Key Concept: Pattern of Circles

c) 120
[Solution Description]
Substitute $k = 10$ into the expression $k^2 + 2k$:
$10^2 + 2 \times 10 = 100 + 20 = 120$
Thus, the number of circles in Step 10 is 120.

Key Concept: Increments in Products

c) $a + b + 1$
[Solution Description]
According to the given pattern, increasing both numbers $a$ and $b$ by 1 increases the product by $a + b + 1$. For example, with $a = 23$ and $b = 27$, the increment is $23 + 27 + 1 = 51$.

Key Concept: Difference of Squares

c) $100^2 - 8^2$
[Solution Description]
To find $108 \times 92$ using the difference of squares identity $(a+b)(a-b)=a^2-b^2$, we need to express it in terms of $(a+b)(a-b)$.
Let's find $a$ and $b$ such that:
$a + b = 108$
$a - b = 92$
Adding these equations:
$2a = 200 \Rightarrow a = 100$
Subtracting them:
$2b = 16 \Rightarrow b = 8$
Now apply the identity:
$108 \times 92 = (100+8)(100-8) = 100^2 - 8^2 = 10000 - 64 = 9936$
We can also verify:
$108 \times 90 = 9720$
$108 \times 2 = 216$
Total $9720 + 216 = 9936$
Thus, the equivalent expression is $100^2 - 8^2$.

Key Concept: Algebraic Pattern - Circles

c) 63
[Solution Description]
To find the number of circles in Step $k$, use the formula:
$k^2 + 2k$.
Substitute $k = 7$:
$7^2 + 2 \times 7 = 49 + 14 = 63$.
Therefore, there are 63 circles in Step 7.

Key Concept: Difference of Squares Identity

c) 125
[Solution Description]
Apply the difference of squares identity:
$a^2 - b^2 = (a + b)(a - b)$.
Here, $a = 15$ and $b = 10$.
So, $15^2 - 10^2 = (15 + 10)(15 - 10) = 25 \times 5 = 125$.
Therefore, the value is 125.

Key Concept: Sum of Squares, Difference of Squares

c) 361
[Solution Description]
First, identify $a$ and $b$ from the given expression $2(12^2 + 7^2)$. Here $a=12$ and $b=7$. Using the identity:
$2(a^2 + b^2) = (a+b)^2 + (a-b)^2$
Substitute the values:
$2(12^2 + 7^2) = (12+7)^2 + (12-7)^2$
Calculate each term:
$(12+7)^2 = 19^2 = 361$
$(12-7)^2 = 5^2 = 25$
Therefore, one of the terms in the sum is 361.

Key Concept: Sum of Squares Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let us expand both $(a + b)^2$ and $(a - b)^2$ using the algebraic identities:
Step 1: Expand $(a + b)^2$:
$(a + b)^2 = a^2 + 2ab + b^2$
Step 2: Expand $(a - b)^2$:
$(a - b)^2 = a^2 - 2ab + b^2$
Step 3: Add the two expansions:
$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)$
Step 4: Simplify the result:
$2a^2 + 2b^2 = 2(a^2 + b^2)$
Thus, the assertion is true because the equation holds for any natural numbers $a$ and $b$. The reason correctly explains how the identity is derived by adding the expansions of $(a + b)^2$ and $(a - b)^2$.

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, evaluate the Assertion:
For $n = 10$, using the formula $n^2 + 2n$, we get $10^2 + 2 \times 10 = 100 + 20 = 120$. So, the Assertion is true.
Now, check the Reason:
Expand $(n+1)^2 - 1$:
$(n+1)^2 - 1 = n^2 + 2n + 1 - 1 = n^2 + 2n$. This matches the formula in the Assertion. Hence, the Reason is true and correctly explains the Assertion.

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's expand both sides of the given equation using algebraic identities:
Right-hand side (RHS):
$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2$
Left-hand side (LHS):
$2(a^2 + b^2) = 2a^2 + 2b^2$
Since LHS = RHS, the assertion is true.
The reason provides the correct expansion formulas for $(a + b)^2$ and $(a - b)^2$, which are directly used to prove the assertion. Hence, the reason is the correct explanation of the assertion.

