Find the value of determinant-i cos- -sin- sin- cos-ii x 2 -x-1 x-1 x-1 x-1

(i) nbsp; cosθ nbsp; amp; sinθ nbsp; sinθ nbsp; amp; cosθ nbsp; =(cosθ )(cosθ ) ( sinθ )(sinθ )=cos^ 2 θ +sin^ 2 θ =1 (ii) nbsp; ...

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Byju's Answer

Standard XII

Mathematics

Differentiation of Inverse Trigonometric Functions

Find the valu...

Question

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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Solution

(i) $∣ ∣ ∣ \begin{matrix} cos θ & - sin θ sin θ & cos θ \end{matrix} ∣ ∣ ∣$
$= (cos θ) (cos θ) - (- sin θ) (sin θ) = {cos}^{2} θ + {sin}^{2} θ = 1$
(ii) $∣ ∣ ∣ \begin{matrix} x^{2} - x + 1 & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$
$= (x^{2} - x + 1) (x + 1) - (x - 1) (x + 1)$
$= x^{3} - x^{2} + x + x^{2} - x + 1 - (x^{2} - 1)$
$= x^{3} + 1 - x^{2} + 1$
$= x^{3} - x^{2} + 2$

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Similar questions

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

Q. If

x = 3 - 2 \sqrt{2}

, find the value of:
i)

\frac{1}{x}

ii)

x + \frac{1}{x}

iii)

x - \frac{1}{x}

iv)

\sqrt{x} + \frac{1}{\sqrt{x}}

\sqrt{x} - \frac{1}{\sqrt{x}}

vi)

x^{2} + \frac{1}{x^{2}}

vii)

x^{2} - \frac{1}{x^{2}}

Q. If

\frac{1}{\sqrt{2}} < x < 1

, then find the value of

{cos}^{- 1} x + {cos}^{- 1} (\frac{x + \sqrt{1 - x^{2}}}{\sqrt{2}})

Q. Evaluate:

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

Factorise: x⁴-1

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Find the value of determinant.(i) ∣∣∣cosθ−sinθsinθcosθ∣∣∣(ii) ∣∣∣x2−x+1x−1x+1x+1∣∣∣

(i) ∣∣∣cosθ−sinθsinθcosθ∣∣∣=(cosθ)(cosθ)−(−sinθ)(sinθ)=cos2θ+sin2θ=1(ii) ∣∣∣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

Find the value of determinant.
(i) $∣ ∣ ∣ \begin{matrix} cos θ & - sin θ sin θ & cos θ \end{matrix} ∣ ∣ ∣$
(ii) $∣ ∣ ∣ \begin{matrix} x^{2} - x + 1 & x - 1 x + 1 & x + 1 \end{matrix} ∣ ∣ ∣$