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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Sridharacharya’s method of fast squaring

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify, let us take $a = 10$ and $b = 2$.
Using the identity:
$10^2 = (10 + 2)(10 - 2) + 2^2 = 12 \times 8 + 4 = 96 + 4 = 100$, which matches $10^2 = 100$.
The assertion is true.
The reason states that this identity is derived from $(a + b)(a - b) = a^2 - b^2$ by rearranging terms. This is also correct, as:
$(a + b)(a - b) = a^2 - b^2$ implies $a^2 = (a + b)(a - b) + b^2$.
Thus, the Reason correctly explains the Assertion.

Key Concept: Distributive Property and Algebraic Identities

d) 5184
[Solution Description]
To find $72^2$ using Sridharacharya's method: Let $a = 72$. Choose $b = 2$ so that $(a + b)(a - b) + b^2$ simplifies the calculation.
Step 1: Compute $(72 + 2)(72 - 2) = 74 \times 70$.
Step 2: Calculate $74 \times 70 = 5180$.
Step 3: Add $b^2 = 2^2 = 4$ to the result from Step 2: $5180 + 4 = 5184$.
Therefore, $72^2 = 5184$.

Key Concept: Sridharacharya’s Method for Fast Squaring

c) Assertion is true, but Reason is false.
[Solution Description]
To verify the assertion, let us compute $48^2$ using Sridharacharya's method:
Choose $a = 48$ and $b = 2$.
Substituting into the identity:
$48^2 = (48 + 2)(48 - 2) + 2^2$
$48^2 = 50 \times 46 + 4$
$48^2 = 2300 + 4 = 2304$
The assertion is true as the calculation yields $2304$.
Now, examine the reason:
The given identity is $a^2 = (a + b)(a - b) + b^2$, not derived from $(a + 1)^2$.
$(a + 1)^2$ expands to $a^2 + 2a + 1$, which is different from the given identity.
Thus, the reason is false.
Therefore, Assertion is true, but Reason is false.

Key Concept: Application of Identity 1C

d) 9996
[Solution Description]
Let $a = 100$, $b = 2$
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$
Thus,
$(100 + 2)(100 - 2) = 100^2 - 2^2$
$100^2 = 10{,}000$ and $2^2 = 4$
So, $10{,}000 - 4 = 9996$
Hence, $98 \times 102 = 9996$.

Key Concept: Distributive Property and Algebraic Identities

d) 21025
[Solution Description]
To find $145^2$ using Sridharacharya's method: Let $a = 145$. Choose $b = 5$ to simplify the calculation.
Step 1: Compute $(145 + 5)(145 - 5) = 150 \times 140$.
Step 2: Calculate $150 \times 140 = 21000$.
Step 3: Add $b^2 = 5^2 = 25$ to the result from Step 2: $21000 + 25 = 21025$.
Therefore, $145^2 = 21025$.

Key Concept: Sridharacharya’s method of fast squaring

c) $(102 + 2)(102 - 2) + 2^2$
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 102$, $b = 2$.
Step 1: Compute $(a + b)(a - b) = (102 + 2)(102 - 2) = 104 \times 100 = 10400$.
Step 2: Compute $b^2 = 2^2 = 4$.
Step 3: Add the results from Step 1 and Step 2: $10400 + 4 = 10404$.
The correct expression is $(102 + 2)(102 - 2) + 2^2$.

Key Concept: Distributive Property, Sridharacharya’s Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(a + b)(a - b) + b^2 = a^2$. Let's verify this step by step:
Step 1: Expand $(a + b)(a - b)$. Using the distributive property, we get:
$(a + b)(a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^2 - b^2$.
Step 2: Add $b^2$ to the result from Step 1:
$a^2 - b^2 + b^2 = a^2$.
Therefore, the assertion is true.
The reason states that $(a + b)(a - b) = a^2 - b^2$, which is correct as shown in Step 1. Adding $b^2$ indeed gives $a^2$, which matches the assertion. Hence, the reason correctly explains the assertion.

Key Concept: Sridharacharya’s method of fast squaring

b) 2401
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 49$, $b = 1$.
Step 1: Compute $(a + b)(a - b) = (49 + 1)(49 - 1) = 50 \times 48 = 2400$.
Step 2: Compute $b^2 = 1^2 = 1$.
Step 3: Add the results from Step 1 and Step 2: $2400 + 1 = 2401$.
Therefore, $49^2 = 2401$.

