1
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

Let $$A$$ be a $$3 \times 3$$ real matrix such that

$$A\left(\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right)=2\left(\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right), A\left(\begin{array}{l} -1 \\ 0 \\ 1 \end{array}\right)=4\left(\begin{array}{l} -1 \\ 0 \\ 1 \end{array}\right), A\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=2\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right) \text {. }$$

Then, the system $$(A-3 I)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$$ has :

A
exactly two solutions
B
infinitely many solutions
C
unique solution
D
no solution
2
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

If the system of linear equations

\begin{aligned} & x-2 y+z=-4 \\ & 2 x+\alpha y+3 z=5 \\ & 3 x-y+\beta z=3 \end{aligned}

has infinitely many solutions, then $$12 \alpha+13 \beta$$ is equal to

A
60
B
54
C
64
D
58
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$$. For a square matrix $$M$$, let trace $$(M)$$ denote the sum of all the diagonal entries of $$M$$. Then, among the statements:

(I) Trace $$(R)=0$$

(II) If trace $$(\operatorname{adj}(\operatorname{adj}(R))=0$$, then $$R$$ has exactly one non-zero entry.

A
Only (I) is true
B
Only (II) is true
C
Both (I) and (II) are true
D
Neither (I) nor (II) is true
4
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?

A
System is consistent if $$\lambda \neq 1$$ and $$\mu=13$$
B
System is inconsistent if $$\lambda=1$$ and $$\mu \neq 13$$
C
System has unique solution if $$\lambda \neq 1$$ and $$\mu \neq 13$$
D
System has infinite number of solutions if $$\lambda=1$$ and $$\mu=13$$
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