it makes $(a + b)$ and $(a - b)$ easy to multiply.
Step 1: Apply the identity:
$298^2 = (298 + 2)(298 - 2) + 2^2$
Step 2: Calculate $(298 + 2) = 300$ and $(298 - 2) = 296$
Step 3: Multiply them: $300 \times 296 = 88800$
Step 4: Add $2^2 = 4$ to the result: $88800 + 4 = 88804$
Therefore, $298^2 = 88804$.

Key Concept: Multiplication with 11, 101, etc.

d) 732553
[Solution Description]
To multiply $7253 \times 101$, we use the identity for multiplying by 101 ($100 + 1$):
Step 1: Break down the multiplication:
$7253 \times 101 = 7253 \times (100 + 1) = 7253 \times 100 + 7253 \times 1$
Step 2: Compute $7253 \times 100 = 725300$
Step 3: Compute $7253 \times 1 = 7253$
Step 4: Add the results: $725300 + 7253 = 732553$
Therefore, $7253 \times 101 = 732553$.

Key Concept: Quick multiplication using identities, distributive property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Solution:
Step 1: Identify $a$ and $b$ in the given problem. Here, $197 = 200 - 3$ and $203 = 200 + 3$, so $a = 200$ and $b = 3$.
Step 2: Apply the identity $(a + b)(a - b) = a^2 - b^2$. Substituting $a = 200$ and $b = 3$, we get:
$(200 + 3)(200 - 3) = 200^2 - 3^2 = 40000 - 9 = 39991$
Step 3: Therefore, $197 \times 203 = 39991$.
The Assertion correctly identifies the application of the identity $(a + b)(a - b) = a^2 - b^2$, and the Reason provides the correct explanation for why this method works (i.e., the numbers are equidistant from $200$).

Key Concept: Quick multiplication using identities

c) 495
[Solution Description]
To find $45 \times 11$, use the distributive property:
$45 \times 11 = 45 \times (10 + 1) = 45 \times 10 + 45 \times 1 = 450 + 45 = 495$.

Key Concept: Sridharacharya's method for squares

c) 2401
[Solution Description]
Using the identity $(a + b)(a - b) + b^2 = a^2$, choose $b = 1$ for $a = 49$.
Therefore, $(49 + 1)(49 - 1) + 1^2 = 50 \times 48 + 1 = 2400 + 1 = 2401$.

Key Concept: Quick multiplication using identities

b) 627
[Solution Description]
Using the distributive property for multiplying by 11:
Break down $57 \times 11 = 57 \times (10 + 1) = 57 \times 10 + 57 \times 1 = 570 + 57 = 627$.

Key Concept: Pattern Recognition in Products

c) $x^2 - 9$
[Solution Description]
Using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$, substitute $a = x$ and $b = 3$:
$(x - 3)(x + 3) = x^2 - 3^2 = x^2 - 9$.

Key Concept: Pattern recognition in identities

b) $x^3 - y^3$
[Solution Description]
The given expression follows the pattern $(a - b)(a^2 + ab + b^2) = a^3 - b^3$.
Thus, $(x - y)(x^2 + xy + y^2) = x^3 - y^3$.

Key Concept: Sridharacharya's Method for Squares

c) 2704
[Solution Description]
Using the identity $(a^2 = (a + b)(a - b) + b^2)$, set $a = 52$ and $b = 2$:
$52^2 = (52 + 2)(52 - 2) + 2^2 = 54 \times 50 + 4 = 2700 + 4 = 2704$.

Key Concept: Quick multiplication using identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion with an example:
Take a 3-digit number, say $123$. Multiplying it by $101$:
$123 \times 101 = 123 \times (100 + 1) = 12300 + 123 = 12423$
As seen, the result is obtained by writing the number $123$ twice ($123$ followed by $23$ for a 3-digit case).
The reason correctly explains the assertion using the distributive property $(a \times (100 + 1) = 100a + a)$. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Quick multiplication using identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To multiply a two-digit number $ab$ by 11, we use the distributive property:
$ab \times 11 = ab \times (10 + 1) = ab \times 10 + ab \times 1$.
This results in shifting the number left by one digit and adding the original number:
$ab0 + ab = a \quad (a + b) \quad b$.
For example, $42 \times 11 = 420 + 42 = 462$.
The Assertion is true, and the Reason correctly explains it.

