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, 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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 - 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: 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: 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, 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, 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: 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: 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, 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: Square of the Sum of Two Numbers

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that $(3x + 4y)^2 = 9x^2 + 16y^2$. Let us verify this using the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 3x$ and $b = 4y$. Substituting these values into the identity, we get: $(3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2$. The assertion misses the middle term $24xy$, so it is false. However, the reason correctly states that the expansion should include the term $24xy$ as per the identity, which makes the reason true. Therefore, Assertion is false, but Reason is true.

Key Concept: Special Cases of the Distributive Property

b) 21
[Solution Description]
Use the identity $(a + b)(a - b) = a^2 - b^2$.
Given $a + b = 7$ and $a - b = 3$, multiply them:
$a^2 - b^2 = (a + b)(a - b) = 7 \times 3 = 21$.
The value of $a^2 - b^2$ is $21$.

Key Concept: Square of the Sum of Two Numbers

b) $4x^2 + 12xy + 9y^2$
[Solution Description]
To expand $(2x + 3y)^2$, we use the identity:
$(a + b)^2 = a^2 + 2ab + b^2$
Here, $a = 2x$ and $b = 3y$. Substituting these values into the identity:
$(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2$

Key Concept: Special Cases of the Distributive Property

b) $25 - 10y + y^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 5$ and $b = y$:
$(5 - y)^2 = 5^2 - 2 \times 5 \times y + y^2 = 25 - 10y + y^2$.
The result is $25 - 10y + y^2$.

Key Concept: Special Cases of the Distributive Property

a) $x^2 + 6x + 9$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, where $a = x$ and $b = 3$:
$(x + 3)^2 = x^2 + 2 \times x \times 3 + 3^2 = x^2 + 6x + 9$.
The expanded form is $x^2 + 6x + 9$.

Key Concept: Special Cases of the Distributive Property

a) $34m^2 - 6mn - 3n^2$
[Solution Description]
First, expand $(5m + 2n)(5m - 2n)$ using the identity $(a + b)(a - b) = a^2 - b^2$:
$(5m)^2 - (2n)^2 = 25m^2 - 4n^2$.
Next, expand $(3m - n)^2$ using $(a - b)^2 = a^2 - 2ab + b^2$:
$(3m)^2 - 2 \times 3m \times n + n^2 = 9m^2 - 6mn + n^2$.
Add the two results:
$25m^2 - 4n^2 + 9m^2 - 6mn + n^2 = 34m^2 - 6mn - 3n^2$.
The equivalent expression is $34m^2 - 6mn - 3n^2$.

Key Concept: Special Cases of the Distributive Property

a) $48xy$
[Solution Description]
First, expand both squares using the identity $(a \pm b)^2 = a^2 \pm 2ab + b^2$.
For $(3x + 4y)^2$:
$(3x)^2 + 2 \times 3x \times 4y + (4y)^2 = 9x^2 + 24xy + 16y^2$.
For $(3x - 4y)^2$:
$(3x)^2 - 2 \times 3x \times 4y + (4y)^2 = 9x^2 - 24xy + 16y^2$.
Now subtract the second expansion from the first:
$(9x^2 + 24xy + 16y^2) - (9x^2 - 24xy + 16y^2) = 48xy$.
The simplified form is $48xy$.

Key Concept: Special Cases of the Distributive Property

c) $4a^2 - 49$
[Solution Description]
Using the identity $(a + b)(a - b) = a^2 - b^2$, where $a = 2a$ and $b = 7$:
$(2a + 7)(2a - 7) = (2a)^2 - 7^2 = 4a^2 - 49$.
The simplified form is $4a^2 - 49$.

