2 Marks Questions

Question 1012 Marks

Factorise:
4x² - 9y² - 2x - 3y

Answer

4x² - 9y² - 2x - 3y
= (2x)² - (3y)² - (2x + 3y) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= (2x + 3y)(2x - 3y) - (2x + 3y)
= (2x + 3y)(2x - 3y - 1)

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Question 1022 Marks

Factorise:
a - b - a² + b²

Answer

a - b - a² + b²
= (a - b) - (a² - b²)
= (a - b) - (a - b)(a + b) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= (a - b)(1 - a - b)

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Question 1032 Marks

Factorise:
4a² - 9b² - 2a - 3b

Answer

4a² - 9b² - 2a - 3b
= (2a)² - (3b)² - (2a + 3b)
= (2a - 3b)(2a + 3b) - (2a + 3b)
= (2a + 3b)(2a - 3b - 1)

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Question 1042 Marks

Factorise:
6 - x - x²

Answer

6 - x - x²
= 6 + 2x - 3x - x²
= 2(3 + x) - x(3 + x)
= (3 + x)(2 - x)

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Question 1052 Marks

Factorise:
x² + y² - z² - 2xy

Answer

x² + y² - z² - 2xy
= (x² + y² - 2xy) - z²
= (x - y)² - z²
= (x - y - z)(x - y + z)

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Question 1062 Marks

Factorise:
x⁵ + x²

Answer

x⁵ + x²
= x²(x³ + 1)
= x²(x + 1)[(x)² - x × 1 + (1)²] Since a³ + b³ = (a + b)(a² - a × b + b²)
= x²(x + 1)(x² - x + 1)

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Question 1072 Marks

Factorise:
x² + 2xy + y² - a² + 2ab - b²

Answer

x² + 2xy + y² - a² + 2ab - b²
= (x² + 2xy + y²) - (a² - 2ab + b²)
= (x + y)² - (a - b)²
= [(x + y) - (a - b)][(x + y) + (a - b)]
= (x + y - a + b)(x + y + a - b)

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Question 1082 Marks

Factorise:
a² + ab(b + 1) + b³

Answer

a² + ab(b + 1) + b³
= a² + ab² + ab + b³
= a² + ab + ab² + b³
= a(a + b) + b²(a + b)
= (a + b)(a + b²)

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Question 1092 Marks

Expand:
$\Big(\frac{1}{2}\text{a}-\frac{1}{4}\text{b}+2\Big)^2$

Answer

$\Big(\frac{1}{2}\text{a}-\frac{1}{4}\text{b}+2\Big)^2=\Big[\Big(\frac{\text{a}}{2}\Big)+\Big(-\frac{\text{b}}{4}\Big)+(2)\Big]^2$
$=\Big(\frac{\text{a}}{2}\Big)^2+\Big(-\frac{\text{b}}{4}\Big)^2+(2)^2\\+2\Big(\frac{\text{a}}{2}\Big)\times\Big(\frac{-\text{b}}{4}\Big)(2)+2\Big(\frac{\text{a}}{2}\Big)(2)$
$=\frac{\text{a}^2}{4}+\frac{\text{b}^2}{16}+4-\frac{\text{ab}}{4}-\text{b}+2\text{a}$

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Question 1102 Marks

Factorise:
3a⁷b - 81a⁴b⁴

Answer

3a⁷b - 81a⁴b⁴
= 3a⁴b(a³ - 27b³)
= 3a⁴b[(a)³ - (3b)³]
= 3a⁴b(a - 3b)[(a)² + a × 3b + (3b)²] Since a³ - b³ = (a - b)(a² + a × b + b²)
= 3a⁴b(a - 3b)(a² + 3ab + 9b²)

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Question 1112 Marks

Factorise:
9a² + 6a + 1 - 36b²

Answer

9a² + 6a + 1 - 36b²
= (9a² + 6a + 1) - 36b²
= [(3a)² + 2(3a)(1) + (1)²] - (6b)²
= (3a + 1)² - (6b)²
= (3a + 1 - 6b)(3a + 1 + 6b)

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Question 1122 Marks

Factorise:
8 - 27b³ - 343c³ - 126bc

Answer

8 - 27b³ - 343c³ - 126bc
= (2)³ + (-3b)³ + (-7c)³ - 3 × (2) × (-3b) × (-7c)
= [2 + (-3b) + (-7c)[(2)² + (-3b)² + (-7c)² - (2)(-3b) - (-3b)(-7c) - (2)(-7c)]
= (2 - 3b - 7c)(4 + 9b² + 49c² + 6b - 21bc + 14c)