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: 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

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: 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: 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: 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 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: 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: 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

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: Fast Multiplication Using Distributive Property

c) 8679
[Solution Description]
Using the distributive property for $789 \times 11$:
$789 \times 11 = 789 \times (10 + 1) = 7890 + 789 = 8679$

Key Concept: Fast Multiplication Using Distributive Property

a) 5016
[Solution Description]
Apply the distributive property to $456 \times 11$:
$456 \times 11 = 456 \times (10 + 1) = 4560 + 456 = 5016$

Key Concept: Fast Multiplication Using Distributive Property

b) 461267
[Solution Description]
To multiply $4567 \times 101$, apply the distributive property:
$4567 \times 101 = 4567 \times (100 + 1) = 456700 + 4567 = 461267$

Key Concept: Distributive Property, Fast Multiplication, Error Analysis

a) Used $1357 \times (10 + 1)$ instead of $1357 \times (100 + 1)$
[Solution Description]
The correct calculation for $1357 \times 101$ using the distributive property is:
$1357 \times 101 = 1357 \times (100 + 1) = 135700 + 1357 = 137057$
The student's answer was $136907$, which is incorrect. The mistake likely occurred because the student might have added $1357$ to $13570$ instead of $135700$, i.e., they used $1357 \times (10 + 1)$ instead of $1357 \times (100 + 1)$.

Key Concept: Distributive Property, Fast Multiplication

b) $abcde \times (100 + 1) = abcde \times 100 + abcde \times 1$
[Solution Description]
To multiply a 5-digit number $abcde$ by 101, we can use the distributive property:
$abcde \times 101 = abcde \times (100 + 1) = abcde \times 100 + abcde \times 1$
This simplifies to:
$abcde00 + abcde = abcde00 + abcde$
The result is obtained by shifting the original number left by two digits and adding the original number.

Key Concept: Fast Multiplications Using the Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the product of 2468 × 11 using the distributive property:
Step 1: Multiply 2468 by 10: $2468 × 10 = 24680$
Step 2: Multiply 2468 by 1: $2468 × 1 = 2468$
Step 3: Add the partial results: $24680 + 2468 = 27148$
The assertion states that this can also be done by adding adjacent digits:
2 (thousands place), (2+4=6) (hundreds place), (4+6=10) (tens place), (6+8=14) (units place), and 8 (last digit). However, carrying over is needed here:
2 → 6 → (10 makes 0 with carryover 1 to next digit) → (14 + 1 = 15 makes 5 with carryover 1 to next digit) → (8 + 1 = 9).
Final result: 27148, which matches previous calculation. The reason correctly explains the method using distributive property.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Fast Multiplication Using Distributive Property

b) 1353
[Solution Description]
To solve $123 \times 11$, we can use the distributive property:
$123 \times 11 = 123 \times (10 + 1) = 1230 + 123 = 1353$

Key Concept: Fast Multiplication Using Distributive Property

b) 8679
[Solution Description]
Multiply $789 \times 11$ by breaking it down as:
$789 \times 11 = 789 \times (10 + 1) = 7890 + 789 = 8679$
Alternatively, applying digit-wise addition for a 3-digit number:
$(7)\ (8 + 7)\ (9 + 8)\ (9) = (7)\ (15)\ (17)\ (9)$
Carry over the excess:
$7\ (15 + 1)\ (7 + 1)\ 9 = 7\ 16\ 8\ 9 = 8679$

Key Concept: Distributive Property, Fast Multiplication

c) 27148
[Solution Description]
To find $2468 \times 11$, we can break it down using the distributive property:
$2468 \times 11 = 2468 \times (10 + 1) = 24680 + 2468$
Now, perform the addition:
$24680 + 2468 = 27148$
So, $2468 \times 11 = 27148$.

Key Concept: Fast Multiplication Using Distributive Property

c) 13574
[Solution Description]
To multiply $1234 \times 11$, we use the distributive property:
$1234 \times 11 = 1234 \times (10 + 1) = 12340 + 1234 = 13574$
Alternatively, using digit-wise addition:
$1\ (2+1)\ (3+2)\ (4+3)\ 4 = 1\ 3\ 5\ 7\ 4 = 13574$

Key Concept: Fast Multiplications Using the Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because using the distributive property simplifies multiplication of large numbers by breaking them into addition of shifted numbers (e.g., $3874 \times 11 = 38740 + 3874$). The reason is also true as it correctly defines the distributive property. Moreover, the reason provides the mathematical basis for the assertion, making it the correct explanation.

Key Concept: Distributive Property, Fast Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion step by step:
1. Using the distributive property: $2468 \times 11 = 2468 \times (10 + 1) = 24680 + 2468$
2. Add these two results:
$$ \\ 24680 + 2468 = 27148$$
3. Hence, the assertion is true as $2468 \times 11 = 27148$.
4. Now let's check the reason: For a 4-digit number $abcd \times 11$, the calculation follows the pattern:
$$ \\ a \, (a + b) \, (b + c) \, (c + d) \, d$$
5. For $2468 \times 11$:
$2$ (thousands), $2 + 4 = 6$ (hundreds), $4 + 6 = 10$ (tens, carry over 1 to hundreds), $6 + 8 = 14$ (units, carry over 1 to tens). Final digits are adjusted with carry-over:
Thousands: $2$
Hundreds: $6 + 1 (\text{carry}) = 7$
Tens: $0 + 1 (\text{carry}) = 1$
Units: $4$
Extending it further with the last digit $8$ gives $27148$.
6. The reason correctly explains the pattern and how the digits are derived, including handling of carry-over. Thus, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Pattern Formula Application

b) 255
[Solution Description]
To find the number of circles at Step 15, substitute $k = 15$ into the formula:
$15^2 + 2 \times 15 = 225 + 30 = 255$
Therefore, the number of circles is 255.

