Key Concept: Generalization of Diagonal Pattern

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

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

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

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

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

Key Concept: Verification of Patterns, Mathematical Exploration

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

Key Concept: Calendar Structure Impact

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

Key Concept: Interpreting calendar-based 2×2 patterns

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

Key Concept: Mathematical Exploration, Application

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

Key Concept: Diagonal Products in 2×2 Calendar Grid

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

Key Concept: Diagonal Product Difference

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

Key Concept: Generalization of Calendar Pattern

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

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

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

Key Concept: Verification of Diagonal Pattern

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

Key Concept: 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: 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: 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: 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: 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: Area Calculation

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

Key Concept: Pattern 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: 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 Generalization

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

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: 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: Correcting misconceptions in algebraic expressions

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

Key Concept: Algebraic Expansion

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

Key Concept: Simplifying Expressions

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

Key Concept: Correcting algebraic misconceptions

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

Key Concept: Number patterns, Algebraic identities

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

Key Concept: Algebraic identities, Error correction

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

Key Concept: Algebraic Identities

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

Key Concept: Simplification of Expressions

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

Key Concept: Expanding and Simplifying Expressions

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

Key Concept: Verification of Identity

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

Key Concept: Difference of squares, Remainder theorem

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

Key Concept: Algebraic Identities

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

Key Concept: 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

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

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: 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

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, 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

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

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, 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: 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

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) 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: Sridharacharya’s Method

d) 5184
[Solution Description]
Using Sridharacharya's identity:
$a^2 = (a + b)(a - b) + b^2$
Here, $a = 72$, $b = 2$.
$(72 + 2)(72 - 2) + 2^2 = 74 \times 70 + 4$
$74 \times 70 = 5180$
Adding $4$ gives $5180 + 4 = 5184$.
Therefore, $72^2 = 5184$.

Key Concept: Sridharacharya’s method of fast squaring

b) 625
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 25$, $b = 5$.
Step 1: Compute $(a + b)(a - b) = (25 + 5)(25 - 5) = 30 \times 20 = 600$.
Step 2: Compute $b^2 = 5^2 = 25$.
Step 3: Add the results from Step 1 and Step 2: $600 + 25 = 625$.
Therefore, $25^2 = 625$.

Key Concept: Application of Identity 1C

d) 9996
[Solution Description]
Let $a = 100$, $b = 2$
Using Identity 1C: $(a + b)(a - b) = a^2 - b^2$
Thus,
$(100 + 2)(100 - 2) = 100^2 - 2^2$
$100^2 = 10{,}000$ and $2^2 = 4$
So, $10{,}000 - 4 = 9996$
Hence, $98 \times 102 = 9996$.

Key Concept: Distributive Property and Algebraic Identities

d) 1203409
[Solution Description]
To find $1097^2$ using Sridharacharya's method: Let $a = 1097$ and $b = 3$.
Step 1: Compute $(1097 + 3)(1097 - 3) = 1100 \times 1094$.
Step 2: Calculate $1100 \times 1094 = 1203400$.
Step 3: Add $b^2 = 3^2 = 9$ to the result from Step 2: $1203400 + 9 = 1203409$.
Therefore, $1097^2 = 1203409$.

Key Concept: Distributive Property and Algebraic Identities

d) 21025
[Solution Description]
To find $145^2$ using Sridharacharya's method: Let $a = 145$. Choose $b = 5$ to simplify the calculation.
Step 1: Compute $(145 + 5)(145 - 5) = 150 \times 140$.
Step 2: Calculate $150 \times 140 = 21000$.
Step 3: Add $b^2 = 5^2 = 25$ to the result from Step 2: $21000 + 25 = 21025$.
Therefore, $145^2 = 21025$.

Key Concept: Pattern Verification

b) $a^2 - b^2 = (a + b)(a - b)$
[Solution Description]
The given pattern shows that:
$9 \times 9 - 1 \times 1 = 10 \times 8$ is equivalent to $9^2 - 1^2 = (9 + 1)(9 - 1) = 10 \times 8$
Similarly, $a^2 - b^2 = (a + b)(a - b)$ is an established algebraic identity derived from the distributive property.
This confirms the correctness of the pattern.

Key Concept: Distributive Property, Sridharacharya’s Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(a + b)(a - b) + b^2 = a^2$. Let's verify this step by step:
Step 1: Expand $(a + b)(a - b)$. Using the distributive property, we get:
$(a + b)(a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^2 - b^2$.
Step 2: Add $b^2$ to the result from Step 1:
$a^2 - b^2 + b^2 = a^2$.
Therefore, the assertion is true.
The reason states that $(a + b)(a - b) = a^2 - b^2$, which is correct as shown in Step 1. Adding $b^2$ indeed gives $a^2$, which matches the assertion. Hence, the reason correctly explains the assertion.

