2 Marks Questions

Question 12 Marks

Factorise:
21x² + 5x - 6

Answer

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

Question 22 Marks

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

Answer

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

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Question 32 Marks

Factorise:
x² - 4x + 3

Answer

x² - 4x + 3
= x² - 3x - x + 3
= x(x - 3) - 1(x - 3)
= (x - 3)(x - 1)

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Question 42 Marks

Factorise:
x² + 19x - 150

Answer

x² + 19x - 150
= x² + 25x - 6x - 150
= x(x + 25) - 6(x + 25)
= (x + 25)(x - 6)

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Question 52 Marks

Factorise:
2x² - 7x - 15

Answer

2x² - 7x - 15
= 2x² - 10x + 3x - 15
= 2x(x - 5) + 3(x - 5)
= (x - 5)(2x + 3)

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Question 62 Marks

Factorise:
40 + 3x - x²

Answer

40 + 3x - x²
= 40 + 8x - 5x - x²
= 8(5 + x) - x(5 + x)
= (5 + x)(8 - x)

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Question 72 Marks

Expand:
(2a + 5b + 7c)²

Answer

(2a + 5b + 7c)² = [(2a) + (-5b) + (-7c)]²

= (2a)² + (-5b)² + (-7c)² + 2(2a)(-5b) + 2(-5b)(-7c) + 2(2a)(-7c)

= 4a² + 25b² + 49c² - 20ab + 70bc - 28ac

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Question 82 Marks

Find the product.
(x - 2y + 3)(x² + 4y² + 2xy + 6y - 3x + 9)

Answer

(x - 2y + 3)(x² + 4y² + 2xy + 6y - 3x + 9)

= (x - 2y + 3)(x² + 4y² + 9 + 2xy + 6y - 3x)

= [x + (-2y) + 3][x² + (-2y)² + (3)² - x × (-2y) - (-2y) × 3 - 3 × x]

= (x)³ + (-2y)³ + (3)³ - 3(x)(-2y)(3)

= x³ - 8y³ + 27 + 18xy

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Question 92 Marks

Expand:

(5a - 3b)³

Answer

(5a - 3b)³

= (5a)³ - (3b)³ - 3(5a)²(3b) + 3(5a)(3b)²

= 125a³ - 27b³ - 225a²b + 135ab²

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Question 102 Marks

Expand:

$\Big(3\text{x}-\frac{5}{\text{x}}\Big)^3$

Answer

$\Big(3\text{x}-\frac{5}{\text{x}}\Big)^3$
$=(3\text{x})^3-\Big(\frac{5}{\text{x}}\Big)^3-3(3\text{x})^2\Big(\frac{5}{\text{x}}\Big)+3(3\text{x})\Big(\frac{5}{\text{x}}\Big)^2$
$=27\text{x}^3-\frac{125}{\text{x}^3}-135\text{x}+\frac{225}{\text{x}}$

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Question 112 Marks

Expand:

$\Big(\frac{4}{5}\text{a}-2\Big)^3$

Answer

$\Big(\frac{4}{5}\text{a}-2\Big)^3$
$\Big(\frac{4}{5}\text{a}\Big)^3-(2)^3-3\Big(\frac{4}{5}\text{a}\Big)^2(2)+3\Big(\frac{4}{5}\text{a}\Big)(2)^2$
$=\frac{64}{125}\text{a}^3-8-\frac{96}{25}\text{a}^2+\frac{48}{5}\text{a}$

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Question 122 Marks

Factorise:
2x² + 11x - 21

Answer

2x² + 11x - 21
= 2x² + 14x - 3x - 21
= 2x(x + 7) - 3(x + 7)
= (x + 7)(2x - 3)

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Question 132 Marks

Factorise:
8x²y³ - x⁵

Answer

8x²y³ - x⁵
= x²(8y³ - x³)
= x²[(2y)³- x³]
= x²[(2y - x)[(2y)² + (2y)(x) + x²]
= x²(2y - x)(4y² + 2xy + x²)