Key Concept: Consistency of expressions across different methods

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion is true because when we simplify both expressions:
$k \times (k + 2) = k^2 + 2k$
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
Both expressions simplify to the same form, which proves their consistency. The reason is also true and correctly explains the assertion, as different algebraic methods indeed lead to the same simplified result.

Key Concept: Pattern recognition, algebraic simplification

b) $k^2 + 5k + 8$
[Solution Description]
First, expand $(k + 3)^2$ using the formula $(a + b)^2 = a^2 + 2ab + b^2$. This gives $k^2 + 6k + 9$. Next, subtract $(k + 1)$ from this result: $k^2 + 6k + 9 - k - 1 = k^2 + 5k + 8$. Therefore, the simplified form is $k^2 + 5k + 8$.

Key Concept: Pattern Formula Verification

c) $k^2 + 2k$
[Solution Description]
Expand the given expression:
$k \times (k + 2) = k^2 + 2k$
Now compare it with other expressions. The option that matches $k^2 + 2k$ is correct.

Key Concept: Consistency of expressions, Algebraic identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's simplify both expressions:
1. For $k \times (k + 2)$, we can expand it as follows:
$k \times (k + 2) = k^2 + 2k$
2. For $(k + 1)^2 - 1$, we use the formula $(a + b)^2 = a^2 + 2ab + b^2$:
$(k + 1)^2 - 1 = (k^2 + 2k + 1) - 1 = k^2 + 2k$
Both expressions simplify to $k^2 + 2k$, confirming they are equivalent for all integer values of $k$. Therefore, Assertion is true.
The Reason states that both expressions simplify to the same form $k^2 + 2k$, which is also true and correctly explains the Assertion.

Key Concept: Expression Simplification

b) $k^2 + 2k$
[Solution Description]
Expand and simplify the expression:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
The simplified form is $k^2 + 2k$.

Key Concept: Real-world application of patterns, step calculation

c) 70
[Solution Description]
Substitute $n = 7$ into the expression $n(n + 3)$: $7 \times (7 + 3) = 7 \times 10 = 70$. Thus, 70 tiles are needed for Step 7.

Key Concept: Multiple methods for pattern representation, algebraic equivalence

c) Yes, they simplify to $k^2 + 4k$
[Solution Description]
First, simplify each expression:
1) $(k + 2)^2 - 4 = k^2 + 4k + 4 - 4 = k^2 + 4k$.
2) $k(k + 4) = k^2 + 4k$.
3) $4k + k^2 = k^2 + 4k$ (rearranged).
All three expressions simplify to $k^2 + 4k$, confirming their equivalence.

Key Concept: Pattern analysis

c) 48
[Solution Description]
Given the formula for the number of circles at Step k is $k^2 + 2k$. For Step 6, substitute $k=6$ into the formula:
$6^2 + 2 \times 6 = 36 + 12 = 48$ circles.

Key Concept: Consistency of expressions across different methods

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's evaluate Assertion (A): The expression provided is $k \times (k + 2)$, which simplifies to $k^2 + 2k$. This matches the general form given in the syllabus for the number of circles at Step k.
Next, let's evaluate Reason (R): The simplified form of $k \times (k + 2)$ indeed gives $k^2 + 2k$, and the syllabus confirms that this matches other alternative forms like $(k + 1)^2 - 1$ or $k \times (k + 1) + k$ when simplified further.
Since both Assertion and Reason are true, and Reason correctly explains why Assertion holds, the correct answer is option a).

Key Concept: Consistency of expressions

a) $(k+1)^2 -1$
[Solution Description]
To find an expression equal to $k^2 + 2k$, let's simplify each option:
1. Expanding $(k+1)^2 -1$ gives $k^2 + 2k + 1 -1 = k^2 + 2k$. This matches.
2. Simplifying $k(k + 1) + k$ gives $k^2 + k + k = k^2 + 2k$. This matches.
3. Factorizing $k(k + 2)$ directly gives $k^2 + 2k$. This matches.
4. The expression $(k + 2)^2 -4$ simplifies to $k^2 + 4k +4 -4 = k^2 +4k$ which doesn't match.
Therefore, options a), b), and c) are equivalent to $k^2 + 2k$.