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Question 1132 Marks

Factorise:
x³ - 3x²+ 3x + 7

Answer

x³ - 3x²+ 3x + 7
= x³ - 3x²+ 3x - 1 + 8
= (x³ - 3x²+ 3x - 1) + 8
= (x - 1)³ + 2³
= (x - 1 + 2)[(x - 1)² - (x - 1)(2) + 2²]
= (x + 1)(x² - 2x + 1 - 2x + 2 + 4)
= (x + 1)(x² - 4x + 7)

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Question 1142 Marks

Factorise:
(x + 2)³ - (x - 2)³

Answer

(x + 2)³ - (x - 2)³
= [(x + 2) - (x - 2)][(x + 2)² + (x + 2)(x - 2) + (x - 2)²]
= 4(x² + 4x + 4 + x² - 4 + x² - 4x + 4)
= 4(3x² + 4)

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Question 1152 Marks

Factorise:
x(x + y)³ - 3x²y(x + y)

Answer

x(x + y)³ - 3x²y(x + y)
= x(x + y)[(x + y)² - 3xy]
= x(x + y)(x² + y² + 2xy - 3xy)
= x(x + y)(x² + y² - xy)

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Question 1162 Marks

Factorise:
16x² + 4y² + 9z² - 16xy - 12yz + 24xz

Answer

We have:
16x² + 4y² + 9z² - 16xy - 12yz + 24xz
= (4x)² + (-2y)² + (3z)² + 2(4x)(-2y) + 2(-2y)(3z) + 2(3z)(4x)
= (4x - 2y + 3z)² [using a² + b² + c² + 2ab + 2bc + 2ca = (a + b + c)²]
Hence, 16x² + 4y² + 9z² - 16xy - 12yz + 24xz = (4x - 2y + 3z)²

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Question 1172 Marks

Factorise:
108a² - 3(b - c)²

Answer

108a² - 3(b - c)²
= 3[(36a² - (b -c)²]
= 3[(6a)² - (b - c)²] $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= 3(6a + b - c)(6a - b + c)

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Question 1182 Marks

Factorise:
1 + 2ab - (a² + b²)

Answer

1 + 2ab - (a² + b²)
= 1 - (a² + b² - 2ab)
= (1)² - (a - b)²
= [1 - (a - b)][1 + (a - b)]
= (1 - a + b)(1 + a - b)

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Question 1192 Marks

Factorise:
216 + 27b³ + 8c³ - 108bc

Answer

216 + 27b³ + 8c³ - 108bc
= (6)³ + (3b)³ + (2c)³ - 3 × 6 × 3b × 2c
= (6 + 3b + 2c)[6² + (3b)² + (2c)² - 6 × 3b - 3b × 2c - 2c × 6]
= (6 + 3b + 2c)(36 + 9b² + 4c² - 18b - 6bc - 12c)

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Question 1202 Marks

Factorise:
x⁴y⁴ - xy

Answer

x⁴y⁴ - xy
= xy(x³y³ - 1)
= xy[(xy)³ - (1)³]
= xy{(xy - 1)[(xy)² + (xy)(1) + (1)²]}
= xy(xy - 1)(x²y² + xy + 1)

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Question 1212 Marks

Factorise:
x² - y² + 6y - 9

Answer

x² - y² + 6y - 9
= x² - (y² - 6y + 9)
= x² - (y² - 2 × y × 3 + 3²)
= x² - (y - 3)² $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= [x + (y - 3)][x - (y - 3)]
= (x + y - 3)(x - y + 3)

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Question 1222 Marks

Expand:
(3x + 2)³

Answer

(3x + 2)³
= (3x)³ + 3 × (3x)²x² + 3 × 3x × (2)² + (2)³
= 27x³ + 54x² + 36x + 8

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Question 1232 Marks

Factorise:
$\text{x}^2-\sqrt{3}\text{x}-6$

Answer

$\text{x}^2-\sqrt{3}\text{x}-6$
$=\text{x}^2-2\sqrt{3}\text{x}+\sqrt{3}\text{x}-6$
$=\text{x}(\text{x}-2\sqrt{3})+\sqrt{3}(\text{x}-2\sqrt{3})$
$=(\text{x}-2\sqrt{3})(\text{x}+\sqrt{3})$

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Question 1242 Marks

Factorise:
(a + 2b)² + 101(a + 2b) + 100

Answer

Given equation: (a + 2b)² + 101(a + 2b) + 100
Let (a + 2b) = x
Then, we have
x² + 101x + 100
= x² + 100x + x + 100
= x(x + 100) + 1(x + 100)
= (x + 100)(x + 1)
= (a + 2b + 100)(a + 2b + 1)

