Key Concept: Pattern Recognition

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

Key Concept: Distributive Property, Multi-step Simplification

b) $116$
[Solution Description]
Substitute $a = 7$ and $b = 3$ into the given identity:
$2(7^2 + 3^2) = (7 + 3)^2 + (7 - 3)^2$
Calculate each part step-by-step:
$7^2 = 49$, $3^2 = 9$, so $2(49 + 9) = 2(58) = 116$.
$(7 + 3)^2 = 10^2 = 100$.
$(7 - 3)^2 = 4^2 = 16$.
Verify: $100 + 16 = 116$.
The value is $116$.

Key Concept: Distributive Property, Pattern Recognition

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

Key Concept: Distributive Property and Algebraic Identities

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, expand $(a + b)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$. Similarly, expand $(a - b)^2$ using $(a - b)^2 = a^2 - 2ab + b^2$. Now, subtract $(a - b)^2$ from $(a + b)^2$: $(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)$. Simplify by combining like terms: $a^2 - a^2 + 2ab + 2ab + b^2 - b^2 = 4ab$. Thus, Assertion is true. Reason uses the identity $(a + b)(a - b) = a^2 - b^2$, which is not directly applicable here since we used expansion of squares, not difference of squares. Hence, Reason is true but does not correctly explain the Assertion.

Key Concept: Distributive Property

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

Key Concept: Sum of Squares, Difference of Squares

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

Key Concept: Special Case Expansion

b) $k^2 + 2k$
[Solution Description]
The expression $k(k + 2)$ can be expanded using the distributive property:
$k(k + 2) = k \cdot k + k \cdot 2 = k^2 + 2k$
Therefore, $k(k + 2) = k^2 + 2k$.

Key Concept: Algebraic Identity - Difference of Squares

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

Key Concept: Distributive Property

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

Key Concept: Distributive Property - Basic Application

b) 35
[Solution Description]
Substitute $k = 5$ into the formula $k^2 + 2k$. The calculation is: $5^2 + 2 \times 5 = 25 + 10 = 35$

Key Concept: Distributive Property, Algebraic Identities

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

Key Concept: Pattern Recognition - Sum of Squares

c) 50
[Solution Description]
Substitute $a = 3$ and $b = 4$ into the formula: $(3 + 4)^2 + (3 - 4)^2 = 49 + 1 = 50$

Key Concept: Square of Sum

c) $9x^2 + 24xy + 16y^2$
[Solution Description]
Using Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 3x$ and $b = 4y$. Substituting into the identity:
$(3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2$

Key Concept: Numerical Calculation

c) 625
[Solution Description]
Write $25$ as $(20 + 5)$ and apply $(a + b)^2 = a^2 + 2ab + b^2$:
$25^2 = (20 + 5)^2 = 20^2 + 2(20)(5) + 5^2 = 400 + 200 + 25 = 625$

Key Concept: Square of Sum Identity

d) Assertion is false, but Reason is true.
[Solution Description]
Let us analyze the assertion and reason:
1. The assertion claims $(a + b)^2 > a^2 + b^2$ only when both $a$ and $b$ are positive.
2. The reason states that $2ab$ is always non-negative for all integers $a$ and $b$.
Testing the assertion:
- If $a = 2$ and $b = 3$, $(2+3)^2 = 25$ while $2^2 + 3^2 = 13$, so it holds ($25 > 13$).
- If $a = -2$ and $b = -3$, $(-2-3)^2 = 25$ while $(-2)^2 + (-3)^2 = 13$, still $25 > 13$.
- If one is positive and other negative, say $a=2$, $b=-3$, then $(2-3)^2 = 1$ vs $4 + 9 = 13$, now $1 a^2 + b^2$ can be true even when both numbers are negative.
Testing the reason:
- $2ab$ is not always non-negative. Example: $a=1$, $b=-1$ gives $2(1)(-1) = -2$ which is negative.
- Hence, the reason is false.
Therefore, Assertion is false and Reason is also false.

Key Concept: Square of Sum

d) Assertion is false, but Reason is true.
[Solution Description]
First, calculate the left-hand side: $(3 + 4)^2 = 7^2 = 49$.
Next, apply the formula for the square of the sum: $(3 + 4)^2 = 3^2 + 2(3)(4) + 4^2 = 9 + 24 + 16 = 49$.
Thus, both Assertion and Reason are true.
However, the value in Assertion is incorrect ($25$ instead of $49$), while the Reason correctly states the formula.

