[5 marks sum]

Question 15 Marks

Factorise$: 5x^2 - y^2 - 4xy + 3x - 3y$

Answer

$5x^2 - y^2 - 4xy + 3x - 3y$
$= x^2 + 4x^2 - y^2 - 4xy + 3x - 3y$
$= (x^2 - y^2) + (4x^2 - 4xy) + (3x - 3y)$
$= (x - y)(x + y) + 4x (x - y) + 3(x - y)$
$= (x - y)[(x + y) + 4x + 3]$
$= (x - y)(x + y + 4x + 3)$
$= (x - y)(5x + y + 3).$

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Question 25 Marks

Factorise the following:$\ (t^2 - t)(4t^2 - 4t - 5) - 6$

Answer

$(t^2 - t)(4t^2 - 4t - 5) - 6$
$= (t^2 - t)[4(t^2 - t) - 5] - 6$
$= a[4a - 5] - 6[$Taking $(t^2 - t) = a]$
$= 4a^2 - 5a - 6$
$= 4a^2 - 8a + 3a - 6$
$= 4a(a - 2) + 3(a - 2)$
$= (a - 2)(4a + 3)$
$= (t^2 - t - 2)[4(t^2 - t) + 3]$
$= (t^2 - 2t + t - 2)(4t^2 - 4t + 3)$
$= [t(t - 2) + 1(t - 2)](4t^2 - 4t + 3)$
$= [(t - 2)(t + 1)](4t^2 - 4t + 3)$
$= (t + 1)(t - 2)(4t^2 - 4t + 3).$

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Question 35 Marks

Factorise the following:$\ (y^2 - 3y)(y^2 - 3y + 7) + 10$

Answer

$(y^2 - 3y)(y^2 - 3y + 7) + 10$
$= a(a + 7) + 10[$taking $(y^2 - 3y) = a]$
$= a^2 + 7a + 10$
$= a^2+ 5a + 2a + 10$
$= a(a + 5) + 2(a + 5)$
$= (a + 5)(a + 2)$
$= (y^2 - 3y + 5)(y^2 - 3y + 2)$
$= (y^2 - 3y + 5)(y^2 - 2y - y + 2)$
$= (y^2 - 3y + 5)[y(y - 2) - 1(y - 2)]$
$= (y^2 - 3y + 5)[(y - 2)(y - 1)]$
$= (y - 1)(y - 2)(y^2 - 3y + 5).$

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Question 45 Marks

Factorise the following:$\ (p^2 + p)^2 - 8(p^2 + p) + 12$

Answer

$(p^2 + p)^2 - 8(p^2 + p) + 12$
$= (p^2 + p)^2 - 6(p^2 + p) - 2(p^2 + p) + 12$
$= (p^2 + p)(p^2 + p - 6) - 2(p^2 + p - 6)$
$= (p^2 + p - 6)(p^2 + p - 2)$
$= (p^2+ 3p - 2p - 6)(p^2 + 2p - p - 2)$
$= [p(p + 3) - 2(p + 3)][p(p + 2) - 1(p + 2)]$
$= [(p + 3)(p - 2)][(p + 2)(p - 1)]$
$= (p + 3)(p - 2)(p + 2)(p - 1).$

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Question 55 Marks

Factorise the following:$\ 12 - (y + y^2)(8 - y - y^2)$

Answer

$12 - (y + y^2)(8 - y - y^2)$
$= 12 - a(8 - a)[$Taking $y + y^2 = a]$
$= 12 - 8a + a^2$
$= 12 - 6a - 2a + a^2$
$= 6(2 - a) - a(2 - a)$
$= (2 - a)(6 - a)$
$= [2 - (y + y^2)][6 - (y + y^2)]$
$= (2 - y - y^2)(6 - y - y^2)$
$= (2 - 2y + y - y^2)(6 - 3y + 2y - y^2)$
$= [2(1 - y) + y(1 - y)][3(2 - y) + y(2 - y)]$
$= [(1 - y)(2 + y)][(2 - y)(3 + y)]$
$= (1 - y)(2 + y)(2 - y)(3 + y)$
$= (y - 1)(y + 2)(y - 2)(y + 3).$

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Question 65 Marks

Factorise the following$: \ (a^2 - 2a)^2 - 23(a^2 - 2a) + 120$

Answer

$(a^2 - 2a)^2 - 23(a^2 - 2a) + 120$
$= (a^2 - 2a)^2 - 15(a^2 - 2a) - 8(a^2 - 2a) + 120$
$= (a^2 - 2a)(a^2 - 2a - 15) - 8(a^2 - 2a - 15)$
$= (a^2 - 2a - 15)(a^2 - 2a - 8)$
$= (a^2 - 5a + 3a - 15)(a^2 - 4a + 2a - 8)$
$= [a(a - 5) + 3(a - 5)][a(a - 4) + 2(a - 4)]$
$= [(a - 5)(a + 3)][(a - 4)(a + 2)]$
$= (a - 5)(a + 3)(a - 4)(a + 2)$
$= (a + 2)(a + 3)(a - 4)(a - 5).$

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Question 75 Marks

Factorise the following by grouping the terms:$\frac{1}{25 x^2}+16 x^2+\frac{8}{5}-12 x-\frac{3}{5 x}$

Answer

$\frac{1}{25 x^2}+16 x^2+\frac{8}{5}-12 x-\frac{3}{5 x}$
$ =\left(\frac{1}{25 x^2}+16 x^2+\frac{8}{5}\right)-\left(12 x+\frac{3}{5 x}\right) $
$=\left(\left(\frac{1}{5} x\right)^2+(4 x)^2+2 \times \frac{1}{5 x} \times 4 x\right)-\left(12 x+\frac{3}{5 x}\right) $
$=\left(\frac{1}{5 x}+4 x\right)^2-3\left(4 x+\frac{1}{5 x}\right) $
$=\left(\frac{1}{5 x}+4 x\right)^2-3\left(\frac{1}{5 x}+4 x\right) $
$ =\left(\frac{1}{5 x}+4 x\right)\left(\frac{1}{5 x}+4 x-3\right) .$

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MATHEMATICS [5 marks sum]

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