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Question
The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4 x ≤ pi/4` is ______.
Options
0
2
1
3
Solution
The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4 x ≤ pi/4` is 1.
Explanation:
We have, `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0
Applying C1 → C1 + C2 + C3
⇒ `|(2cosx + sinx, cosx, cxosx),(2cosx + sinx, sinx, cosx),(2cosx + sinx, cosx, sinx)|`
⇒ `(2cosx + sinx) |(1, cosx, cosx),(1, sinx, cosx),(1, cosx, six)|` = 0
Applying R2 → R2 – R1 and R3 → R3 – R1
⇒ `(2cosx + sinx)|(1, cosx, cosx),(0, sinx - cosx, 0),(0, 0, sinx - cosx)|`
⇒ `(2 cosx + sinx)[1 * (sin x - cos x)^2]` = 0 ...(Expanding along C1)
⇒ `(2 cosx + sinx)(sinx - cos x)^2` = 0
⇒ 2 cos x = –sin x or sin x = cos x
⇒ tan x = –2, which s not possible as for `pi/4 x ≤ pi/4`
We get –1 tan x ≤ 1.
or tan x = 1
∴ x = `p/4`
So, only one real root exist.
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