Key Concept: Algebraic Identities

b) 9996
[Solution Description]
Rewrite $98 \times 102$ as $(100 - 2)(100 + 2)$. Using the identity $a^2 - b^2 = (a + b)(a - b)$, we get:
$(100 - 2)(100 + 2) = 100^2 - 2^2 = 10000 - 4 = 9996$
So, $98 \times 102 = 9996$.

Key Concept: Sum of Squares, Difference of Squares

b) 8
[Solution Description]
Using the identity $2(a^2 + b^2) = (a+b)^2 + (a-b)^2$ with $a=9$ and $b=4$:
$2(9^2 + 4^2) = (9+4)^2 + (9-4)^2 = 13^2 + 5^2 = 169 + 25 = 194$
The two squared terms obtained are $169$ and $25$.
Their square roots are $\sqrt{169} = 13$ and $\sqrt{25} = 5$.
The difference between these square roots is $13 - 5 = 8$.
Alternatively, note that $(a+b)-(a-b) = 2b = 8$ which matches our calculation.

Key Concept: Algebraic Pattern - Square Tiles

c) 35
[Solution Description]
The formula for the number of tiles is $(n + 1)^2 - 1$.
Substitute $n = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$.
Therefore, there are 35 tiles in Step 5.

Key Concept: Difference of squares, Algebraic identities

a) 4891
[Solution Description]
We can express $73 \times 67$ as $(70 + 3)(70 - 3)$. Using the difference of squares identity, this becomes:
$70^2 - 3^2 = 4900 - 9 = 4891$

Key Concept: Algebraic Identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the identity $(a + b)(a - b) = a^2 - b^2$ simplifies multiplication, which is true as shown in examples like $98 \times 102 = 100^2 - 2^2 = 9996$. The reason explains that this identity comes from the distributive property, which is also correct because expanding $(a + b)(a - b)$ gives $a^2 - ab + ab - b^2 = a^2 - b^2$, confirming the identity. Thus, the reason correctly explains the assertion.

Key Concept: Using identities to simplify expressions

d) 64
[Solution Description]
To solve $(5 + 3)^2$, apply the identity $(a + b)^2 = a^2 + 2ab + b^2$.
Here, $a = 5$ and $b = 3$.
Substituting these values into the identity:
$(5 + 3)^2 = 5^2 + 2 \times 5 \times 3 + 3^2$
Calculate each term step-by-step:
$5^2 = 25$
$2 \times 5 \times 3 = 30$
$3^2 = 9$
Now add all terms:
$25 + 30 + 9 = 64$

Key Concept: Using the identity $a^2 - b^2 = (a + b

b) 2200
[Solution Description]

We know that
$a^2 - b^2 = (a-b)(a+b)$

Here,
$a = 47,\quad b = 3$

So,
$47^2 - 3^2 = (47-3)(47+3)$

$= 44 \times 50$
$= 2200$

Therefore, the value of
$47^2 - 3^2$
is 2200.

Key Concept: Difference of squares

d) 64
[Solution Description]
To find $10^2 - 6^2$, use the identity $a^2 - b^2 = (a + b)(a - b)$.
Here, $a = 10$ and $b = 6$.
Substituting these values into the identity:
$10^2 - 6^2 = (10 + 6)(10 - 6)$
Calculate each part step-by-step:
$10 + 6 = 16$
$10 - 6 = 4$
Now multiply them:
$16 \times 4 = 64$

Key Concept: Using the identity $(a + b

c) 25
[Solution Description]

We know that
$(a+b)^2 = a^2 + 2ab + b^2$

Here, $((x+5)^2)$ means $(a=x)$ and $(b=5).$ So,
$(x+5)^2 = x^2 + 2(x)(5) + 5^2$
$= x^2 + 10x + 25$

