1
JEE Main 2024 (Online) 9th April Morning Shift
+4
-1

Let $$\lambda, \mu \in \mathbf{R}$$. If the system of equations

\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}

has infinitely many solutions, then $$\mu+2 \lambda$$ is equal to :

A
24
B
25
C
27
D
22
2
JEE Main 2024 (Online) 8th April Evening Shift
+4
-1

If $$\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$$ and $$\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$$, then $$\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$$ is equal to :

A
2
B
3
C
1
D
0
3
JEE Main 2024 (Online) 8th April Evening Shift
+4
-1

If the system of equations $$x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$$ has infinitely many solutions, then $$(2 \mu+3 \lambda)$$ is equal to :

A
$$-2$$
B
2
C
3
D
$$-3$$
4
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
$$-9$$
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