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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Square of Sum Identity

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

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

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

Key Concept: Pattern recognition, Squares

c) 120
[Solution Description]
For Step $k = 10$, substitute into the equation:
$(10 + 1)^2 - 1 = 11^2 - 1 = 121 - 1 = 120$
Alternatively, using the other form of the identity:
$10^2 + 2 \times 10 = 100 + 20 = 120$
Both methods yield the same result.

Key Concept: Squares of numbers, Difference of squares

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

Key Concept: Square of a Number Using Identity

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

Key Concept: Sum of Squares Pattern

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

Key Concept: Difference of Squares Identity

c) $(5 + y)(5 - y)$
[Solution Description]
The difference of squares identity states that $a^2 - b^2 = (a + b)(a - b)$. For $25 - y^2$, we can write 25 as $5^2$, so $a = 5$ and $b = y$. Applying the identity:
$25 - y^2 = 5^2 - y^2 = (5 + y)(5 - y)$.

Key Concept: Modified Sridharacharya's Identity

b) 2025
[Solution Description]
Modified Sridharacharya's Identity states that $a^2 = (a + b)(a - b) + b^2$. Let $a = 45$ and choose $b = 5$ for simplicity. Applying the identity:
$45^2 = (45 + 5)(45 - 5) + 5^2 = 50 \times 40 + 25 = 2000 + 25 = 2025$.

Key Concept: Difference of squares, Pattern recognition

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

Key Concept: Difference of Squares Pattern

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

Key Concept: Special Cases of the Distributive Property: Difference of Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The difference of squares formula is given by $a^2 - b^2 = (a + b)(a - b)$. Here, $a = 25$ and $b = 9$. Applying the formula, we get $25^2 - 9^2 = (25 + 9)(25 - 9) = 34 \times 16$. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Square of a Number Using Identity

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

Key Concept: Sridharacharya's method for squares

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

Key Concept: Squaring numbers using identities

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

Key Concept: Quick multiplication using identities, distributive property

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

Key Concept: Quick multiplication using identities

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: 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: Multiplication with 11, 101, etc.

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

Key Concept: Sridharacharya's Method for Squares

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

Key Concept: Quick multiplication using identities

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

Key Concept: Quick multiplication using identities

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: Pattern recognition in identities

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

Key Concept: Quick multiplication using identities

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

Key Concept: 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: Area of Shaded Regions

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

Key Concept: Algebraic expressions for visual patterns

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

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

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

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: 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: 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

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

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

Key Concept: Algebraic expressions for visual patterns, Multiple methods

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

Key Concept: 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: 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, 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: 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: 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

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

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

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: 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

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: 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: 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: Algebraic Identities

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

Key Concept: Difference of Squares Identity

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

Key Concept: Algebraic Pattern Investigation

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

Key Concept: Algebraic Pattern Investigation

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

Key Concept: Algebraic Pattern - Square Tiles

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

Key Concept: Increments in Products

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

Key Concept: Sum of Squares, Difference of Squares

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

Key Concept: Pattern of Circles

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

Key Concept: Sum of Squares Identity

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

Key Concept: Sum of Squares, Difference of Squares

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

Key Concept: Algebraic Pattern - Circles

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

Key Concept: Difference of Squares

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

Key Concept: Special Case Identity

c) -6
[Solution Description]
Using the identity for change in product when one term decreases and another increases:
$\text{Change} = an + bm + mn$
Here, $a = 2$, $b = 3$, $m = -2$, $n = 3$. Plugging in the values:
$2 \times 3 + 3 \times (-2) + (-2) \times 3 = 6 - 6 - 6 = -6$

Key Concept: Product Change Identity

a) $3b - 2a - 6$
[Solution Description]
Using Identity 1: $(a + m)(b + n) = ab + mb + an + mn$. Here, $m = 3$ and $n = -2$.
Substituting, we get:
$(a + 3)(b - 2) = ab + 3b - 2a - 6$
The change in product is $3b - 2a - 6$, which is derived from subtracting the original product $ab$ from the new product.

Key Concept: Identity Application

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

Key Concept: Product Change When Both Numbers Are Altered, Special Cases

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, when both numbers are increased by 1, the new product is $(a + 1)(b + 1) = ab + a + b + 1$. Comparing this with the original product $ab$, the increase in the product is $(ab + a + b + 1) - ab = a + b + 1$. Thus, the Assertion is true. The Reason correctly states the algebraic expansion of $(a + 1)(b + 1)$, which also matches the syllabus. Moreover, the Reason explains why the Assertion is true because it shows how the increase ($a + b + 1$) arises from the identity.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Distributive Property, Algebraic Identities

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

Key Concept: Expansion of Binomials

b) $xy - 4y + 5x - 20$
[Solution Description]
Use the identity $(a - u)(b + v) = ab - ub + av - uv$. Here, $a = x$, $u = 4$, $b = y$, and $v = 5$.
Substituting, we get:
$(x - 4)(y + 5) = xy - 4y + 5x - 20$

Key Concept: Special Identities, Binomial Expansion

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

Key Concept: Product Change, Distributive Property

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

Key Concept: Distributive Property

c) $xy - 4x + 3y - 12$
[Solution Description]
We expand the expression $(x + 3)(y - 4)$ using the distributive property:
$(x + 3)(y - 4) = x(y) + x(-4) + 3(y) + 3(-4)$
Simplifying each term gives:
$xy - 4x + 3y - 12$

Key Concept: Special Case Identity

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

Key Concept: General Identity for Product Change

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

Key Concept: General Identity for Product Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we use the general identity for product change:
$(a + m)(b + n) = ab + mb + an + mn$
Here, $m = 3$ and $n = -2$. Substituting these values into the identity:
$(a + 3)(b - 2) = ab + 3b + a(-2) + 3(-2) = ab + 3b - 2a - 6$
The assertion correctly applies the identity. The reason explains the process of expanding the product by multiplying each term in the brackets and considering their signs, which is accurate. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: 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

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: 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, 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: 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: 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: 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: 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: 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: 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 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: 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: 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: 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: 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 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

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: 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 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 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 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) 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: 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 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 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: 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 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: 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 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: 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 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Distributive Property and Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, we apply the distributive property:
$-3p(-5p + 2q) = (-3p)(-5p) + (-3p)(2q) = 15p^2 - 6pq$
This matches the assertion, so Assertion (A) is true.
The Reason (R) correctly states the distributive property in algebra and explains why the simplification in (A) is valid. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Error spotting in algebraic simplification, distribution

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