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BITSAT 2022
Exam
BITSAT
Year
2022
Subject
Mathematics
Chapter
Matrices

BITSAT 2022 · Mathematics · Matrices

If [ array cc 1 & - θ θ & 1 array ] [ array cc 1 & θ - θ & 1 array...

 If \text{ If }\left[1−tan⁡θtan⁡θ1\begin{array}{cc} 1 & -\tan \theta \\ \tan \theta & 1 \end{array}\right]\left[1tan⁡θ−tan⁡θ1\begin{array}{cc} 1 & \tan \theta \\ -\tan \theta & 1 \end{array}\right]^{-1}=\left[a−bba\begin{array}{cc} a & -b \\ b & a \end{array}\right], then

Options

  1. Aa=1,b=1a=1, b=1
  2. Ba=sin⁡2θ,b=cos⁡2θa=\sin 2 \theta, b=\cos 2 \theta
  3. Ca=cos⁡2θ,b=sin⁡2θa=\cos 2 \theta, b=\sin 2 \theta
  4. DNone of these

Correct answer

C. a=cos⁡2θ,b=sin⁡2θa=\cos 2 \theta, b=\sin 2 \theta

Step-by-step solution

Given, \left[1−tan⁡θtan⁡θ1\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[1tan⁡θ−tan⁡θ1\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1}=\left[a−bba\begin{array}{cc}a & -b \\ b & a\end{array}\right]\Rightarrow\left[1−tan⁡θtan⁡θ1\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right] \cdot \frac{1}{1+\tan ^2 \theta}\left[1−tan⁡θtan⁡θ1\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]=\left[a−bba\begin{array}{cc}a & -b \\ b & a\end{array}\right]\Rightarrow \frac{1}{1+\tan ^2 \theta}\left[1−tan⁡2θ−2tan⁡θ2tan⁡θ1−tan⁡2θ\begin{array}{cc}1-\tan ^2 \theta & -2 \tan \theta \\ 2 \tan \theta & 1-\tan ^2 \theta\end{array}\right]=\left[a−bba\begin{array}{cc}a & -b \\ b & a\end{array}\right]\Rightarrow \quad\left[1−tan⁡2θ1+tan⁡2θ2tan⁡θ1+tan⁡2θ2tan⁡θ1+tan⁡2θ1−tan⁡2θ1+tan⁡2θ\begin{array}{cc}\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta} & \frac{2 \tan \theta}{1+\tan ^2 \theta} \\ \frac{2 \tan \theta}{1+\tan ^2 \theta} & \frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\end{array}\right]=\left[a−bba\begin{array}{cc}a & -b \\ b & a\end{array}\right]\Rightarrow\left[cos⁡2θ−sin⁡2θsin⁡2θcos⁡2θ\begin{array}{cc}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{array}\right]=\left[a−bba\begin{array}{cc}a & -b \\ b & a\end{array}\right]\Rightarrow \quad a=\cos 2 \theta, b=\sin 2 \theta
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