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: 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

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: 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) $16x^2 + 24x + 9$
[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

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 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: 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 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 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: 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

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: Square Tiles Pattern

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

Key Concept: Mathematical Identities, Verification

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

Key Concept: 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: Area Calculations

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

Key Concept: Pattern 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 and pattern generalization

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

Key Concept: Algebraic Identities

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

Key Concept: 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: Number Pattern Representation

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

Key Concept: Pattern 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: 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: 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: 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 Application

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

Key Concept: 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, Real-World Application

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

Key Concept: Product of Sum and Difference 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 Verification

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

Key Concept: Pattern Recognition

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

Key Concept: Identity 1C, Multi-concept Application

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

Key Concept: Verification of Identity

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

Key Concept: Identity 1C Application

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

Key Concept: Consistency of expressions across different methods

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

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

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

Key Concept: Consistency of expressions, Algebraic identities

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

Key Concept: Pattern Formula Application

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

Key Concept: Expression Simplification

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

Key Concept: Multiple methods for pattern representation, algebraic equivalence

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

Key Concept: Multiple methods

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

Key Concept: Consistency of expressions across different methods

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

Key Concept: Consistency of expressions

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

Key Concept: Pattern analysis

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

Key Concept: Pattern recognition, algebraic simplification

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

Key Concept: Pattern Formula Verification

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

Key Concept: Application

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

Key Concept: Pattern Interpretation and Algebraic Expression

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

Key Concept: Pattern Analysis

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

Key Concept: Expression Verification

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

Key Concept: Pattern Prediction

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

Key Concept: Multiple Methods to Solve a Problem

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

Key Concept: Expression Evaluation

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

Key Concept: Pattern Generalization, Algebraic Expression

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

Key Concept: Simplification

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

Key Concept: Pattern and Algebraic Expression

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

Key Concept: Pattern Recognition

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

Key Concept: Pattern Interpretation

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

Key Concept: 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, 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 - 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: 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: 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: 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: 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: 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

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

Key Concept: Pattern Recognition - Sum of Squares

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

Key Concept: Distributive Property, 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: Algebraic Expansion and Simplification

a) $12xy - 8xz + 10x - 5y$
[Solution Description]
First, expand each term separately:
1. $4x(3y - 2z) = 12xy - 8xz$
2. $5(2x - y) = 10x - 5y$
Now combine these results:
$12xy - 8xz + 10x - 5y$
There are no like terms to combine further, so this is the simplified form.

Key Concept: Binomial Squaring Error Identification

b) Omitted the middle term $-24ab$
[Solution Description]
The correct expansion using $(x - y)^2 = x^2 - 2xy + y^2$ would be:
$(3a - 4b)^2 = (3a)^2 - 2(3a)(4b) + (4b)^2 = 9a^2 - 24ab + 16b^2$
The student's error was:
1. They correctly squared both terms ($9a^2$ and $16b^2$)
2. But they forgot to include the middle term $-24ab$
3. They also used addition between the squared terms instead of subtraction

Key Concept: Algebraic Simplification, Error Identification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify, let's expand $(5m + 6n)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 5m$ and $b = 6n$. Substituting these values into the identity, we get:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$
The given simplification in the problem statement only includes $25m^2 + 36n^2$, missing the middle term $60mn$. Therefore, both Assertion and Reason are true, and Reason correctly explains why the given simplification is incorrect.

Key Concept: Expanding and Simplifying Algebraic Expressions

c) $15p^2 - 6pq$
[Solution Description] To expand and simplify $–3p (–5p + 2q)$, distribute $-3p$ to each term inside the parentheses:
$-3p \times (-5p) + (-3p) \times 2q = 15p^2 - 6pq$
The simplified form is $15p^2 - 6pq$.

Key Concept: Distributive Property

d) Assertion is false, but Reason is true.
[Solution Description]
The given Assertion states that $(3x - 4)^2 = 9x^2 - 16$. However, using the correct identity $(a - b)^2 = a^2 - 2ab + b^2$, the expansion should be:
$(3x - 4)^2 = (3x)^2 - 2(3x)(4) + (4)^2 = 9x^2 - 24x + 16$.
The Reason correctly states the expansion formula for $(a - b)^2$, which is not applied in the Assertion.
Therefore, the Assertion is false, but the Reason is true.

