1
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

Let $$\mathrm{A}$$ be a square matrix such that $$\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$$. Then $$\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$$ is equal to

A
$$\mathrm{A}^2+\mathrm{A}^{\mathrm{T}}$$
B
$$\mathrm{A}^3+\mathrm{I}$$
C
$$\mathrm{A}^3+\mathrm{A}^{\mathrm{T}}$$
D
$$\mathrm{A}^2+\mathrm{I}$$
2
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval

A
$$(-2,1)$$
B
$$\left(-\frac{3}{2}, \frac{3}{2}\right)$$
C
$$(-3,0)$$
D
$$(0,3)$$
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$.

Given below are two statements :

Statement I : $f(-x)$ is the inverse of the matrix $f(x)$.

Statement II : $f(x) f(y)=f(x+y)$.

In the light of the above statements, choose the correct answer from the options given below :
A
Statement I is false but Statement II is true
B
Both Statement I and Statement II are false
C
Both Statement I and Statement II are true
D
Statement I is true but Statement II is false
4
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
Out of Syllabus
Let the determinant of a square matrix A of order $m$ be $m-n$, where $m$ and $n$

satisfy $4 m+n=22$ and $17 m+4 n=93$.

If $\operatorname{det}(n \operatorname{adj}(\operatorname{adj}(m A)))=3^{a} 5^{b} 6^{c}$ then $a+b+c$ is equal to :
A
96
B
84
C
109
D
101
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