Question 14 Marks
Simplify : $\frac{1-2 x+x^2}{1-x^3} \times \frac{1+x+x^2}{1+x}$
Answer
View full question & answer→$\frac{1-2 x+x^2}{1-x^3} \times \frac{1+x+x^2}{1+x}$
Questions
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Question 24 Marks
Simplify : $\frac{a^3-27}{5 a^2-16 a+3} \div \frac{a^2+3 a+9}{25 a^2-1}$
Answer
View full question & answer→$\frac{a^3-27}{5 a^2-16 a+3} \div \frac{a^2+3 a+9}{25 a^2-1}$
Question 34 Marks
Simplify : $\frac{4 x^2-11 x+6}{16 x^2-9}$
Answer
View full question & answer→$\frac{4 x^2-11 x+6}{16 x^2-9}$
Question 44 Marks
Simplify : $\frac{3 x^2-x-2}{x^2-7 x+12} \div \frac{3 x^2-7 x-6}{x^2-4}$
Answer
View full question & answer→$\frac{3 x^2-x-2}{x^2-7 x+12} \div \frac{3 x^2-7 x-6}{x^2-4}$
Question 54 Marks
Simplify : $\frac{x^2-5 x-24}{(x+3)(x+8)} \times \frac{x^2-64}{(x-8)^2}$
Answer
View full question & answer→$\frac{x^2-5 x-24}{(x+3)(x+8)} \times \frac{x^2-64}{(x-8)^2}$
Question 64 Marks
Simplify : $\frac{8 x^3-27 y^3}{4 x^2-9 y^2}$
Answer
View full question & answer→$\frac{8 x^3-27 y^3}{4 x^2-9 y^2}$
Question 74 Marks
Simplify : $\frac{a^2+10 a+21}{a^2+6 a-7} \times \frac{a^2-1}{a+3}$
Answer
View full question & answer→$\frac{a^2+10 a+21}{a^2+6 a-7} \times \frac{a^2-1}{a+3}$
Question 84 Marks
Simplify : $\frac{m^2-n^2}{(m+n)^2} \times \frac{m^2+m n+n^2}{m^3-n^3}$
Answer
View full question & answer→$\begin{aligned} & \text {} \frac{m^2-n^2}{(m+n)^2} \times \frac{m^2+m n+n^2}{m^3-n^3} \\ & \frac{\mathrm{m}^2-\mathrm{n}^2}{(\mathrm{~m}+\mathrm{n})^2} \times \frac{\mathrm{m}^2+\mathrm{mn}+\mathrm{n}^2}{\mathrm{~m}^3-\mathrm{n}^3} \\ & =\frac{(\mathrm{m}+\mathrm{n})(\mathrm{m}-\mathrm{n})}{(\mathrm{m}+\mathrm{n})(\mathrm{m}+\mathrm{n})} \times \frac{\mathrm{m}^2+\mathrm{mn}+\mathrm{n}^2}{(\mathrm{~m}-\mathrm{n})\left(\mathrm{m}^2+\mathrm{mn}+\mathrm{n}^2\right)} \\ & =\frac{\mathbf{1}}{\mathbf{m}+\mathbf{n}}\end{aligned}$
Question 94 Marks
Simplify :
(3xy – 2ab)³ – (3xy + 2ab)³
Answer
View full question & answer→(3xy – 2ab)³ – (3xy + 2ab)³
Here, A = 3xy – 2ab and B = 3xy + 2ab
∴ (3xy – 2ab)³ – (3xy + 2ab)³
= [(3xy – 2ab) – (3xy + 2ab)] [(3xy – 2ab)² + (3xy – 2ab) (3xy + 2ab) + (3xy + 2ab)²]
…[∵ A³ – B³ = (A – B) (A² + AB + B²)]
= (3xy – 2ab – 3xy – 2ab) [(9x²y² – 12xyab + 4a²b²) + (9x²y² – 4a²b²) + (9x²y² + 12xyab + 4a²b²)]
= (- 4ab) (9x²y² + 9x²y² + 9x²y² – 12xyab + 12xyab + 4a²b² – 4a²b² + 4a²b²)
= (- 4ab) (27 xy² + 4a²b²)
= -108x²y²ab – 16a³b³
Here, A = 3xy – 2ab and B = 3xy + 2ab
∴ (3xy – 2ab)³ – (3xy + 2ab)³
= [(3xy – 2ab) – (3xy + 2ab)] [(3xy – 2ab)² + (3xy – 2ab) (3xy + 2ab) + (3xy + 2ab)²]