Key Concept: Algebraic Simplification

d) Assertion is false, but Reason is true.
[Solution Description]
Let us simplify the given expression step by step:
First, expand the expression:
$2(x - 1) + 3(x + 4) = 2x - 2 + 3x + 12$
Now, combine like terms:
$2x + 3x = 5x$ and $-2 + 12 = 10$
So, the correct simplified form is $5x + 10$, not $5x + 11$.
The Reason states that $-2$ and $+12$ were incorrectly combined as $+3$, which is false because they were actually correctly combined as $+10$.
Therefore, Assertion is false, but Reason is true.

Key Concept: Complex Expression Simplification

a) $2p^2 + 6pq + 6p - 2q$
[Solution Description]
Let's simplify step by step:
1. Expand first term: $2p(p+3q) = 2p^2 + 6pq$
2. Expand second term: $-3(q-2p) = -3q + 6p$
3. Now we have: $2p^2 + 6pq - 3q + 6p + q$
4. Combine like terms: $-3q + q = -2q$
5. Final form: $2p^2 + 6pq + 6p - 2q$

Key Concept: Simplifying Polynomials

b) $5w^2 + 6w$
[Solution Description]
The expression $5w^2 + 6w$ cannot be simplified further because the terms $5w^2$ and $6w$ are not like terms (they have different exponents). Therefore, the correct answer is the expression itself.

Key Concept: Expanding Binomials

b) $ab + 4a + 2b + 8$
[Solution Description]
To expand $(a + 2)(b + 4)$, we use the distributive property (FOIL method):
$a \times b = ab, \quad a \times 4 = 4a, \quad 2 \times b = 2b, \quad 2 \times 4 = 8$
Combining these gives the expanded form $ab + 4a + 2b + 8$.

Key Concept: Expanding and Simplifying Algebraic Expressions

b) $–3p (–5p + 2q) = 15p^2 - 6pq$
[Solution Description]
The mistake is in the expansion. The correct expansion of $-3p(-5p + 2q)$ should use the distributive property:
$-3p \times (-5p) = 15p^2, \quad -3p \times 2q = -6pq$
So the correct simplified form is $15p^2 - 6pq$, not $p - 2q$.

Key Concept: Combining Like Terms

c) $5x + 10$
[Solution Description] First, distribute the coefficients:
$2(x - 1) = 2x - 2$
$3(x + 4) = 3x + 12$
Next, combine like terms:
$2x - 2 + 3x + 12 = (2x + 3x) + (-2 + 12) = 5x + 10$
The correct simplification is $5x + 10$.

Key Concept: Checking Errors in Expansion

a) Missing the term $60mn$
[Solution Description] The expression $(5m + 6n)^2$ should be expanded using the formula $(a+b)^2 = a^2 + 2ab + b^2$.
$(5m + 6n)^2 = (5m)^2 + 2 \times 5m \times 6n + (6n)^2 = 25m^2 + 60mn + 36n^2$
The given simplification misses the middle term $60mn$.

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

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

Key Concept: Square of Difference

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

Key Concept: Square of Difference

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

Key Concept: Square of Difference, Algebraic Identity

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

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

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

Key Concept: Square of Difference Identity

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

Key Concept: Identity 1B: Square of Difference

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

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

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

Key Concept: Square of Difference

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

Key Concept: Application of Square of Difference

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

Key Concept: Verification of Identity

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

Key Concept: Square of Difference Identity

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

Key Concept: Pattern Investigation

c) 24
[Solution Description]
Observing the pattern:
Step 1: $1^2 + 2(1) = 3$
Step 2: $2^2 + 2(2) = 8$
Step 3: $3^2 + 2(3) = 15$
For Step 4: $4^2 + 2(4) = 16 + 8 = 24$

Key Concept: Pattern Generalization

b) 255
[Solution Description]
Substitute $k = 15$ into $k^2 + 2k$:
$15^2 + 2(15) = 225 + 30 = 255$.

