Evaluate the determinants- beginvmatrix cos theta-sinltheta 1sin scoslthetalend vmatrix x- 2-x-18 x-1 x-1 - x-1lend vmatrix

Cosθ amp; sinθ sinθ amp;cosθ=(cosθ)(cosθ) (sinθ)( sinθ) =cos^2θ+sin^2θ=1 (∵ sin^2θ+cos^2θ=1) x^2 x+_1 amp; x 1 x+1 amp; x+1=(x^2 x+1)(x+1) (x+1 ...

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Byju's Answer

Standard XII

Mathematics

Determinant

Evaluate the ...

Question

Evaluate the determinants.

$∣ ∣ ∣ \begin{matrix} c o s θ & - s i n θ s i n θ & c o s θ \end{matrix} ∣ ∣ ∣$

$∣ ∣ ∣ \begin{matrix} x^{2} - x +_{1} & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ \begin{matrix} c o s θ & - s i n θ s i n θ & c o s θ \end{matrix} ∣ ∣ ∣ = (c o s θ) (c o s θ) - (s i n θ) (- s i n θ)$
$= c o s^{2} θ + s i n^{2} θ = 1 (∵ s i n^{2} θ + c o s^{2} θ = 1)$

$∣ ∣ ∣ \begin{matrix} x^{2} - x +_{1} & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$ $= (x^{2} - x + 1) (x + 1) - (x + 1) (x - 1) = x^{3} - x^{2} + x + x^{2} - x + 1 - (x^{2} - 1) = x^{3} + 1 (x^{2} - 1) = x^{3} - x^{2} + 2$

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Similar questions

Evaluate the determinants.

$∣ ∣ ∣ \begin{matrix} x^{2} - x +_{1} & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$

Evaluate the determinant
$(i)$ $∣ ∣ ∣ \begin{matrix} cos θ & - sin θ sin θ & cos θ \end{matrix} ∣ ∣ ∣$
$(i i)$ $∣ ∣ ∣ \begin{matrix} x^{2} - x + 1 & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & x & x^{2} x^{2} & 1 & x x & x^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = (1 - x^{3})^{2}$

Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 3 c o s θ & 1 s i n θ & 1 & 3 c o s θ 1 & s i n θ & 1 \end{matrix} ∣ ∣ ∣ ∣ = R_{1} \to R_{1} - R_{3} Δ = ∣ ∣ ∣ ∣ \begin{matrix} 0 & 3 c o s θ - s i n θ & 0 s i n θ & 1 & 3 c o s θ 1 & s i n θ & 1 \end{matrix} ∣ ∣ ∣ ∣

Q. Find the value of determinant.
(i)

∣ ∣ ∣ \begin{matrix} cos θ & - sin θ sin θ & cos θ \end{matrix} ∣ ∣ ∣

(ii)

∣ ∣ ∣ \begin{matrix} x^{2} - x + 1 & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣

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Evaluate the determinants. ∣∣∣cosθ−sinθsinθcosθ∣∣∣ ∣∣∣x2−x+1x−1x+1x+1∣∣∣

∣∣∣cosθ−sinθsinθcosθ∣∣∣=(cosθ)(cosθ)−(sinθ)(−sinθ) =cos2θ+sin2θ=1 (∵sin2θ+cos2θ=1) ∣∣∣x2−x+1x−1x+1x+1∣∣∣=(x2−x+1)(x+1)−(x+1)(x−1)=x3−x2+x+x2−x+1−(x2−1)=x3+1(x2−1)=x3−x2+2

Evaluate the determinants.

$∣ ∣ ∣ \begin{matrix} c o s θ & - s i n θ s i n θ & c o s θ \end{matrix} ∣ ∣ ∣$

$∣ ∣ ∣ \begin{matrix} x^{2} - x +_{1} & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$