Key Concept: Expression Evaluation

c) 19
[Solution Description]
To find the value of $3m + 7$ when $m = 4$, substitute 4 for $m$ in the expression. First multiply $3 \times 4$ to get 12, then add 7 to get $12 + 7 = 19$.

Key Concept: Calculating Age Using Algebraic Expression

c) 18 years
[Solution Description]
Given the algebraic expression for Shabnam's age is $a + 3$. Substituting $a = 15$ into the expression gives $15 + 3 = 18$ years.

Key Concept: Algebraic Expressions

c) $y = 2x + 5$
[Solution Description]
The problem states Priya has twice as many books as Ravi plus 5 more. If Ravi's books are $x$, then twice that amount is $2x$. Adding 5 more gives $2x + 5$. Therefore, the correct equation is $y = 2x + 5$, where $y$ represents Priya's books.

Key Concept: Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $s = a + 3$ can determine Shabnam's age when Aftab’s age ($a$) is known. This is true because, as per the given example, Shabnam is always 3 years older than Aftab. The reason explains why letters like $a$ and $s$ are used in algebra: they simplify representing mathematical relationships compared to ordinary language. Both statements are true, and the reason correctly explains why the assertion holds, as it describes the fundamental benefit of using letter-numbers.

Key Concept: Algebraic expressions, Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we verify the assertion by substituting $x = 4$ into the expression $3x + 5$:
$3(4) + 5 = 12 + 5 = 17$
Thus, the assertion is true. Next, we check the reason: The rules for simplifying arithmetic expressions (addition, multiplication, etc.) apply to algebraic expressions when letter-numbers are substituted with given values. This is indeed a fundamental principle of algebra, making the reason true and correctly explaining the assertion.

Key Concept: Basic Algebraic Expression

c) $s - 3$
[Solution Description]
According to the syllabus example, Aftab's age ($a$) is Shabnam's age minus 3 years. So the expression is: $a = s - 3$

Key Concept: Expression evaluation, multi-step reasoning

c) 13
[Solution Description]
Step 1: Calculate $p(2)$ by substituting $x = 2$ in $p(x) = 3x^2 - 4x + 7$.
$p(2) = 3(2)^2 - 4(2) + 7 = 12 - 8 + 7 = 11$
Step 2: Calculate $q(1)$ by substituting $x = 1$ in $q(x) = x^2 + 2x - 5$.
$q(1) = (1)^2 + 2(1) - 5 = 1 + 2 - 5 = -2$
Step 3: Subtract $q(1)$ from $p(2)$.
$p(2) - q(1) = 11 - (-2) = 13$
The result is 13.

Key Concept: Cost Calculation Using Letter-Numbers

d) $35c + 60j$
[Solution Description]
Step 1: Calculate the cost of $c$ coconuts: $35 \times c = 35c$.
Step 2: Calculate the cost of $j$ kg jaggery: $60 \times j = 60j$.
Step 3: Add both costs to get the total cost: $35c + 60j$.
Therefore, the total cost is $35c + 60j$.

Key Concept: Algebraic Expression Application

c) 6
[Solution Description]
The equation given is $2x + 5 = 17$. To find the input $x$, subtract 5 from both sides: $2x = 12$. Then divide both sides by 2: $x = 6$.

Key Concept: Forming Algebraic Expressions

b) $2 \times n$
[Solution Description]
Each L-shape requires 2 matchsticks. For $n$ L-shapes, the total number of matchsticks is given by multiplying the number of L-shapes ($n$) by 2. The algebraic expression is: $2 \times n$.

Key Concept: Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $3x + 5$ models the scenario where each book costs \$$x$, and there is a fixed additional cost of \$$5$ (for example, shipping). This relationship is correctly represented algebraically because the variable $x$ stands for the unknown price of the book. The reason explains why variables are used in algebraic expressions—to generalize relationships involving unknowns. Here, both statements are true, and the reason logically justifies the assertion by explaining the role of variables.

Key Concept: Pattern Recognition, Algebraic Expressions

b) 2
[Solution Description]
Substitute the given values into the formula:
$4(3) - k = 10$
$12 - k = 10$
Subtract 10 from both sides:
$12 - 10 = k$
$k = 2$

Key Concept: Ages of Aftab and Shabnam, Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, Shabnam is always 3 years older than Aftab. This relationship is consistently represented by the equation $s = a + 3$.
Step 1: Verify the Assertion.
Given that Aftab’s age ($a$) is 25 years, substitute into the equation:
$s = 25 + 3 = 28$ years.
Thus, when Aftab is 25, Shabnam is indeed 28 years old, so the Assertion is true.
Step 2: Verify the Reason.
The equation $s = a + 3$ directly follows from the given condition that Shabnam is 3 years older than Aftab. Therefore, the Reason is also true.
Step 3: Check if the Reason explains the Assertion.
The Reason provides the general formula that justifies the specific case in the Assertion. Hence, the Reason correctly explains the Assertion.
Conclusion: Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Letter-numbers, Multi-step solutions, Advanced reasoning

b) 15
[Solution Description]
Let Aftab's age be $a$ and Shabnam's age be $s$. From the problem, $s = 2a - 5$ and $a + s = 40$. Substitute $s$ from the first equation into the second equation: $a + (2a - 5) = 40$. Simplify to get $3a - 5 = 40$. Add 5 to both sides: $3a = 45$. Divide by 3: $a = 15$. So, Aftab is 15 years old.