Key Concept: Square of Sum Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's evaluate the assertion and reason step by step.
1) **Assertion Evaluation**:
Using the given identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = 3x$ and $b = 4$.
$(3x + 4)^2 = (3x)^2 + 2 \times (3x \times 4) + 4^2 = 9x^2 + 24x + 16$
This matches the assertion, so the assertion is true.
2) **Reason Evaluation**:
The reason states the correct algebraic identity for the square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. This is mathematically accurate, so the reason is also true.
3) **Explanation Check**:
The reason correctly explains the assertion because it provides the identity used to derive the expanded form in the assertion. Thus, the reason is the correct explanation of the assertion.

Key Concept: Square of Sum

c) 1369
[Solution Description]
To find $37^2$ using Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$, decompose 37 as $30 + 7$. Then, apply the identity:
$37^2 = (30 + 7)^2 = 30^2 + 2 \times (30 \times 7) + 7^2 = 900 + 420 + 49 = 1369.$

Key Concept: Square of Sum

c) $16x^2 + 24x + 9$
[Solution Description]
To find $(4x + 3)^2$, we use Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 4x$ and $b = 3$. Applying the identity, we get:
$(4x + 3)^2 = (4x)^2 + 2 \times (4x \times 3) + 3^2 = 16x^2 + 24x + 9.$

Key Concept: Square of Sum Application

b) $9y^2 + 42y + 49$
[Solution Description]
Apply the identity $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 3y$ and $b = 7$:
$(3y + 7)^2 = (3y)^2 + 2(3y)(7) + 7^2 = 9y^2 + 42y + 49$

Key Concept: Square of Sum Identity

b) $x^2 + 8x + 16$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = x$ and $b = 4$:
$(x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$

Key Concept: Difference of Squares

c) 2800
[Solution Description]
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$. Here, $a = 107$, $b = 93$.
$107^2 - 93^2 = (107 + 93)(107 - 93) = 200 \times 14 = 2800$

Key Concept: Square of Sum

c) 7
[Solution Description]
The given expression is $(m + n)^2 = m^2 + 14m + n^2$. Using Identity 1A:
$(m + n)^2 = m^2 + 2mn + n^2.$
Comparing with the given expression, we get:
$2mn = 14m \implies n = \frac{14m}{2m} = 7.$

Key Concept: Square of Difference

c) $25p^2 - 70pq + 49q^2$
[Solution Description]
Using Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 5p$ and $b = 7q$. Substituting into the identity:
$(5p - 7q)^2 = (5p)^2 - 2(5p)(7q) + (7q)^2 = 25p^2 - 70pq + 49q^2$

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: 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: 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

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 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

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

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: 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: 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: 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 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: 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 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: 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

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: 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: 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: 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 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: 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: 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: 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: Step-based algebraic patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify if the assertion and reason are true:
1. Assertion claims that $(k + 1)^2 - 1$ represents the circles in Step $k$.
From Method 1 in the syllabus:
- Step 1: $(1 + 1)^2 - 1 = 3$ circles
- Step 2: $(2 + 1)^2 - 1 = 8$ circles
- Step 3: $(3 + 1)^2 - 1 = 15$ circles
The formula works for these steps, so the Assertion is true.
2. Reason states that for Step 3, $4^2 - 1 = 15$ matches the pattern.
As shown in Step 3 above, this calculation is correct, so the Reason is true.
3. The Reason provides a specific example confirming the general formula stated in the Assertion. Hence, the Reason correctly explains the Assertion.

Key Concept: Algebraic Patterns

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

Key Concept: Algebraic Pattern Interpretation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that using Method 1, the expression for the number of circles in Step $k$ is $(k + 1)^2 - 1$. This matches the given pattern where:
- For Step 1: $(1 + 1)^2 - 1 = 4 - 1 = 3$
- For Step 2: $(2 + 1)^2 - 1 = 9 - 1 = 8$
- For Step 3: $(3 + 1)^2 - 1 = 16 - 1 = 15$
The reason explains that Method 1 works by taking the step number, adding 1, squaring it, and subtracting 1. This is a valid explanation for why the assertion is true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Step-based pattern, Algebraic forms

c) 35
[Solution Description]
Both expressions represent the same pattern, so we can use either to find the number of circles in Step 5. Let's use $(k + 1)^2 - 1$:
For Step 5, set $k = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$
Verification using the other expression $k^2 + 2k$:
$5^2 + 2 \times 5 = 25 + 10 = 35$
Both methods give the same result.

