Let R=\left(\begin{array}{ccc}x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z)\end{array} be a non-zero 3 \times 3 matrix, where {\x} \sin \theta= {\y} \sin (\theta+\frac{2 \pi}{3}) = {\z} \sin (\theta+\frac{4 \pi}{3}) \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}({\adj}({\adj}( {\R}))=0, then {\R} has exactly one non-zero entry. [30-Jan-2024 Shift 2]

Correct Answer is : Only (II) is true

JEE Main 30-Jan-2024 Shift 2 Solved Paper

Section: Mathematics

Question : 20 of 90

Marks: +1, -0

Let R=(

x	0	0
0	y	0
0	0	z

) be a non-zero 3×3 matrix, where xsin‌θ=ysin‌(θ+‌

2π

)=zsin‌(θ+‌

4π

) ≠0,θ∈(0,2π). 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(adj(adj(R))=0, then R has exactly one non-zero entry.

[30-Jan-2024 Shift 2]

Both (I) and (II) are true
Neither (I) nor (II) is true
Only (II) is true
Only (I) is true

Validate

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