मराठी

The number of distinct real roots of |sinxcosxcosxcosxsinxcosxcosxcosxsinx| = 0 in the interval π4 x≤π4 is ______. - Mathematics

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प्रश्न

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 ______.

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उत्तर

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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पाठ 4: Determinants - Exercise [पृष्ठ ८१]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
पाठ 4 Determinants
Exercise | Q 28 | पृष्ठ ८१

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