Key Concept: Basic Concept: Relationship between Ages

c) $s = a + 3$
[Solution Description]
The problem states that Shabnam is always 3 years older than Aftab. So, the correct algebraic expression is:
$s = a + 3$

Key Concept: Finding Aftab’s age

a) 32 years
[Solution Description]
The problem states that Aftab is 3 years younger than Shabnam. Let $s$ represent Shabnam’s age and $a$ represent Aftab’s age. The relationship is given by $a = s - 3$. Substituting Shabnam’s current age ($s = 35$) into the equation: $a = 35 - 3 = 32$. Thus, Aftab’s age is 32 years.

Key Concept: Verification of Formula

c) 11
[Solution Description]
We use the formula $2y + 1$ to verify the number of matchsticks for Step 5:
Substitute y = 5 into the formula:
$2 \times 5 + 1 = 10 + 1 = 11$.
From the table, Step 5 indeed has 11 matchsticks, confirming the formula's correctness.

Key Concept: Matchstick patterns, General formula

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we need to verify the assertion. According to the general formula provided in the syllabus, the number of matchsticks for Step $y$ is given by $2y + 1$. For $y = 10$, substituting into the formula: $2(10) + 1 = 21$. This matches the assertion. The reason states the correct formula, and it correctly explains why the assertion is true since applying the formula leads to the claimed result.

Key Concept: Matchstick patterns, General rule derivation

b) $2y + 1$
[Solution Description]
We are given the expression for the number of matchsticks in Step $y$ as $3 + 2 \times (y - 1)$. Let us simplify this expression step-by-step:
First, expand the multiplication: $3 + 2 \times y - 2 \times 1 = 3 + 2y - 2$.
Next, combine like terms: $(3 - 2) + 2y = 1 + 2y$.
Rearranging, we get $2y + 1$. This simplified form must be equivalent to the original expression.

Key Concept: Matchstick patterns, Multi-step solution

c) The new pattern grows twice as fast.
[Solution Description]
To compare the growth rates, let's analyze the difference between the two patterns at any step $y$:
Original pattern: $2y + 1$
New pattern: $4y - 1$
Difference = $(4y - 1) - (2y + 1) = 2y - 2$.
For $y = 1$, difference = $0$.
For $y = 2$, difference = $2$.
For larger $y$, the difference increases linearly.
This means the new pattern grows twice as fast as the original pattern since the coefficient of $y$ is double (4 vs 2).

Key Concept: Substitution, Evaluating Expressions

c) $87$
[Solution Description]
Substitute $p = 5$ and $q = 2$ into the expression:
$3(5)^2 - 4(2) + 2(5)(2)$. Calculate each term step-by-step:
$3(25) - 8 + 20 = 75 - 8 + 20$. Now add/subtract left to right:
$75 - 8 = 67$, then $67 + 20 = 87$. The final value is $87$.

Key Concept: Applying distributive property

c) 40
[Solution Description]
The distributive property states that $a \times (b - c) = a \times b - a \times c$. Applying this to the given expression:
$5 \times (12 - 4) = 5 \times 12 - 5 \times 4$
Now perform the multiplications:
$5 \times 12 = 60,\quad 5 \times 4 = 20$
Subtract the results:
$60 - 20 = 40$
So the simplified form is $40$.

Key Concept: Arithmetic Expressions and Grouping

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To evaluate the expression $42 + 15 - (8 - 7)$, we first solve the bracket $(8 - 7)$.
Step 1: Calculate $(8 - 7) = 1$.
Step 2: Substitute back into the expression: $42 + 15 - 1$.
Step 3: Perform addition: $42 + 15 = 57$.
Step 4: Finally, subtract: $57 - 1 = 56$.
Thus, the assertion is true because the expression simplifies correctly. The reason is also true because evaluating brackets first follows the standard order of operations (BODMAS/PEMDAS). Moreover, the reason correctly explains why the assertion holds.

Key Concept: Swapping and Grouping Terms

c) 130
[Solution Description]
Group the terms conveniently: $(83 - 13) + (28 + 32) = 70 + 60 = 130$.

Key Concept: Practical Application

c) $20x + 52y$
[Solution Description]
Total cost before refund: $25x + 60y$.
Total refund: $5x + 8y$.
Net amount paid: $(25x + 60y) - (5x + 8y)$.
Distribute the subtraction: $25x + 60y - 5x - 8y$.
Group like terms: $(25x - 5x) + (60y - 8y)$.
Simplify: $20x + 52y$.
The net amount paid is $20x + 52y$.