Comparing this with
$x^2 + 10x + k$

we get:
$k = 25$

So, the value of $k$ is $(\boxed{25}\).$

Key Concept: Verifying patterns

a) $100 + 16 = 116$
[Solution Description]
The given pattern is $(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$. Substitute $a = 7$ and $b = 3$ into the equation.
First calculate $(a + b)^2$:
$(7 + 3)^2 = 10^2 = 100$
Next calculate $(a - b)^2$:
$(7 - 3)^2 = 4^2 = 16$
Add the two results:
$100 + 16 = 116$
Now compute $2(a^2 + b^2)$:
$a^2 = 7^2 = 49$
$b^2 = 3^2 = 9$
Sum them:
$49 + 9 = 58$
Multiply by 2:
$2 \times 58 = 116$
Both sides are equal, confirming the pattern holds.

Key Concept: Sum of squares, Algebraic identities

a) 100
[Solution Description]
Plugging in $a = 8$ and $b = 6$ into the identity:
$2(8^2 + 6^2) = (8 + 6)^2 + (8 - 6)^2$
$2(64 + 36) = 14^2 + 2^2$
$200 = 196 + 4$
Now, solving for $a^2 + b^2$:
$2(a^2 + b^2) = 200 \implies a^2 + b^2 = 100$

Key Concept: Square of binomial, Expanding expressions

c) $16x^2 + 56x + 49$
[Solution Description]
Applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 4x$ and $b = 7$, we get:
$(4x)^2 + 2(4x)(7) + 7^2 = 16x^2 + 56x + 49$

Key Concept: Algebraic Pattern Investigation: Sum of Squares and Difference of Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any two natural numbers $a$ and $b$, $2(a^2 + b^2)$ can be written as $(a + b)^2 + (a - b)^2$. This is supported by the algebraic identity provided in the reason. By expanding $(a + b)^2$ and $(a - b)^2$:
$(a + b)^2 = a^2 + 2ab + b^2 \\ \\ (a - b)^2 = a^2 - 2ab + b^2$
Adding these two equations gives:
$(a + b)^2 + (a - b)^2 = 2a^2 + 2b^2 = 2(a^2 + b^2)$
Thus, the assertion is true, and the reason correctly explains it using the derived identity.

Key Concept: Using identities to explain numerical puzzles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(10 + 5)^2$ can be expanded using the identity $(a + b)^2 = a^2 + 2ab + b^2$. This is correct because the identity holds for any real numbers $a$ and $b$, including 10 and 5. Applying it, we get:
$(10 + 5)^2 = 10^2 + 2 \times 10 \times 5 + 5^2 = 100 + 100 + 25 = 225$
The reason correctly explains why the identity works: the square of a binomial indeed equals the sum of the squares of the terms and twice their product. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Applying $(a + b

a) 95
[Solution Description]

We know that
$(a+b)(a-b)=a^2-b^2$

Here,
$a=12,\quad b=7$

So,
$(12+7)(12-7)=12^2-7^2$

$=144-49$

$=95$

Therefore, the value of
$(12+7)(12-7)$
is 95.

Key Concept: Difference of Squares, Multi-step Solution

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

c) $x^2 + 8x + 16$
[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 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 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: Application of Square of Sum/Difference

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

Key Concept: Square of the 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 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, 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: 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 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: 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: 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 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 Pattern

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

Key Concept: Squares of numbers, Difference of squares

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

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

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

Key Concept: Difference of Squares 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: 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 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: 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: 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: 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 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: 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: 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 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) 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: 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: 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 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

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) 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

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: Special Cases of the Distributive Property

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

Key Concept: Special Cases of the Distributive Property

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Special Cases of the Distributive Property

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

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

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

Key Concept: Product of Sum and Difference of Two Numbers

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

Key Concept: Special Cases of the Distributive Property

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

Key Concept: Special Cases of the Distributive Property

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

Key Concept: Special Cases of the Distributive Property

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

Key Concept: Quick multiplication using identities

b) 627
[Solution Description]
Using the distributive property for multiplying by 11:
Break down $57 \times 11 = 57 \times (10 + 1) = 57 \times 10 + 57 \times 1 = 570 + 57 = 627$.