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Question 142 Marks

Factorise:
6x² + 17x + 12

Answer

6x² + 17x + 12
= 6x² + 9x + 8x + 12
= 3x(2x + 3) + 4(2x + 3)
= (2x + 3)(3x + 4)

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Question 152 Marks

Factorise:
$2\sqrt{3}\text{x}^2+\text{x}-5\sqrt{3}$

Answer

$2\sqrt{3}\text{x}^2+\text{x}-5\sqrt{3}$
$=2\sqrt{3}\text{x}^2+6\text{x}-5\text{x}-5\sqrt{3}$
$=2\sqrt{3}\text{x}\big(\text{x}+\sqrt{3}\big)-5\big(\text{x}+\sqrt{3}\Big)$
$=\big(\text{x}+\sqrt{3}\big)\big(2\sqrt{3}\text{x}-5\big)$

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Question 162 Marks

Factorise:
a² - b² - 4ac + 4c²

Answer

a² - 4ac + 4c² - b²
= a² - 4ac + 4c² - b²
= a² - 2 × a × 2c + (2c)² - b²
= (a - 2c)² - b² $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= (a - 2c + b)(a - 2c - b)

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Question 172 Marks

Factorise:
x² + 7x - 98

Answer

x² + 7x - 98
= x² + 14x - 7x - 98
= x(x + 14) - 7(x + 14)
= (x + 14)(x - 7)

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Question 182 Marks

Factorise:
(3a - 2b)³ + (2b - 5c)³ + (5c - 3a)³

Answer

Put (3a - 2b) = x, (2b - 5c) = y and (5c - 3a) = z.
We have:
x + y + z = 3a - 2b + 2b - 5c + 5c - 3a = 0
Now,
(3a - 2b)³ + (2b - 5c)³ + (5c - 3a)³ = x³ + y³ + z³
= 3xyz [Here, x + y + z = 0. So, x³ + y³ + z³]
= (3a - 2b)(2b - 5c)(5c - 3a)

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Question 192 Marks

Factorise:
2x⁴ - 32

Answer

2x⁴ - 32
= 2(x⁴ - 16)
= 2[(x²)² - (4)²]
= 2[(x² - 4)(x² + 4)]
= 2[(x² - 2²)(x² + 4)]
= 2[(x - 2)(x + 2)(x² + 4)]
= 2(x - 2)(x + 2)(x² + 4)

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Question 202 Marks

Factorise:
25x²+ 4y² + 9z² - 20xy - 12yz + 30xz.

Answer

We have:
25x² + 4y² + 9z² - 20xy - 12yz + 30xz
= (5x)² + (-2y)² + (3z)² + 2(5x)(-2y) + 2(-2y)(3z) + 2(3z)(5x)
= [(5x) + (-2y) + (3z)]²
= (5x - 2y + 3z)²

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Question 212 Marks

Expand:
$\Big(3\text{a}+\frac{1}{4\text{b}}\Big)^3$

Answer

$\Big(3\text{a}+\frac{1}{4\text{b}}\Big)^3$
$=(3\text{a})^3+\Big(\frac{1}{4\text{b}}\Big)^3+3(3\text{a})^2\Big(\frac{1}{4\text{b}}\Big)+3(3\text{a})\Big(\frac{1}{4\text{b}}\Big)^2$
$=27\text{a}^3+\frac{1}{64\text{b}^3}+\frac{27\text{a}^2}{4\text{b}}+\frac{9\text{a}}{16\text{b}^2}$

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Question 222 Marks

Factorise:
64a³ - 343

Answer

64a³ - 343
= (4a)³ - (7)³
= (4a - 7)[(4a)² + (4a)(7) + (7)²]
= (4a - 7)(16a² + 28a + 49)
= 64 + 112 + 196a - 112 - 196a - 343
= 64a³ - 343

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Question 232 Marks

Factorise:
18x² + 3x - 10

Answer

18x² + 3x - 10
= 18x² - 12x + 15x - 10
= 6x(3x - 2) + 5(3x - 2)
= (6x + 5)(3x - 2)