Key Concept: Square of the Difference of Two Numbers

b) $25p^2 - 40pq + 16q^2$
[Solution Description]
Using the identity for the square of the difference:
$(a - b)^2 = a^2 - 2ab + b^2$
Here, $a = 5p$ and $b = 4q$. Substituting these values:
$(5p - 4q)^2 = (5p)^2 - 2(5p)(4q) + (4q)^2 = 25p^2 - 40pq + 16q^2$

Key Concept: Product of Sum and Difference of Two Numbers

d) $49m^2 - 4n^2$
[Solution Description]
The given expression can be expanded using the identity:
$(a + b)(a - b) = a^2 - b^2$
Here, $a = 7m$ and $b = 2n$. Substituting these values:
$(7m + 2n)(7m - 2n) = (7m)^2 - (2n)^2 = 49m^2 - 4n^2$

Key Concept: Square of the Sum, Product of Sum and Difference

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this Assertion-Reason question, let's evaluate both statements step by step.
1. **Assertion**:
- Calculate $(x + y)^2 = x^2 + 2xy + y^2$.
- Calculate $(x - y)^2 = x^2 - 2xy + y^2$.
- Subtract the second equation from the first:
$(x + y)^2 - (x - y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy$
- Thus, the Assertion is true.
2. **Reason**:
- The Reason correctly states the expansions of $(x + y)^2$ and $(x - y)^2$ using Identity 1A and 2 from the syllabus.
- It also correctly explains why the difference results in $4xy$ by cancellation of $x^2$ and $y^2$ terms.
3. **Conclusion**: Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Square of the Sum of Two Numbers

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that $(x + y)^2 = x^2 + y^2$, which is incorrect because the correct expansion is $(x + y)^2 = x^2 + 2xy + y^2$ (Identity 1A). The reason claims that the square of the sum equals the sum of squares, which is also false as it misses the $2xy$ term. Thus, Assertion is false but Reason is true.

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: 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: 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: 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: 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: 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: 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: 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: 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 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: 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

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, 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

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

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: 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

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: 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

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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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

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: 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: Identity 1C Application

c) 375
[Solution Description]
Let us express 15 as (20 - 5) and 25 as (20 + 5). Using Identity 1C:
$(a + b)(a - b) = a^2 - b^2$
Here, $a = 20$ and $b = 5$. Substituting these values:
$15 \times 25 = (20 - 5)(20 + 5) = 20^2 - 5^2 = 400 - 25 = 375$

Key Concept: Identity 1C: Product of Sum and Difference

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify both the Assertion and Reason.
Expansion of $(a + b)^2$ gives:
$$ (a + b)^2 = a^2 + 2ab + b^2$$
Expansion of $(a - b)^2$ gives:
$$ (a - b)^2 = a^2 - 2ab + b^2$$
Now, add both expansions:
$$ (a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2 = 2(a^2 + b^2)$$
This confirms that the Assertion is true as it directly follows from algebraic manipulation, and the Reason correctly explains it by providing the underlying identity.

Key Concept: Identity 1C, Multi-concept Application

d) $44$
[Solution Description]
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$. Here, $a = 7$ and $b = \sqrt{5}$.
Step 1: Apply the identity: $7^2 - (\sqrt{5})^2$
Step 2: Calculate squares: $49 - 5$
Step 3: Subtract: $44$
The simplified form is 44.

Key Concept: Identity 1C, Real-World Application

b) $2xy + 2y^2$
[Solution Description]
The total area of the square field is $(x + y)^2$. The area of the first rectangular plot is $(x + y)(x - y)$. Using Identity 1C: $(x + y)(x - y) = x^2 - y^2$.
Step 1: Total area = $(x + y)^2$
Step 2: Area of first plot = $x^2 - y^2$
Step 3: Area of second plot = Total area - First plot area
Step 4: Substitute values: $(x + y)^2 - (x^2 - y^2)$
Step 5: Expand $(x + y)^2$: $x^2 + 2xy + y^2 - x^2 + y^2$
Step 6: Simplify: $2xy + 2y^2$
The area of the second plot is $2xy + 2y^2$.

