10 Question MCQ (including Assertion)

Let \(\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d}\end{array}\right]\) be a square matrix such that adj \(\mathrm{A}=\mathrm{A}\). Then, \((\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})\) is equal to :

The maximum value of \(\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1+\cos \theta & 1 & 1\end{array}\right|\) is \((\theta\) is real number \()\)

The number of distinct real roots of \(\left|\begin{array}{ccc}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) is

Let \(\Delta=\left|\begin{array}{ccc}\mathrm{A} x & x^2 & 1 \\ \mathrm{~B} y & y^2 & 1 \\ \mathrm{C} z & z^2 & 1\end{array}\right|\) and \(\Delta_1=\left|\begin{array}{ccc}\mathrm{A} & \mathrm{B} & \mathrm{C} \\ x & y & z \\ z y & z x & x y\end{array}\right|\), then

If \(\mathrm{A}=\left[\begin{array}{ccc}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right]\), then \(\mathrm{A}^{-1}\) exists if

If A, B and C are angles of a triangle, then the determinant \(\left|\begin{array}{ccc}-1 & \cos \mathrm{C} & \cos \mathrm{B} \\ \cos \mathrm{C} & -1 & \cos \mathrm{~A} \\ \cos \mathrm{~B} & \cos \mathrm{~A} & -1\end{array}\right|\) is equal to

If \(\mathrm{A}(\operatorname{adj} \mathrm{A})=\left[\begin{array}{ccc}2026 & 0 & 0 \\ 0 & 2026 & 0 \\ 0 & 0 & 2026\end{array}\right]\), then the value of \(|\operatorname{adj} \mathrm{A}|\) is equal to :

There are two values of \(a\) which makes determinant, \(\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86\), then sum of these number is

If A and B are invertible matrices, then which of the following is not correct?