1. Problem 2: Find the unknown sides of rectangles given one side and area. (a) Given breadth $= (x - y)$ m and area $= (x^2 - y^2)$ m$^2$. Step 1: Recall area of rectangle $= \text{length} \times \text{breadth}$. Step 2: Substitute known values: $$ (x^2 - y^2) = \text{length} \times (x - y) $$ Step 3: Factorize $x^2 - y^2$ as $(x - y)(x + y)$. Step 4: So, $$ (x - y)(x + y) = \text{length} \times (x - y) $$ Step 5: Divide both sides by $(x - y)$ (assuming $x \neq y$): $$ \text{length} = x + y $$ Answer: Length $= x + y$ m. (b) Given length $= (2x + 5y)$ m and area $= (2x^2 + 3xy - 5y^2)$ m$^2$. Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let breadth $= b$. Step 3: So, $$ 2x^2 + 3xy - 5y^2 = (2x + 5y) \times b $$ Step 4: Solve for $b$ by dividing: $$ b = \frac{2x^2 + 3xy - 5y^2}{2x + 5y} $$ Step 5: Factor numerator: $$ 2x^2 + 3xy - 5y^2 = (2x - y)(x + 5y) $$ Step 6: So, $$ b = \frac{(2x - y)(x + 5y)}{2x + 5y} $$ No common factor with denominator, so breadth $= \frac{(2x - y)(x + 5y)}{2x + 5y}$ m. (c) Given breadth $= (2a - b)$ m and area $= (2a^2 + 5ab - 3b^2)$ m$^2$. Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let length $= l$. Step 3: So, $$ 2a^2 + 5ab - 3b^2 = l \times (2a - b) $$ Step 4: Factor numerator: $$ 2a^2 + 5ab - 3b^2 = (2a - b)(a + 3b) $$ Step 5: So, $$ l = \frac{(2a - b)(a + 3b)}{2a - b} = a + 3b $$ Answer: Length $= a + 3b$ m. (d) Given length $= (a + 3)$ m and area $= (a^2 + 27)$ m$^2$. Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let breadth $= b$. Step 3: So, $$ a^2 + 27 = (a + 3) \times b $$ Step 4: Solve for $b$: $$ b = \frac{a^2 + 27}{a + 3} $$ Step 5: Try to factor numerator: $a^2 + 27$ does not factor nicely over integers. Step 6: Use polynomial division: Divide $a^2 + 27$ by $a + 3$: $$ a^2 + 27 = (a + 3)(a - 3) + 36 $$ So, $$ b = a - 3 + \frac{36}{a + 3} $$ Breadth is $b = \frac{a^2 + 27}{a + 3}$ m. --- 2. Problem 3: Find unknown sides given area and one side. (a) Given area $= 35x^2 - xy - 12y^2$ cm$^2$, length $= (7x + 4)$ cm, breadth $= ?$ Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let breadth $= b$. Step 3: So, $$ 35x^2 - xy - 12y^2 = (7x + 4) \times b $$ Step 4: Factor area: Try to factor $35x^2 - xy - 12y^2$. Step 5: Factor by grouping or trial: $$ 35x^2 - xy - 12y^2 = (7x + 4y)(5x - 3y) $$ Step 6: So, $$ (7x + 4) \times b = (7x + 4y)(5x - 3y) $$ Since length is $7x + 4$, breadth is $5x - 3y$ (assuming $4y$ matches $4$ in length, likely a typo, but proceed with given). Answer: Breadth $= 5x - 3y$ cm. (b) Given area $= 2x^2 - 7x + 6$ cm$^2$, breadth $= (x - 2)$ cm, length $= ?$ Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let length $= l$. Step 3: So, $$ 2x^2 - 7x + 6 = l \times (x - 2) $$ Step 4: Factor numerator: Try to factor $2x^2 - 7x + 6$. Step 5: Factors of $2x^2 - 7x + 6$ are $(2x - 3)(x - 2)$. Step 6: So, $$ l = \frac{(2x - 3)(x - 2)}{x - 2} = 2x - 3 $$ Answer: Length $= 2x - 3$ cm. (c) Given area $= 8a^3 + 27$ cm$^2$, breadth $= (2a + 3)$ cm, length $= ?$ Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let length $= l$. Step 3: Recognize $8a^3 + 27$ as sum of cubes: $$ 8a^3 + 27 = (2a)^3 + 3^3 $$ Step 4: Factor sum of cubes: $$ (2a + 3)(4a^2 - 6a + 9) $$ Step 5: So, $$ l = \frac{8a^3 + 27}{2a + 3} = 4a^2 - 6a + 9 $$ Answer: Length $= 4a^2 - 6a + 9$ cm. (d) Given area $= a^2 - b^2$ cm$^2$, length $= (a^2 + b^2)$ cm, breadth $= ?$ Step 1: Area $= \text{length} \times \text{breadth}$. Step 2: Let breadth $= b$. Step 3: So, $$ a^2 - b^2 = (a^2 + b^2) \times b $$ Step 4: Solve for $b$: $$ b = \frac{a^2 - b^2}{a^2 + b^2} $$ Breadth is $b = \frac{a^2 - b^2}{a^2 + b^2}$ cm. --- 3. Problem 4: (a) Product of two expressions is $2a^2 + 13a + 24$. One expression is $(a + 8)$. Find the other. Step 1: Let other expression be $x$. Step 2: So, $$ (a + 8) \times x = 2a^2 + 13a + 24 $$ Step 3: Solve for $x$ by dividing: $$ x = \frac{2a^2 + 13a + 24}{a + 8} $$ Step 4: Factor numerator: Try to factor $2a^2 + 13a + 24$. Step 5: Factors are $(2a + 3)(a + 8)$. Step 6: So, $$ x = \frac{(2a + 3)(a + 8)}{a + 8} = 2a + 3 $$ Answer: Other expression is $2a + 3$. (b) Divide $(9x^2 - 4y^2)$ by $(3x^2 - 2y^2)$. Step 1: Recognize difference of squares: $$ 9x^2 - 4y^2 = (3x - 2y)(3x + 2y) $$ Step 2: Denominator is $3x^2 - 2y^2$, which does not factor nicely. Step 3: Perform polynomial division: Divide $9x^2 - 4y^2$ by $3x^2 - 2y^2$. Step 4: Divide leading terms: $$ \frac{9x^2}{3x^2} = 3 $$ Step 5: Multiply denominator by 3: $$ 3(3x^2 - 2y^2) = 9x^2 - 6y^2 $$ Step 6: Subtract: $$ (9x^2 - 4y^2) - (9x^2 - 6y^2) = 2y^2 $$ Step 7: Remainder is $2y^2$, so quotient is $3$ with remainder $2y^2$. Answer: Quotient $= 3 + \frac{2y^2}{3x^2 - 2y^2}$. --- Final answers summarized: 2(a) Length $= x + y$ 2(b) Breadth $= \frac{(2x - y)(x + 5y)}{2x + 5y}$ 2(c) Length $= a + 3b$ 2(d) Breadth $= \frac{a^2 + 27}{a + 3}$ 3(a) Breadth $= 5x - 3y$ 3(b) Length $= 2x - 3$ 3(c) Length $= 4a^2 - 6a + 9$ 3(d) Breadth $= \frac{a^2 - b^2}{a^2 + b^2}$ 4(a) Other expression $= 2a + 3$ 4(b) Quotient $= 3 + \frac{2y^2}{3x^2 - 2y^2}$