Let a, b and c be three real numbers satisfying
[a b c] 
= [0 0 0] ...........(E)
(i) If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is
Text Solution
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(i) (ii)a (iii)b
Sol. ( i to iii)
a + 8b + 7c = 0 ........... (i)
9a + 2b + 3c = 0 ........... (ii)
a + b + c = 0 ............. (iii)
Δ = 
= 1.(–1) – 8(6) + 7(7) = 0
Let c = λ
∴ a + 8b = –7 λ ⇒ a + b = – λ ⇒ b = 
λ & a = 
∴ (a, b, c) ≡ 
where λ ∈ R
(i) P(a, b, c) lies on the plane 2x + y + z = 1
∴ 
= 1 ⇒ 
= 1 ⇒ λ = –7
∴ 7a + b +c = 7 + 6 – 7 = 6
(ii) a = 2 ⇒ λ = –14
∴ b = 12 & c = –14
Now = 
= 
3 ω + 1 + 3 ω 2 = 3( ω + ω 2 ) + 1 = – 2
(iii) b = 6 ⇒ λ = –7
⇒ a = 1 & c = – 7
now ax 2 + bx + c = 0 ⇒ x 2 + 6x – 7 = 0 ⇒ x = –7 , 1
∴ 
= 
= 1 + 
+ 
+ ....... ∞ = 
= 7
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