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Question 1252 Marks

Factorise:
1 + b³ + 8c³ - 6bc

Answer

1 + b³ + 8c³ - 6bc
= (1)³ + (b)³ + (2c)³ - 3 × 1 × b × 2c
= (1 + b + 2c)[1² + b² + (2c)² - 1 × b - b × 2c - 1 × 2c]
= (1 + b + 2c)(1² + b² + 4c² - b - 2bc - 2c)

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Question 1262 Marks

Factorise:
16x⁴ - 1

Answer

16x⁴ - 1
= (4x²)² - (1)²
= (4x² - 1)(4x² + 1)
= [(2x)² - (1)²](4x² + 1)
= (2x - 1)(2x + 1)(4x² + 1)

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Question 1272 Marks

Factorise:
16x⁴ + 54x

Answer

16x⁴ + 54x
= 2x(8x 3 + 27)
= 2x[(2x)³ + (3)³] Since a³ + b³ = (a + b)(a² - a × b + b²)
= 2x(2x + 3)[(2x)² - 2x × 3 + 3²]
= 2x(2x+3)(4x² -6x +9)

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Question 1282 Marks

Factorise:
4a² - 4b² + 4a + 1

Answer

4a² - 4b² + 4a + 1
= (4a² + 4a + 1) - 4b²
= [(2a)² + 2 × 2a × 1 + (1)²] - (2b)²
= (2a + 1)² - (2b)²
= (2a + 1 - 2b)(2a + 1 + 2b)
= (2a - 2b + 1)(2a + 2b + 1)

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Question 1292 Marks

Factorise:
$\text{x}^2+2\sqrt{3}\text{x}-24$

Answer

$\text{x}^2+2\sqrt{3}\text{x}-24$
$=\text{x}^2+4\sqrt{3}\text{x}-2\sqrt{3}\text{x}-24$
$=\text{x}(\text{x}+4\sqrt{3})-2\sqrt{3}(\text{x}+4\sqrt{3})$
$=(\text{x}+4\sqrt{3})(\text{x}-2\sqrt{3})$

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Question 1302 Marks

Expand:
(a - 2b - 3c)²

Answer

(a - 2b - 3c)² = [a + (-2b) + (-3c)]²
=(a)²+ (-2b)² + (-3c)² + 2(a)(-2b) + 2(-2b)(-3c) + 2(a)(-3c)
=a² + 4b² + 9c² - 4ab + 12bc - 6ac

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Question 1312 Marks

Factorise:
a³ + 3a²b + 3ab² + b³ - 8

Answer

a³ + 3a²b + 3ab² + b³ - 8
= (a + b)³ - 2³
=[(a + b) - 2][(a + b)² + (a + b)2 + 2²]
=(a + b - 2)[(a + b)² + 2(a + b) + 4]

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Question 1322 Marks

Factorise:
x³ - 512

Answer

x³ - 512
= (x)³ - (8)³
= (x - 8)[(x)² + x × 8 + (8)²] Since a³ - b³ = (a - b)(a² + a × b + b²)
= (x - 8)(x² + 8x + 64)
= x³ + 8x² + 64x - 8x² - 64x - 512
= x³ - 512

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Question 1332 Marks

Factorise:
x³ - x² + ax + x - a - 1

Answer

x³ - x² + ax + x - a - 1
= x³ - x² + ax - a + x - 1
= x²(x - 1) + a(x - 1) + 1(x - 1)
= (x - 1)(x² + a + 1)

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Question 1342 Marks

Factorise:
$216\text{x}^3+\frac{1}{125}$

Answer

$216\text{x}^3+\frac{1}{125}$
We know that:
Since a² + b³ = (a + b)(a² - a × b + b²)
Let us rewrite
$216\text{x}^3+\frac{1}{125}$
$=(6\text{x})^3+\Big(\frac{1}{5}\Big)^3$
$=\Big(6\text{x}+\frac{1}{5}\Big)\bigg[(6\text{x})^2-6\text{x}\times\frac{1}{5}+\Big(\frac{1}{5}\Big)^2\bigg]$
$=\Big(6\text{x}+\frac{1}{5}\Big)\Big(36\text{x}^2-\frac{6\text{x}}{5}+\frac{1}{25}\Big)$

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Question 1352 Marks

Factorise:
x² + 20x - 69

Answer

x² + 20x - 69
= x² + 23x - 3x - 69
= x(x + 23) - 3(x + 23)
= (x + 23)(x - 3)