Key Concept: Multiple methods

d) Method 4: $(k + 2)^2 - 4$
[Solution Description]
Let's verify each method:
1. Method 1: $k \times (k + 2) = k^2 + 2k$ — correct.
2. Method 2: $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$ — correct.
3. Method 3: $k \times (k + 1) + k = k^2 + k + k = k^2 + 2k$ — correct.
4. Method 4: $(k + 2)^2 - 4 = k^2 + 4k + 4 - 4 = k^2 + 4k$ — incorrect as it gives $k^2 +4k$ instead of $k^2 +2k$.
Thus, Method 4 is incorrect.

Key Concept: Square of Sum Identity

d) Assertion is false, but Reason is true.
[Solution Description]
Let us analyze the assertion and reason:
1. The assertion claims $(a + b)^2 > a^2 + b^2$ only when both $a$ and $b$ are positive.
2. The reason states that $2ab$ is always non-negative for all integers $a$ and $b$.
Testing the assertion:
- If $a = 2$ and $b = 3$, $(2+3)^2 = 25$ while $2^2 + 3^2 = 13$, so it holds ($25 > 13$).
- If $a = -2$ and $b = -3$, $(-2-3)^2 = 25$ while $(-2)^2 + (-3)^2 = 13$, still $25 > 13$.
- If one is positive and other negative, say $a=2$, $b=-3$, then $(2-3)^2 = 1$ vs $4 + 9 = 13$, now $1 a^2 + b^2$ can be true even when both numbers are negative.
Testing the reason:
- $2ab$ is not always non-negative. Example: $a=1$, $b=-1$ gives $2(1)(-1) = -2$ which is negative.
- Hence, the reason is false.
Therefore, Assertion is false and Reason is also false.

Key Concept: Square of Sum

c) 1369
[Solution Description]
To find $37^2$ using Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$, decompose 37 as $30 + 7$. Then, apply the identity:
$37^2 = (30 + 7)^2 = 30^2 + 2 \times (30 \times 7) + 7^2 = 900 + 420 + 49 = 1369.$

Key Concept: Square of Sum

c) $16x^2 + 24x + 9$
[Solution Description]
To find $(4x + 3)^2$, we use Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 4x$ and $b = 3$. Applying the identity, we get:
$(4x + 3)^2 = (4x)^2 + 2 \times (4x \times 3) + 3^2 = 16x^2 + 24x + 9.$

Key Concept: Difference of Squares

c) 2800
[Solution Description]
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$. Here, $a = 107$, $b = 93$.
$107^2 - 93^2 = (107 + 93)(107 - 93) = 200 \times 14 = 2800$

Key Concept: Square of Sum

d) Assertion is false, but Reason is true.
[Solution Description]
First, calculate the left-hand side: $(3 + 4)^2 = 7^2 = 49$.
Next, apply the formula for the square of the sum: $(3 + 4)^2 = 3^2 + 2(3)(4) + 4^2 = 9 + 24 + 16 = 49$.
Thus, both Assertion and Reason are true.
However, the value in Assertion is incorrect ($25$ instead of $49$), while the Reason correctly states the formula.

Key Concept: Square of Sum Identity

b) $x^2 + 8x + 16$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = x$ and $b = 4$:
$(x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$

Key Concept: Square of Difference

c) $25p^2 - 70pq + 49q^2$
[Solution Description]
Using Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 5p$ and $b = 7q$. Substituting into the identity:
$(5p - 7q)^2 = (5p)^2 - 2(5p)(7q) + (7q)^2 = 25p^2 - 70pq + 49q^2$

Key Concept: Square of Sum Application

b) $9y^2 + 42y + 49$
[Solution Description]
Apply the identity $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 3y$ and $b = 7$:
$(3y + 7)^2 = (3y)^2 + 2(3y)(7) + 7^2 = 9y^2 + 42y + 49$

Key Concept: Square of Sum Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's evaluate the assertion and reason step by step.
1) **Assertion Evaluation**:
Using the given identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = 3x$ and $b = 4$.
$(3x + 4)^2 = (3x)^2 + 2 \times (3x \times 4) + 4^2 = 9x^2 + 24x + 16$
This matches the assertion, so the assertion is true.
2) **Reason Evaluation**:
The reason states the correct algebraic identity for the square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. This is mathematically accurate, so the reason is also true.
3) **Explanation Check**:
The reason correctly explains the assertion because it provides the identity used to derive the expanded form in the assertion. Thus, the reason is the correct explanation of the assertion.

