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: 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: 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 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: 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

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) 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: 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

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

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: 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

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

Key Concept: Square of a Number Using Identity

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

Key Concept: Square of Sum Identity

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

Key Concept: Pattern recognition, Squares

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

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

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

Key Concept: Difference of squares, Pattern recognition

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

Key Concept: Difference of Squares Pattern

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

Key Concept: Modified Sridharacharya's Identity

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

Key Concept: Difference of Squares Identity

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

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

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

Key Concept: Sum of Squares Pattern

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

Key Concept: Squares of numbers, Difference of squares

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

Key Concept: Square of a Number Using Identity

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

Key Concept: Algebraic Patterns

b) 35
[Solution Description]
According to Method 1, the number of circles in Step k is given by $(k + 1)^2 - 1$. For Step 5, substitute $k = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$
Therefore, the number of circles in Step 5 is 35.

Key Concept: Step-based pattern, Algebraic equivalence

b) $k^2 + 2k$
[Solution Description]
Start with the given expression:
$k \times (k + 1) + k$
Simplify step-by-step:
$k^2 + k + k = k^2 + 2k$
Factorize:
$k(k + 2)$
Therefore, $k^2 + 2k$ and $k(k + 2)$ are equivalent to the given expression.

Key Concept: Pattern recognition and algebraic expression

c) 26
[Solution Description]
Using Method 2, the formula for the number of circles in Step $k$ is $k^2 + 2 \times k$.
Total circles up to Step 3:
$1^2 + 2 \times 1 = 1 + 2 = 3 \quad (\text{Step } 1)$
$2^2 + 2 \times 2 = 4 + 4 = 8 \quad (\text{Step } 2)$
$3^2 + 2 \times 3 = 9 + 6 = 15 \quad (\text{Step } 3)$
Total: $3 + 8 + 15 = 26$.

Key Concept: Algebraic Patterns

d) All three methods
[Solution Description]
We verify each method for $k = 4$:
\begin{itemize}

Key Concept: Step-based algebraic patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify if the assertion and reason are true:
1. Assertion claims that $(k + 1)^2 - 1$ represents the circles in Step $k$.
From Method 1 in the syllabus:
- Step 1: $(1 + 1)^2 - 1 = 3$ circles
- Step 2: $(2 + 1)^2 - 1 = 8$ circles
- Step 3: $(3 + 1)^2 - 1 = 15$ circles
The formula works for these steps, so the Assertion is true.
2. Reason states that for Step 3, $4^2 - 1 = 15$ matches the pattern.
As shown in Step 3 above, this calculation is correct, so the Reason is true.
3. The Reason provides a specific example confirming the general formula stated in the Assertion. Hence, the Reason correctly explains the Assertion.

Key Concept: Pattern interpretation, Algebraic equivalence

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the algebraic equivalence of the three expressions:
1. Expand $(k + 1)^2 - 1$:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
2. Simplify $k(k + 1) + k$:
$k(k + 1) + k = k^2 + k + k = k^2 + 2k$
All three expressions simplify to $k^2 + 2k$, proving they are equivalent forms. Thus, the assertion about any one form being valid is correct, and the reason correctly explains their equivalence through algebraic simplification.

Key Concept: Algebraic Pattern Interpretation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that using Method 1, the expression for the number of circles in Step $k$ is $(k + 1)^2 - 1$. This matches the given pattern where:
- For Step 1: $(1 + 1)^2 - 1 = 4 - 1 = 3$
- For Step 2: $(2 + 1)^2 - 1 = 9 - 1 = 8$
- For Step 3: $(3 + 1)^2 - 1 = 16 - 1 = 15$
The reason explains that Method 1 works by taking the step number, adding 1, squaring it, and subtracting 1. This is a valid explanation for why the assertion is true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Step-based pattern, Algebraic forms

c) 35
[Solution Description]
Both expressions represent the same pattern, so we can use either to find the number of circles in Step 5. Let's use $(k + 1)^2 - 1$:
For Step 5, set $k = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$
Verification using the other expression $k^2 + 2k$:
$5^2 + 2 \times 5 = 25 + 10 = 35$
Both methods give the same result.