…[∵ A³ – B³ = (A – B) (A² + AB + B²)]
= (3xy – 2ab – 3xy – 2ab) [(9x²y² – 12xyab + 4a²b²) + (9x²y² – 4a²b²) + (9x²y² + 12xyab + 4a²b²)]
= (- 4ab) (9x²y² + 9x²y² + 9x²y² – 12xyab + 12xyab + 4a²b² – 4a²b² + 4a²b²)
= (- 4ab) (27 xy² + 4a²b²)
= -108x²y²ab – 16a³b³
Question 104 Marks
Simplify :
p³ – (p + 1)³
Answer
View full question & answer→p³ – (p + 1)³
= p³ – (p³ + 3p² + 3p + 1) …[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= p³ – p³ – 3p² – 3p – 1
= – 3p² – 3p – 1
= p³ – (p³ + 3p² + 3p + 1) …[∵ (a + b)³ = a³ + 3a²b + 3ab² + b³]
= p³ – p³ – 3p² – 3p – 1
= – 3p² – 3p – 1
Question 114 Marks
Simplify :
(a + b)³ – a³ – b³
Answer
View full question & answer→(a + b)³ – a³ – b³
= a³ + 3a²b + 3ab² + b³ – a³ – b³
= 3a²b + 3ab²
= a³ + 3a²b + 3ab² + b³ – a³ – b³
= 3a²b + 3ab²
Question 124 Marks
Simplify :
(3a + 5b)³ – (3a – 5b)³
Answer
View full question & answer→(3a + 5b)³ – (3a – 5b)³
Here, A = 3a + 5b and B = 3a – 5b
= [(3a + 5b) – (3a – 5b)] [(3a + 5b)² + (3a + 5b) (3a – 5b) + (3a – 5b)²]
…[∵ A³ – B³ = (A – B)(A² + AB + B²)]
= (3a + 5b – 3a + 5b) [(9a² + 30ab + 25b²) + (9a² – 25b²) + (9a² – 30ab + 25b²)]
= 10b (9a² + 9a² + 9a² + 30ab – 30ab + 25b² – 25b² + 25b²)
= 10b (27a² + 25b²)
= 270a²b + 250b³
Here, A = 3a + 5b and B = 3a – 5b
= [(3a + 5b) – (3a – 5b)] [(3a + 5b)² + (3a + 5b) (3a – 5b) + (3a – 5b)²]
…[∵ A³ – B³ = (A – B)(A² + AB + B²)]
= (3a + 5b – 3a + 5b) [(9a² + 30ab + 25b²) + (9a² – 25b²) + (9a² – 30ab + 25b²)]
= 10b (9a² + 9a² + 9a² + 30ab – 30ab + 25b² – 25b² + 25b²)
= 10b (27a² + 25b²)
= 270a²b + 250b³
Question 134 Marks
Simplify :
(x + y)³ – (x – y)³
Answer
View full question & answer→(x + y)³ – (x – y)³
Here, a = x + y and b = x – y
(x + y)³ – (x – y)³
= [(x + y) – (x – y)] [(x + y)² + (x + y) (x – y) + (x – y)]
…[a³ – b³ = (a – b)(a² + ab + b²)]
= (x + y – x + y) [(x² + 2xy + y²) + (x² – y²) + (x² – 2xy + y²)]
= 2y(x² + x² + x² + 2xy – 2xy + y² – y² + y²)
= 2y (3x² + y²)
= 6x²y + 2y³
Here, a = x + y and b = x – y
(x + y)³ – (x – y)³
= [(x + y) – (x – y)] [(x + y)² + (x + y) (x – y) + (x – y)]
…[a³ – b³ = (a – b)(a² + ab + b²)]
= (x + y – x + y) [(x² + 2xy + y²) + (x² – y²) + (x² – 2xy + y²)]
= 2y(x² + x² + x² + 2xy – 2xy + y² – y² + y²)
= 2y (3x² + y²)
= 6x²y + 2y³
Question 144 Marks
Factorize :
44x² – x – 3
Answer
View full question & answer→44x² – x – 3 44 × (-3) = -132
= 44x² – 12x + 11x – 3
= 4x(11x – 3) + 1(11x – 3)
= (11x – 3) (4x + 1)
= 44x² – 12x + 11x – 3
= 4x(11x – 3) + 1(11x – 3)
= (11x – 3) (4x + 1)
Question 154 Marks
Factorize :
20x² – 26x + 8
Answer
View full question & answer→20x² – 26x + 8
= 2(10x² – 13x + 4) 10 × 4 = 40