Key Concept: Numerical Calculation

c) 625
[Solution Description]
Write $25$ as $(20 + 5)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$:
$25^2 = (20 + 5)^2 = 20^2 + 2(20)(5) + 5^2 = 400 + 200 + 25 = 625$

Key Concept: Square of Sum

c) $9x^2 + 24xy + 16y^2$
[Solution Description]
Using Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 3x$ and $b = 4y$. Substituting into the identity:
$(3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2$

Key Concept: Square of Sum

c) 7
[Solution Description]
The given expression is $(m + n)^2 = m^2 + 14m + n^2$. Using Identity 1A:
$(m + n)^2 = m^2 + 2mn + n^2.$
Comparing with the given expression, we get:
$2mn = 14m \implies n = \frac{14m}{2m} = 7.$

Key Concept: Special Case Identity

c) -6
[Solution Description]
Using the identity for change in product when one term decreases and another increases:
$\text{Change} = an + bm + mn$
Here, $a = 2$, $b = 3$, $m = -2$, $n = 3$. Plugging in the values:
$2 \times 3 + 3 \times (-2) + (-2) \times 3 = 6 - 6 - 6 = -6$

Key Concept: Identity Application

c) 13
[Solution Description]
The original product is $ab = 5 \times 7 = 35$.
The new product is $(5 + 1)(7 + 1) = 6 \times 8 = 48$.
The increase is $48 - 35 = 13$.

Key Concept: General Identity for Product Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we use the general identity for product change:
$(a + m)(b + n) = ab + mb + an + mn$
Here, $m = 3$ and $n = -2$. Substituting these values into the identity:
$(a + 3)(b - 2) = ab + 3b + a(-2) + 3(-2) = ab + 3b - 2a - 6$
The assertion correctly applies the identity. The reason explains the process of expanding the product by multiplying each term in the brackets and considering their signs, which is accurate. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Special Case Identity

c) $4x^2 + 12x + 9$
[Solution Description]
The identity $(a + b)^2 = a^2 + 2ab + b^2$ is used here with $a = 2x$ and $b = 3$.
Substituting, we get:
$(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$

Key Concept: Distributive Property

c) $xy - 4x + 3y - 12$
[Solution Description]
We expand the expression $(x + 3)(y - 4)$ using the distributive property:
$(x + 3)(y - 4) = x(y) + x(-4) + 3(y) + 3(-4)$
Simplifying each term gives:
$xy - 4x + 3y - 12$

Key Concept: Product Change Identity

a) $3b - 2a - 6$
[Solution Description]
Using Identity 1: $(a + m)(b + n) = ab + mb + an + mn$. Here, $m = 3$ and $n = -2$.
Substituting, we get:
$(a + 3)(b - 2) = ab + 3b - 2a - 6$
The change in product is $3b - 2a - 6$, which is derived from subtracting the original product $ab$ from the new product.

Key Concept: Product Change, Distributive Property

b) $3a - 2b - 6$
[Solution Description]
Let the initial numbers be $a$ and $b$. The new numbers are $a - 2$ and $b + 3$. Using the distributive property:
$(a - 2)(b + 3) = ab + 3a - 2b - 6$
The change in the product is $(ab + 3a - 2b - 6) - ab = 3a - 2b - 6$.

Key Concept: General Identity for Product Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's expand $(a + 1)(b - 1)$ using the general identity $(a + m)(b + n) = ab + mb + an + mn$. Substitute $m = 1$ and $n = -1$ into the identity:
$(a + 1)(b - 1) = ab + (1)(b) + (a)(-1) + (1)(-1) = ab + b - a - 1$
This matches the given assertion, so the Assertion (A) is true. The Reason (R) correctly explains the assertion by referring to the general identity used for expanding the product.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Special Identities, Binomial Expansion

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

Key Concept: Expansion of Binomials

b) $xy - 4y + 5x - 20$
[Solution Description]
Use the identity $(a - u)(b + v) = ab - ub + av - uv$. Here, $a = x$, $u = 4$, $b = y$, and $v = 5$.
Substituting, we get:
$(x - 4)(y + 5) = xy - 4y + 5x - 20$

Key Concept: Product Change When Both Numbers Are Altered, Special Cases

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, when both numbers are increased by 1, the new product is $(a + 1)(b + 1) = ab + a + b + 1$. Comparing this with the original product $ab$, the increase in the product is $(ab + a + b + 1) - ab = a + b + 1$. Thus, the Assertion is true. The Reason correctly states the algebraic expansion of $(a + 1)(b + 1)$, which also matches the syllabus. Moreover, the Reason explains why the Assertion is true because it shows how the increase ($a + b + 1$) arises from the identity.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Distributive Property, Algebraic Identities

a) $xy - 3y + 4x - 12$
[Solution Description]
Using the identity $(a + m)(b + n) = ab + mb + an + mn$, let $m = -3$ and $n = 4$. Substituting these values into the identity:
$(x - 3)(y + 4) = xy + (-3)y + x(4) + (-3)(4) = xy - 3y + 4x - 12$

Key Concept: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we expand $(k + 1)^2 - 1$ step by step:
Step 1: Apply the identity $(a + b)^2 = a^2 + 2ab + b^2$.
So, $(k + 1)^2 = k^2 + 2 \times k \times 1 + 1^2 = k^2 + 2k + 1$.
Step 2: Subtract 1 from the expanded form.
Therefore, $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$.
This matches the given expression $k^2 + 2k$. Hence, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Distributive Property, Pattern Recognition

b) $9951$
[Solution Description]
Express $107 \times 93$ as $(100 + 7)(100 - 7)$.
Using the identity $(a + b)(a - b) = a^2 - b^2$, substitute $a = 100$ and $b = 7$:
$(100 + 7)(100 - 7) = 100^2 - 7^2 = 10000 - 49 = 9951$.
The result is $9951$.