Key Concept: Simplification, Brackets

b) $-2a + 2b$
[Solution Description]
First innermost bracket: $(b - 3a) = b - 3a$
Next level: $[2a - (b - 3a)] = 2a - b + 3a = 5a - b$
Another part: $(3b + 2a) = 3b + 2a$
Now substitute: $[5a - (3b + 2a)] = 5a - 3b - 2a = 3a - 3b$
And $\{4b - [5a - b]\} = 4b - 5a + b = 5b - 5a$
Combine both parts: $(3a - 3b) + (5b - 5a) = -2a + 2b = 2b - 2a$

Key Concept: Addition and Subtraction of Like Terms

c) $12n$
[Solution Description]
Step 1: Simplify the expressions inside the parentheses first.
$12n - 4n = 8n$ and $6n - 2n = 4n$
Step 2: Now add the simplified terms.
$8n + 4n = 12n$
So, the simplified form is $12n$.

Key Concept: Like/Unlike terms, Simplification

b) $(8m + 4n)x + (6m - n)y$
[Solution Description]
First calculate initial sales: $x(12m + 5n) + y(7m - 3n) = 12mx + 5nx + 7my - 3ny$
Then calculate returns: $x(4m + n) + y(m - 2n) = 4mx + nx + my - 2ny$
Subtract returns from sales: $(12mx + 5nx + 7my - 3ny) - (4mx + nx + my - 2ny) = 8mx + 4nx + 6my - ny$
Combine like terms: $(8m + 4n)x + (6m - n)y$

Key Concept: Removing Brackets

c) 25
[Solution Description]
Remove the brackets by distributing the negative sign inside:
$45 - (12 + 8) = 45 - 12 - 8$.
Now perform the subtraction step-by-step:
$(45 - 12) - 8 = 33 - 8 = 25$.
Therefore, the value of the expression is $25$.

Key Concept: Order of Operations, Brackets

b) 6
[Solution Description]
First, solve the bracket $(34 + 19) = 53$. Then multiply by 2: $53 \times 2 = 106$. Rewrite the expression as $100 - 106 + 12$. Subtract 106 from 100 to get $-6$. Finally, add 12 to $-6$ to get 6.

Key Concept: Expression evaluation

b) 82
[Solution Description]
Follow the order of operations (multiplication before addition):
$7 \times 4 = 28 \quad \text{and} \quad 9 \times 6 = 54$
Now add the results:
$28 + 54 = 82$
The value of the expression is $82$.

Key Concept: Order of Operations

b) 10
[Solution Description]
Follow the order of operations (PEMDAS/BODMAS). Multiplication comes before subtraction.
First, multiply $5 \times 4 = 20$.
Then subtract from $30$: $30 - 20 = 10$.
Therefore, the value of the expression is $10$.

Key Concept: Using Brackets and Signs

b) 37
[Solution Description]
There are two ways to solve this expression:
1. First solve the brackets: $(18 + 13) = 31$. Then subtract from 68: $68 - 31 = 37$.
2. Distribute the negative sign: $68 - 18 - 13 = 50 - 13 = 37$.
Both methods give the same result.

Key Concept: Distributive Property

c) 35
[Solution Description]
Using the distributive property:
$5 \times (3 + 4) = (5 \times 3) + (5 \times 4) = 15 + 20 = 35$.

Key Concept: Using Brackets with Negative Sign

b) 15
[Solution Description]
To evaluate $45 - (12 + 18)$, first solve the bracket:
$12 + 18 = 30$. Now subtract this from 45:
$45 - 30 = 15$.

Key Concept: Swapping and grouping terms, negative sign outside brackets

c) 59
[Solution Description]
Start by solving the first bracket: $35 + 18 = 53$. Now, the expression becomes $92 - 53 + 25 - (12 - 7)$. Solve the second bracket: $12 - 7 = 5$. The expression now simplifies to $92 - 53 + 25 - 5$. Subtract $92 - 53 = 39$. Then add $39 + 25 = 64$. Finally, subtract $64 - 5 = 59$.

Key Concept: Distributive property, simplification

b) $14x + 39y - 31z$
[Solution Description]
First, we apply the distributive property to each term:
$7(3x + 4y - 2z) = 21x + 28y - 14z$
$-5(2x - y + z) = -10x + 5y - 5z$
$+3(x + 2y - 4z) = 3x + 6y - 12z$
Now combine all terms:
$(21x - 10x + 3x) + (28y + 5y + 6y) + (-14z - 5z - 12z)$
$= (14x) + (39y) + (-31z)$
$= 14x + 39y - 31z$

Key Concept: Distributive property over addition

d) 55
[Solution Description]
Using distributive property of multiplication over addition, we can write:
$5 \times (8 + 3) = 5 \times 8 + 5 \times 3$
Now, multiply each term:
$5 \times 8 = 40$ and $5 \times 3 = 15$
Add the results together:
$40 + 15 = 55$

Key Concept: Distributive Property in Arithmetic Expressions

b) 12p
[Solution Description]
The distributive property allows us to combine like terms by adding the coefficients and keeping the variable the same. Here, $7p + 5p = (7 + 5)p = 12p$.

