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JEE MATHEMATICS MCQ: MATRICES

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Application +4 | -1
Inverse of the matrix [cosā”2Īøā€…ā€Šāˆ’āˆ’sinā”2Īøsinā”2Īøā€…ā€Šcosā”2Īø]\left[ {\begin{array}{cccccccccccccccccccc} {\cos 2\mathrm{\theta }\;--\sin 2\mathrm{\theta }}\\ {\sin 2\mathrm{\theta }\;\cos 2\mathrm{\theta }} \end{array}} \right] is
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Correct option: D
[cosā”2Īøā€…ā€Šsinā”2Īøāˆ’sinā”2Īøā€…ā€Šcosā”2Īø]\left[ {\begin{array}{cccccccccccccccccccc} {\cos 2\mathrm{\theta }\;\sin 2\mathrm{\theta }}\\ { - \sin 2\mathrm{\theta }\;\cos 2\mathrm{\theta }} \end{array}} \right]
Option D is correct because the inverse of a rotation matrix is its transpose, which involves changing the sign of the sine term.

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If āˆ£abāˆ’cb+cc+abcāˆ’aaāˆ’ba+bcāˆ£=0\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0 and a,b,ca, b, c are real numbers then the line ax+by+c=0a x+b y+c=0 passes through the fixed point is The number of solutions of the system of equations x2āˆ’x3=1,ā€…ā€Šāˆ’x1+2x3=āˆ’2,ā€…ā€Šx1āˆ’2,ā€…ā€Šx1āˆ’2x2=3{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3 is If the equations 2x+3y+1=0,3x+yāˆ’2=02x + 3y + 1 = 0,3x + y - 2 = 0 and ax+2yāˆ’b=0ax + 2y - b = 0 are consistent, then The value of the determinant āˆ£1(Ī±2xāˆ’Ī±āˆ’2x)2(Ī±2x+Ī±āˆ’2x)21(Ī²2xāˆ’Ī²āˆ’2x)2(Ī²2x+Ī²āˆ’2x)21(Ī³2xāˆ’Ī³āˆ’2x)2(Ī³2x+Ī³āˆ’2x)2āˆ£\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|, is If Ļ‰\omega is the cube root of unity, then āˆ£1Ļ‰Ļ‰2Ļ‰Ļ‰21Ļ‰21Ļ‰āˆ£\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right| is equal to For any matrix AA of order 2Ɨ22 \times 2, if A(adjā”A)=[100010]A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right], then āˆ£Aāˆ£|A| is equal to
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