Key Concept: Distributive Property

b) 91
[Solution Description]
We can rewrite $13 \times 7$ as $(10 + 3)(10 - 3)$. Using the identity $(a + b)(a - b) = a^2 - b^2$, we get:
$(10 + 3)(10 - 3) = 10^2 - 3^2 = 100 - 9 = 91$
Thus, $13 \times 7 = 91$.

Key Concept: Pattern Recognition - Sum of Squares

c) 50
[Solution Description]
Substitute $a = 3$ and $b = 4$ into the formula: $(3 + 4)^2 + (3 - 4)^2 = 49 + 1 = 50$

Key Concept: Algebraic Identity - Difference of Squares

c) 9996
[Solution Description]
Express $98 \times 102$ as $(100 - 2)(100 + 2)$. Apply the difference of squares identity: $100^2 - 2^2 = 10000 - 4 = 9996$

Key Concept: Distributive Property - Basic Application

b) 35
[Solution Description]
Substitute $k = 5$ into the formula $k^2 + 2k$. The calculation is: $5^2 + 2 \times 5 = 25 + 10 = 35$

Key Concept: Distributive Property and Algebraic Identities

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, expand $(a + b)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$. Similarly, expand $(a - b)^2$ using $(a - b)^2 = a^2 - 2ab + b^2$. Now, subtract $(a - b)^2$ from $(a + b)^2$: $(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)$. Simplify by combining like terms: $a^2 - a^2 + 2ab + 2ab + b^2 - b^2 = 4ab$. Thus, Assertion is true. Reason uses the identity $(a + b)(a - b) = a^2 - b^2$, which is not directly applicable here since we used expansion of squares, not difference of squares. Hence, Reason is true but does not correctly explain the Assertion.

Key Concept: Distributive Property, Multi-step Simplification

b) $116$
[Solution Description]
Substitute $a = 7$ and $b = 3$ into the given identity:
$2(7^2 + 3^2) = (7 + 3)^2 + (7 - 3)^2$
Calculate each part step-by-step:
$7^2 = 49$, $3^2 = 9$, so $2(49 + 9) = 2(58) = 116$.
$(7 + 3)^2 = 10^2 = 100$.
$(7 - 3)^2 = 4^2 = 16$.
Verify: $100 + 16 = 116$.
The value is $116$.

Key Concept: Special Case Expansion

b) $k^2 + 2k$
[Solution Description]
The expression $k(k + 2)$ can be expanded using the distributive property:
$k(k + 2) = k \cdot k + k \cdot 2 = k^2 + 2k$
Therefore, $k(k + 2) = k^2 + 2k$.

Key Concept: Pattern Recognition

c) 130
[Solution Description]
Using the given pattern, we substitute $a = 7$ and $b = 4$ into the identity:
$2(a^2 + b^2) = (a + b)^2 + (a - b)^2$
So,
$2(7^2 + 4^2) = (7 + 4)^2 + (7 - 4)^2 = 11^2 + 3^2 = 121 + 9 = 130$
Thus, $2(4^2 + 7^2) = 130$.

Key Concept: Distributive Property, Algebraic Identities

b) $16k$
[Solution Description]
First, expand both squares using the identity $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$.
$(k + 4)^2 = k^2 + 8k + 16$
$(k - 4)^2 = k^2 - 8k + 16$
Subtract the second expansion from the first:
$(k^2 + 8k + 16) - (k^2 - 8k + 16) = 16k$.
The simplified form is $16k$.

Key Concept: Sum of Squares, Difference of Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion by expanding both sides:
\textbf{Left Side:}
$2(a^2 + b^2) = 2a^2 + 2b^2$
\textbf{Right Side:}
$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2$
Both sides simplify to $2a^2 + 2b^2$, confirming the assertion is true. The reason provides the correct derivation steps, proving it is not only true but also the correct explanation.

Key Concept: Difference of Squares Identity

b) $a^2 - b^2$
[Solution Description]
Using the distributive property:
$(a + b)(a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^2 - b^2$

Key Concept: Special Cases of Distributive Property, Geometry-based explanation

b) 1089
[Solution Description]
We can visualize this problem using geometry. The area of the inner square (side length 33) is $(40 - 7)^2 = 33^2$. According to the identity, we can write:
$(40 - 7)^2 = 40^2 - 2(40 \times 7) + 7^2$
Calculating step-by-step:
$40^2 = 1600$
$2(40 \times 7) = 560$
$7^2 = 49$
Now, substitute these values:
$1600 - 560 + 49 = 1089$
Hence, $(40 - 7)^2 = 1089$.

