Key Concept: Identity 1C Application

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

Key Concept: Product of Sum and Difference Identity

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

Key Concept: Pattern Recognition

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

Key Concept: Identity 1C Verification

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

Key Concept: Pattern Recognition

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

Key Concept: Identity 1C, Multi-concept Application

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

Key Concept: Verification of Identity

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

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

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

Key Concept: Identity 1C, Multi-step Solution

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

Key Concept: Identity 1C, Real-World Application

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

Key Concept: Product of Sum and Difference

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

Key Concept: Identity 1C Application

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

Key Concept: 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: 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: 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 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

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 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 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 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: 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: 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) 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: 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: 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: 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) 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

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: 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: 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 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: 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 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: 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) $5x + 10$
[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

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: 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: 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 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: 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: 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 and Increments in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(a + 3)(b - 2)$ expands to $ab - 2a + 3b - 6$, which is correct based on the distributive property. The reason states the general identity $(a + m)(b - n) = ab - an + bm - mn$, where substituting $m = 3$ and $n = 2$ gives the same result as the assertion. Both are true, and the reason correctly explains the assertion.

Key Concept: Increments in Products

c) 70
[Solution Description]
According to increments in products, if one number is increased by 1, the product increases by the other number. For $7 \times 9$, increasing 9 by 1 gives:
$7 \times (9 + 1) = 7 \times 9 + 7 = 63 + 7 = 70$

Key Concept: Increments in Products, Complex Multiplication

a) $-2p + 3q - 6$
[Solution Description]
The new product after increasing $p$ by 3 and decreasing $q$ by 2 is $(p + 3)(q - 2)$. Expanding this using the distributive property:
$(p + 3)(q - 2) = pq - 2p + 3q - 6$
To find the increase, subtract the original product $pq$:
$(pq - 2p + 3q - 6) - pq = -2p + 3q - 6$
The increase is $-2p + 3q - 6$. Note that a negative increase implies a decrease.

Key Concept: Identity Application, Expansion with Subtraction

b) $xy + 2x - 5y - 10$
[Solution Description]
To expand $(x - 5)(y + 2)$, we use the distributive property:
$(x - 5)(y + 2) = x(y + 2) - 5(y + 2)$
We further distribute:
$xy + 2x - 5y - 10$
Thus, the expanded form is $xy + 2x - 5y - 10$.

Key Concept: General Case for Increments

a) $10 + 5m + 2n + mn$
[Solution Description]
Using the general case formula for increments:
$(2 + m)(5 + n) = 2 \cdot 5 + m \cdot 5 + 2 \cdot n + m \cdot n$
Simplifying:
$10 + 5m + 2n + mn$

Key Concept: Increments in Products

b) $ab - a + b - 1$
[Solution Description]
Using the distributive property:
$(a + 1)(b - 1) = a \cdot b + a \cdot (-1) + 1 \cdot b + 1 \cdot (-1)$
Simplifying:
$ab - a + b - 1$

Key Concept: Distributive Property of Multiplication over Addition

c) $5x + 15$
[Solution Description]
The distributive property states that $a(b + c) = ab + ac$. Applying this to $5(x + 3)$, we multiply 5 with both terms inside the parentheses:
$5(x + 3) = 5 \times x + 5 \times 3 = 5x + 15$

Key Concept: Expansion of Algebraic Expressions

c) $ab - 2a + 4b - 8$
[Solution Description]
Using the formula for expanding $(a + m)(b - n) = ab - an + bm - mn$:
$(a + 4)(b - 2) = ab - 2a + 4b - 8$

Key Concept: Distributive Property

b) $3v - 3u + uv - 9$
[Solution Description]
To expand $(3 + u)(v - 3)$, we apply the distributive property:
$(3 + u)(v - 3) = 3 \cdot v + 3 \cdot (-3) + u \cdot v + u \cdot (-3)$
Simplifying each term:
$3v - 9 + uv - 3u$
Rearranging terms:
$3v - 3u + uv - 9$

Key Concept: Distributive Property of Multiplication over Addition

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion demonstrates the distributive property: $3 \times (4 + 5) = 3 \times 9 = 27$ and $3 \times 4 + 3 \times 5 = 12 + 15 = 27$. Both are equal, so the Assertion is true. The Reason correctly explains the distributive property used in the Assertion. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Distributive Property, Increments in Products

a) $4a + 3b + 12$
[Solution Description]
To find how much $(a + 3)(b + 4)$ exceeds $ab$, we first expand the expression using the distributive property:
$(a + 3)(b + 4) = ab + 4a + 3b + 12$
The excess is calculated by subtracting the original product $ab$:
$(ab + 4a + 3b + 12) - ab = 4a + 3b + 12$
Therefore, the product increases by $4a + 3b + 12$.

