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: 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: 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

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

Key Concept: Difference of squares, Pattern recognition

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

Key Concept: Difference of Squares Pattern

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

Key Concept: Sum of Squares 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: 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: Square of a Number Using Identity

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

Key Concept: Square of Sum Identity

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

Key Concept: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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 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: 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: 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

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

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

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

Key Concept: Error spotting in algebraic simplification, distribution

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

Key Concept: Error in Expansion

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Distributive property, Like terms

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

Key Concept: Error in Distributive Property

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

Key Concept: Error 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: 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, 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: Visualizing Squares and Rectangles

c) $50^2 - 2 \times (50 \times 3) + 3^2$
[Solution Description]
Using the geometric approach:
Step 1: The area of the larger square of side 50 is $50^2 = 2500$.
Step 2: Subtract the areas of the two rectangles formed by removing 3 units from the side: $50 \times 3 + 50 \times 3 = 300$.
Step 3: Add back the small square of side 3 once to adjust for double subtraction: $3^2 = 9$.
Final expression: $50^2 - 2 \times (50 \times 3) + 3^2 = 2500 - 300 + 9 = 2209$.

Key Concept: Expansion of Two Binomials

c) $xy + 3x + 2y + 6$
[Solution Description]
Expand using the distributive property:
$(x + 2)(y + 3) = x \cdot y + x \cdot 3 + 2 \cdot y + 2 \cdot 3 = xy + 3x + 2y + 6$

Key Concept: Geometry-based identities, Distributive Property

c) $x^2 - 9$
[Solution Description]
The area of a rectangle is given by the product of its length and width. Here, the sides are $(x + 3)$ and $(x - 3)$. Using the identity $(a + b)(a - b) = a^2 - b^2$, we can find the area as:
$(x + 3)(x - 3) = x^2 - 9$
Therefore, the area is $x^2 - 9$.

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(60 - 5)^2$ can be evaluated using the formula $60^2 - 2 \times 60 \times 5 + 5^2$, which aligns with the geometric interpretation where we subtract the areas of two rectangles ($60 \times 5$ each) and add back the area of the small square ($5^2$) to avoid double subtraction. The reason explains this geometric process correctly, confirming the assertion. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because $(a - b)^2$ can indeed be expanded to $a^2 - 2ab + b^2$, which represents the geometric process of subtracting two rectangles of area $ab$ and adding back the overlapping square of area $b^2$ to get the area of the smaller square. The reason correctly explains this geometric interpretation, as it describes the steps involved in deriving the identity $(a - b)^2 = a^2 - 2ab + b^2$ using areas.

Key Concept: Geometry-based distributive property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the identity $(a + b)(a - b) = a^2 - b^2$ can be proven geometrically. This is true because we can visualize it as follows:
Step 1: Draw a large square with side length $a$. Its area is $a^2$.
Step 2: Subtract a smaller square with side length $b$ from one corner of the large square. Now, the remaining shape has an area of $a^2 - b^2$.
Step 3: Alternatively, divide the remaining shape into two rectangles with dimensions $(a + b)$ and $(a - b)$. Their combined area is $(a + b)(a - b)$, which matches $a^2 - b^2$.
The reason correctly explains how the geometric method verifies the algebraic identity by subtracting areas. Therefore, both assertion and reason are true, and the reason is the correct explanation.

Key Concept: Special Cases of Distributive Property, Geometry-based explanation

b) 1089
[Solution Description]
We can visualize this problem using geometry. The area of the inner square (side length 33) is $(40 - 7)^2 = 33^2$. According to the identity, we can write:
$(40 - 7)^2 = 40^2 - 2(40 \times 7) + 7^2$
Calculating step-by-step:
$40^2 = 1600$
$2(40 \times 7) = 560$
$7^2 = 49$
Now, substitute these values:
$1600 - 560 + 49 = 1089$
Hence, $(40 - 7)^2 = 1089$.