Key Concept: Step-based pattern, Multi-concept

b) $(n + 1)^2 - 1$
[Solution Description]
We can derive the general expression for the pattern:
Step 1: 3 circles
Step 2: Step 1 + (2 $\times$ 2 + 1) = 3 + 5 = 8
Step 3: Step 2 + (2 $\times$ 3 + 1) = 8 + 7 = 15
Observing the pattern, we see that the number of circles follows the formula $(n + 1)^2 - 1$.
Verification:
For Step 1: $(1 + 1)^2 - 1 = 4 - 1 = 3$
For Step 2: $(2 + 1)^2 - 1 = 9 - 1 = 8$
For Step 3: $(3 + 1)^2 - 1 = 16 - 1 = 15$
Thus, the correct expression is $(n + 1)^2 - 1$.

Key Concept: Pattern recognition and algebraic expression

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

Key Concept: Step-based pattern, Algebraic equivalence

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

Key Concept: Pattern interpretation, Algebraic equivalence

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the algebraic equivalence of the three expressions:
1. Expand $(k + 1)^2 - 1$:
$(k + 1)^2 - 1 = k^2 + 2k + 1 - 1 = k^2 + 2k$
2. Simplify $k(k + 1) + k$:
$k(k + 1) + k = k^2 + k + k = k^2 + 2k$
All three expressions simplify to $k^2 + 2k$, proving they are equivalent forms. Thus, the assertion about any one form being valid is correct, and the reason correctly explains their equivalence through algebraic simplification.

Key Concept: Pattern recognition and algebraic expression

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

Key Concept: Algebraic Patterns

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

Key Concept: Pattern recognition and algebraic expression

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

Key Concept: Algebraic Patterns

b) 35
[Solution Description]
According to Method 1, the number of circles in Step k is given by $(k + 1)^2 - 1$. For Step 5, substitute $k = 5$:
$(5 + 1)^2 - 1 = 6^2 - 1 = 36 - 1 = 35$
Therefore, the number of circles in Step 5 is 35.

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: 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: 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, Difference of Squares

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

Key Concept: Application of 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: Square of the Difference of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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: 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: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication, Error Analysis

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Area Calculations

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

Key Concept: Algebraic Identities

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

Key Concept: Number Patterns

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

Key Concept: Pattern Analysis

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

Key Concept: Number Pattern Representation

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

Key Concept: Mathematical Identities, Verification

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

Key Concept: Pattern Representation, Simplification

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

Key Concept: Algebraic Identities

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

Key Concept: Pattern Recognition and Generalization, Algebraic Simplification

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

Key Concept: Algebraic identities and pattern generalization

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

Key Concept: Sridharacharya's Method for Squares, Application

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

Key Concept: Square Tiles Pattern

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

Key Concept: Pattern recognition 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: 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: 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: 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: 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: 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: 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]
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: 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: Expanding Binomial Expressions

b) $4m^2 + 12mn + 9n^2$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$,
$(2m + 3n)^2 = (2m)^2 + 2(2m)(3n) + (3n)^2$
$= 4m^2 + 12mn + 9n^2$

Key Concept: Simplification using Distributive Property

a) $\frac{5x^2}{3} - \frac{10xy}{3} + \frac{5x}{12}$
[Solution Description]
To simplify, distribute $\frac{5x}{3}$ to each term inside the parentheses:
$\frac{5x}{3} \times x = \frac{5x^2}{3}$
$\frac{5x}{3} \times (-2y) = -\frac{10xy}{3}$
$\frac{5x}{3} \times \frac{1}{4} = \frac{5x}{12}$
So, the simplified form is $\frac{5x^2}{3} - \frac{10xy}{3} + \frac{5x}{12}$

Key Concept: Simplifying using exponent notation

b) $3a^3$
[Solution Description]
To simplify $3a^2 \times a$, multiply the coefficients and add the exponents of like bases:
$3a^2 \times a = 3a^{2 + 1} = 3a^3$

Key Concept: Simplification of algebraic expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion is true because expanding $(a + b)^2$ gives $a^2 + 2ab + b^2$ as per the algebraic identity. The Reason is also true because the distributive property is applied to expand the expression, and like terms ($ab + ab$) are combined to get $2ab$. Moreover, the Reason correctly explains the Assertion as it details the method used for simplification.