Key Concept: Pattern recognition and algebraic expression

c) 24
[Solution Description]
Using Method 1, the formula for the number of circles in Step $k$ is $(k + 1)^2 - 1$. For Step 4, substitute $k = 4$:
$(4 + 1)^2 - 1 = 5^2 - 1 = 25 - 1 = 24$
So, the number of circles in Step 4 is 24.

Key Concept: Pattern recognition and algebraic expression

b) $k^2 + 2k$
[Solution Description]
According to Method 3, the number of circles in Step $k$ is given by the expression $k \times (k + 1) + k$, which simplifies to $k^2 + 2k$.

Key Concept: Algebraic Patterns

b) $k^2 + 2k$
[Solution Description]
Method 3 states that for Step k, the number of circles is given by $k \times (k + 1) + k$. Simplifying this:
$k(k + 1) + k = k^2 + k + k = k^2 + 2k$
Among the options, only one matches this simplified form.

Key Concept: Step-based pattern, Multi-concept

b) $(n + 1)^2 - 1$
[Solution Description]
We can derive the general expression for the pattern:
Step 1: 3 circles
Step 2: Step 1 + (2 $\times$ 2 + 1) = 3 + 5 = 8
Step 3: Step 2 + (2 $\times$ 3 + 1) = 8 + 7 = 15
Observing the pattern, we see that the number of circles follows the formula $(n + 1)^2 - 1$.
Verification:
For Step 1: $(1 + 1)^2 - 1 = 4 - 1 = 3$
For Step 2: $(2 + 1)^2 - 1 = 9 - 1 = 8$
For Step 3: $(3 + 1)^2 - 1 = 16 - 1 = 15$
Thus, the correct expression is $(n + 1)^2 - 1$.

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: 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 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: 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: 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: 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: 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: 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: 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: 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

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: 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 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 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: 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: 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: 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 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

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 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 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: 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: Verification of Patterns, Mathematical Exploration

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

Key Concept: Generalization of Diagonal Pattern

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

Key Concept: Verification of Diagonal Pattern

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

Key Concept: Generalization of Calendar Pattern

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

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

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

Key Concept: Diagonal Products in 2×2 Calendar Grid

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

Key Concept: 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: 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: 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: 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: 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

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: 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: 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: 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: 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: 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: 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, 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: 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: 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

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: 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

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: 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: 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) $x^2 + 6x + 9$
[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

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: 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: 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: 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: Area calculation, Algebraic identities

a) $(m - n)^2$
[Solution Description]
Total area of large square: $(m + n)^2$.
Total area of four rectangles: $4mn$.
Shaded area = $(m + n)^2 - 4mn$.
Simplifying: $m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$.

Key Concept: Generalizing Patterns

d) $ab + 2b + 3a + 6$
[Solution Description]
Substitute $m = 2$ and $n = 3$ into the identity:
$(a + 2)(b + 3) = ab + 2b + 3a + 6$

Key Concept: Pattern Analysis Using Algebra

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. The given general expression for Step $k$ is $k^2 + 2k$.
2. Substitute $k = 3$ into the expression: $(3)^2 + 2(3) = 9 + 6 = 15$.
3. The calculation matches the stated reason.
4. Thus, Assertion (A) is true, and Reason (R) is also true. However, the reason merely verifies the correctness of the formula for a specific step and does not explain why the general form of the expression holds.

Key Concept: Area Calculation

b) $(n - m)^2$
[Solution Description]
Expand $(m + n)^2$: $m^2 + 2mn + n^2$.
Subtract $4mn$: $m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$.

Key Concept: Area Calculations Using Algebraic Identities

c) $(n - m)^2$
[Solution Description]
Expand $(m + n)^2 - 4mn$:
$(m + n)^2 - 4mn = m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$
This matches Yusuf’s method.