Key Concept: Combining like terms

b) 15n
[Solution Description]
All terms are like terms as they have the same variable $n$. Combine them by adding/subtracting coefficients:
First, subtract 4n from 12n:
$12n - 4n = 8n$
Next, add 7n to the result:
$8n + 7n = 15n$

Key Concept: Swapping and Grouping Terms

c) 85
[Solution Description]
First, group the terms conveniently: $(45 - 15) + (23 + 32)$.
Calculate $45 - 15 = 30$.
Calculate $23 + 32 = 55$.
Now add the results: $30 + 55 = 85$.

Key Concept: Multiple Operations, Grouping

b) 22
[Solution Description]
Solve the terms inside the brackets first:
$(12 - 5) = 7$,
$(4 + 2) = 6$.
Now, substitute these back into the original expression:
$40 - 3 \times 7 + 18 \div 6$.
Perform the multiplication and division:
$3 \times 7 = 21$,
$18 \div 6 = 3$.
Now, rewrite the expression as $40 - 21 + 3$.
Finally, perform the operations from left to right:
$40 - 21 = 19$,
$19 + 3 = 22$.

Key Concept: Bracket Evaluation with Negative Sign

c) 20
[Solution Description]
Solve the bracket first: $(12 + 18) = 30$.
Now subtract this from 50: $50 - 30 = 20$.
Alternatively, remove the brackets: $50 - 12 - 18$.
Calculate $50 - 12 = 38$.
Subtract the remaining term: $38 - 18 = 20$.

Key Concept: Algebraic Expression Evaluation

c) 17
[Solution Description]
Given $x = 5$, substitute it into the expression $3x + 7 - x$:
1. Replace $x$ with $5$:
\begin{align*}
3(5) + 7 - 5 &= 15 + 7 - 5
\end{align*}
2. Perform addition and subtraction from left to right:
\begin{align*}
15 + 7 &= 22
22 - 5 &= 17
\end{align*}
The value of the expression is $17$.

Key Concept: Brackets in Arithmetic Expressions

b) 35
[Solution Description]
The expression $75 - (25 + 15)$ can be solved in two ways:
1. **Method 1**: Solve the bracket first.
\begin{align*}
25 + 15 &= 40
75 - 40 &= 35
\end{align*}
2. **Method 2**: Remove the bracket by distributing the negative sign.
\begin{align*}
75 - 25 - 15 &= 50 - 15
&= 35
\end{align*}
Both methods give the same result, which is $35$.

Key Concept: Order of operations, Algebraic expressions

c) $10(x - y)$[Solution Description]Correct calculation: \$50 × x + \$30 × y = $50x + 30y$
Incorrect calculation: (x + y) × \$40 = $40(x + y)$
Difference = $(50x + 30y) - 40(x + y)$
= $50x + 30y - 40x - 40y$
= $10x - 10y$
= $10(x - y)$

Key Concept: Evaluating Algebraic Expression

b) 30
[Solution Description]
The expression $6k$ means $6 \times k$. Substituting $k = 5$ gives $6 \times 5 = 30$. Therefore, the value is 30.

Key Concept: Omission of Multiplication Symbol

c) 40
[Solution Description]
The sequence of multiples of 4 is given by $4n$, where $n$ is the term number. For the 10th term, substitute $n = 10$ into the expression: $4 \times 10 = 40$. Thus, the 10th term is 40.

Key Concept: Omission of Multiplication Symbol in Algebraic Expressions

c) 28
[Solution Description]
To find the area of the rectangle when $v = 4$, we substitute $v$ with 4 in the expression $7v$.
Step 1: Replace $v$ with 4 in the expression $7v$.
$7v = 7 \times v$
$7 \times 4 = 28$
The area of the rectangle is 28 square units.

Key Concept: Finding nth term of a sequence

b) $4n$
[Solution Description]
The given sequence is 4, 8, 12, 16, ..., which are multiples of 4.
The first term is $4 \times 1 = 4$, the second term is $4 \times 2 = 8$, and so on.
Therefore, the nth term of the sequence is $4n$.

Key Concept: Finding a specific term in a sequence

b) 56
[Solution Description]
To find the 8th term, substitute $n = 8$ into the expression $7n$. So, $7 \times 8 = 56$. Therefore, the 8th term is 56.

Key Concept: nth term of a sequence

c) $5n$
[Solution Description]
The given sequence is multiples of 5. The first term is $5 \times 1$, the second term is $5 \times 2$, and so on. Therefore, the nth term can be expressed as $5n$.

Key Concept: Identifying mistakes in simplification

b) 10
[Solution Description]
Substitute $x = -3$ into $7 - x$. Since subtracting a negative number is the same as adding its positive counterpart, $7 - (-3)$ becomes $7 + 3 = 10$.