Key Concept: Distributive Property - Geometric Visualization

b) 3025
[Solution Description]
The area of the larger square is $60^2 = 3600$.
The area of two rectangles is $2 \times (60 \times 5) = 600$.
The area of the small square is $5^2 = 25$.
So, the required area is $3600 - 600 + 25 = 3025$.

Key Concept: Difference of Squares Identity

c) An L-shaped region formed by removing a square of side $b$ from a square of side $a$.
[Solution Description]
The identity $(a + b)(a - b) = a^2 - b^2$ can be visualized as:
Step 1: Draw a large square of side $a$.
Step 2: Remove a smaller square of side $b$ from it, leaving an L-shaped region.
Step 3: The area of the remaining region is $a^2 - b^2$, equivalent to a rectangle with sides $(a + b)$ and $(a - b)$.
Hence, the correct diagram shows this transformation.

Key Concept: Squares of Numbers, Modified Identity

b) 3136
[Solution Description]
To compute $56^2$, let us choose $b = 4$ (since $56 + 4 = 60$ and $56 - 4 = 52$, which are easy to multiply). Using the identity:
$56^2 = (56 + 4)(56 - 4) + 4^2$
Substituting the values:
$(56 + 4) = 60, \quad (56 - 4) = 52$
$4^2 = 16$
Now, multiply $(60 \times 52)$:
$60 \times 50 = 3000$
$60 \times 2 = 120$
$3000 + 120 = 3120$
Finally, add $b^2$:
$3120 + 16 = 3136$
Thus, $56^2 = 3136$.

Key Concept: Distributive Property Application

c) 2025
[Solution Description]
Using Sridharacharya's identity $a^2 = (a + b)(a - b) + b^2$:
Step 1: Let $a = 45$, $b = 5$.
Step 2: Compute $(a + b)(a - b) = (45 + 5)(45 - 5) = 50 \times 40 = 2000$.
Step 3: Add $b^2 = 25$.
Final answer: $45^2 = 2000 + 25 = 2025$.

Key Concept: Difference of Squares Identity

b) $a^2 - b^2$
[Solution Description]
Using the distributive property:
$(a + b)(a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^2 - b^2$

Key Concept: Mathematical Exploration, Application

a) No solution exists
[Solution Description]
Using the standard 2×2 pattern:
$\begin{matrix} \\ a & a+1 \\ \\ a+7 & a+8 \\ \end{matrix}$.
The diagonal products are $P_1 = a(a+8)$ and $P_2 = (a+1)(a+7)$.
Given $P_1 = P_2 + 6$:
$a(a+8) = (a+1)(a+7) + 6$.
Expanding both sides:
$a^2 + 8a = a^2 + 8a + 7 + 6$.
Simplifying: $0 = 13$, which implies no solution exists.
But this contradicts the question, so rechecking:
Alternatively, if $P_2 = P_1 + 6$:
$(a+1)(a+7) = a(a+8) + 6$.
Expanding: $a^2 + 8a + 7 = a^2 + 8a + 6$.
Simplifying: $7 = 6$, again no solution.
Therefore, the condition is impossible unless the difference is -6.
Let $P_1 = P_2 - 6$:
$a^2 + 8a = a^2 + 8a + 7 - 6$
Simplifies to $0 = 1$, still invalid.
Hence, the only possible valid case is when $a = 3$ (empirical check):
For $a = 3$, $P_1 = 33$, $P_2 = 40$, difference is 7 (not matching).
For $a = 2$, $P_1 = 20$, $P_2 = 24$, difference is 4.
For $a = 1$, $P_1 = 9$, $P_2 = 16$, difference is 7.
Thus, no integer $a$ satisfies the given condition.

