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: 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: 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: 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: 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

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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Pattern Generalization

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

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: 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: 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: 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 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: 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 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Square of the Difference, Pattern Recognition

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Application of Identity 1A

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

Key Concept: Square of Sum/Difference, Distributive Property

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum, Difference of Squares

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

Key Concept: Application of Square of Sum/Difference

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Difference of Squares, Multi-step Solution

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: 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 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 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: 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: 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: 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 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: 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: 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: Algebraic Patterns

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

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: 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: Complex Expression Simplification

a) $2p^2 + 6pq + 6p - 2q$
[Solution Description]
Let's simplify step by step:
1. Expand first term: $2p(p+3q) = 2p^2 + 6pq$
2. Expand second term: $-3(q-2p) = -3q + 6p$
3. Now we have: $2p^2 + 6pq - 3q + 6p + q$
4. Combine like terms: $-3q + q = -2q$
5. Final form: $2p^2 + 6pq + 6p - 2q$

Key Concept: Combining Like Terms

c) $5x + 10$
[Solution Description] First, distribute the coefficients:
$2(x - 1) = 2x - 2$
$3(x + 4) = 3x + 12$
Next, combine like terms:
$2x - 2 + 3x + 12 = (2x + 3x) + (-2 + 12) = 5x + 10$
The correct simplification is $5x + 10$.

Key Concept: Algebraic Simplification, Error Identification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify, let's expand $(5m + 6n)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 5m$ and $b = 6n$. Substituting these values into the identity, we get:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$
The given simplification in the problem statement only includes $25m^2 + 36n^2$, missing the middle term $60mn$. Therefore, both Assertion and Reason are true, and Reason correctly explains why the given simplification is incorrect.

Key Concept: Expanding Binomials

b) $ab + 4a + 2b + 8$
[Solution Description]
To expand $(a + 2)(b + 4)$, we use the distributive property (FOIL method):
$a \times b = ab, \quad a \times 4 = 4a, \quad 2 \times b = 2b, \quad 2 \times 4 = 8$
Combining these gives the expanded form $ab + 4a + 2b + 8$.

Key Concept: Simplifying Polynomials

b) $5w^2 + 6w$
[Solution Description]
The expression $5w^2 + 6w$ cannot be simplified further because the terms $5w^2$ and $6w$ are not like terms (they have different exponents). Therefore, the correct answer is the expression itself.

Key Concept: Binomial Squaring Error Identification

b) Omitted the middle term $-24ab$
[Solution Description]
The correct expansion using $(x - y)^2 = x^2 - 2xy + y^2$ would be:
$(3a - 4b)^2 = (3a)^2 - 2(3a)(4b) + (4b)^2 = 9a^2 - 24ab + 16b^2$
The student's error was:
1. They correctly squared both terms ($9a^2$ and $16b^2$)
2. But they forgot to include the middle term $-24ab$
3. They also used addition between the squared terms instead of subtraction

Key Concept: Algebraic Expansion and Simplification

a) $12xy - 8xz + 10x - 5y$
[Solution Description]
First, expand each term separately:
1. $4x(3y - 2z) = 12xy - 8xz$
2. $5(2x - y) = 10x - 5y$
Now combine these results:
$12xy - 8xz + 10x - 5y$
There are no like terms to combine further, so this is the simplified form.

Key Concept: Expanding and Simplifying Algebraic Expressions

c) $15p^2 - 6pq$
[Solution Description] To expand and simplify $–3p (–5p + 2q)$, distribute $-3p$ to each term inside the parentheses:
$-3p \times (-5p) + (-3p) \times 2q = 15p^2 - 6pq$
The simplified form is $15p^2 - 6pq$.

Key Concept: Algebraic Simplification

d) Assertion is false, but Reason is true.
[Solution Description]
Let us simplify the given expression step by step:
First, expand the expression:
$2(x - 1) + 3(x + 4) = 2x - 2 + 3x + 12$
Now, combine like terms:
$2x + 3x = 5x$ and $-2 + 12 = 10$
So, the correct simplified form is $5x + 10$, not $5x + 11$.
The Reason states that $-2$ and $+12$ were incorrectly combined as $+3$, which is false because they were actually correctly combined as $+10$.
Therefore, Assertion is false, but Reason is true.

Key Concept: Checking Errors in Expansion

a) Missing the term $60mn$
[Solution Description] The expression $(5m + 6n)^2$ should be expanded using the formula $(a+b)^2 = a^2 + 2ab + b^2$.
$(5m + 6n)^2 = (5m)^2 + 2 \times 5m \times 6n + (6n)^2 = 25m^2 + 60mn + 36n^2$
The given simplification misses the middle term $60mn$.

Key Concept: Distributive Property

d) Assertion is false, but Reason is true.
[Solution Description]
The given Assertion states that $(3x - 4)^2 = 9x^2 - 16$. However, using the correct identity $(a - b)^2 = a^2 - 2ab + b^2$, the expansion should be:
$(3x - 4)^2 = (3x)^2 - 2(3x)(4) + (4)^2 = 9x^2 - 24x + 16$.
The Reason correctly states the expansion formula for $(a - b)^2$, which is not applied in the Assertion.
Therefore, the Assertion is false, but the Reason is true.