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Question 242 Marks

Factorise:
a²- b² + 2bc - c²

Answer

a²- b² + 2bc - c²
= a² - (b² - 2bc + c²)
= a² - (b - c)²
= [a - (b - c)][a + (b - c)]
= (a - b + c)(a + b - c)

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Question 252 Marks

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

Answer

We have:
9x² + 16y² + 4z² - 24xy + 16yz - 12xz
= (2x)² + (3y)² + (-4z)² + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x)
= [(2x) + (3y) + (-4z)]²
= (2x + 3y - 4z)²

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Question 262 Marks

Factorise:
$\text{x}^2+7\sqrt{6}+60$

Answer

$\text{x}^2+7\sqrt{6}+60$
$=\text{x}^2+2\sqrt{6}\text{x}+5\sqrt{6}\text{x}+60$
$=\text{x}(\text{x}+2\sqrt{6})+5\sqrt{6}(\text{x}+2\sqrt{6})$
$=(\text{x}+2\sqrt{6})(\text{x}+5\sqrt{6})$

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Question 272 Marks

Factorise:
$\frac{3}{5}\text{x}^2-\frac{19}{5}\text{x}+4$

Answer

$\frac{3}{5}\text{x}^2-\frac{19}{5}\text{x}+4$
$=\frac{3}{5}\text{x}^2+\frac{4}{5}\text{x}-3\text{x}+4$
$=\frac{\text{x}}{5}(3\text{x}-4)-1(3\text{x}-4)$
$=\Big(\frac{\text{x}}{5}-1\Big)(3\text{x}-4)$

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Question 282 Marks

Factorise:
$\text{x}^2-2\sqrt{2}\text{x}-30$

Answer

$\text{x}^2-2\sqrt{2}\text{x}-30$
$=\text{x}^2-5\sqrt{2}\text{x}+3\sqrt{2}\text{x}-30$
$=\text{x}(\text{x}+5\sqrt{2})-3\sqrt{2}(\text{x}+5\sqrt{2})$
$=(\text{x}-5\sqrt{2})(\text{x}+3\sqrt{2})$

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Question 292 Marks

Factorise:
2a³ + 16b³ - 5a - 10b

Answer

2a³ + 16b³ - 5a - 10b
= 2(a³ + 8b³) - 5(a + 2b)
= 2[(a)³ + (2b)³] - 5(a + 2b) Since a³ + b³ = (a + b (a² - a × b + b²)
= 2(a + 2b)[(a)² - a × 2b + (2b)²] - 5(a + 2b)
= (a + 2b)[2(a² - 2ab + 4b²) - 5]

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Question 302 Marks

Evaluate:
(995)²

Answer

(995)² = (1000 - 5)²
= [(1000) + (-5)]²
= (1000)² + 2 × (1000) × (-5) + (-5)²
= 1000000 - 10000 + 25
= 990025

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Question 312 Marks

Factorise:
x⁶ - 7x³ - 8

Answer

Given equation is x⁶ - 7x³ - 8.
Putting x³ = y, we get
y² - 7y - 8
= y² - 8y + y - 8
= y(y - 8) + 1(y - 8)
= (y - 8)(y + 1)
= (x³ - 8)(x³ + 1)
= (x³ - 2³)(x³ + 1³)
= (x - 2)(x² + 2x + 4)(x + 1)(x² - x + 1)
= (x - 2)(x + 1)(x² + 2x + 4)(x² - x + 1)

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Question 322 Marks

Factorise:
$7\text{x}^2+2\sqrt{14}\text{x}+2$

Answer

$7\text{x}^2+2\sqrt{14}\text{x}+2$
$=7\text{x}^2+\sqrt{2}\big(\sqrt{7}\text{x}\big)+\sqrt{2}\big(\sqrt{7}\text{x}\big)+2$
$=\sqrt{7}\text{x}\big(\sqrt{7}\text{x}+\sqrt{2}\big)+\sqrt{2}\big(\sqrt{7}\text{x}+\sqrt{2}\big)$
$=\big(\sqrt{7}\text{x}+\sqrt{2}\big)\big(\sqrt{7}\text{x}+\sqrt{2}\big)=\big(\sqrt{7}\text{x}+\sqrt{2}\big)^2$

