1
JEE Main 2024 (Online) 8th April Morning Shift
+4
-1

Let $$A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$$. If $$A^3=4 A^2-A-21 I$$, where $$I$$ is the identity matrix of order $$3 \times 3$$, then $$2 a+3 b$$ is equal to

A
$$-10$$
B
$$-12$$
C
$$-13$$
D
$$-9$$
2
JEE Main 2024 (Online) 6th April Evening Shift
+4
-1

If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}+2 \mathrm{n}$$ is equal to :

A
2
B
4
C
3
D
6
3
JEE Main 2024 (Online) 6th April Morning Shift
+4
-1

For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$$. Then $$2 A_{10}-A_8$$ is

A
$$4 \alpha+2 \beta$$
B
0
C
$$2 n$$
D
$$2 \alpha+4 \beta$$
4
JEE Main 2024 (Online) 5th April Evening Shift
+4
-1

The values of $$m, n$$, for which the system of equations

\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}

has infinitely many solutions, satisfy the equation :

A
$$\mathrm{m}^2+\mathrm{n}^2-\mathrm{m}-\mathrm{n}=46$$
B
$$\mathrm{m}^2+\mathrm{n}^2+\mathrm{mn}=68$$
C
$$\mathrm{m}^2+\mathrm{n}^2-\mathrm{mn}=39$$
D
$$\mathrm{m}^2+\mathrm{n}^2+\mathrm{m}+\mathrm{n}=64$$
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