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

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: 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, 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: 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 for visual patterns

c) 16
[Solution Description]
Substitute $m = 3$ and $n = 7$ into the formula:
$(n - m)^2 = (7 - 3)^2 = 4^2 = 16$

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, 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: 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: 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:
$k^2 + 2k = 5^2 + 2(5) = 25 + 10 = 35$

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: 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: 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: 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: 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 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: 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: Pattern Generalization

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

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: 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 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: Verifying patterns

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

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

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

Key Concept: Square of binomial, Expanding expressions

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

Key Concept: Using the identity $(a + b

c) 25
[Solution Description]

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

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

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

we get:
$k = 25$

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

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

b) 2200
[Solution Description]

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

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

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

$= 44 \times 50$
$= 2200$

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

Key Concept: Applying $(a + b

a) 95
[Solution Description]

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

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

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

$=144-49$

$=95$

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

Key Concept: Difference of squares

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

Key Concept: Sum of squares, Algebraic identities

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

Key Concept: Using identities to simplify expressions

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

Key Concept: Using identities to explain numerical puzzles

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

Key Concept: Algebraic Identities

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

Key Concept: Difference of squares, Algebraic identities

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

Key Concept: Square of 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

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 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) $9x^2 + 24xy + 16y^2$
[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

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

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 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

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 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

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: 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 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: 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

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

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

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

Key Concept: Identity 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, Algebraic Identity

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

Key Concept: Identity 1B: Square of Difference

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

Key Concept: Square of Difference

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: 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

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: Squaring numbers using identities

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

Key Concept: Quick multiplication using identities

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: Sridharacharya's Method for Squares

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

Key Concept: Quick multiplication using identities

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

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

Key Concept: Multiplication with 11, 101, etc.

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

Key Concept: 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: 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: 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: Distributive Property, Algebraic Identities

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

Key Concept: Distributive Property 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: Pattern Recognition - Sum of Squares

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

Key Concept: Distributive Property - Basic Application

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

Key Concept: Special Case Expansion

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

Key Concept: Distributive Property

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

Key Concept: Pattern Recognition

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

Key Concept: 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: Sum of Squares, Difference of Squares

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

Key Concept: Distributive Property, Multi-step Simplification

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

Key Concept: 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: Multiplicative Increment with Variables

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

Key Concept: Generalization Using Distributive Property

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

Key Concept: Increment in Products

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: 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

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

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: 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

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

Key Concept: Increments in Products

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

Key Concept: Negative Numbers, Increments in Products

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

Key Concept: Increments in Products

b) 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: 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: Distributive Property of Multiplication over Addition

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

Key Concept: Distributive Property of Multiplication over Addition

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

Key Concept: Increments in Products

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

Key Concept: General Case for Increments

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

Key Concept: Identity Application, Expansion with Subtraction

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

Key Concept: Expansion of Algebraic Expressions

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

Key Concept: Increments in Products

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

Key Concept: Distributive Property, Increments in Products

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

Key Concept: Distributive Property and Increments in Products

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

Key Concept: Distributive Property

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

Key Concept: Distributive Property

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

Key Concept: Increments in Products, Complex Multiplication

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

Key Concept: Combining Like Terms

b) $3x^2 + 4x$
[Solution Description]
Combine like terms step-by-step:
$5x^2 - 2x^2 = 3x^2,$
$3x + x = 4x.$
Thus, the correct simplification is:
$3x^2 + 4x.$
The given incorrect simplification matches the correct answer, so there is no error here.

Key Concept: Incorrect Expansion

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, let's expand $(3x + 4)^2$ correctly:
$(3x + 4)^2 = (3x)^2 + 2 \times 3x \times 4 + 4^2 = 9x^2 + 24x + 16$.
The given assertion claims the simplified form is $9x^2 + 16$, which is incorrect because the middle term $24x$ is missing.
The reason states that the middle term is missing, which is true and explains why the assertion is false.
Therefore, Assertion is false, but Reason is true.

Key Concept: Like Terms Error

b) $4p^2 + 6p$
[Solution Description]
Combine like terms:
$5p^2 - p^2 = 4p^2$
$2p + 4p = 6p$
The simplified form is $4p^2 + 6p$.