Key Concept: Distributive Property - Geometric Visualization

b) 3025
[Solution Description]
The area of the larger square is $60^2 = 3600$.
The area of two rectangles is $2 \times (60 \times 5) = 600$.
The area of the small square is $5^2 = 25$.
So, the required area is $3600 - 600 + 25 = 3025$.

Key Concept: Geometry-based identities, Distributive Property

c) $x^2 - 9$
[Solution Description]
The area of a rectangle is given by the product of its length and width. Here, the sides are $(x + 3)$ and $(x - 3)$. Using the identity $(a + b)(a - b) = a^2 - b^2$, we can find the area as:
$(x + 3)(x - 3) = x^2 - 9$
Therefore, the area is $x^2 - 9$.

Key Concept: Visualizing Squares and Rectangles

c) $50^2 - 2 \times (50 \times 3) + 3^2$
[Solution Description]
Using the geometric approach:
Step 1: The area of the larger square of side 50 is $50^2 = 2500$.
Step 2: Subtract the areas of the two rectangles formed by removing 3 units from the side: $50 \times 3 + 50 \times 3 = 300$.
Step 3: Add back the small square of side 3 once to adjust for double subtraction: $3^2 = 9$.
Final expression: $50^2 - 2 \times (50 \times 3) + 3^2 = 2500 - 300 + 9 = 2209$.

Key Concept: Expansion of Two Binomials

c) $xy + 3x + 2y + 6$
[Solution Description]
Expand using the distributive property:
$(x + 2)(y + 3) = x \cdot y + x \cdot 3 + 2 \cdot y + 2 \cdot 3 = xy + 3x + 2y + 6$

Key Concept: Distributive Property Application

c) 2025
[Solution Description]
Using Sridharacharya's identity $a^2 = (a + b)(a - b) + b^2$:
Step 1: Let $a = 45$, $b = 5$.
Step 2: Compute $(a + b)(a - b) = (45 + 5)(45 - 5) = 50 \times 40 = 2000$.
Step 3: Add $b^2 = 25$.
Final answer: $45^2 = 2000 + 25 = 2025$.

Key Concept: Squares of Numbers, Modified Identity

b) 3136
[Solution Description]
To compute $56^2$, let us choose $b = 4$ (since $56 + 4 = 60$ and $56 - 4 = 52$, which are easy to multiply). Using the identity:
$56^2 = (56 + 4)(56 - 4) + 4^2$
Substituting the values:
$(56 + 4) = 60, \quad (56 - 4) = 52$
$4^2 = 16$
Now, multiply $(60 \times 52)$:
$60 \times 50 = 3000$
$60 \times 2 = 120$
$3000 + 120 = 3120$
Finally, add $b^2$:
$3120 + 16 = 3136$
Thus, $56^2 = 3136$.

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because $(a - b)^2$ can indeed be expanded to $a^2 - 2ab + b^2$, which represents the geometric process of subtracting two rectangles of area $ab$ and adding back the overlapping square of area $b^2$ to get the area of the smaller square. The reason correctly explains this geometric interpretation, as it describes the steps involved in deriving the identity $(a - b)^2 = a^2 - 2ab + b^2$ using areas.

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(60 - 5)^2$ can be evaluated using the formula $60^2 - 2 \times 60 \times 5 + 5^2$, which aligns with the geometric interpretation where we subtract the areas of two rectangles ($60 \times 5$ each) and add back the area of the small square ($5^2$) to avoid double subtraction. The reason explains this geometric process correctly, confirming the assertion. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Difference of Squares Identity

c) An L-shaped region formed by removing a square of side $b$ from a square of side $a$.
[Solution Description]
The identity $(a + b)(a - b) = a^2 - b^2$ can be visualized as:
Step 1: Draw a large square of side $a$.
Step 2: Remove a smaller square of side $b$ from it, leaving an L-shaped region.
Step 3: The area of the remaining region is $a^2 - b^2$, equivalent to a rectangle with sides $(a + b)$ and $(a - b)$.
Hence, the correct diagram shows this transformation.

Key Concept: Geometry-based distributive property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the identity $(a + b)(a - b) = a^2 - b^2$ can be proven geometrically. This is true because we can visualize it as follows:
Step 1: Draw a large square with side length $a$. Its area is $a^2$.
Step 2: Subtract a smaller square with side length $b$ from one corner of the large square. Now, the remaining shape has an area of $a^2 - b^2$.
Step 3: Alternatively, divide the remaining shape into two rectangles with dimensions $(a + b)$ and $(a - b)$. Their combined area is $(a + b)(a - b)$, which matches $a^2 - b^2$.
The reason correctly explains how the geometric method verifies the algebraic identity by subtracting areas. Therefore, both assertion and reason are true, and the reason is the correct explanation.