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Question 1362 Marks

Factorise:
15x² - x - 28

Answer

15x² - x - 28
= 15x² + 20x - 21x - 28
= 5x(3x + 4) - 7(3x + 4)
= (3x + 4)(5x - 7)

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Question 1372 Marks

Factorise:
a³ + 8b³ + 64c³ - 24abc

Answer

a³ + 8b³ + 64c³ - 24abc
= a³ + (2b)³ + (4c)³ - 3 × a × 2b × 4c
= (a + 2b + 4c)[a² + (2b)² + (4c)² - a × 2b - 2b × 4c - 4c × a]
= (a + 2b + 4c)(a² + 4b² + 16c² - 2ab - 8bc - 4ca)

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Question 1382 Marks

Evaluate:
(99)²

Answer

(99)² = (100 - 1)²
= [(100) + (-1)]²
= (100)² + 2 × (100) × (-1) + (-1)²
= 10000 - 200 + 1
= 9801

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Question 1392 Marks

Factorise:
$8\text{x}^3-\frac{1}{27\text{y}^3}$

Answer

We know that:
Since a³ - b³ = (a - b)(a² + a × b + b²)
Let us rewrite
$8\text{x}^3-\frac{1}{27\text{y}^3}$
$=(2\text{x})^3-\Big(\frac{1}{3\text{y}}\Big)^3$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\bigg[(2\text{x})^2+2\text{x}\times\frac{1}{3\text{y}}+\Big(\frac{1}{3\text{y}}\Big)^2\bigg]$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\Big(4\text{x}^2+\frac{2\text{x}}{3\text{y}}+\frac{1}{9\text{y}^2}\Big)$

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Question 1402 Marks

Factorise:
32x⁴ - 500x

Answer

32x⁴ - 500x
= 4x(8x³ - 125)
= 4x[(2x)³ - (5)³]
= 4x[(2x - 5)[(2x)² + 2x × 5 + (5)²] Since a³ - b³ = (a - b)(a² + a × b + b²)
= 4x(2x - 5)(4x² + 10x + 25)

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Question 1412 Marks

Factorise:
x² - (a + b)x + ab

Answer

x² - (a + b)x + ab
= x² - ax - bx + ab
= x(x - a) - b(x - a)
= (x - a)(x - b)

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Question 1422 Marks

Factorise:
5x² - 16x - 21

Answer

5x² - 16x - 21
= 5x² + 5x - 21x - 21
= 5x(x + 1) -21(x + 1)
= (x + 1)(5x - 21)

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Question 1432 Marks

Factorise:
x² - 26x + 133

Answer

x² - 26x + 133
= x² - 19x - 7x + 133
= x(x - 19) - 7(x - 19)
= (x - 19)(x - 7)

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Question 1442 Marks

Factorise:
x² - 22x + 120

Answer

x² - 22x + 120
= x² - 10x - 12x + 120
= x(x - 10) - 12(x - 10)
= (x - 10)(x - 12)

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Question 1452 Marks

Factorise:

Evaluate {(999)² - 1}

Answer

{(999)² - 1}
= {(999)² - (1)²}
= {(999 - 1)(999 + 1)}
= 998 × 1000
= 998000

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Question 1462 Marks

Factorise:
6x² - 5x - 21

Answer

6x² - 5x - 21
= 6x² + 9x - 14x - 21
= 3x(2x + 3) - 7(2x + 3)
= (3x - 7)(2x + 3)

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Question 1472 Marks

Factorise:
4(a + b) - 6(a + b)²

Answer

4(a + b) - 6(a + b)²
= (a + b)[4 - 6(a + b)]
= 2(a + b)(2 - 3a - 3b)
= 2(a + b)(2 - 3a - 3b)

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Question 1482 Marks

Factorise:
9x² - 3x - 20

Answer

9x² - 3x - 20
= 9x² - 15x + 12x - 20
= 3x(3x - 5) + 4(3x - 5)
= (3x - 5)(3x + 4)

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Question 1492 Marks

Factorise:
$\text{x}^2+\frac{12}{35}\text{x}+\frac{1}{35}$

Answer

$\text{x}^2+\frac{12}{35}\text{x}+\frac{1}{35}$
$=\text{x}^2+\frac{5\text{x}}{35}+\frac{\text{x}}{5}+\frac{1}{35}$
$=5\text{x}\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)+1\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)$
$=(5\text{x}+1)\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)$

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Question 1502 Marks

Factorise:
x² - 24x - 180

Answer

x² - 24x - 180
= x² - 30x + 6x - 180
= x(x - 30) + 6(x - 30)
= (x - 30)(x + 6)

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Maths 2 Marks Questions