Key Concept: Substitution in Algebraic Expressions

c) 14
[Solution Description]
First, substitute k with 4:
$3k + 2 = 3 \times 4 + 2 = 12 + 2 = 14$

Key Concept: Substitution, Simplification

b) 47
[Solution Description]
First, substitute $p = 2$ and $q = 5$ into the expression: $3(2)^2 + 4(2)(5) - 5$.
Calculate each term step-by-step:
1. $3(2)^2 = 3 \times 4 = 12$
2. $4(2)(5) = 8 \times 5 = 40$
3. $-q = -5$
Now add them together: $12 + 40 - 5 = 47$.

Key Concept: Simplification of Algebraic Expressions

a) $4a + 5b + 10$
[Solution Description]
First, group like terms: $(3a + 8a - 7a) + (9b - 4b) + (-6 + 16)$. Then simplify each group: $4a + 5b + 10$.

Key Concept: Simplification, Subtraction of Expressions

a) $19f + 20s - 42$
[Solution Description]
The problem requires subtracting the first expression from the second:
$(-23 + 13f + 12s) - (-6f + 19 - 8s)$
Remove the parentheses carefully:
$-23 + 13f + 12s + 6f - 19 + 8s$
Combine like terms:
$(13f + 6f) + (12s + 8s) + (-23 - 19) = 19f + 20s - 42$

Key Concept: Subtracting Algebraic Expressions

a) $18x + 16y - 23$
[Solution Description]
Rewrite subtraction as addition of the opposite: $7y - 10 + 3x - (-15x + 13 - 9y) = 7y - 10 + 3x + 15x - 13 + 9y$. Combine like terms: $(3x + 15x) + (7y + 9y) + (-10 - 13) = 18x + 16y - 23$.

Key Concept: Simplifying algebraic expressions

b) $15m + 2n$
[Solution Description]
First, remove the brackets:
$12m - 5n + 3m + 7n$.
Group like terms together:
$12m + 3m - 5n + 7n$.
Now, combine the coefficients of like terms:
$(12 + 3)m + (-5 + 7)n = 15m + 2n$.
The simplified form is $15m + 2n$.

Key Concept: Combining like terms

a) $4x + 3y$
[Solution Description]
To simplify the expression, first identify and group the like terms (terms with the same variable part). Combine the coefficients of like terms while keeping the variables unchanged.
1. Group the like terms for $x$: $(7x - 3x) = (7 - 3)x = 4x$.
2. Group the like terms for $y$: $(5y - 2y) = (5 - 2)y = 3y$.
3. Combine the simplified terms: $4x + 3y$.
The simplified form of the expression is $4x + 3y$.

Key Concept: Simplification with unlike terms

a) $9x + 6y$
[Solution Description]
Remove the brackets carefully, remembering to distribute the negative sign:
$7x + 4y - 3x + 2y + 5x$.
Group like terms together:
$7x - 3x + 5x + 4y + 2y$.
Combine the coefficients:
$(7 - 3 + 5)x + (4 + 2)y = 9x + 6y$.
The simplified form is $9x + 6y$.

Key Concept: Perimeter of a rectangle, simplification of algebraic expressions

b) 4 units
[Solution Description]
Given $p = 24$ and $l = 2b$. Substitute these into the perimeter formula:
$24 = 2(2b) + 2b$.
Simplify the equation step-by-step:
$24 = 4b + 2b$.
Combine like terms:
$24 = 6b$.
Solve for $b$:
$b = 24 / 6 = 4$.
Therefore, the breadth is 4 units.

Key Concept: Perimeter Calculation

c) 16 cm
[Solution Description]
Given, length ($l$) = 5 cm and breadth ($b$) = 3 cm.
The formula for the perimeter of a rectangle is:
$p = 2l + 2b$
Substituting the values:
$p = 2 \times 5 + 2 \times 3$
$p = 10 + 6$
$p = 16$ cm
Therefore, the perimeter is 16 cm.

Key Concept: Perimeter of a rectangle, simplification of algebraic expressions

c) $16x$
[Solution Description]
Substitute $l = 5x$ and $b = 3x$ into the simplified perimeter formula:
$p = 2(5x) + 2(3x) = 10x + 6x = 16x$.
Alternatively, using the original expression:
$p = 5x + 5x + 3x + 3x = 16x$.
Both give the same result, confirming the simplification is correct.
Therefore, the perimeter is $16x$.

Key Concept: Simplification of like terms

a) $14m$
[Solution Description]
Step 1: Calculate the total number of pencils sold on both days.
$4m + 2m = 6m$
Step 2: Calculate the total number of erasers sold on both days.
$5m + 3m = 8m$
Step 3: Add the totals to find the combined expression.
$6m + 8m = 14m$
So, the simplified expression is $14m$.

Key Concept: Combining Like Terms

b) $6p + 7q$
[Solution Description]
Combine the like terms with p and q separately. For p terms: 9p - 3p = 6p. For q terms: 5q + 2q = 7q. The simplified form is $6p + 7q$.

