Question

[tex] If D= \left|\begin{array}{ccc}1&cosθ&1\\-cosθ &1&cosθ \\-1 &cosθ&1\end{array}\right|

then value of D lies in the interval........,Select Proper option from the given options.

(a) (2,∞)

(b) (2,4)

(c) [2,4]

(d) [-2,2]
[/tex]

Answer 1

Given,$D= \left|\begin{array}{ccc}1&cos\theta&1\\-cos\theta &1&cos\theta \\-1 &cos\theta&1\end{array}\right|$

solving Determinant,
D = 1(1 × 1 - cosθ × cosθ) - cosθ(-cosθ × 1 - cosθ × -1) + 1(-cosθ × cosθ - 1 × -1 )

D = (1 - cos²θ) - cosθ(-cosθ + cosθ) + (-cos²θ + 1)

D = sin²θ + sin²θ

D = 2sin²θ

we know, -1 ≤ sinθ ≤ 1
0 ≤ sin²θ ≤ 1
0 ≤ 2sin²θ ≤ 2

hence, D belongs to [0, 2]
here in option (d) [-2, 2] means to say [-2,0) U [0, 2]
2nd interval is same of value of D.

hence, we can say that value of D lies in the interval [-2, 2]
therefore , option (d) is correct.


[note :- it's not matter [-2,0) is not the value of D. we just considered [0, 2] , as you can see no any options which are considered [0,2] so, I can choose option (d) ]