… [Taking out the common factor 2]
= 2(10x² – 8x – 5x + 4)
= 2[2x(5x – 4) – 1(5x – 4)]
= 2 (5x – 4) (2x – 1)
= 2(10x² – 13x + 4) 10 × 4 = 40
… [Taking out the common factor 2]
= 2(10x² – 8x – 5x + 4)
= 2[2x(5x – 4) – 1(5x – 4)]
= 2 (5x – 4) (2x – 1)
Question 164 Marks
Factorize :
2x² + x – 45
Answer
View full question & answer→2x² + x – 45 2 × (- 45) = -90
= 2x² + 10x – 9x – 45
= 2x(x + 5) – 9 (x + 5)
= (x + 5) (2x – 9)
= 2x² + 10x – 9x – 45
= 2x(x + 5) – 9 (x + 5)
= (x + 5) (2x – 9)
Question 174 Marks
Factorize :
3x² + 14x + 15
Answer
View full question & answer→3x² + 14x + 15 3 × 15 = 45
= 3x² + 9x + 5x + 15
= 3x(x + 3) + 5(x + 3)
= (x + 3) (3x + 5)
= 3x² + 9x + 5x + 15
= 3x(x + 3) + 5(x + 3)
= (x + 3) (3x + 5)
Question 184 Marks
Factorize :
m² – 25m + 100
Answer
View full question & answer→m² – 25m + 100
= m² – 20m – 5m + 100
= m(m – 20) – 5(m – 20)
= (m – 20) (m – 5)
= m² – 20m – 5m + 100
= m(m – 20) – 5(m – 20)
= (m – 20) (m – 5)
Question 194 Marks
Factorize :
m² – 23m + 120
Answer
View full question & answer→m² – 23m + 120
= m² – 15m – 8m + 120
= m (m – 15) – 8 (m – 15)
= (m – 15) (m – 8)
= m² – 15m – 8m + 120
= m (m – 15) – 8 (m – 15)
= (m – 15) (m – 8)
Question 204 Marks
Factorize :
p² – 7p – 44
Answer
View full question & answer→p² – 7p – 44
= p² – 11p + 4p – 44
= p(p – 11) + 4(p – 11)
= (p – 11)(p + 4)
= p² – 11p + 4p – 44
= p(p – 11) + 4(p – 11)
= (p – 11)(p + 4)
Question 214 Marks
Factorize :
p² – 2p – 35
Answer
View full question & answer→p² – 2p – 35
= p² – 7p + 5p – 35
= p(p – 7) + 5(p – 7)
= (p – 7)(p + 5)
= p² – 7p + 5p – 35
= p(p – 7) + 5(p – 7)
= (p – 7)(p + 5)
Question 224 Marks
Factorize :
5y² + 5y – 10
Answer
View full question & answer→5y² + 5y – 10
= 5(y² + y – 2)
… [Taking out the common factor 5]
= 5(y² + 2y – y – 2)
= 5[y(y + 2) – 1(y + 2)]
= 5 (p + 2)(y- 1)
= 5(y² + y – 2)
… [Taking out the common factor 5]
= 5(y² + 2y – y – 2)
= 5[y(y + 2) – 1(y + 2)]
= 5 (p + 2)(y- 1)
Question 234 Marks
Factorize :
y² + 24y + 144
Answer
View full question & answer→y² + 24y + 144
= y² + 12y + 12y + 144
= y(y + 12) + 12(y + 12)
= (y + 12)(y + 12)
= y² + 12y + 12y + 144
= y(y + 12) + 12(y + 12)
= (y + 12)(y + 12)
Question 244 Marks
Factorize :
x² – 10x + 9
Answer
View full question & answer→x² – 10x + 9
= x² – 9x – x + 9
= x (x – 9) – 1(x – 9)
= (x – 9)(x – 1)
= x² – 9x – x + 9
= x (x – 9) – 1(x – 9)
= (x – 9)(x – 1)
Question 254 Marks
Factorize :
x² + 9x + 18
Answer
View full question & answer→x² + 9x + 18
= x² + 6x + 3x + 18
= x (x + 6) + 3(x + 6)
= (x + 6) (x + 3)
= x² + 6x + 3x + 18
= x (x + 6) + 3(x + 6)
= (x + 6) (x + 3)