Test Index

CBSE Class 12 Math 2012 Solved Paper

© examsnet.com
Question : 8 of 29
Marks: +1, -0
Simplify: cos θ ∣cos⁡θsin⁡θ−sin⁡θcos⁡θ∣\begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix} + sin θ ∣sin⁡θ−cos⁡θcos⁡θsin⁡θ∣\begin{vmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{vmatrix}
Solution:  
cos θ ∣cos⁡θsin⁡θ−sin⁡θcos⁡θ∣\begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix} + sin θ ∣sin⁡θ−cos⁡θcos⁡θsin⁡θ∣\begin{vmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{vmatrix}
= ∣cos⁡2θcos⁡θsin⁡θ−sin⁡θcos⁡θcos⁡2θ∣\begin{vmatrix} \cos^2\theta & \cos\theta\sin\theta \\ -\sin\theta\cos\theta & \cos^2\theta \end{vmatrix} + ∣sin⁡2θ−sin⁡θcos⁡θsin⁡θcos⁡θsin⁡2θ∣\begin{vmatrix} \sin^2\theta & -\sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2\theta \end{vmatrix}
=
∣cos⁡2θ+sin⁡2θcos⁡θsin⁡θ−sin⁡θcos⁡θ−sin⁡θcos⁡θ+cos⁡θsin⁡θcos⁡2θ+sin⁡2θ∣\begin{vmatrix} \cos^2\theta+\sin^2\theta & \cos\theta\sin\theta-\sin\theta\cos\theta \\ -\sin\theta\cos\theta+\cos\theta\sin\theta & \cos^2\theta+\sin^2\theta \end{vmatrix}
= ∣1001∣\begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} [Since cos⁡2θ+sin⁡2θ\cos^2\theta+\sin^2\theta = 1]
© examsnet.com
Go to Question:
Prev Question
Next Question