Key Concept: Pattern Verification

b) $a^2 - b^2 = (a + b)(a - b)$
[Solution Description]
The given pattern shows that:
$9 \times 9 - 1 \times 1 = 10 \times 8$ is equivalent to $9^2 - 1^2 = (9 + 1)(9 - 1) = 10 \times 8$
Similarly, $a^2 - b^2 = (a + b)(a - b)$ is an established algebraic identity derived from the distributive property.
This confirms the correctness of the pattern.

Key Concept: Sridharacharya’s Method

d) 5184
[Solution Description]
Using Sridharacharya's identity:
$a^2 = (a + b)(a - b) + b^2$
Here, $a = 72$, $b = 2$.
$(72 + 2)(72 - 2) + 2^2 = 74 \times 70 + 4$
$74 \times 70 = 5180$
Adding $4$ gives $5180 + 4 = 5184$.
Therefore, $72^2 = 5184$.

Key Concept: Distributive Property and Algebraic Identities

d) 1203409
[Solution Description]
To find $1097^2$ using Sridharacharya's method: Let $a = 1097$ and $b = 3$.
Step 1: Compute $(1097 + 3)(1097 - 3) = 1100 \times 1094$.
Step 2: Calculate $1100 \times 1094 = 1203400$.
Step 3: Add $b^2 = 3^2 = 9$ to the result from Step 2: $1203400 + 9 = 1203409$.
Therefore, $1097^2 = 1203409$.

Key Concept: Sridharacharya’s method of fast squaring

b) 625
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 25$, $b = 5$.
Step 1: Compute $(a + b)(a - b) = (25 + 5)(25 - 5) = 30 \times 20 = 600$.
Step 2: Compute $b^2 = 5^2 = 25$.
Step 3: Add the results from Step 1 and Step 2: $600 + 25 = 625$.
Therefore, $25^2 = 625$.

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

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, 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: 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 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

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 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 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: Error in Expansion

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

Key Concept: Error Spotting in Algebraic Simplification

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: 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: 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: 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: Application of Identity 1A

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

Key Concept: Square of the 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 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

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: 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: Difference of Squares, Multi-step Solution

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

Key Concept: Square of the Sum, Difference of Squares

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of 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: 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 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: 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: 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, 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: 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: 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: 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: 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

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: Area Calculations

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

Key Concept: Pattern Analysis

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

Key Concept: Number Pattern Representation

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

Key Concept: Pattern Representation, Simplification

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

Key Concept: Pattern Recognition and Generalization, Algebraic Simplification

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

Key Concept: Square Tiles Pattern

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

Key Concept: Mathematical Identities, Verification

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

Key Concept: Sridharacharya's Method for Squares, Application

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

Key Concept: Algebraic Identities

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

Key Concept: Number Patterns

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

Key Concept: Algebraic identities and pattern generalization

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

Key Concept: Algebraic Identities

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

Key Concept: 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: 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: 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

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: 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: 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 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: 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: 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: 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: Expression Simplification

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

Key Concept: Consistency of expressions, Algebraic identities

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

Key Concept: Multiple methods for pattern representation, algebraic equivalence

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

Key Concept: Pattern analysis

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

Key Concept: Pattern recognition, algebraic simplification

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

Key Concept: Multiple methods

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

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

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

Key Concept: Consistency of expressions across different methods

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

Key Concept: Pattern Formula Application

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

Key Concept: Pattern Formula Verification

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

Key Concept: Consistency of expressions across different methods

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

Key Concept: Consistency of expressions

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

Key Concept: 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: Square of a Number Using Identity

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

Key Concept: Difference of Squares 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: 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: 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: 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: 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: 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: 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: 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: 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: Product Change, Distributive Property

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

Key Concept: General Identity for Product Change

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

Key Concept: 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: Special Case Identity

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

Key Concept: Identity Application

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

Key Concept: 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: 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: 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

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

Key Concept: General Identity for Product Change

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

Key Concept: 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: Simplifying Expressions

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

Key Concept: Correcting algebraic misconceptions

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

Key Concept: Simplification of Expressions

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

Key Concept: Correcting misconceptions in algebraic expressions

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

Key Concept: Expanding and Simplifying Expressions

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

Key Concept: Algebraic Expansion

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

Key Concept: Algebraic identities, Error correction

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

Key Concept: Algebraic Identities

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

Key Concept: Difference of squares, Remainder theorem

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

Key Concept: Algebraic Identities

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

Key Concept: Number patterns, Algebraic identities

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

Key Concept: Verification of Identity

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

Key Concept: Generalizing Patterns

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