Key Concept: Sridharacharya’s method of fast squaring

c) $(102 + 2)(102 - 2) + 2^2$
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 102$, $b = 2$.
Step 1: Compute $(a + b)(a - b) = (102 + 2)(102 - 2) = 104 \times 100 = 10400$.
Step 2: Compute $b^2 = 2^2 = 4$.
Step 3: Add the results from Step 1 and Step 2: $10400 + 4 = 10404$.
The correct expression is $(102 + 2)(102 - 2) + 2^2$.

Key Concept: Distributive Property and Algebraic Identities

d) 5184
[Solution Description]
To find $72^2$ using Sridharacharya's method: Let $a = 72$. Choose $b = 2$ so that $(a + b)(a - b) + b^2$ simplifies the calculation.
Step 1: Compute $(72 + 2)(72 - 2) = 74 \times 70$.
Step 2: Calculate $74 \times 70 = 5180$.
Step 3: Add $b^2 = 2^2 = 4$ to the result from Step 2: $5180 + 4 = 5184$.
Therefore, $72^2 = 5184$.

Key Concept: Sridharacharya’s Method for Fast Squaring

c) Assertion is true, but Reason is false.
[Solution Description]
To verify the assertion, let us compute $48^2$ using Sridharacharya's method:
Choose $a = 48$ and $b = 2$.
Substituting into the identity:
$48^2 = (48 + 2)(48 - 2) + 2^2$
$48^2 = 50 \times 46 + 4$
$48^2 = 2300 + 4 = 2304$
The assertion is true as the calculation yields $2304$.
Now, examine the reason:
The given identity is $a^2 = (a + b)(a - b) + b^2$, not derived from $(a + 1)^2$.
$(a + 1)^2$ expands to $a^2 + 2a + 1$, which is different from the given identity.
Thus, the reason is false.
Therefore, Assertion is true, but Reason is false.

Key Concept: Sridharacharya’s method of fast squaring

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify, let us take $a = 10$ and $b = 2$.
Using the identity:
$10^2 = (10 + 2)(10 - 2) + 2^2 = 12 \times 8 + 4 = 96 + 4 = 100$, which matches $10^2 = 100$.
The assertion is true.
The reason states that this identity is derived from $(a + b)(a - b) = a^2 - b^2$ by rearranging terms. This is also correct, as:
$(a + b)(a - b) = a^2 - b^2$ implies $a^2 = (a + b)(a - b) + b^2$.
Thus, the Reason correctly explains the Assertion.

Key Concept: Sridharacharya’s method of fast squaring

b) 2401
[Solution Description]
Using Sridharacharya's identity: $a^2 = (a + b)(a - b) + b^2$. Here, $a = 49$, $b = 1$.
Step 1: Compute $(a + b)(a - b) = (49 + 1)(49 - 1) = 50 \times 48 = 2400$.
Step 2: Compute $b^2 = 1^2 = 1$.
Step 3: Add the results from Step 1 and Step 2: $2400 + 1 = 2401$.
Therefore, $49^2 = 2401$.

Key Concept: Simplification of Algebraic Expressions

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

Key Concept: Distributive Property, Simplification

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

Key Concept: Multiplying Trinomials, Simplification

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

Key Concept: Simplification using Distributive Property

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

Key Concept: Distributive Property, Special Identities

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

Key Concept: Special identities

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

Key Concept: Special Identities, Simplification

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

Key Concept: Simplifying using exponent notation

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

Key Concept: Simplifying with Like Terms

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

Key Concept: Distributive property

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

Key Concept: Expanding Binomial Expressions

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

Key Concept: Simplification of algebraic expressions

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

Key Concept: 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, 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 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: 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: 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 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: 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: 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: 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: 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: 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: Square of Sum

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

Key Concept: Difference of Squares

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

Key Concept: Square of Sum 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) $9x^2 + 24xy + 16y^2$
[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 Identity

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

Key Concept: Square of Sum

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

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

Key Concept: Square of Sum

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

Key Concept: Square of Sum

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: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, apply the distributive property correctly:
$-3p(-5p + 2q) = (-3p)(-5p) + (-3p)(2q) = 15p^2 - 6pq$.
The reason correctly states the distributive property formula.
Thus, both Assertion and Reason are true, and Reason explains the Assertion.

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

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

Key Concept: Error spotting in algebraic simplification, distribution

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

Key Concept: Distributive property, Like terms

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

Key Concept: Error in Expansion

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

Key Concept: Error in Combining Like Terms

d) $5x + 10$
[Solution Description]
Expand and simplify step-by-step:
$2(x - 1) + 3(x + 4) = 2x - 2 + 3x + 12$
Combine like terms:
$2x + 3x - 2 + 12 = 5x + 10$
The correct simplified form is $\boxed{5x + 10}$.