Key Concept: Quick multiplication using identities

c) 495
[Solution Description]
To find $45 \times 11$, use the distributive property:
$45 \times 11 = 45 \times (10 + 1) = 45 \times 10 + 45 \times 1 = 450 + 45 = 495$.

Key Concept: Quick multiplication using identities

c) 88804
[Solution Description]
To find $298^2$ using Sridharacharya's identity $a^2 = (a + b)(a - b) + b^2$, we choose $b = 2$ since it makes $(a + b)$ and $(a - b)$ easy to multiply.
Step 1: Apply the identity:
$298^2 = (298 + 2)(298 - 2) + 2^2$
Step 2: Calculate $(298 + 2) = 300$ and $(298 - 2) = 296$
Step 3: Multiply them: $300 \times 296 = 88800$
Step 4: Add $2^2 = 4$ to the result: $88800 + 4 = 88804$
Therefore, $298^2 = 88804$.

Key Concept: Quick multiplication using identities, distributive property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Solution:
Step 1: Identify $a$ and $b$ in the given problem. Here, $197 = 200 - 3$ and $203 = 200 + 3$, so $a = 200$ and $b = 3$.
Step 2: Apply the identity $(a + b)(a - b) = a^2 - b^2$. Substituting $a = 200$ and $b = 3$, we get:
$(200 + 3)(200 - 3) = 200^2 - 3^2 = 40000 - 9 = 39991$
Step 3: Therefore, $197 \times 203 = 39991$.
The Assertion correctly identifies the application of the identity $(a + b)(a - b) = a^2 - b^2$, and the Reason provides the correct explanation for why this method works (i.e., the numbers are equidistant from $200$).

Key Concept: Multiplication with 11, 101, etc.

d) 732553
[Solution Description]
To multiply $7253 \times 101$, we use the identity for multiplying by 101 ($100 + 1$):
Step 1: Break down the multiplication:
$7253 \times 101 = 7253 \times (100 + 1) = 7253 \times 100 + 7253 \times 1$
Step 2: Compute $7253 \times 100 = 725300$
Step 3: Compute $7253 \times 1 = 7253$
Step 4: Add the results: $725300 + 7253 = 732553$
Therefore, $7253 \times 101 = 732553$.

Key Concept: Pattern Recognition in Products

c) $x^2 - 9$
[Solution Description]
Using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$, substitute $a = x$ and $b = 3$:
$(x - 3)(x + 3) = x^2 - 3^2 = x^2 - 9$.

Key Concept: Pattern recognition in identities

b) $x^3 - y^3$
[Solution Description]
The given expression follows the pattern $(a - b)(a^2 + ab + b^2) = a^3 - b^3$.
Thus, $(x - y)(x^2 + xy + y^2) = x^3 - y^3$.

Key Concept: Quick multiplication using identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To multiply a two-digit number $ab$ by 11, we use the distributive property:
$ab \times 11 = ab \times (10 + 1) = ab \times 10 + ab \times 1$.
This results in shifting the number left by one digit and adding the original number:
$ab0 + ab = a \quad (a + b) \quad b$.
For example, $42 \times 11 = 420 + 42 = 462$.
The Assertion is true, and the Reason correctly explains it.

Key Concept: Sridharacharya's Method for Squares

c) 2704
[Solution Description]
Using the identity $(a^2 = (a + b)(a - b) + b^2)$, set $a = 52$ and $b = 2$:
$52^2 = (52 + 2)(52 - 2) + 2^2 = 54 \times 50 + 4 = 2700 + 4 = 2704$.

Key Concept: Sridharacharya's method for squares

c) 2401
[Solution Description]
Using the identity $(a + b)(a - b) + b^2 = a^2$, choose $b = 1$ for $a = 49$.
Therefore, $(49 + 1)(49 - 1) + 1^2 = 50 \times 48 + 1 = 2400 + 1 = 2401$.

