HCU MCA 2013 Paper

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Question : 53 of 75

Marks: +1, -0

Consider the determinent

△=|\table1,\sinθ,1;-\sinθ,1,\sinθ;-1,-\sinθ,1|

Then,

{△}/2

for all values of

θ

, lies between

0 and 1
2 and 4
1 and 2
> 4

Solution:

(c) Given that,

△=|\table1,\sinθ,1;-\sinθ,1,\sinθ;-1,-\sinθ,1|

Applying operation

C_1→C_1+C_3

△=|\table2,\sinθ,1;0,1,\sinθ;0,-\sinθ,1|

Expanding along

C_1

;
△ = 2 {1 + sin

^2θ

}

⇒{△}/2

= 1 + sin

^2θ\;\;\;\;\;\;\;\;\;...(\i)

∴

\;\;\;\;\;\;\;-1≤\sinθ≤1

⇒

\;\;\;\;\;\;\;-0≤\sin^2θ≤1

⇒

\;\;\;\;\;\;\;\;\;\;\;1≤1+\sin^2θ≤2

⇒

\;\;\;\;\;\;\;\;\;\;\;1≤{△}/2≤2\;\;\;\;\;\;\;\;\;\[\From \Eq.(\i)

∴

{△}/2

∈ [1,2]

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