Key Concept: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion (A) claims that $(a + 1)(b - 1) = ab + b - a - 1$. Let us verify this using the distributive property mentioned in reason (R). Expanding $(a + 1)(b - 1)$ using the distributive property:
Step 1: Apply the distributive property: $(a + 1)(b - 1) = a(b) + a(-1) + 1(b) + 1(-1)$.
Step 2: Simplify each term: $ab - a + b - 1$.
Step 3: Combine like terms: $ab + b - a - 1$.
This matches the assertion, so assertion (A) is true. The reason (R) correctly states the general form of the distributive property, and it is the correct explanation for assertion (A). Therefore, both assertion and reason are true, and the reason explains the assertion.

Key Concept: Pattern Generalization, Algebraic Expression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let us simplify all four methods:
\textbf{Method 1}: $k \times (k + 2) = k^2 + 2k$
\textbf{Method 2}: $k \times (k + 1) + k = k^2 + k + k = k^2 + 2k$
\textbf{Method 3}: $k^2 + 2 \times k = k^2 + 2k$
\textbf{Method 4}: $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
All four methods simplify to the same expression $k^2 + 2k$, confirming that the assertion is true. The reason correctly explains why the assertion holds, as it demonstrates that multiple valid methods converge to the same result.

Key Concept: Multiple Methods to Solve a Problem

c) 35
[Solution Description]
The first method gives $5 \times (5 + 2) = 5 \times 7 = 35$. According to the problem statement, all correct methods must yield the same result, so the other method must also give 35.

Key Concept: Pattern Analysis

b) 35
[Solution Description]
To find the number of circles in Step 5, substitute $k = 5$ into the formula $k^2 + 2k$.
$5^2 + 2 \times 5 = 25 + 10 = 35$
Therefore, there are 35 circles in Step 5.

Key Concept: Pattern Interpretation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the expression $k^2 + 2k$ correctly gives the number of circles in Step k of the pattern, which is confirmed by the syllabus material. The Reason states that all four methods lead to this same expression, which is also true as shown in the syllabus. Furthermore, the Reason correctly explains why the Assertion is valid since it demonstrates multiple approaches converging to the same result.

Key Concept: Pattern Recognition

c) 15
[Solution Description]
To find the number of circles in Step 3, substitute $k = 3$ into the given expression:
$k^2 + 2k = 3^2 + 2 \times 3 = 9 + 6 = 15$
So, there are 15 circles in Step 3.

Key Concept: Pattern and Algebraic Expression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $k^2 + 2k$ correctly gives the number of circles at any step $k$ in the pattern. This is true as verified by all four methods (Method 1: $(k + 1)^2 - 1$, Method 2: $k^2 + 2 \times k$, Method 3: $k \times (k + 1) + k$, and Method 4: $k \times (k + 2)$), which simplify to $k^2 + 2k$. Thus, the assertion is true.
The reason states that all four methods lead to the same simplified form $k^2 + 2k$. This is also true, as shown in the syllabus. Moreover, the reason correctly explains why the assertion is true because the equivalence of all methods confirms the validity of the expression $k^2 + 2k$ for the pattern.
Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Pattern Prediction

b) 7
[Solution Description]
Given $k^2 + 2k = 63$, we solve for $k$:
$k^2 + 2k - 63 = 0$
$(k + 9)(k - 7) = 0$
Solutions are $k = -9$ (invalid since step cannot be negative) and $k = 7$.
Therefore, $n = 7$.

Key Concept: Application

c) 24
[Solution Description]
Substitute $n = 4$ into the expression:
$n^2 + 2n = 4^2 + 2 \times 4 = 16 + 8 = 24$
There are 24 circles in Step 4.

Key Concept: Simplification

b) $k^2 + 2k$
[Solution Description]
Simplify the given expression step-by-step:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
The simplified form is $k^2 + 2k$.

Key Concept: Pattern Interpretation and Algebraic Expression

b) 440
[Solution Description]
To find the number of circles in Step 20, substitute $k = 20$ into the given formula:
$\text{Number of circles} = k^2 + 2k = 20^2 + 2 \times 20 = 400 + 40 = 440$

Key Concept: Expression Evaluation

b) $(k + 1)^2 - 1$
[Solution Description]
The given formula is $k^2 + 2k$. Let's verify each option:
- Option 1: $k(k + 3) = k^2 + 3k$ → Not equivalent.
- Option 2: $(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$ → Equivalent.
- Option 3: $k^2 + k + 2$ → Not equivalent.
- Option 4: $2k^2 + k$ → Not equivalent.
Hence, $(k + 1)^2 - 1$ is the correct equivalent expression.