Key Concept: Squaring numbers using identities

c) 27225
[Solution Description]
To compute $165^2$ using the given identity, let us choose $b = 5$:
Step 1: Apply the identity:
$165^2 = (165 + 5)(165 - 5) + 5^2$
Step 2: Compute $(165 + 5) = 170$ and $(165 - 5) = 160$
Step 3: Multiply them: $170 \times 160 = 27200$
Step 4: Add $5^2 = 25$: $27200 + 25 = 27225$
Therefore, $165^2 = 27225$.

Key Concept: Quick multiplication using identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion with an example:
Take a 3-digit number, say $123$. Multiplying it by $101$:
$123 \times 101 = 123 \times (100 + 1) = 12300 + 123 = 12423$
As seen, the result is obtained by writing the number $123$ twice ($123$ followed by $23$ for a 3-digit case).
The reason correctly explains the assertion using the distributive property $(a \times (100 + 1) = 100a + a)$. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Identity Application

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

Key Concept: Special Case Identity

c) $4x^2 + 12x + 9$
[Solution Description]
The identity $(a + b)^2 = a^2 + 2ab + b^2$ is used here with $a = 2x$ and $b = 3$.
Substituting, we get:
$(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$

Key Concept: Distributive Property, Algebraic Identities

a) $xy - 3y + 4x - 12$
[Solution Description]
Using the identity $(a + m)(b + n) = ab + mb + an + mn$, let $m = -3$ and $n = 4$. Substituting these values into the identity:
$(x - 3)(y + 4) = xy + (-3)y + x(4) + (-3)(4) = xy - 3y + 4x - 12$

Key Concept: 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: General Identity for Product Change

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

Key Concept: Product Change, Distributive Property

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

Key Concept: 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: 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 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: 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: 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: 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

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

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: 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: 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: 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: 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: 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, Algebraic Identity

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

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: 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: Fast Multiplication

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

Key Concept: Multiplication by Parts

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

Key Concept: Special Case: Square of Sum

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

Key Concept: Distributive Property, Square of Sum

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

Key Concept: Distributive Property

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

Key Concept: Distributive Property, Product of Sum and Difference

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

Key Concept: Distributive Property, Special Cases

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Distributive Property Application

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

Key Concept: Distributive Property

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

Key Concept: Square of Sum Identity

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

Key Concept: Distributive Property

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

Key Concept: Increments in Products

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, let's verify the Assertion (A). Consider the product $(a + 1)(b - 1)$. Increasing both numbers by 1 gives $(a + 2)(b)$. Using the distributive property:
$(a + 2)(b) = ab + 2b$
The increase in the product is $(ab + 2b) - (a + 1)(b - 1)$. Expanding $(a + 1)(b - 1)$ as per the given formula yields:
$(a + 1)(b - 1) = ab + b - a - 1$
Thus, the difference is:
$(ab + 2b) - (ab + b - a - 1) = b + a + 1$
So, the Assertion (A) is true.
Now, consider the Reason (R). The identity $(a + m)(b + n) = ab + mb + an + mn$ is a standard algebraic identity that holds true for all integers $a, b, m,$ and $n$, making the Reason (R) true as well. However, the Reason (R) is not the correct explanation for the Assertion (A), because the Assertion deals with a specific case of increasing both numbers in the product $(a + 1)(b - 1)$ by 1, whereas the Reason provides a general identity.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Increments in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when both numbers in a product are increased by 1, the increase in the product is equal to the sum of the original numbers plus 1. The reason explains this using the distributive property of multiplication, which expands $(a + 1)(b + 1)$ to $ab + a + b + 1$. Here, the increase in the product is $a + b + 1$, validating the assertion. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Distributive Property, Increments in Products

c) 82
[Solution Description]
Original product: $P = 15 \times 24$.
New product after increment: $(15 + 2)(24 + 2) = 17 \times 26$.
Increase in product = $(17 \times 26) - (15 \times 24)$.
First, calculate $17 \times 26$:
$17 \times 20 = 340$ and $17 \times 6 = 102$, so $340 + 102 = 442$.
Next, calculate $15 \times 24$:
$15 \times 20 = 300$ and $15 \times 4 = 60$, so $300 + 60 = 360$.
Increase = $442 - 360 = 82$.
Alternatively, using the formula for increment when both numbers change:
Increase = $a \cdot n + b \cdot m + m \cdot n = 15 \times 2 + 24 \times 2 + 2 \times 2 = 30 + 48 + 4 = 82$.

