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: 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: Increments in Products

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

Key Concept: Algebraic Pattern - 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: 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: Sum of Squares, Difference of Squares

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

Key Concept: Pattern of Circles

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

Key Concept: 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: 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 Identities

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

Key Concept: Algebraic Pattern 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 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: 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: 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: 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: 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: 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: 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: Like Terms Error

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

Key Concept: Algebraic Expansion and Simplification

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

Key Concept: Binomial Squaring Error

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

Key Concept: 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: 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, 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: 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: 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: 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: 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: 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: 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) 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: 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 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: 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: 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: Quick multiplication using identities

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

Key Concept: Quick multiplication using identities

c) 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 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

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

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: 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: 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: Squaring numbers using identities

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

Key Concept: Pattern recognition in 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Distributive Property, Multi-step Simplification

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

Key Concept: Special Case Expansion

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

Key Concept: Distributive Property, Pattern Recognition

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

Key Concept: Pattern Recognition - Sum of Squares

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

Key Concept: Distributive Property

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

Key Concept: Pattern Recognition

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

Key Concept: Sum of Squares, Difference of Squares

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

Key Concept: Distributive Property and Algebraic Identities

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

Key Concept: Distributive Property - Basic Application

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

Key Concept: Distributive Property, Algebraic Identities

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

Key Concept: Distributive Property

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

Key Concept: Algebraic Identity - Difference of Squares

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication, Error Analysis

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Distributive Property, Fast Multiplication

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Fast Multiplications Using the Distributive Property

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

Key Concept: Fast Multiplication Using Distributive Property

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

Key Concept: Geometry-based identities, Distributive Property

c) $x^2 - 9$
[Solution Description]
The area of a rectangle is given by the product of its length and width. Here, the sides are $(x + 3)$ and $(x - 3)$. Using the identity $(a + b)(a - b) = a^2 - b^2$, we can find the area as:
$(x + 3)(x - 3) = x^2 - 9$
Therefore, the area is $x^2 - 9$.

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(60 - 5)^2$ can be evaluated using the formula $60^2 - 2 \times 60 \times 5 + 5^2$, which aligns with the geometric interpretation where we subtract the areas of two rectangles ($60 \times 5$ each) and add back the area of the small square ($5^2$) to avoid double subtraction. The reason explains this geometric process correctly, confirming the assertion. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Squares of Numbers, Modified Identity

b) 3136
[Solution Description]
To compute $56^2$, let us choose $b = 4$ (since $56 + 4 = 60$ and $56 - 4 = 52$, which are easy to multiply). Using the identity:
$56^2 = (56 + 4)(56 - 4) + 4^2$
Substituting the values:
$(56 + 4) = 60, \quad (56 - 4) = 52$
$4^2 = 16$
Now, multiply $(60 \times 52)$:
$60 \times 50 = 3000$
$60 \times 2 = 120$
$3000 + 120 = 3120$
Finally, add $b^2$:
$3120 + 16 = 3136$
Thus, $56^2 = 3136$.

Key Concept: Geometry-based distributive property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the identity $(a + b)(a - b) = a^2 - b^2$ can be proven geometrically. This is true because we can visualize it as follows:
Step 1: Draw a large square with side length $a$. Its area is $a^2$.
Step 2: Subtract a smaller square with side length $b$ from one corner of the large square. Now, the remaining shape has an area of $a^2 - b^2$.
Step 3: Alternatively, divide the remaining shape into two rectangles with dimensions $(a + b)$ and $(a - b)$. Their combined area is $(a + b)(a - b)$, which matches $a^2 - b^2$.
The reason correctly explains how the geometric method verifies the algebraic identity by subtracting areas. Therefore, both assertion and reason are true, and the reason is the correct explanation.

Key Concept: Difference of Squares Identity

b) $a^2 - b^2$
[Solution Description]
Using the distributive property:
$(a + b)(a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b = a^2 - b^2$

Key Concept: Geometry-based explanation for $(a - b

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because $(a - b)^2$ can indeed be expanded to $a^2 - 2ab + b^2$, which represents the geometric process of subtracting two rectangles of area $ab$ and adding back the overlapping square of area $b^2$ to get the area of the smaller square. The reason correctly explains this geometric interpretation, as it describes the steps involved in deriving the identity $(a - b)^2 = a^2 - 2ab + b^2$ using areas.

Key Concept: Special Cases of Distributive Property, Geometry-based explanation

b) 1089
[Solution Description]
We can visualize this problem using geometry. The area of the inner square (side length 33) is $(40 - 7)^2 = 33^2$. According to the identity, we can write:
$(40 - 7)^2 = 40^2 - 2(40 \times 7) + 7^2$
Calculating step-by-step:
$40^2 = 1600$
$2(40 \times 7) = 560$
$7^2 = 49$
Now, substitute these values:
$1600 - 560 + 49 = 1089$
Hence, $(40 - 7)^2 = 1089$.

Key Concept: Distributive Property - Geometric Visualization

b) 3025
[Solution Description]
The area of the larger square is $60^2 = 3600$.
The area of two rectangles is $2 \times (60 \times 5) = 600$.
The area of the small square is $5^2 = 25$.
So, the required area is $3600 - 600 + 25 = 3025$.

Key Concept: Difference of Squares Identity

c) An L-shaped region formed by removing a square of side $b$ from a square of side $a$.
[Solution Description]
The identity $(a + b)(a - b) = a^2 - b^2$ can be visualized as:
Step 1: Draw a large square of side $a$.
Step 2: Remove a smaller square of side $b$ from it, leaving an L-shaped region.
Step 3: The area of the remaining region is $a^2 - b^2$, equivalent to a rectangle with sides $(a + b)$ and $(a - b)$.
Hence, the correct diagram shows this transformation.