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Question 332 Marks

Factorise:
$\frac{\text{x}^3}{216}-8\text{y}^3$

Answer

$\frac{\text{x}^3}{216}-8\text{y}^3$
$=\Big(\frac{\text{x}}{6}\Big)^3-(2\text{y})^3$
$=\Big(\frac{\text{x}}{6}-2\text{y}\Big)\bigg[\Big(\frac{\text{x}}{6}\Big)^2+\Big(\frac{\text{x}}{6}\Big)(2\text{y})+(2\text{y})^2\bigg]$
$=\Big(\frac{\text{x}}{6}-2\text{y}\Big)\Big(\frac{\text{x}^2}{36}+\frac{\text{xy}}{3}+4\text{y}^2\Big)$

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Question 342 Marks

Factorise:
2x² + 3x - 90

Answer

2x² + 3x - 90
= 2x² - 12x + 15x - 90
= 2x(x - 6) + 15(x - 6)
= (x - 6)(2x + 15)

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Question 352 Marks

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

Answer

$\sqrt{2}\text{x}^2+3\text{x}+\sqrt{2}$
$=\sqrt{2}\text{x}^2+\text{x}+2\text{x}+\sqrt{2}$
$=\text{x}\big(\sqrt{2}\text{x}+1\big)+\sqrt{2}\big(\sqrt{2}\text{x}+1\big)$
$=\big(\sqrt{2}\text{x}+1\big)\big(\text{x}+\sqrt{2}\big)$

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Question 362 Marks

Factorise:
15x² + 2x - 8

Answer

15x² + 2x - 8
= 15x² - 10x + 12x - 8
= 5x(3x - 2) + 4(3x - 2)
= (3x - 2)(5x + 4)

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Question 372 Marks

Factorise:
(a + b)³ - (a - b)³

Answer

We know that, Since a³ - b³ = (a - b)(a² + a × b + b²)
Therefore,
(a + b)³ - (a - b)³
= [a + b - (a - b)][(a + b)² + (a + b)(a - b) + (a - b)²
= (a + b - a + b)[a² + b² + 2ab + a² - b² + a² + b² - 2ab]
= 2b(3a² + b²)

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Question 382 Marks

Factorise:
x² + 18x + 32

Answer

x² + 18x + 32
= x² + 16x + 2x + 32
= x(x + 16) + 2(x + 16)
= (x + 16)(x + 2)

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Question 392 Marks

Factorise:
8(x + y)³ - 27(x - y)³

Answer

8(x + y)³ - 27(x - y)³
= [2³(x + y)³] - [3³ (x - y)³]
= [2(x + y) - 3(x - y)]{[2(x + y)]² + 2(x + y)3(x - y) + [3(x - y)]²}
= (2x + 2y - 3x + 3y){[4(x² + y² + 2xy)] + 6(x² - y²) + [9(x² + y² - 2xy]}
= (-x + 5y){4x² + 4y² + 8xy + 6x² - 6y² + 9x² + 9y² - 18xy}
= (-x + 5y)(19x² + 7y² - 10xy)

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Question 402 Marks

Factorise:
(2a + 1)³ + (a - 1)³

Answer

(2a + 1)³ + (a - 1)³
= (2a + 1 + a - 1)[(2a + 1)² - (2a + 1)(a - 1) + (a - 1)²]
= (3a)[4a² + 4a + 1 - 2a² + 2a - a + 1 + a² - 2a + 1]
= 3a(3a² + 3a + 3)
= 9a(a² + a + 1)

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Question 412 Marks

Factorise:
ab(x² + 1) + x(a² + b²)

Answer

ab(x² + 1) + x(a² + b²)
= abx² + ab + a²x + b²x
= abx² + a²x + ab + b²x
= ax(bx + a) + b(bx + a)
= (bx + a)(ax + b)

