1
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 $$\mu \neq 1$$
2
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
3
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

$$\text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$$

A
9
B
17
C
3
D
5
4
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
$$\mathrm{A}^2+\mathrm{I}$$
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