Key Concept: Expression Verification

a) $(k + 1)^2 - 1$
[Solution Description]
We need to verify which expression simplifies to $k^2 + 2k$:
\begin{itemize}

Key Concept: Distributive Property

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

Key Concept: Pattern Recognition

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

Key Concept: Distributive Property 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: Algebraic Identity - Difference of Squares

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

Key Concept: Distributive Property, Pattern Recognition

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

Key Concept: Distributive Property, Algebraic Identities

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

Key Concept: Distributive Property

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

Key Concept: 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: 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: Sum of Squares, Difference of Squares

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

Key Concept: Distributive Property, 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: 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: 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: 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: 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: 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: 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: Pattern recognition, algebraic simplification

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

Key Concept: Pattern 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: 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: Pattern Formula Application

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

Key Concept: Consistency of expressions across different methods

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

Key Concept: 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: Square of Difference

a) $x^2 - 6x + 9$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, where $a = x$ and $b = 3$:
$(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$

Key Concept: Identity 1B: Square of Difference, Algebraic Manipulation

a) $9x^2 - 24xy + 16y^2$
[Solution Description]
To find $(3x - 4y)^2$, we use Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 3x$ and $b = 4y$.
Substitute these values into the identity:
$(3x - 4y)^2 = (3x)^2 - 2 \times 3x \times 4y + (4y)^2$
$= 9x^2 - 24xy + 16y^2$.

Key Concept: Square of Difference

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

Key Concept: Identity 1C: Difference of Squares, Algebraic Manipulation

b) $25m^2 - 49n^2$
[Solution Description]
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$, where $a = 5m$ and $b = 7n$.
$(5m + 7n)(5m - 7n) = (5m)^2 - (7n)^2$
$= 25m^2 - 49n^2$.

Key Concept: Verification of Identity

d) $25a^2 - 70ab + 49b^2$
[Solution Description]
Apply the square of difference identity $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 5a$ and $b = 7b$:
$(5a - 7b)^2 = (5a)^2 - 2(5a)(7b) + (7b)^2$
$= 25a^2 - 70ab + 49b^2$

Key Concept: Identity 1B: Square of Difference, Numerical Application

b) 9604
[Solution Description]
Express $98$ as $100 - 2$. Using Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 100$ and $b = 2$.
$(100 - 2)^2 = 100^2 - 2 \times 100 \times 2 + 2^2$
$= 10000 - 400 + 4$
$= 9604$.

Key Concept: Identity 1B: Square of Difference

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we expand $(3x - 4y)^2$ using Identity 1B:
$(3x - 4y)^2 = (3x)^2 - 2 \times 3x \times 4y + (4y)^2 = 9x^2 - 24xy + 16y^2$
The assertion matches the expanded form. The reason correctly states the identity used in the expansion and explains the assertion. Hence, both are true, and the reason is the correct explanation.

Key Concept: Square of Difference Identity

c) $9x^2 - 24xy + 16y^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 3x$ and $b = 4y$, we substitute:
$(3x - 4y)^2 = (3x)^2 - 2(3x)(4y) + (4y)^2$
$= 9x^2 - 24xy + 16y^2$

Key Concept: Square of Difference Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(x - 3)^2 = x^2 - 6x + 9$. Using the square of difference identity $(a - b)^2 = a^2 - 2ab + b^2$, we can verify this by substituting $a = x$ and $b = 3$. This gives $(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Square of Difference

b) 2401
[Solution Description]
We can write $49$ as $(50 - 1)$. Using $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 50$ and $b = 1$:
$49^2 = (50 - 1)^2 = 50^2 - 2(50)(1) + 1^2 = 2500 - 100 + 1 = 2401$

Key Concept: Square of Difference, Algebraic Identity

d) Assertion is false, but Reason is true.
[Solution Description]

Key Concept: Application of Square of Difference

d) 9801
[Solution Description]
Express $99^2$ as $(100 - 1)^2$. Using the square of difference identity:
$(100 - 1)^2 = 100^2 - 2(100)(1) + 1^2$
$= 10000 - 200 + 1 = 9801$

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: 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: Pattern recognition and algebraic expression

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

Key Concept: Pattern recognition and algebraic expression

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

Key Concept: Algebraic Patterns

b) 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: Algebraic Patterns

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

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

Key Concept: Algebraic Patterns

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

Key Concept: Step-based pattern, Algebraic equivalence

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

Key Concept: Pattern 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: 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: 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

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: 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: Square Tiles Pattern

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

Key Concept: Pattern 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: 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: Mathematical Identities, Verification

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

Key Concept: Area Calculations

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

Key Concept: 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: 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 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: 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: Distributive Property

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