Factorise:
(i) 4x 2 + 9y 2 + 16z 2 + 12xy – 24yz – 16xz
(ii) 2x 2 + y 2 + 8z 2 – 2√2 xy + 4√2 yz – 8xz
(i) It is known that,
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz : (2x)2 + (3y)2 + (-4z)2 + 2(2x)(3y) + 2(3y)(-4z) + 2(2x)(-4z)
= (2x + 3y - 4z)2 = (2x + 3y - 4z)(2x + 3y - 4z)
(ii) 2x2 + y2 + 8z2 – 2√2xy + 4√2 yz – 8xz
= (-√2x)2 + (y)2 + (2√2z)2 + 2(-√2x) (y) + 2(y)(2√2z) + 2 (-√2x)(2√2z)
= (-√2x + y + 2√2z)2 = (-√2x + y + 2√2z) (-√2x + y + 2√2z)
Factorise each of the following:
(i) 8a 3 + b 3 + 12a 2b + 6ab2
(ii) 8a 3 – b 3 – 12a 2b + 6ab2
(iii) 27 – 125a 3 – 135a + 225a 2
(iv) 64a 3 – 27b 3 – 144a 2b + 108ab2
(v) 27p 3 – \(\frac{1}{ 216}\) – \(\frac{9 }{ 2}\) p2 + \(\frac{1 }{4}\) p
Expand each of the following, using suitable identities:
(i) (x + 2y + 4z) 2 (ii) (2x – y + z) 2 (iii) (–2x + 3y + 2z) 2
(iv) (3a – 7b – c) 2 (v) (–2x + 5y – 3z) 2 (vi) [ \(\frac{1 }{ 4}\) a - \(\frac{1 }{ 2}\) b + 1]2
In Fig. 9.26, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.