Key Concept: Calendar Structure Impact

c) Because each row in the calendar is offset by 7 days
[Solution Description] The rows in a calendar are offset by exactly 7 days due to weekly structure. For the labeling $\begin{array}{cc} a & (a + 1) \\ (a + 7) & (a + 8) \end{array}$, the first diagonal product is $a(a + 8)$ and the second diagonal product is $(a + 1)(a + 7)$. Expanding both:
First diagonal: $a^2 + 8a$
Second diagonal: $a^2 + 8a + 7$
The difference $(a^2 + 8a + 7) - (a^2 + 8a) = 7$ arises because of the fixed 7-day row offset in calendars.

Key Concept: Verification of Diagonal Pattern

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

Key Concept: Generalization of Calendar Pattern

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

Key Concept: Generalization of Diagonal Pattern

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

Key Concept: Verification of Patterns, Mathematical Exploration

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

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in a 2×2 calendar square, the difference between diagonal products is always 7. This is true because, for any 2×2 square labeled as $a$, $(a + 1)$, $(a + 7)$, and $(a + 8)$, the diagonal products are:
$a \times (a + 8) = a^2 + 8a$
and
$(a + 1) \times (a + 7) = a^2 + 8a + 7$
The difference between these two products is $(a^2 + 8a + 7) - (a^2 + 8a) = 7$. Therefore, the assertion is correct.
The reason explains why this happens: the numbers in a calendar row increase by 1 and the numbers in a column increase by 7. This explanation is also correct and directly justifies the assertion.
Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Diagonal Product Difference

c) 7
[Solution Description] The diagonal products are calculated as follows:
First diagonal product: $a \times (a + 8) = a^2 + 8a$
Second diagonal product: $(a + 1) \times (a + 7) = a^2 + 8a + 7$
The difference between the two products is:
$(a^2 + 8a + 7) - (a^2 + 8a) = 7$
Therefore, the difference is always 7.

Key Concept: Interpreting calendar-based 2×2 patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- **Assertion (A)**: The difference between the diagonal products in a 2×2 calendar square is calculated as $(a + 1)(a + 7) - a(a + 8) = a^2 + 8a + 7 - a^2 - 8a = 7$. Therefore, the assertion is true.
- **Reason (R)**: A weekly calendar has numbers increasing by 1 in rows and by 7 in columns, making the labeling $\begin{array}{cc} a & (a + 1) \\ a + 7 & (a + 8) \end{array}$ valid. This spacing causes the diagonal product difference to be 7. So, the reason is true and correctly explains the assertion.
Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Diagonal Products in 2×2 Calendar Grid

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

Key Concept: Calendar-Based 2×2 Square Patterns, Verification of Patterns

c) 17
[Solution Description]
Given the 2×2 square pattern:
$\begin{matrix} \\ a & a+1 \\ \\ a+7 & a+8 \\ \end{matrix}$.
The diagonals are $(a, a+8)$ and $(a+1, a+7)$.
Product of first diagonal: $P_1 = a(a+8)$.
Product of second diagonal: $P_2 = (a+1)(a+7)$.
Given $P_1 = 72$, so $a(a+8) = 72$.
Solving $a^2 + 8a - 72 = 0$ gives $a = 4$ or $a = -12$ (invalid for calendar).
Thus, $a = 4$.
Now, $P_2 = (4+1)(4+7) = 5 \times 11 = 55$.
Difference $= P_1 - P_2 = 72 - 55 = 17$.

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

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

Key Concept: 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 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 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: Difference of Squares, Multi-step Solution

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

Key Concept: Square of the Sum, Difference of Squares

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

Key Concept: Square of 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: 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

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: 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

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: 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: 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: 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: 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 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: 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: 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 + 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: 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 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: 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: 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: 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: 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]
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: 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: 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: 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

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) 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, 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: 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: 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: 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) 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 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

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

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: 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: 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: 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: 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

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: 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: 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: 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: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum 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: 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: 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

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

Key Concept: Special Cases of the Distributive Property

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: 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

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

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: 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: 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: 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: 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: 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: 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-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: 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: 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 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:
$15 \times 25 = (20 - 5)(20 + 5) = 20^2 - 5^2 = 400 - 25 = 375$