Key Concept: Product of Sum and Difference, Special Cases

a) $16p^2 - 81q^2$
[Solution Description]
This is a special case of $(a + b)(a - b) = a^2 - b^2$. Here, $a = 4p$ and $b = 9q$.
So,
$(4p + 9q)(4p - 9q) = (4p)^2 - (9q)^2 = 16p^2 - 81q^2$

Key Concept: Distributive Property in Expansion

b) $6xy - 8x + 15y - 20$
[Solution Description]
To expand $(2x + 5)(3y - 4)$, we use the distributive property:
$(2x + 5)(3y - 4) = 2x \times 3y + 2x \times (-4) + 5 \times 3y + 5 \times (-4)$
Simplify each term:
$6xy - 8x + 15y - 20$

Key Concept: Distributive Property

a) $x^2 + 7x + 12$
[Solution Description]
To expand $(x + 3)(x + 4)$, we use the distributive property:
$(x + 3)(x + 4) = x(x + 4) + 3(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$

Key Concept: Square of Sum

a) $a^2 + 10a + 25$
[Solution Description]
The square of the sum $(a + b)^2$ is given by $a^2 + 2ab + b^2$. Applying this to $(a + 5)^2$:
$(a + 5)^2 = a^2 + 2(a)(5) + 5^2 = a^2 + 10a + 25$

Key Concept: Distributive Property, Expansion of Products

c) $15x^3 - x^2 - 73x + 63$
[Solution Description]
To expand $(5x - 7)(3x^2 + 4x - 9)$, we use the distributive property:
First term: $5x \times 3x^2 = 15x^3$
Second term: $5x \times 4x = 20x^2$
Third term: $5x \times (-9) = -45x$
Fourth term: $-7 \times 3x^2 = -21x^2$
Fifth term: $-7 \times 4x = -28x$
Sixth term: $-7 \times (-9) = 63$
Now combine like terms:
$15x^3 + (20x^2 - 21x^2) + (-45x - 28x) + 63$
$= 15x^3 - x^2 - 73x + 63$

Key Concept: Distributive Property, Special Cases of Expansion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this assertion-reason question, we first evaluate the Assertion and Reason separately.
Step 1: Check the Assertion. Expand $(x + 2y)^2$ using the formula for the square of a sum:
$(x + 2y)^2 = x^2 + 2 \times x \times 2y + (2y)^2 = x^2 + 4xy + 4y^2$
Therefore, the Assertion is true.
Step 2: Check the Reason. The Reason states that $(a + b)^2 = a^2 + 2ab + b^2$, which is indeed the correct formula for the square of a sum, a special case of the distributive property. Hence, the Reason is also true.
Step 3: Determine if the Reason explains the Assertion. Since the Assertion directly follows from applying the formula mentioned in the Reason, the Reason correctly explains the Assertion.

Key Concept: Distributive Property of Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To expand $(x + 2)(y - 3)$, we use the distributive property, which states that $a(b + c) = ab + ac$. Here, we apply it twice: first, multiply $x$ by each term in the second bracket, giving $xy - 3x$, and then multiply $2$ by each term in the second bracket, giving $2y - 6$. Combining these results, we get $xy - 3x + 2y - 6$. Both Assertion and Reason are true, and Reason correctly explains how the expansion is derived using the distributive property.

Key Concept: Square of a Sum, Fast Multiplication

b) 11449
[Solution Description]
We can write $107^2$ as $(100 + 7)^2$. Using the identity $(a + b)^2 = a^2 + 2ab + b^2$:
$a = 100$, $b = 7$
$a^2 = 10000$
$2ab = 2 \times 100 \times 7 = 1400$
$b^2 = 49$
Now add them together:
$10000 + 1400 = 11400$
$11400 + 49 = 11449$

Key Concept: Square of a Sum

c) $49p^2 + 56pq + 16q^2$
[Solution Description]
We apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ where $a = 7p$ and $b = 4q$:
$(7p + 4q)^2 = (7p)^2 + 2 \times 7p \times 4q + (4q)^2$
Simplify each term:
$49p^2 + 56pq + 16q^2$

Key Concept: Product of Sum and Difference

b) $81m^2 - 25n^2$
[Solution Description]
The formula for the product of sum and difference is $(a + b)(a - b) = a^2 - b^2$. Here, $a = 9m$ and $b = 5n$:
$(9m + 5n)(9m - 5n) = (9m)^2 - (5n)^2$
Simplify each term:
$81m^2 - 25n^2$

Key Concept: Like Terms

a) $3x^2$ and $5x^2$
[Solution Description]
Like terms must have the same variable part (same letter-numbers). Checking each option:

Key Concept: Distributive Property of Multiplication Over Addition

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any numbers $x$, $y$, and $z$, the expression $(x + y) \cdot z$ can be expanded as $xz + yz$. This is correct according to the distributive property of multiplication over addition. The reason explains the distributive property accurately: multiplying a sum by a number is equivalent to multiplying each addend by that number and then adding the results. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Distributive Property