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Question 422 Marks

Factorise:
$\text{x}^2+\frac{1}{\text{x}^2}-2-3\text{x}+\frac{3}{\text{x}}$

Answer

$\text{x}^2+\frac{1}{\text{x}^2}-2-3\text{x}+\frac{3}{\text{x}}$
$=\Big(\text{x}-\frac{1}{\text{x}}\Big)^2-3\Big(\text{x}-\frac{1}{\text{x}}\Big)$
$=\Big(\text{x}-\frac{1}{\text{x}}\Big)\Big(\text{x}-\frac{1}{\text{x}}-3\Big)$

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Question 432 Marks

Factorise:
$\text{x}^2-3\sqrt{5}\text{x}-20$

Answer

$\text{x}^2-3\sqrt{5}\text{x}-20$
$=\text{x}^2-4\sqrt{5}\text{x}+\sqrt{5}\text{x}-20$
$=\text{x}(\text{x}-4\sqrt{5})+\sqrt{5}(\text{x}-4\sqrt{5})$
$=\text{x}(\text{x}-4\sqrt{5})(\text{x}+\sqrt{5})$

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Question 442 Marks

Factorise:
1 - 27a³

Answer

1 - 27a³
= (1)³ - (3a)³
= (1 - 3a)[(1)² + 1 × 3a + (3a)²] Since a³ - b³ = (a - b)(a² + a × b + b²)
= (1 - 3a)(1 + 3a + 9a²)

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Question 452 Marks

Factorise:
$2\text{x}^2-\text{x}+\frac{1}{8}$

Answer

$2\text{x}^2-\text{x}+\frac{1}{8}$
$=2\text{x}^2-\frac{1}{2}\text{x}-\frac{1}{2}\text{x}+\frac{1}{8}$
$=\frac{\text{x}}{2}(4\text{x}-1)-\frac{1}{8}(4\text{x}-1)$
$=\Big(\frac{\text{x}}{2}-\frac{1}{8}\Big)(4\text{x}-1)$

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Question 462 Marks

Factorise:
ab(x² + y²) - xy(a² + b²)

Answer

ab(x² + y²) - xy(a² + b²)
= abx² + aby² - a²xy - b²xy
= abx² - a²xy + aby² - b²xy
= ax(bx - ay) + by(ay - bx)
= (bx - ay)(ax - by)

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Question 472 Marks

Factorise:
x² – 32x - 105

Answer

x² – 32x - 105
= x² – 35x + 3x - 105
= x (x - 35) + 3(x - 35)
= (x - 35)(x + 3)

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Question 482 Marks

Factorise:
$\text{x}^2+6\sqrt{6}\text{x}+48$

Answer

$\text{x}^2+6\sqrt{6}\text{x}+48$
$=\text{x}^2+4\sqrt{6}\text{x}+2\sqrt{6}\text{x}+48$
$=\text{x}(\text{x}+4\sqrt{6})+2\sqrt{6}(\text{x}+4\sqrt{6})$
$=(\text{x}+4\sqrt{6})(\text{x}+2\sqrt{6})$

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Question 492 Marks

Factorise:
$\text{x}^2-2+\frac{1}{\text{x}^2}-\text{y}^2$

Answer

$\text{x}^2-2+\frac{1}{\text{x}^2}-\text{y}^2$
$=\Big(\text{x}^2-2(\text{x}^2)\Big(\frac{1}{\text{x}^2}\Big)+\frac{1}{\text{x}^2}\Big)-\text{y}^2$
$=\Big(\text{x}-\frac{1}{\text{x}}\Big)^2-\text{y}^2$
$=\Big(\text{x}-\frac{1}{\text{x}}+\text{y}\Big)\Big(\text{x}-\frac{1}{\text{x}}-\text{y}\Big)$

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Question 502 Marks

Factorise:
7a³ + 56b³

Answer

7a³ + 56b³
= 7(a³ + 8b³)
= 7[(a)³ + (2b)³]
= 7(a + 2b)[a² - a × 2b + (2b)²] Since a³ + b³ = (a + b)(a² - a × b + b²)
= 7(a + 2b)(a² - 2ab + 4b²)

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