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Question

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


Solution

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

Mathematics
NCERT
Standard XII

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