Key Concept: Identity 1C Verification

b) $x^2 - y^2$
[Solution Description]
The identity for $(x + y)(x - y)$ is derived as follows:
$(x + y)(x - y) = x^2 - xy + xy - y^2 = x^2 - y^2.$

Key Concept: Pattern Recognition

c) 116
[Solution Description]
Using the given identity,
$2(a^2 + b^2) = (a + b)^2 + (a - b)^2.$
Substituting $a = 7$ and $b = 3$,
$2(7^2 + 3^2) = (7 + 3)^2 + (7 - 3)^2 = 10^2 + 4^2 = 100 + 16 = 116.$

Key Concept: Product of Sum and Difference

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states the algebraic identity $(a + b)(a - b) = a^2 - b^2$ which is true.
The reason provides justification using the distributive property:
1) Expand $(a + b)(a - b)$ using distributive property: $a(a - b) + b(a - b) = a^2 - ab + ba - b^2$.
2) Simplify $-ab + ba$ to 0 because multiplication is commutative ($ab = ba$).
3) Final simplified form is $a^2 - b^2$, which matches the assertion.
Thus, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Pattern Recognition

b) 63
[Solution Description]
Given $7 \times 9 = (8 + 1)(8 - 1)$. Applying Identity 1C:
$(a + b)(a - b) = a^2 - b^2$
Here, $a = 8$ and $b = 1$. So,
$7 \times 9 = 8^2 - 1^2 = 64 - 1 = 63$

Key Concept: Verification of Identity

c) $(12 + 8)(12 - 8)$
[Solution Description]
According to Identity 1C:
$a^2 - b^2 = (a + b)(a - b)$
For $a = 12$ and $b = 8$, we get:
$12^2 - 8^2 = (12 + 8)(12 - 8) = 20 \times 4 = 80$

Key Concept: Product of Sum and Difference Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(x + y)(x - y)$ simplifies to $x^2 - y^2$. This is correct because applying the distributive property (also known as the FOIL method for binomials), we get:
Step 1: Multiply the first terms: $x \cdot x = x^2$.
Step 2: Multiply the outer terms: $x \cdot (-y) = -xy$.
Step 3: Multiply the inner terms: $y \cdot x = xy$.
Step 4: Multiply the last terms: $y \cdot (-y) = -y^2$.
Step 5: Combine all terms: $x^2 - xy + xy - y^2$.
Step 6: Simplify by combining like terms ($-xy + xy = 0$): $x^2 - y^2$.
The reason provided is that $(a + b)(a - b) = a^2 - b^2$ is an algebraic identity, which is also true. Moreover, the reason correctly explains why the assertion holds, as demonstrated above.

Key Concept: Identity 1C Application

b) 2475
[Solution Description]
To simplify $45 \times 55$, we can rewrite it using the identity $(a + b)(a - b) = a^2 - b^2$. Here, let $a = 50$ and $b = 5$. Then,
$45 \times 55 = (50 - 5)(50 + 5) = 50^2 - 5^2 = 2500 - 25 = 2475.$

Key Concept: Identity 1C, Multi-step Solution

a) $-2$
[Solution Description]
According to Identity 1C, $(a + b)(a - b) = a^2 - b^2$. Here, $a = 3$ and $b = x$. So, $3^2 - x^2 = 9 - x^2$. Given that this equals 5:
Step 1: $9 - x^2 = 5$
Step 2: Subtract 5 from both sides: $4 - x^2 = 0$
Step 3: Add $x^2$ to both sides: $4 = x^2$
Step 4: Take square root of both sides: $x = \pm 2$
Therefore, $x$ can be either 2 or -2.

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: 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: 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, 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 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 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: 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: 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, 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: 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: Algebraic expressions for visual patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $k^2 + 2k$ correctly represents the number of circles in Step k, which is true as verified by multiple methods in the syllabus. The reason states that all simplified forms of different algebraic expressions for this pattern lead to the same expression, which is also true and explains why the assertion is correct because all valid interpretations converge to the same algebraic form.