Key Concept: Simplification of Algebraic Expressions, Like Terms

a) $3xp + \frac{3yq}{2}$
[Solution Description]
First, calculate earnings from pencils:
Monday: $x \times p = xp$
Tuesday: $2x \times p = 2xp$
Total pencils: $xp + 2xp = 3xp$
Next, calculate earnings from erasers:
Monday: $y \times q = yq$
Tuesday: $\frac{y}{2} \times q = \frac{yq}{2}$
Total erasers: $yq + \frac{yq}{2} = \frac{3yq}{2}$
Combined total earnings: $3xp + \frac{3yq}{2}$

Key Concept: Area of split rectangles, Like terms

b) $10m$
[Solution Description]
First, find the total area of rectangle ABCD:
$\text{Area of ABCD} = 15 \times m = 15m.$
Next, find the area of rectangle EBCF:
$\text{Area of EBCF} = 5 \times m = 5m.$
Subtract the area of EBCF from the total area to get the area of AEFD:
$\text{Area of AEFD} = 15m - 5m = 10m.$

Key Concept: Area of split rectangles

c) $8n$
[Solution Description]
The area of rectangle AEFD can be found using its side lengths, which are $(12 - 4)$ and $n$.
Step 1: Simplify $(12 - 4)$.
$12 - 4 = 8$.
Step 2: Multiply to find the area.
Area = $8 \times n = 8n$.
Thus, the simplified expression for the area is $8n$.

Key Concept: Rectangle Area Subtraction

b) $8k$
[Solution Description]
Total area of ABCD: $12k \times n = 12kn$. Width of AEFD is $8$ (since $12 - 4 = 8$). Area of AEFD can be calculated directly as $8 \times k = 8k$ (assuming the height remains $k$), or by subtracting the area of EBCF ($4k$) from ABCD ($12k$): $12k - 4k = 8k$.

Key Concept: Like Terms Simplification

b) $2x + 4y$
[Solution Description]
First, group the like terms together: $(4x - 2x) + (3y + y)$.
Next, add the coefficients of like terms:
$4x - 2x = (4 - 2)x = 2x$ and
$3y + y = (3 + 1)y = 4y$.
So, the simplified form is $2x + 4y$.

Key Concept: Like Terms Simplification, Brackets Handling

a) $10m + 13n$
[Solution Description]
The initial expenditure is $(15m + 20n)$.
After selling books for $(8m + 12n)$, the net expenditure becomes $(15m + 20n) - (8m + 12n) = (15m - 8m) + (20n - 12n) = 7m + 8n$.
Then he buys more books for $(3m + 5n)$.
So, total expenditure now is $(7m + 8n) + (3m + 5n) = (7m + 3m) + (8n + 5n) = 10m + 13n$.

Key Concept: Simplification of Like Terms

b) $3x + 7y$
[Solution Description]
To simplify $(5x + 3y) - (2x - 4y)$, first remove the parentheses by distributing the negative sign to the terms inside the second set of parentheses:
$5x + 3y - 2x + 4y$
Next, group like terms together:
$(5x - 2x) + (3y + 4y)$
Combine the coefficients of like terms:
$3x + 7y$
The simplified form is $3x + 7y$.

Key Concept: Unlike Terms Identification

b) $7y$ and $4z$
[Solution Description]
Unlike terms have different variable parts. Here:
- $5x$ and $3x$ are like terms (same variable $x$).
- $7y$ and $4z$ are unlike terms (different variables $y$ and $z$).
- $2a$ and $9a$ are like terms.
- $6b$ and $8c$ are unlike terms.
Thus, $7y$ and $4z$ is an example of unlike terms.

Key Concept: Expression Evaluation

b) $10$
[Solution Description]
Substitute $a = 4$ and $b = 1$ into $3a - 2b$:
$(3 \times 4) - (2 \times 1) = 12 - 2 = 10$.

Key Concept: Simplification of Algebraic Expressions

c) $13m + 11n$
[Solution Description]
Total initial cost for $x$ notebooks and $y$ pens: $3m x + 4n y$.
Amount refunded due to returning 2 notebooks and 1 pen: $2m + n$.
Final amount paid: $(3m x + 4n y) - (2m + n)$.
Assuming $x = 5$ and $y = 3$ for calculation,
Initial cost: $3m \times 5 + 4n \times 3 = 15m + 12n$.
Subtract refund: $15m + 12n - 2m - n = 13m + 11n$.
The final amount paid is $13m + 11n$.

Key Concept: Matchstick Patterns

c) 46
[Solution Description]
First, observe the pattern:
1 square = 4 matchsticks
2 squares = 7 matchsticks
3 squares = 10 matchsticks
The pattern shows that for each additional square, 3 more matchsticks are needed. So, the formula is $M = 3n + 1$, where $M$ is the number of matchsticks and $n$ is the number of squares.
For 15 squares:
$M = 3(15) + 1 = 45 + 1 = 46$ matchsticks.

Key Concept: Rope Folding and Cutting Pattern

c) 8
[Solution Description]

Key Concept: Traffic Signal Colour Pattern

a) Red
[Solution Description]
The sequence repeats every 3 positions:
1 - Red, 2 - Yellow, 3 - Green, 4 - Red, etc.
To find the colour at position 97, divide 97 by 3 and check the remainder.
$97 \div 3 = 32$ with a remainder of 1.
Remainder 1 corresponds to Red.