Key Concept: Algebraic Expansion and Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expansion of $(5m + 6n)^2$ is $25m^2 + 60mn + 36n^2$. This is correct because:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$
The reason states that the cross-term in the expansion of a binomial square $(a + b)^2$ is given by $2ab$, which is also correct and explains why the assertion is true. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Binomial Squaring Error

c) $4a^2 + 12ab + 9b^2$
[Solution Description]
Use the formula for squaring a binomial:
$(2a + 3b)^2 = (2a)^2 + 2 \times 2a \times 3b + (3b)^2 = 4a^2 + 12ab + 9b^2$

Key Concept: Distributive Property Error

c) $6xy - 12xz$
[Solution Description]
To expand $3x(2y - 4z)$, apply the distributive property:
$3x \times 2y = 6xy$
$3x \times (-4z) = -12xz$
Therefore, the correct expansion is $6xy - 12xz$.

Key Concept: Distributive Property

c) $12x - 20y + 8$
[Solution Description]
First, apply the distributive property to $4(3x - 5y + 2)$. Multiply each term inside the parentheses by 4:
$4 \times 3x = 12x, \quad 4 \times (-5y) = -20y, \quad 4 \times 2 = 8.$
Combine these results to get the correct expansion:
$12x - 20y + 8.$

Key Concept: Distributive property, Combining like terms

d) Combining the a terms
[Solution Description]
First distribute: $5(2a - 3b) = 10a - 15b$ and $2(a + 4b) = 2a + 8b$.
Combine these: $10a - 15b + 2a + 8b$.
Combine like terms: $(10a + 2a) + (-15b + 8b) = 12a - 7b$.
The error was in combining $10a + 2a$ as 13a instead of 12a.

Key Concept: Distributive property, Squaring binomials

c) $24xy$
[Solution Description]
When squaring a binomial $(a + b)^2$, the correct expansion is $a^2 + 2ab + b^2$. For $(4x + 3y)^2$, the correct expansion is:
$(4x)^2 + 2(4x)(3y) + (3y)^2 = 16x^2 + 24xy + 9y^2$. The missing term is 24xy.

Key Concept: Algebraic fractions, Distributive property

a) $7p + 2q$
[Solution Description]
First distribute $\frac{1}{3}$: $\frac{1}{3}(9p) - \frac{1}{3}(6q) = 3p - 2q$.
Then distribute 4: $4(p) + 4(q) = 4p + 4q$.
Combine all terms: $3p - 2q + 4p + 4q = (3p + 4p) + (-2q + 4q) = 7p + 2q$.
The mistake was in not distributing the 4 to both terms inside the second parentheses.

Key Concept: Analyzing incorrect expansions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expansion of $(5m + 6n)^2$ is $25m^2 + 60mn + 36n^2$. This is correct because applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ gives:
$(5m + 6n)^2 = (5m)^2 + 2(5m)(6n) + (6n)^2 = 25m^2 + 60mn + 36n^2$.
The reason states that the identity $(a + b)^2 = a^2 + 2ab + b^2$ was applied correctly, which is true and explains why the assertion is correct.

Key Concept: Binomial Expansion

c) $4a^2 - 12ab + 9b^2$
[Solution Description]
Use the formula for squaring a binomial: $(A - B)^2 = A^2 - 2AB + B^2$. Here, $A = 2a$ and $B = 3b$.
$(2a - 3b)^2 = (2a)^2 - 2 \times 2a \times 3b + (3b)^2 = 4a^2 - 12ab + 9b^2.$

Key Concept: Incremental Changes in Products

a) $a$
[Solution Description]
When the second number $b$ is increased by 1, the new product becomes $a(b + 1) = ab + a$. The increase in the product is $(ab + a) - ab = a$.

Key Concept: Incremental Changes in Products

c) 26
[Solution Description]
The original product is $8 \times 17 = 136$. After increasing both numbers by 1, the new product is $(8 + 1)(17 + 1) = 9 \times 18 = 162$. The change in product is $162 - 136 = 26$.
Using the formula $(a + 1)(b + 1) = ab + (a + b + 1)$, we substitute $a = 8$, $b = 17$, which gives:
$8 \times 17 + (8 + 17 + 1) = 136 + 26 = 162$
Therefore, the product increases by 26.

Key Concept: Fast Multiplication, Distributive Property

b) $249268$
[Solution Description]
Rewrite 101 as $(100 + 1)$ and apply the distributive property:
$2468 \times 101 = 2468 \times (100 + 1)$
$= 2468 \times 100 + 2468 \times 1$
$= 246800 + 2468$
$= 249268$.