Key Concept: Special identities

b) $a^2 + 2ab + b^2$
[Solution Description]
Using the special identity $(a + b)^2 = a^2 + 2ab + b^2$, the expansion is straightforward:
$(a + b)^2 = a^2 + 2ab + b^2$.

Key Concept: Distributive Property, Simplification

a) $6x^2 - 7x - 20$
[Solution Description]
To simplify $(3x + 4)(2x - 5)$, we use the distributive property:
$(3x + 4)(2x - 5) = 3x \times 2x + 3x \times (-5) + 4 \times 2x + 4 \times (-5)$
Simplify each term:
$3x \times 2x = 6x^2$
$3x \times (-5) = -15x$
$4 \times 2x = 8x$
$4 \times (-5) = -20$
Combine the terms:
$6x^2 - 15x + 8x - 20 = 6x^2 - 7x - 20$

Key Concept: Special Identities, Simplification

a) $4a^2 - 12ab + 9b^2$
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, we substitute $a = 2a$ and $b = 3b$:
$(2a - 3b)^2 = (2a)^2 - 2 \times 2a \times 3b + (3b)^2$
Simplify each term:
$(2a)^2 = 4a^2$
$2 \times 2a \times 3b = 12ab$
$(3b)^2 = 9b^2$
Combine the terms:
$4a^2 - 12ab + 9b^2$

Key Concept: Simplifying with Like Terms

a) $20p^2 - 5pq$
[Solution Description]
First, distribute $-4p$ inside the parentheses:
$-4p \times (-5p) = 20p^2$
$-4p \times 2q = -8pq$
Now, add $3pq$ to the result:
$20p^2 - 8pq + 3pq = 20p^2 - 5pq$

Key Concept: Distributive Property, Special Identities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, recognize that the expression $\left(\frac{x}{2} - \frac{y}{3}\right)\left(\frac{x}{2} + \frac{y}{3}\right)$ is of the form $(a - b)(a + b)$, where $a = \frac{x}{2}$ and $b = \frac{y}{3}$. Using the identity $(a - b)(a + b) = a^2 - b^2$, we substitute the values of $a$ and $b$:
$\left(\frac{x}{2}\right)^2 - \left(\frac{y}{3}\right)^2 = \frac{x^2}{4} - \frac{y^2}{9}$
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Multiplying Trinomials, Simplification

a) $x^3 - x^2 - x + 10$
[Solution Description]
We use the distributive property to multiply $(x + 2)$ with each term in $(x^2 - 3x + 5)$:
$(x + 2)(x^2 - 3x + 5) = x \times x^2 + x \times (-3x) + x \times 5 + 2 \times x^2 + 2 \times (-3x) + 2 \times 5$
Simplify each term:
$x \times x^2 = x^3$
$x \times (-3x) = -3x^2$
$x \times 5 = 5x$
$2 \times x^2 = 2x^2$
$2 \times (-3x) = -6x$
$2 \times 5 = 10$
Combine the terms:
$x^3 - 3x^2 + 5x + 2x^2 - 6x + 10 = x^3 - x^2 - x + 10$

Key Concept: Simplification of Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify $3x(2x - 5y + 1)$, we use the distributive property:
Step 1: Multiply $3x$ by each term inside the bracket:
$3x \times 2x = 6x^2$
$3x \times (-5y) = -15xy$
$3x \times 1 = 3x$
Step 2: Combine the results: $6x^2 - 15xy + 3x$.
The Assertion is true as it correctly applies the distributive property.
The Reason is also true and correctly explains why the assertion holds.