Key Concept: Age-Based Relationship

c) $2m$
[Solution Description]
Riya's age is twice Meena's age. If Meena's age is $m$, then Riya's age is $2m$. Thus, the correct expression is $2m$.

Key Concept: Letter-Numbers

c) $a + 3$
[Solution Description]
According to the problem statement, Shabnam is 3 years older than Aftab. This means Shabnam's age can be found by adding 3 to Aftab's age.
Step 1: Let Aftab's age be represented by $a$.
Step 2: Add 3 to Aftab's age to find Shabnam's age.
Algebraic expression: $a + 3$
The correct algebraic expression representing Shabnam's age is $a + 3$.

Key Concept: Matchstick Pattern

b) $2n$
[Solution Description]
Each L-shaped pattern requires 2 matchsticks. Therefore, the number of matchsticks needed is directly proportional to the number of Ls made.
Step 1: Let the number of Ls be represented by $n$.
Step 2: Multiply the number of Ls by 2 to get the total matchsticks needed.
Algebraic expression: $2n$
The correct algebraic expression is $2n$.

Key Concept: Deriving number machine formula

a) $2a + b$
[Solution Description]
Let's test each option:
Option a) $2a + b$ → $(2×2)+3=7$ ✔️ but $(2×4)+1=9 ≠13$ ✗
Option b) $3a + b$ → $(3×2)+3=9≠7$ ✗
Option c) $3a - b$ → $(3×2)-3=3≠7$ ✗
Option d) $4a - b$ → $(4×2)-3=5≠7$ ✗
Testing pattern manually:
Between (2,3)→7 and (4,1)→13, difference in outputs (6) equals ($4a-2a=8$) minus ($b_2-b_1=-2$), suggesting formula might be $3a + b + (some pattern)$. However none exactly match all cases, indicating possible error in question design (all options fail). But closest matching some cases is option a).

Key Concept: Identifying number machine formula

c) 17
[Solution Description]
Given formula is $3a + b$ where $a$ is first input (4) and $b$ is second input (5). Substitute values: $3 \times 4 + 5 = 12 + 5 = 17$

Key Concept: Verifying the formula with given inputs

b) (7, 0)
[Solution Description]
To verify which input pair $(a, b)$ gives the output 14 using $2a - b = 14$, test each option:
Option 1: $(5, 4)$
$2 \times 5 - 4 = 10 - 4 = 6 \neq 14$. Incorrect.
Option 2: $(7, 0)$
$2 \times 7 - 0 = 14 - 0 = 14$. Correct.
Option 3: $(8, 5)$
$2 \times 8 - 5 = 16 - 5 = 11 \neq 14$. Incorrect.
Option 4: $(10, 6)$
$2 \times 10 - 6 = 20 - 6 = 14$. Also correct, but since only one option must be marked, we take the first correct option in the list.
Therefore, the correct answer is $(7, 0)$.

Key Concept: Position and Design, Algebraic Expression

d) 13
[Solution Description]
To find the difference in positions:
1. Calculate the position of the 20th occurrence of Design A: $3(20) - 2 = 60 - 2 = 58$.
2. Calculate the position of the 15th occurrence of Design C: $3(15) = 45$.
3. Subtract the two positions: $58 - 45 = 13$.

Key Concept: Remainder Method, Multiple Concepts

c) 74
[Solution Description]
1. Find the position of the 25th occurrence of Design B: $3(25) - 1 = 75 - 1 = 74$.
2. Verify using the remainder method: Divide 74 by 3. Quotient = 24, Remainder = 2 (since $3 \times 24 = 72$ and $74 - 72 = 2$).
3. According to the remainder rule, remainder 2 corresponds to Design B, which matches.

Key Concept: Determining the design from a given position

a) Design A
[Solution Description]
To determine which design appears at a given position, divide the position number by 3 and observe the remainder:
Step 1: Divide 98 by 3.
$98 \div 3 = 32 \text{ with a remainder of } 2$
Step 2: Interpret the remainder:
- Remainder 0: Design C
- Remainder 1: Design B
- Remainder 2: Design A
Since the remainder is 2, Design A appears at position 98.

Key Concept: Generalization, algebraic modeling

c) $4y + 16$
[Solution Description]
The numbers in the $2 \times 2$ square are $y$, $y + 1$, $y + 7$, and $y + 8$. To find the total sum $S$, add them together:
$S = y + (y + 1) + (y + 7) + (y + 8)$
Combine like terms:
$S = 4y + 16$
Factor out the common term:
$S = 4(y + 4)$
Thus, the sum $S$ is $4y + 16$.

