1
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
2
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 $$\mu=13$$
3
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

Consider the system of linear equations $$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$$ where $$\lambda, \mu \in \mathbf{R}$$. Which one of the following statements is NOT correct ?

A
The system has unique solution if $$\lambda \neq \frac{1}{2}$$ and $$\mu \neq 1,15$$
B
The system has infinite number of solutions if $$\lambda=\frac{1}{2}$$ and $$\mu=15$$
C
The system is consistent if $$\lambda \neq \frac{1}{2}$$
D
The system is inconsistent if $$\lambda=\frac{1}{2}$$ and $$\mu \neq 1$$
4
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

Let $$A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$$ and $$P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]$$. The sum of the prime factors of $$\left|P^{-1} A P-2 I\right|$$ is equal to

A
66
B
27
C
23
D
26
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