Key Concept: Generalizing Patterns

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

Key Concept: Visual Patterns and Algebraic Expressions

b) $n^2 + 2n$
[Solution Description]
Observing the pattern:
- Step 1: $1^2 + 2 \times 1 = 3$
- Step 2: $2^2 + 2 \times 2 = 8$
- Step 3: $3^2 + 2 \times 3 = 15$
Thus, the general expression for Step $n$ is $n^2 + 2n$.

Key Concept: Algebraic Pattern Investigation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of tiles in Step n is $n^2 + 2n$. This can be verified using the formula derived from the pattern, which matches the given expression. The reason explains that the pattern grows by adding a row and a column at each step, which correctly justifies the algebraic expression in the assertion. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Area Calculations Using Algebraic Identities

c) $(n - m)^2$
[Solution Description]
Expand $(m + n)^2 - 4mn$:
$(m + n)^2 - 4mn = m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$
This matches Yusuf’s method.

Key Concept: Visual and Numerical Growth Patterns, Algebraic Expression

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify, let's analyze the pattern:
Step 1: $1^2 + 2(1) = 3$ tiles (matches given method $1 \times 3$)
Step 2: $2^2 + 2(2) = 8$ tiles (matches given method $2 \times 4$)
Step 3: $3^2 + 2(3) = 15$ tiles (matches given method $3 \times 5$)
The assertion correctly states the algebraic expression for the pattern. The reason explains that the quadratic growth comes from adding rows and columns each time. However, while both are true, the reason doesn't fully explain why the specific expression $n^2 + 2n$ emerges - it just describes a general quadratic pattern.

Key Concept: Area calculation, Algebraic identities

a) $(m - n)^2$
[Solution Description]
Total area of large square: $(m + n)^2$.
Total area of four rectangles: $4mn$.
Shaded area = $(m + n)^2 - 4mn$.
Simplifying: $m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$.

Key Concept: Area Calculation

b) $(n - m)^2$
[Solution Description]
Expand $(m + n)^2$: $m^2 + 2mn + n^2$.
Subtract $4mn$: $m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n - m)^2$.

Key Concept: Pattern analysis, Multiple methods

a) 255
[Solution Description]
Original formula gives $15^2 + 2(15) = 225 + 30 = 255$ circles.
However, the question specifies 2 extra circles per step beyond the formula.
This creates an arithmetic sequence: 255, 257, 259,...
But since it's only Step 15, no additional steps means total remains 255.

Key Concept: Pattern recognition, Algebraic expression

c) $n^2 + 2n$
[Solution Description]
First, observe the pattern:
Step 1: $1^2 + 2(1) = 3$ tiles,
Step 2: $2^2 + 2(2) = 8$ tiles,
Step 3: $3^2 + 2(3) = 15$ tiles.
The general formula for Step n is $n^2 + 2n$.

Key Concept: Pattern Analysis Using Algebra

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. The given general expression for Step $k$ is $k^2 + 2k$.
2. Substitute $k = 3$ into the expression: $(3)^2 + 2(3) = 9 + 6 = 15$.
3. The calculation matches the stated reason.
4. Thus, Assertion (A) is true, and Reason (R) is also true. However, the reason merely verifies the correctness of the formula for a specific step and does not explain why the general form of the expression holds.

Key Concept: Expanding and Simplifying Expressions

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

Key Concept: Difference of squares, Remainder theorem

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

Key Concept: Algebraic Expansion

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

Key Concept: Verification of Identity

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

Key Concept: Correcting misconceptions in algebraic expressions

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

Key Concept: Algebraic identities, Error correction

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

Key Concept: Simplifying Expressions

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

Key Concept: Correcting algebraic misconceptions

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

Key Concept: Simplification of Expressions

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

Key Concept: Number patterns, Algebraic identities

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

Key Concept: Algebraic Identities

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

Key Concept: Algebraic Identities

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

Key Concept: Generalization Using Distributive Property

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

Key Concept: Negative Numbers, Increments in Products

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

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

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

Key Concept: Increments in Products

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

Key Concept: Distributive Property, Increments in Products

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

Key Concept: Increments in Products

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

Key Concept: Increment in Products

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

Key Concept: Multiplicative Increment with Variables

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

Key Concept: Increment in Products

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

Key Concept: Increments in Products

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

Key Concept: Increment in Products

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

Key Concept: Increments in Products

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

Key Concept: 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, 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.