Key Concept: Pattern Investigation

c) 24
[Solution Description]
Observing the pattern:
Step 1: $1^2 + 2(1) = 3$
Step 2: $2^2 + 2(2) = 8$
Step 3: $3^2 + 2(3) = 15$
For Step 4: $4^2 + 2(4) = 16 + 8 = 24$

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of tiles in Step n is $n^2 + 2n$. This can be verified using the formula derived from the pattern, which matches the given expression. The reason explains that the pattern grows by adding a row and a column at each step, which correctly justifies the algebraic expression in the assertion. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Pattern Generalization

b) 255
[Solution Description]
Substitute $k = 15$ into $k^2 + 2k$:
$15^2 + 2(15) = 225 + 30 = 255$.

Key Concept: Visual Patterns and Algebraic Expressions

b) $n^2 + 2n$
[Solution Description]
Observing the pattern:
- Step 1: $1^2 + 2 \times 1 = 3$
- Step 2: $2^2 + 2 \times 2 = 8$
- Step 3: $3^2 + 2 \times 3 = 15$
Thus, the general expression for Step $n$ is $n^2 + 2n$.

Key Concept: Visual and Numerical Growth Patterns, Algebraic Expression

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify, let's analyze the pattern:
Step 1: $1^2 + 2(1) = 3$ tiles (matches given method $1 \times 3$)
Step 2: $2^2 + 2(2) = 8$ tiles (matches given method $2 \times 4$)
Step 3: $3^2 + 2(3) = 15$ tiles (matches given method $3 \times 5$)
The assertion correctly states the algebraic expression for the pattern. The reason explains that the quadratic growth comes from adding rows and columns each time. However, while both are true, the reason doesn't fully explain why the specific expression $n^2 + 2n$ emerges - it just describes a general quadratic pattern.

Key Concept: Pattern recognition, Algebraic expression

c) $n^2 + 2n$
[Solution Description]
First, observe the pattern:
Step 1: $1^2 + 2(1) = 3$ tiles,
Step 2: $2^2 + 2(2) = 8$ tiles,
Step 3: $3^2 + 2(3) = 15$ tiles.
The general formula for Step n is $n^2 + 2n$.

Key Concept: Pattern analysis, Multiple methods

a) 255
[Solution Description]
Original formula gives $15^2 + 2(15) = 225 + 30 = 255$ circles.
However, the question specifies 2 extra circles per step beyond the formula.
This creates an arithmetic sequence: 255, 257, 259,...
But since it's only Step 15, no additional steps means total remains 255.

Key Concept: Special identities

b) $a^2 + 2ab + b^2$
[Solution Description]
Using the special identity $(a + b)^2 = a^2 + 2ab + b^2$, the expansion is straightforward:
$(a + b)^2 = a^2 + 2ab + b^2$.

Key Concept: Simplifying with Like Terms

a) $20p^2 - 5pq$
[Solution Description]
First, distribute $-4p$ inside the parentheses:
$-4p \times (-5p) = 20p^2$
$-4p \times 2q = -8pq$
Now, add $3pq$ to the result:
$20p^2 - 8pq + 3pq = 20p^2 - 5pq$

Key Concept: Simplification using Distributive Property

a) $\frac{5x^2}{3} - \frac{10xy}{3} + \frac{5x}{12}$
[Solution Description]
To simplify, distribute $\frac{5x}{3}$ to each term inside the parentheses:
$\frac{5x}{3} \times x = \frac{5x^2}{3}$
$\frac{5x}{3} \times (-2y) = -\frac{10xy}{3}$
$\frac{5x}{3} \times \frac{1}{4} = \frac{5x}{12}$
So, the simplified form is $\frac{5x^2}{3} - \frac{10xy}{3} + \frac{5x}{12}$

Key Concept: Expanding Binomial Expressions

b) $4m^2 + 12mn + 9n^2$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$,
$(2m + 3n)^2 = (2m)^2 + 2(2m)(3n) + (3n)^2$
$= 4m^2 + 12mn + 9n^2$

Key Concept: Simplifying using exponent notation

b) $3a^3$
[Solution Description]
To simplify $3a^2 \times a$, multiply the coefficients and add the exponents of like bases:
$3a^2 \times a = 3a^{2 + 1} = 3a^3$

Key Concept: Simplification of algebraic expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion is true because expanding $(a + b)^2$ gives $a^2 + 2ab + b^2$ as per the algebraic identity. The Reason is also true because the distributive property is applied to expand the expression, and like terms ($ab + ab$) are combined to get $2ab$. Moreover, the Reason correctly explains the Assertion as it details the method used for simplification.