Key Concept: Decrease in One Number and Increase in the Other

a) $-a + b - 1$
[Solution Description]
The original product is $ab$. The new product after changes is $(a + 1)(b - 1)$.
Expanding this using the distributive property:
$(a + 1)(b - 1) = ab - a + b - 1.$
The change in the product is $(ab - a + b - 1) - ab = -a + b - 1$.

Key Concept: Negative Numbers, Increments in Products

b) Decreases by 86
[Solution Description]
Original product: $P = (-7) \times 12 = -84$.
New product: $(-7 - 3)(12 + 5) = (-10) \times 17 = -170$.
Change in product = $-170 - (-84) = -86$.
Using the formula for increments:
Change = $a \cdot n + b \cdot m + m \cdot n = (-7) \times 5 + 12 \times (-3) + (-3) \times 5 = -35 - 36 - 15 = -86$.
The product decreases by 86.

Key Concept: Increments in Products

b) 27
[Solution Description]
The original product is $23 \times 27$. Increasing the first number by 1 gives $(23 + 1) \times 27 = 24 \times 27$. The difference between the new and original product is:
$24 \times 27 - 23 \times 27 = (24 - 23) \times 27 = 1 \times 27 = 27.$
So, the product increases by 27.

Key Concept: Increments in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us first evaluate the Assertion:
Original product = $5 \times 7 = 35$.
New product when 5 is increased by 1: $(5 + 1) \times 7 = 6 \times 7 = 42$.
The product increases by $42 - 35 = 7$, so the Assertion is true.
Now, let's check the Reason:
The formula given is $(a + 1)(b) = ab + b$, which is correct. However, according to the Assertion, the increase is equal to b (which is 7 here). Therefore, the Reason correctly explains the Assertion.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Increment in Products

b) Decreases by 1
[Solution Description]
The initial product is $25 \times 40 = 1000$. After changing the numbers, the new product is $(25 + 2) \times (40 - 3) = 27 \times 37 = 999$. The change in the product is $999 - 1000 = -1$, meaning it decreases by 1.

Key Concept: Increment in Products

b) 20
[Solution Description]
The initial product is $15 \times 20 = 300$. If the first number is increased by 1, the new product becomes $(15 + 1) \times 20 = 16 \times 20 = 320$. The increase in the product is $320 - 300 = 20$.

Key Concept: Multiplicative Increment with Variables

b) $k(y - x) - k^2$
[Solution Description]
Original product: $P = xy$.
New product: $(x + k)(y - k) = xy - xk + ky - k^2$.
Change in product = $(xy - xk + ky - k^2) - xy = -xk + ky - k^2$.
This can also be written as $k(y - x) - k^2$.
Depending on the values of $x$, $y$, and $k$, the change could be positive, negative, or zero.

Key Concept: Generalization Using Distributive Property

b) $a + b + 1$
[Solution Description]
The original product is $ab$. After increasing both numbers by 1, the new product is $(a + 1)(b + 1)$. Using the distributive property:
$(a + 1)(b + 1) = ab + a + b + 1.$
The increase in the product is $(ab + a + b + 1) - ab = a + b + 1$.

Key Concept: Increment in Products

c) 23
[Solution Description]
The initial product is $10 \times 12 = 120$. When both numbers are increased by 1, the new product becomes $(10 + 1) \times (12 + 1) = 11 \times 13 = 143$. The increase in the product is $143 - 120 = 23$.