beginvmatrix 0 -sin- x -cos - X -sin - X -cos- xsin -2-sin- X -cos- Xcos-2-cos- X -sin - Xlend vmatrix is equal toA- sin xB- cos x-sin xC- 1D- -1

The correct option is D 10 amp; sin x cos x amp;sin x+cos x nbsp; nbsp;sin^2x amp;sin x nbsp; amp;cos x nbsp; nbsp;cos^2x amp; co ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

beginvmatrix ...

Question

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & sin x - cos x & sin x + cos x {sin}^{2} x & sin x & cos x {cos}^{2} x & - cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ ∣$ is equal to

sin x

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cos x + sin x

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Solution

The correct option is C $- 1$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & sin x - cos x & sin x + cos x {sin}^{2} x & sin x & cos x {cos}^{2} x & - cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ ∣$
Using row operations,
$R_{1} \to R_{1} - (R_{2} + R_{3})$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} - 1 & 0 & 0 {sin}^{2} x & sin x & cos x {cos}^{2} x & - cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 1 ({sin}^{2} x + {cos}^{2} x) = - 1$

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Similar questions

Q. The number of distinct real roots of

∣ ∣ ∣ ∣ \begin{matrix} sin x & cos x & cos x cos x & sin x & cos x cos x & cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ = 0

in the interval

- \frac{π}{4} \leq x \leq \frac{π}{4}

Q. The integral

\int \frac{s i n^{2} x c o s^{2} x}{(s i n^{5} x + c o s^{3} x s i n^{2} x + s i n^{3} x c o s^{2} x + c o s^{5} x)^{2}} d x

is equal to

Q. The number of distinct real roots of

∣ ∣ ∣ ∣ \begin{matrix} s i n x & c o s x & c o s x c o s x & s i n x & c o s x c o s x & c o s x & s i n x \end{matrix} ∣ ∣ ∣ ∣ = 0

in the interval

- \frac{π}{4} \leq x \leq \frac{π}{4}

The number of distinct real roots of $∣ ∣ ∣ ∣ \begin{matrix} s i n x & c o s x & c o s x c o s x & s i n x & c o s x c o s x & c o s x & s i n x \end{matrix} ∣ ∣ ∣ ∣ = 0$ in the interval $- \frac{π}{4} \leq x \leq \frac{π}{4}$ is

(a) 0
(b) 2
(c) 1
(d) 3

∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & sin x - cos x & sin x + cos x {sin}^{2} x & sin x & cos x {cos}^{2} x & - cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ ∣

is equal to

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∣∣ ∣ ∣∣0sin x−cos xsin x+cos xsin2xsin xcos xcos2x−cos xsin x∣∣ ∣ ∣∣ is equal to

The correct option is C −1∣∣ ∣ ∣∣0sin x−cos xsin x+cos xsin2xsin xcos xcos2x−cos xsin x∣∣ ∣ ∣∣ Using row operations, R1→R1−(R2+R3) =∣∣ ∣ ∣∣−100sin2xsin xcos xcos2x−cos xsin x∣∣ ∣ ∣∣=−1(sin2x+cos2x)=−1

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & sin x - cos x & sin x + cos x {sin}^{2} x & sin x & cos x {cos}^{2} x & - cos x & sin x \end{matrix} ∣ ∣ ∣ ∣ ∣$ is equal to