Key Concept: Incremental Changes in Products

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the Assertion and Reason, let us analyze the expression $(a + 1)(b - 1)$. Expanding this gives:
$(a + 1)(b - 1) = ab - a + b - 1 = ab + (b - a - 1).$
The Assertion states that $(a + 1)(b - 1) < ab$ for all $a$ and $b$, which implies $(b - a - 1) ab$. The Reason is also false because $(b - a - 1)$ is not always negative. Therefore, the correct answer is (d).

Key Concept: Incremental Changes in Products, Distributive Property

c) $p + 99$
[Solution Description]
Using the property $(a + m)(b + n) = ab + an + bm + mn$, where $a = 12$, $b = 18$, and $m = n = 3$:
$(12 + 3)(18 + 3) = 12 \times 18 + 12 \times 3 + 18 \times 3 + 3 \times 3$
$= p + 36 + 54 + 9$
$= p + 99$.

Key Concept: Incremental Changes in Products

b) $b$
[Solution Description]
When the first number $a$ is increased by 1, the new product becomes $(a + 1)b = ab + b$. The increase in the product is $(ab + b) - ab = b$.

Key Concept: Incremental Changes in Products

c) 30
[Solution Description]
The new product after increasing the first number by 3 and decreasing the second number by 2 is $(15 + 3)(22 - 2) = 18 \times 20 = 360$. The change in product is calculated as $360 - 330 = 30$.
Alternatively, using the formula for incremental changes:
$(a + m)(b + n) = ab + an + bm + mn$
Here, $a = 15$, $b = 22$, $m = 3$, and $n = -2$. Substituting these values, we get:
$15 \times (-2) + 22 \times 3 + 3 \times (-2) = -30 + 66 - 6 = 30$
Thus, the product increases by 30.

Key Concept: Incremental Changes in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that increasing both numbers in a product by 1 results in an increase of $a + b + 1$ in the product. This is correct because:
1. Starting with two numbers $a$ and $b$, their product is $ab$.
2. After increasing both by 1, we get $(a+1)(b+1)$.
3. Expanding this using the distributive property:
$(a+1)(b+1) = ab + a + b + 1$.
4. The difference between the new product and the original product is $(ab + a + b + 1) - ab = a + b + 1$.
The reason correctly explains the assertion by showing the algebraic expansion that justifies the increase. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Incremental Changes in Products

b) $a + b + 1$
[Solution Description]
When both numbers are increased by 1, the new product becomes $(a + 1)(b + 1) = ab + a + b + 1$. The increase in the product is $(ab + a + b + 1) - ab = a + b + 1$.

Key Concept: Distributive Property

b) 312
[Solution Description]
We can express $13 \times 24$ as $(12 + 1)(25 - 1)$. Using the formula $(a + m)(b + n) = ab + an + bm + mn$, where $a = 12$, $b = 25$, $m = 1$, and $n = -1$, we get:
$12 \times 25 + 12 \times (-1) + 25 \times 1 + 1 \times (-1) = 300 - 12 + 25 - 1 = 312$
Thus, $13 \times 24 = 312$.

Key Concept: Incremental Changes in Products, Mixed Operations

a) $P + xy + 15x + 9y + 5$
[Solution Description]
Let $x' = x + 1$ and $y' = y - 2$. The new product is:
$(5(x + 1) + 2(y - 2))(3(x + 1) - (y - 2))$
Simplify each part:
$= (5x + 5 + 2y - 4)(3x + 3 - y + 2)$
$= (5x + 2y + 1)(3x - y + 5)$.
Expand using the distributive property:
$= 5x \cdot 3x + 5x \cdot (-y) + 5x \cdot 5 + 2y \cdot 3x + 2y \cdot (-y) + 2y \cdot 5 + 1 \cdot 3x + 1 \cdot (-y) + 1 \cdot 5$
$= 15x^2 - 5xy + 25x + 6xy - 2y^2 + 10y + 3x - y + 5$
Combine like terms:
$= 15x^2 + xy + 28x - 2y^2 + 9y + 5$.
The original product $P = 15x^2 - xy + 13x - 2y^2$.
The difference is $(xy + 15x + 9y + 5)$.

Key Concept: Incremental Changes in Products

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the distributive property, when one of the numbers in a product (e.g., $a$) is increased by 1, the new product becomes $(a + 1)b = ab + b$. This means the increase in the product is equal to the other number ($b$). The reason correctly explains the assertion using the distributive property of multiplication.

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

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.