Key Concept: Error spotting in algebraic simplification, binomial expansion

c) $4y^2 - 28y + 49$
[Solution Description]
Use the identity $(a - b)^2 = a^2 - 2ab + b^2$:
$(2y)^2 - 2 \times 2y \times 7 + 7^2 = 4y^2 - 28y + 49$.

Key Concept: Error in Distributive Property

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

Key Concept: Distributive Property and Simplification

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: Error Spotting in Algebraic Simplification

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

Key Concept: 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: 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: 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: 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: 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: 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, 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: 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: 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: 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 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: 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: Distributive Property, Fast Multiplication

a) 24543
[Solution Description]
We can express 101 as $(100 + 1)$. Using the distributive property:
$243 \times 101 = 243 \times (100 + 1) = (243 \times 100) + (243 \times 1) = 24300 + 243 = 24543$

Key Concept: Distributive Property Application

b) $12 + 3x + 4y + xy$
[Solution Description]
To expand $(4 + x)(3 + y)$, we use the distributive property:
$(4 + x)(3 + y) = 4 \times 3 + 4 \times y + x \times 3 + x \times y$
Simplifying each term gives:
$12 + 4y + 3x + xy$
So, the expanded form is $12 + 3x + 4y + xy$.

Key Concept: Fast Multiplication

b) $2323$
[Solution Description]
We can write $23 \times 101$ as $23 \times (100 + 1)$. Using the distributive property:
$23 \times 101 = 23 \times 100 + 23 \times 1$
Calculating each term:
$2300 + 23 = 2323$
Hence, the product is $2323$.

Key Concept: Square of Sum Identity

d) $16 + 40 + 25$
[Solution Description]
Applying the identity, $(a + b)^2 = a^2 + 2ab + b^2$. Substituting $a = 4$ and $b = 5$, we get $(4 + 5)^2 = 4^2 + 2 \times 4 \times 5 + 5^2 = 16 + 40 + 25 = 81$.

Key Concept: Special Case: Square of Sum

b) $49$
[Solution Description]
Applying the identity $(a + b)^2 = a^2 + 2ab + b^2$ where $a = 5$ and $b = 2$:
$(5 + 2)^2 = 5^2 + 2 \times 5 \times 2 + 2^2$
Calculating each term:
$25 + 20 + 4 = 49$
Thus, the value is $49$.

Key Concept: Distributive Property, Square of Sum

a) $25x^2 + 70xy + 49y^2$
[Solution Description]
To solve this, we use the square of sum formula: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 5x$ and $b = 7y$. Applying the formula:
$(5x)^2 + 2(5x)(7y) + (7y)^2 = 25x^2 + 70xy + 49y^2$

Key Concept: Distributive Property, Special Cases

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To expand $(x + y)^2$, we use the distributive property as follows:
$(x + y)^2 = (x + y)(x + y) = x \times x + x \times y + y \times x + y \times y = x^2 + 2xy + y^2$
Here, the assertion is true because $(x + y)^2$ indeed expands to $x^2 + 2xy + y^2$. The reason is also true and correctly explains the assertion, as the distributive property is used to break down the expression and perform the expansion.

Key Concept: Distributive Property

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To expand $(2x + 3)(4y - 5)$, we use the distributive property (also known as FOIL method for binomials). Multiply each term in the first bracket by each term in the second bracket:
$(2x)(4y) + (2x)(-5) + (3)(4y) + (3)(-5) = 8xy - 10x + 12y - 15$
The assertion is correct. The reason correctly states the distributive property, but it does not explicitly explain the expansion of two binomials (which requires applying the distributive property twice). Thus, while both are true, the reason does not fully explain the assertion.

Key Concept: Distributive Property

c) $15 + 40$
[Solution Description]
Using the distributive property, $a(b + c) = ab + ac$. Here, $a = 5$, $b = 3$, and $c = 8$. Therefore, $5(3 + 8) = (5 \times 3) + (5 \times 8) = 15 + 40 = 55$.

Key Concept: Distributive Property, Product of Sum and Difference

a) $16a^2 - 81b^2$
[Solution Description]
This uses the product of sum and difference identity: $(a + b)(a - b) = a^2 - b^2$. Here, $a = 4a$ and $b = 9b$.
$(4a)^2 - (9b)^2 = 16a^2 - 81b^2$

Key Concept: Multiplication by Parts

b) $1212$
[Solution Description]
We can write $101$ as $(100 + 1)$. Thus, $12 \times 101 = 12 \times (100 + 1) = (12 \times 100) + (12 \times 1) = 1200 + 12 = 1212$.

Key Concept: Distributive Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because applying the distributive property to $5(3 + 4)$ indeed gives $5 \times 3 + 5 \times 4 = 15 + 20 = 35$. The reason correctly explains the distributive property, which is $a(b + c) = ab + ac$, and justifies the assertion.

Key Concept: Square of a Sum, Fast Multiplication