Key Concept: Expanding and Simplifying Algebraic Expressions

b) $–3p (–5p + 2q) = 15p^2 - 6pq$
[Solution Description]
The mistake is in the expansion. The correct expansion of $-3p(-5p + 2q)$ should use the distributive property:
$-3p \times (-5p) = 15p^2, \quad -3p \times 2q = -6pq$
So the correct simplified form is $15p^2 - 6pq$, not $p - 2q$.

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Distributive property, Like terms

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Distributive Property and Simplification

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

Key Concept: Error in Distributive Property

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

Key Concept: Error spotting in algebraic simplification, binomial expansion

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

Key Concept: Distributive Property

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

Key Concept: Error Spotting in Algebraic Simplification

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

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

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

Key Concept: Error spotting in algebraic simplification, distribution

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

Key Concept: Error in Combining Like Terms

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

Key Concept: Error in Expansion

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication, Error Analysis

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property

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

Key Concept: Product Change, 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: 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: 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: 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 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: 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: 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: 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: 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: 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: 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: Distributive property, Combining like terms

d) Combining the a terms
[Solution Description]
First distribute: $5(2a - 3b) = 10a - 15b$ and $2(a + 4b) = 2a + 8b$.
Combine these: $10a - 15b + 2a + 8b$.
Combine like terms: $(10a + 2a) + (-15b + 8b) = 12a - 7b$.
The error was in combining $10a + 2a$ as 13a instead of 12a.

Key Concept: Combining Like Terms

b) $3x^2 + 4x$
[Solution Description]
Combine like terms step-by-step:
$5x^2 - 2x^2 = 3x^2,$
$3x + x = 4x.$
Thus, the correct simplification is:
$3x^2 + 4x.$
The given incorrect simplification matches the correct answer, so there is no error here.

Key Concept: Binomial Squaring Error

c) $4a^2 + 12ab + 9b^2$
[Solution Description]
Use the formula for squaring a binomial:
$(2a + 3b)^2 = (2a)^2 + 2 \times 2a \times 3b + (3b)^2 = 4a^2 + 12ab + 9b^2$

Key Concept: Algebraic fractions, Distributive property

a) $7p + 2q$
[Solution Description]
First distribute $\frac{1}{3}$: $\frac{1}{3}(9p) - \frac{1}{3}(6q) = 3p - 2q$.
Then distribute 4: $4(p) + 4(q) = 4p + 4q$.
Combine all terms: $3p - 2q + 4p + 4q = (3p + 4p) + (-2q + 4q) = 7p + 2q$.
The mistake was in not distributing the 4 to both terms inside the second parentheses.

Key Concept: Analyzing incorrect expansions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expansion of $(5m + 6n)^2$ is $25m^2 + 60mn + 36n^2$. This is correct because applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ gives:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$.
The reason states that the identity $(a + b)^2 = a^2 + 2ab + b^2$ was applied correctly, which is true and explains why the assertion is correct.

Key Concept: Binomial Expansion

c) $4a^2 - 12ab + 9b^2$
[Solution Description]
Use the formula for squaring a binomial: $(A - B)^2 = A^2 - 2AB + B^2$. Here, $A = 2a$ and $B = 3b$.
$(2a - 3b)^2 = (2a)^2 - 2 \times 2a \times 3b + (3b)^2 = 4a^2 - 12ab + 9b^2.$

Key Concept: Algebraic Expansion and Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expansion of $(5m + 6n)^2$ is $25m^2 + 60mn + 36n^2$. This is correct because:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$
The reason states that the cross-term in the expansion of a binomial square $(a + b)^2$ is given by $2ab$, which is also correct and explains why the assertion is true. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Distributive Property

c) $12x - 20y + 8$
[Solution Description]
First, apply the distributive property to $4(3x - 5y + 2)$. Multiply each term inside the parentheses by 4:
$4 \times 3x = 12x, \quad 4 \times (-5y) = -20y, \quad 4 \times 2 = 8.$
Combine these results to get the correct expansion:
$12x - 20y + 8.$

Key Concept: Incorrect Expansion

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, let's expand $(3x + 4)^2$ correctly:
$(3x + 4)^2 = (3x)^2 + 2 \times 3x \times 4 + 4^2 = 9x^2 + 24x + 16$.
The given assertion claims the simplified form is $9x^2 + 16$, which is incorrect because the middle term $24x$ is missing.
The reason states that the middle term is missing, which is true and explains why the assertion is false.
Therefore, Assertion is false, but Reason is true.

Key Concept: Distributive Property Error

c) $6xy - 12xz$
[Solution Description]
To expand $3x(2y - 4z)$, apply the distributive property:
$3x \times 2y = 6xy$
$3x \times (-4z) = -12xz$
Therefore, the correct expansion is $6xy - 12xz$.

Key Concept: Distributive property, Squaring binomials

c) $24xy$
[Solution Description]
When squaring a binomial $(a + b)^2$, the correct expansion is $a^2 + 2ab + b^2$. For $(4x + 3y)^2$, the correct expansion is:
$(4x)^2 + 2(4x)(3y) + (3y)^2 = 16x^2 + 24xy + 9y^2$. The missing term is 24xy.

Key Concept: Like Terms Error

b) $4p^2 + 6p$
[Solution Description]
Combine like terms:
$5p^2 - p^2 = 4p^2$
$2p + 4p = 6p$
The simplified form is $4p^2 + 6p$.

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: 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: 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: Difference of squares, Pattern recognition

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

Key Concept: Pattern 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: 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: 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: Difference of Squares Identity

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

Key Concept: Square of a Number Using Identity

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

Key Concept: 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: 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 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: 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