Key Concept: Algebraic expressions for visual patterns

c) 16
[Solution Description]
Substitute $m = 3$ and $n = 7$ into the formula:
$(n - m)^2 = (7 - 3)^2 = 4^2 = 16$

Key Concept: Area Calculation, Algebraic Identities

c) $(b - a)^2$
[Solution Description]
The problem states that the area of the shaded region can be calculated using two methods:
Method 1: $(a + b)^2 - 4ab$
This represents the total area of the large square minus the area of the four rectangles.
Method 2: $(b - a)^2$
This represents the area of the smaller shaded square formed inside the larger square.
Since both expressions are equivalent as per the given identity, the area of the shaded region can be represented by either $(a + b)^2 - 4ab$ or $(b - a)^2$.

Key Concept: Algebraic expressions for visual patterns

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. This is verified by the fact that all four different methods (Method 1: $(k + 1)^2 - 1$, Method 2: $k^2 + 2 \times k$, Method 3: $k \times (k + 1) + k$, and Method 4: $k \times (k + 2)$) simplify to the same expression $k^2 + 2k$. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Algebraic expressions for visual patterns

b) 15
[Solution Description]
Substitute $x = 5$ and $y = 2$ into the formula:
$x^2 - xy = 5^2 - 5 \times 2 = 25 - 10 = 15$

Key Concept: Area of Shaded Regions

b) $(x - y)^2$
[Solution Description]
Total area of square is $(x + y)^2 = x^2 + 2xy + y^2$.
Area of four rectangles is $4xy$.
Shaded area = Total area - Rectangle areas = $x^2 + 2xy + y^2 - 4xy = x^2 - 2xy + y^2 = (x - y)^2$

Key Concept: Algebraic Expressions for Visual Patterns

c) 35
[Solution Description]
To find the number of circles in step 5, we substitute k = 5 into the given formula:
$k^2 + 2k = 5^2 + 2(5) = 25 + 10 = 35$

Key Concept: Algebraic expressions for visual patterns

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

Key Concept: Algebraic expressions for visual patterns, Multiple methods

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we substitute $k = 15$ into the given expression $k^2 + 2k$. Calculation: $15^2 + 2 \times 15 = 225 + 30 = 255$. Thus, the assertion is true. The reason states that all four methods lead to the same expression $k^2 + 2k$, which is correct as shown in the syllabus. Moreover, the reason correctly explains why the assertion holds because the unified expression derived from all methods allows us to compute the number of circles accurately.

Key Concept: Algebraic Expressions for Visual Patterns

c) 12
[Solution Description]
Using the given formula with n = 4:
$n^2 - n = 4^2 - 4 = 16 - 4 = 12$

Key Concept: Algebraic Expressions, Pattern Recognition

c) 46
[Solution Description]
To find the number of tiles in Step 5, substitute $n = 5$ into the given expression $2n^2 - n + 1$.
Step 1: Substitute $n = 5$ into the expression.
$2(5)^2 - 5 + 1$
Step 2: Calculate $5^2 = 25$.
Step 3: Multiply 2 by 25 to get $50$.
Step 4: Subtract 5 from 50 to get $45$.
Step 5: Add 1 to 45 to get $46$.
Therefore, there are 46 tiles in Step 5.

Key Concept: Pattern Recognition, Algebraic Expressions

c) 255
[Solution Description]
To find the number of circles in Step 15, substitute $k = 15$ into the given expression $k^2 + 2k$.
Step 1: Substitute $k = 15$ into the expression.
$(15)^2 + 2(15)$
Step 2: Calculate $15^2 = 225$.
Step 3: Calculate $2 \times 15 = 30$.
Step 4: Add the results from Step 2 and Step 3.
$225 + 30 = 255$.
Therefore, there are 255 circles in Step 15.

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: 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: 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, 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: 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: 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: 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: 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]
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: 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: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because applying the distributive property to $5(3 + 4)$ indeed gives $5 \times 3 + 5 \times 4 = 15 + 20 = 35$. The reason correctly explains the distributive property, which is $a(b + c) = ab + ac$, and justifies the assertion.