Key Concept: Expansion of Two Binomials

c) $xy + 3x + 2y + 6$
[Solution Description]
Expand using the distributive property:
$(x + 2)(y + 3) = x \cdot y + x \cdot 3 + 2 \cdot y + 2 \cdot 3 = xy + 3x + 2y + 6$

Key Concept: Distributive Property Application

c) 2025
[Solution Description]
Using Sridharacharya's identity $a^2 = (a + b)(a - b) + b^2$:
Step 1: Let $a = 45$, $b = 5$.
Step 2: Compute $(a + b)(a - b) = (45 + 5)(45 - 5) = 50 \times 40 = 2000$.
Step 3: Add $b^2 = 25$.
Final answer: $45^2 = 2000 + 25 = 2025$.

Key Concept: Visualizing Squares and Rectangles

c) $50^2 - 2 \times (50 \times 3) + 3^2$
[Solution Description]
Using the geometric approach:
Step 1: The area of the larger square of side 50 is $50^2 = 2500$.
Step 2: Subtract the areas of the two rectangles formed by removing 3 units from the side: $50 \times 3 + 50 \times 3 = 300$.
Step 3: Add back the small square of side 3 once to adjust for double subtraction: $3^2 = 9$.
Final expression: $50^2 - 2 \times (50 \times 3) + 3^2 = 2500 - 300 + 9 = 2209$.

Key Concept: Expression Simplification

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

Key Concept: Multiple methods for pattern representation, algebraic equivalence

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

Key Concept: 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: Multiple methods

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

Key Concept: Consistency of expressions across different methods

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] 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: 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: 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 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: 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 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: Square of Sum/Difference, Distributive Property

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion claims that $(a + b)^2$ is always greater than $a^2 + b^2$. The reason given is that the term $2ab$ ensures this inequality holds for non-zero $a$ and $b$.
Step 1: Expand $(a + b)^2$ using Identity 1A:
$(a + b)^2 = a^2 + 2ab + b^2$
Step 2: Compare $(a + b)^2$ with $a^2 + b^2$. The difference is $2ab$.
Step 3: The term $2ab$ can be positive, negative, or zero depending on the signs of $a$ and $b$:
- If $a$ and $b$ are both positive or both negative, $2ab > 0$, so $(a + b)^2 > a^2 + b^2$.
- If one of them is positive and the other is negative, $2ab < 0$, which may make $(a + b)^2 < a^2 + b^2$ (e.g., $a = 1$, $b = -1$ gives $(1 - 1)^2 = 0 < 1 + 1 = 2$). - If either $a$ or $b$ is zero, $2ab = 0$, making $(a + b)^2 = a^2 + b^2$. Thus, the assertion is false because $(a + b)^2$ is not always greater than $a^2 + b^2$. The reason is true when $a$ and $b$ are both non-zero and have the same sign, but it fails to account for cases where $2ab \leq 0$. Therefore, the reason does not fully explain the assertion, but it is conditionally correct.

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Difference of Squares, Multi-step Solution

c) 9996
[Solution Description]
We use the identity $(a + b)(a - b) = a^2 - b^2$ with $a = 100$ and $b = 2$:
$(100 - 2)(100 + 2) = 100^2 - 2^2$
$= 10{,}000 - 4$
$= 9996$.

Key Concept: Square of the Sum of Two Numbers

c) $x^2 + 8x + 16$
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = x$ and $b = 4$.
Step 1: Square the first term: $x^2$.
Step 2: Multiply the two terms and double the result: $2 \times (x \times 4) = 8x$.
Step 3: Square the second term: $4^2 = 16$.
Final expansion: $x^2 + 8x + 16$.

Key Concept: Square of the Difference of Two Numbers

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

Key Concept: Application of Square of Sum/Difference

c) $42km$
[Solution Description]
Expanding $(7k + 3m)^2$ using the identity $(a + b)^2 = a^2 + 2ab + b^2$, where $a = 7k$ and $b = 3m$:
$(7k + 3m)^2 = (7k)^2 + 2(7k)(3m) + (3m)^2 = 49k^2 + 42km + 9m^2$. The middle term is $42km$.

Key Concept: Square of the Sum, Difference of Squares

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

Key Concept: Square of the Difference, Pattern Recognition

c) 9801
[Solution Description]
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 100$ and $b = 1$:
$(100 - 1)^2 = 100^2 - 2 \times 100 \times 1 + 1^2$
$= 10000 - 200 + 1$
$= 9801$.

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Square of the Sum of Two Numbers

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

Key Concept: Application of Identity 1A

c) 121
[Solution Description]
Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, substitute $a = 10$ and $b = 1$.
Step 1: Square the first term: $10^2 = 100$.
Step 2: Multiply the two terms and double the result: $2 \times (10 \times 1) = 20$.
Step 3: Square the second term: $1^2 = 1$.
Final calculation: $100 + 20 + 1 = 121$.

Key Concept: Square of the Difference of Two Numbers

c) $25y^2 - 20y + 4$
[Solution Description]
The identity $(a - b)^2 = a^2 - 2ab + b^2$ is used here with $a = 5y$ and $b = 2$. Substituting:
$(5y - 2)^2 = (5y)^2 - 2(5y)(2) + 2^2 = 25y^2 - 20y + 4$.

Key Concept: 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: 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: 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: Multiplying Trinomials, Simplification

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

Key Concept: Simplification of algebraic expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
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: 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: 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: 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: 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: 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 $\frac{5x^2}{3} - \frac{10xy}{3} + \frac{5x}{12}$