Key Concept: Distributive property

a) $\frac{3}{2}a^2 - \frac{3}{2}ab + \frac{3}{10}a$
[Solution Description]
Apply the distributive property to expand the expression:
$\frac{3a}{2}(a) = \frac{3}{2}a^2$
$\frac{3a}{2}(-b) = -\frac{3}{2}ab$
$\frac{3a}{2}\left(\frac{1}{5}\right) = \frac{3}{10}a$
Combining these, we get: $\frac{3}{2}a^2 - \frac{3}{2}ab + \frac{3}{10}a$

Key Concept: Algebraic expressions for visual patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $k^2 + 2k$ correctly gives the number of circles in Step $k$ of the pattern. This is verified by the fact that all four different methods (Method 1: $(k + 1)^2 - 1$, Method 2: $k^2 + 2 \times k$, Method 3: $k \times (k + 1) + k$, and Method 4: $k \times (k + 2)$) simplify to the same expression $k^2 + 2k$. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Algebraic Expressions, Pattern Recognition

c) 46
[Solution Description]
To find the number of tiles in Step 5, substitute $n = 5$ into the given expression $2n^2 - n + 1$.
Step 1: Substitute $n = 5$ into the expression.
$2(5)^2 - 5 + 1$
Step 2: Calculate $5^2 = 25$.
Step 3: Multiply 2 by 25 to get $50$.
Step 4: Subtract 5 from 50 to get $45$.
Step 5: Add 1 to 45 to get $46$.
Therefore, there are 46 tiles in Step 5.

Key Concept: Area of Shaded Regions

b) $(x - y)^2$
[Solution Description]
Total area of square is $(x + y)^2 = x^2 + 2xy + y^2$.
Area of four rectangles is $4xy$.
Shaded area = Total area - Rectangle areas = $x^2 + 2xy + y^2 - 4xy = x^2 - 2xy + y^2 = (x - y)^2$

Key Concept: Algebraic expressions for visual patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $k^2 + 2k$ correctly represents the number of circles in Step k, which is true as verified by multiple methods in the syllabus. The reason states that all simplified forms of different algebraic expressions for this pattern lead to the same expression, which is also true and explains why the assertion is correct because all valid interpretations converge to the same algebraic form.

Key Concept: Algebraic expressions for visual patterns, Multiple methods

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we substitute $k = 15$ into the given expression $k^2 + 2k$. Calculation: $15^2 + 2 \times 15 = 225 + 30 = 255$. Thus, the assertion is true. The reason states that all four methods lead to the same expression $k^2 + 2k$, which is correct as shown in the syllabus. Moreover, the reason correctly explains why the assertion holds because the unified expression derived from all methods allows us to compute the number of circles accurately.

Key Concept: Pattern Recognition, Algebraic Expressions

c) 255
[Solution Description]
To find the number of circles in Step 15, substitute $k = 15$ into the given expression $k^2 + 2k$.
Step 1: Substitute $k = 15$ into the expression.
$(15)^2 + 2(15)$
Step 2: Calculate $15^2 = 225$.
Step 3: Calculate $2 \times 15 = 30$.
Step 4: Add the results from Step 2 and Step 3.
$225 + 30 = 255$.
Therefore, there are 255 circles in Step 15.

Key Concept: Algebraic Expressions for Visual Patterns

c) 12
[Solution Description]
Using the given formula with n = 4:
$n^2 - n = 4^2 - 4 = 16 - 4 = 12$

Key Concept: Algebraic expressions for visual patterns

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

Key Concept: Area Calculation, Algebraic Identities

c) $(b - a)^2$
[Solution Description]
The problem states that the area of the shaded region can be calculated using two methods:
Method 1: $(a + b)^2 - 4ab$
This represents the total area of the large square minus the area of the four rectangles.
Method 2: $(b - a)^2$
This represents the area of the smaller shaded square formed inside the larger square.
Since both expressions are equivalent as per the given identity, the area of the shaded region can be represented by either $(a + b)^2 - 4ab$ or $(b - a)^2$.

Key Concept: Algebraic expressions for visual patterns

b) 15
[Solution Description]
Substitute $x = 5$ and $y = 2$ into the formula:
$x^2 - xy = 5^2 - 5 \times 2 = 25 - 10 = 15$

Key Concept: Algebraic Expressions for Visual Patterns

c) 35
[Solution Description]
To find the number of circles in step 5, we substitute k = 5 into the given formula:
$k^2 + 2k = 5^2 + 2(5) = 25 + 10 = 35$