Key Concept: Distributive Property, Product of Sum and Difference

a) $16a^2 - 81b^2$
[Solution Description]
This uses the product of sum and difference identity: $(a + b)(a - b) = a^2 - b^2$. Here, $a = 4a$ and $b = 9b$.
$(4a)^2 - (9b)^2 = 16a^2 - 81b^2$

Key Concept: Fast Multiplication

b) $2323$
[Solution Description]
We can write $23 \times 101$ as $23 \times (100 + 1)$. Using the distributive property:
$23 \times 101 = 23 \times 100 + 23 \times 1$
Calculating each term:
$2300 + 23 = 2323$
Hence, the product is $2323$.

Key Concept: Multiplication by Parts

b) $1212$
[Solution Description]
We can write $101$ as $(100 + 1)$. Thus, $12 \times 101 = 12 \times (100 + 1) = (12 \times 100) + (12 \times 1) = 1200 + 12 = 1212$.

Key Concept: Distributive Property

c) $15 + 40$
[Solution Description]
Using the distributive property, $a(b + c) = ab + ac$. Here, $a = 5$, $b = 3$, and $c = 8$. Therefore, $5(3 + 8) = (5 \times 3) + (5 \times 8) = 15 + 40 = 55$.

Key Concept: Square of Sum Identity

d) $16 + 40 + 25$
[Solution Description]
Applying the identity, $(a + b)^2 = a^2 + 2ab + b^2$. Substituting $a = 4$ and $b = 5$, we get $(4 + 5)^2 = 4^2 + 2 \times 4 \times 5 + 5^2 = 16 + 40 + 25 = 81$.

Key Concept: Distributive Property, Fast Multiplication

a) 24543
[Solution Description]
We can express 101 as $(100 + 1)$. Using the distributive property:
$243 \times 101 = 243 \times (100 + 1) = (243 \times 100) + (243 \times 1) = 24300 + 243 = 24543$

Key Concept: Special Case: Square of Sum

b) $49$
[Solution Description]
Applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ where $a = 5$ and $b = 2$:
$(5 + 2)^2 = 5^2 + 2 \times 5 \times 2 + 2^2$
Calculating each term:
$25 + 20 + 4 = 49$
Thus, the value is $49$.

Key Concept: Distributive Property Application

b) $12 + 3x + 4y + xy$
[Solution Description]
To expand $(4 + x)(3 + y)$, we use the distributive property:
$(4 + x)(3 + y) = 4 \times 3 + 4 \times y + x \times 3 + x \times y$
Simplifying each term gives:
$12 + 4y + 3x + xy$
So, the expanded form is $12 + 3x + 4y + xy$.

Key Concept: Distributive Property, Square of Sum

a) $25x^2 + 70xy + 49y^2$
[Solution Description]
To solve this, we use the square of sum formula: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 5x$ and $b = 7y$. Applying the formula:
$(5x)^2 + 2(5x)(7y) + (7y)^2 = 25x^2 + 70xy + 49y^2$

Key Concept: Distributive Property, Special Cases

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To expand $(x + y)^2$, we use the distributive property as follows:
$(x + y)^2 = (x + y)(x + y) = x \times x + x \times y + y \times x + y \times y = x^2 + 2xy + y^2$
Here, the assertion is true because $(x + y)^2$ indeed expands to $x^2 + 2xy + y^2$. The reason is also true and correctly explains the assertion, as the distributive property is used to break down the expression and perform the expansion.

Key Concept: Distributive Property

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To expand $(2x + 3)(4y - 5)$, we use the distributive property (also known as FOIL method for binomials). Multiply each term in the first bracket by each term in the second bracket:
$(2x)(4y) + (2x)(-5) + (3)(4y) + (3)(-5) = 8xy - 10x + 12y - 15$
The assertion is correct. The reason correctly states the distributive property, but it does not explicitly explain the expansion of two binomials (which requires applying the distributive property twice). Thus, while both are true, the reason does not fully explain 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 $7$.