Key Concept: Verification

d) Numbers: 6, 7, 14, 15
[Solution Description]
The diagonal sum for any valid $2 \times 2$ calendar square must satisfy $2a + 8$, where $a$ is the top-left number.
Option a: Top-left $a = 3$, numbers are $3, 4, 10, 11$.
Diagonal sums: $3 + 11 = 14$ and $4 + 10 = 14$. Also, $2(3) + 8 = 14$. Valid.
Option b: Top-left $a = 10$, numbers are $10, 11, 17, 18$.
Diagonal sums: $10 + 18 = 28$ and $11 + 17 = 28$. Also, $2(10) + 8 = 28$. Valid.
Option c: Top-left $a = 15$, numbers are $15, 16, 22, 23$.
Diagonal sums: $15 + 23 = 38$ and $16 + 22 = 38$. Also, $2(15) + 8 = 38$. Valid.
Option d: Top-left $a = 6$, numbers are $6, 7, 14, 15$.
Here, the bottom-left number should be $6 + 7 = 13$, not 14. This violates the calendar pattern since consecutive days differ by 1 and weeks differ by 7. Thus, the square is invalid and does not satisfy the diagonal sum equality.

Key Concept: Diagonal Sum Equality

c) 18
[Solution Description]
Given that the top-left number is $a = 5$, the four numbers in the $2 \times 2$ square are:
- Top-left: $5$
- Top-right: $5 + 1 = 6$
- Bottom-left: $5 + 7 = 12$
- Bottom-right: $5 + 8 = 13$
The sum of the first diagonal (top-left to bottom-right) is $5 + 13 = 18$.
The sum of the second diagonal (top-right to bottom-left) is $6 + 12 = 18$.
Both sums are equal and can also be calculated using the expression $2a + 8 = 2(5) + 8 = 10 + 8 = 18$.

Key Concept: Verification of Matchstick Pattern

c) $2 \times 7 + 1$
[Solution Description]
According to the pattern, the number of matchsticks in any Step $y$ is calculated using $2y + 1$. For Step 7:
$2 \times 7 + 1 = 14 + 1 = 15$
Alternatively, you can verify from the table that Step 6 has 13 matchsticks. Since each step adds 2 matchsticks, Step 7 should have:
$13 + 2 = 15$
Thus, the correct expression is $2 \times 7 + 1$.

Key Concept: Matchstick figures, 2y + 1 rule

b) $3n + 2$
[Solution Description]
The initial step has 5 matchsticks ($S_1 = 5$). Each subsequent step adds 3 matchsticks ($S_{n} = S_{n-1} + 3$). To find a general formula for Step $n$, observe that for each step beyond 1, 3 matchsticks are added $(n - 1)$ times. Thus, the total number of matchsticks in Step $n$ is $5 + 3(n - 1)$. Simplifying:
$S_n = 5 + 3(n - 1) = 5 + 3n - 3 = 3n + 2$

Key Concept: Matchstick Pattern - Step Identification

b) 7
[Solution Description]
Set the equation $2y + 1 = 15$.
Solve for $y$: Subtract 1 from both sides: $2y = 14$.
Divide by 2: $y = 7$.

Key Concept: Diagonal Sums, Formula Detective

b) $p = -13$
[Solution Description]
The diagonal sum is $2p + 8$. The number machine's output is $2(p+1) - (p+7) = 2p + 2 - p - 7 = p - 5$. Equating the two: $2p + 8 = p - 5$ leads to $p = -13$.

Key Concept: Calendar Number Patterns

c) $5a$
[Solution Description]
According to the syllabus, the sum of numbers in a cross-shaped pattern is always 5 times the central number. Therefore:
$\text{Sum} = 5a.$

Key Concept: Calendar Number Patterns

c) 50
[Solution Description]
For a cross-shaped calendar pattern with center number $a = 10$, the other numbers are $a - 7$, $a - 1$, $a + 1$, and $a + 7$.
Step 1: Calculate the surrounding numbers:
$10 - 7 = 3$, $10 - 1 = 9$, $10 + 1 = 11$, $10 + 7 = 17$.
Step 2: Add all five numbers: $3 + 9 + 10 + 11 + 17 = 50$.
The total sum of the numbers is 50.

Key Concept: Matchstick Patterns, Algebraic Expressions

a) $4n + 1$
[Solution Description]
Given the expression: $5 + 4(n - 1)$.
First, expand the parentheses: $5 + 4n - 4$.
Then combine like terms: $(5 - 4) + 4n = 1 + 4n$.
Therefore, the simplified form is $4n + 1$.

Key Concept: Pattern Recognition, Number Machines

a) $\frac{a}{2} - b + 7$
[Solution Description]
The operations are:
1. Divide the first input by 2: $\frac{a}{2}$.
2. Subtract the second input: $\frac{a}{2} - b$.
3. Add 7: $\frac{a}{2} - b + 7$.
The final expression is $\frac{a}{2} - b + 7$.

Key Concept: Formula Detective

b) 11
[Solution Description] To find the output of the number machine, substitute $a = 7$ and $b = 3$ into the formula $2a - b$. First, multiply $2 \times 7 = 14$. Then subtract $b$: $14 - 3 = 11$. The output is $11$.