If the determinant $\left|\begin{array}{ccc}\cos 2 x & \sin ^2 x & \cos 2 x \\ \sin ^2 x & \cos 2 x & \cos…

If the determinant $\left|\begin{array}{ccc}\cos 2 x & \sin ^2 x & \cos 2 x \\ \sin ^2 x & \cos 2 x & \cos ^2 x \\ \cos 2 x & \cos ^2 x & \cos 2 x\end{array}\right|$ is expanded in powers of $\cos x$, then the constant term in the expansion is
  1. 1
  2. -1
  3. 0
  4. 2

Solution

Let $\Delta=\left|\begin{array}{ccc}\cos 2 x & \sin ^2 x & \cos 2 x \\ \sin ^2 x & \cos 2 x & \cos ^2 x \\ \cos 2 x & \cos ^2 x & \cos 2 x\end{array}\right|$ On expanding along $R_1$. $ \begin{aligned} & \Delta=\cos 2 x\left[\cos ^2 2 x-\cos ^4 x\right]-\sin ^2 x \\ & {\left[\sin ^2 x \cos 2 x-\cos ^2 x \cdot \cos 2 x\right]} \\ & +\cos 2 x\left[\sin ^2 x \cos ^2 x-\cos ^2 2 x\right] \\ & \Delta=\cos 2 x\left[\left(2 \cos ^2 x-1\right)^2-\cos ^4 x\right] \\ & -\sin ^2 x\left[\cos 2 x\left(\sin ^2 x-\cos ^2 x\right)\right] \\ & +\cos 2 x\left[\left(1-\cos ^2 x\right) \cos ^2 x-\left(2 \cos ^2 x-1\right)^2\right] \\ & \Delta=\left(2 \cos ^2 x-1\right)\left[4 \cos ^4 x+1-4 \cos ^2 x-\cos ^4 x\right] \\ & -\left(1-\cos ^2 x\right)[\cos 2 x(-\cos 2 x)] \\ & +\cos 2 x\left[\cos ^2 x-\cos ^4 x-4 \cos ^4 x-1+4 \cos ^2 x\right] \\ & =\left(2 \cos ^2 x-1\right)\left[3 \cos ^4 x+1+5 \cos ^2 x-5 \cos ^4 x-1\right] \\ & +\left(1-\cos ^2 x\right)\left[\left(2 \cos ^2 x-1\right)^2\right] \\ & \Rightarrow\left(2 \cos ^2 x-1\right)\left[5 \cos ^2 x-2 \cos ^4 x\right]+\left(2 \cos ^2 x-1\right) \\ & {\left[2 \cos ^2 x-1-2 \cos ^3 x+\cos ^2 x\right]} \\ & \Rightarrow\left(2 \cos ^2 x-1\right)\left[5 \cos ^2 x-2 \cos ^4 x+3 \cos ^2 x\right. \\ & \left.-2 \cos ^3 x-1\right] \\ & \Rightarrow\left(2 \cos ^2 x-1\right)\left[8 \cos ^2 x-2 \cos ^4 x-2 \cos ^3 x-1\right] \\ & \end{aligned} $ Hence, constant term will be $(-1) \times(-1)=1$

Asked in: AP EAMCET 2021 (25 Aug Shift 2)

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