Key Concept: Simplification of Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify $3x(2x - 5y + 1)$, we use the distributive property:
Step 1: Multiply $3x$ by each term inside the bracket:
$3x \times 2x = 6x^2$
$3x \times (-5y) = -15xy$
$3x \times 1 = 3x$
Step 2: Combine the results: $6x^2 - 15xy + 3x$.
The Assertion is true as it correctly applies the distributive property.
The Reason is also true and correctly explains why the assertion holds.

Key Concept: Distributive property

a) $\frac{3}{2}a^2 - \frac{3}{2}ab + \frac{3}{10}a$
[Solution Description]
Apply the distributive property to expand the expression:
$\frac{3a}{2}(a) = \frac{3}{2}a^2$
$\frac{3a}{2}(-b) = -\frac{3}{2}ab$
$\frac{3a}{2}\left(\frac{1}{5}\right) = \frac{3}{10}a$
Combining these, we get: $\frac{3}{2}a^2 - \frac{3}{2}ab + \frac{3}{10}a$

Key Concept: Distributive Property, Special Identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, recognize that the expression $\left(\frac{x}{2} - \frac{y}{3}\right)\left(\frac{x}{2} + \frac{y}{3}\right)$ is of the form $(a - b)(a + b)$, where $a = \frac{x}{2}$ and $b = \frac{y}{3}$. Using the identity $(a - b)(a + b) = a^2 - b^2$, we substitute the values of $a$ and $b$:
$\left(\frac{x}{2}\right)^2 - \left(\frac{y}{3}\right)^2 = \frac{x^2}{4} - \frac{y^2}{9}$
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Multiplying Trinomials, Simplification

a) $x^3 - x^2 - x + 10$
[Solution Description]
We use the distributive property to multiply $(x + 2)$ with each term in $(x^2 - 3x + 5)$:
$(x + 2)(x^2 - 3x + 5) = x \times x^2 + x \times (-3x) + x \times 5 + 2 \times x^2 + 2 \times (-3x) + 2 \times 5$
Simplify each term:
$x \times x^2 = x^3$
$x \times (-3x) = -3x^2$
$x \times 5 = 5x$
$2 \times x^2 = 2x^2$
$2 \times (-3x) = -6x$
$2 \times 5 = 10$
Combine the terms:
$x^3 - 3x^2 + 5x + 2x^2 - 6x + 10 = x^3 - x^2 - x + 10$

Key Concept: Distributive Property, Simplification

a) $6x^2 - 7x - 20$
[Solution Description]
To simplify $(3x + 4)(2x - 5)$, we use the distributive property:
$(3x + 4)(2x - 5) = 3x \times 2x + 3x \times (-5) + 4 \times 2x + 4 \times (-5)$
Simplify each term:
$3x \times 2x = 6x^2$
$3x \times (-5) = -15x$
$4 \times 2x = 8x$
$4 \times (-5) = -20$
Combine the terms:
$6x^2 - 15x + 8x - 20 = 6x^2 - 7x - 20$

Key Concept: Special Identities, Simplification

a) $4a^2 - 12ab + 9b^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, we substitute $a = 2a$ and $b = 3b$:
$(2a - 3b)^2 = (2a)^2 - 2 \times 2a \times 3b + (3b)^2$
Simplify each term:
$(2a)^2 = 4a^2$
$2 \times 2a \times 3b = 12ab$
$(3b)^2 = 9b^2$
Combine the terms:
$4a^2 - 12ab + 9b^2$

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 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: 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: 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: 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: 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: 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: 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: 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: 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.