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Saturday, 10 October

JEE Main 2024 · MathsMultiple choiceOtherHardMulti-step

JEE Main 30 January 2024, Shift 2, Maths Q3

Question 3 of 90 in this shift, Maths question 3 of 30, Section A.

Let R=(x000y000z)R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} be a non-zero 3×33\times 3 matrix, where xsin⁡θ=ysin⁡(θ+2π3)=zsin⁡(θ+4π3)≠0x\sin\theta = y\sin\left(\theta + \frac{2\pi}{3}\right) = z\sin\left(\theta + \frac{4\pi}{3}\right) \neq 0, θ∈(0,2π)\theta \in (0, 2\pi). For a square matrix MM, let trace (M)(M) denote the sum of all the diagonal entries of MM. Then, among the statements: (I) Trace (R)=0(R) = 0 (II) If trace (adj(adj(R)))=0(\mathrm{adj}(\mathrm{adj}(R))) = 0, then RR has exactly one non-zero entry.
  1. (1)Only (I) is true
  2. (2)Only (II) is trueOfficial answer
  3. (3)Both (I) and (II) are true
  4. (4)Neither (I) nor (II) is true

Official answer

Option 2

NTA final key.

Chapter
Determinants
Topic
Matrix Inverse
Idea tested
Inverse Relation to Double Adjoint

Same idea in other shifts

Asked 8× in all
  1. 4 Apr 2026, Shift 1 · Q6Let A=[112−201135]A=\begin{bmatrix}1&1&2\\-2&0&1\\1&3&5\end{bmatrix}. Then the sum of all elements of the matrix…MediumSingle correct
  2. 6 Apr 2026, Shift 1 · Q21Let A=[−11−1101001]A=\begin{bmatrix}-1&1&-1\\1&0&1\\0&0&1\end{bmatrix} satisfy…MediumNumerical value
  3. 7 Apr 2025, Shift 1 · Q5Let AA be a 3×33 \times 3 matrix such that ∣adj(adj(adj A))∣=81|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\,A))| = 81. If…HardSingle correct
  4. 8 Apr 2025, Shift 2 · Q4Let A=[22+p2+p+q46+2p8+3p+2q612+3p20+6p+3q]A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix}. If…MediumSingle correct
  5. 6 Apr 2024, Shift 2 · Q4If AA is a square matrix of order 3 such that det⁡(A)=3\det(A)=3 and…MediumSingle correct
  6. 9 Apr 2024, Shift 1 · Q23Let A be a non-singular matrix of order 3. If det⁡(3 adj(2 adj((det⁡A)A)))=3−13⋅2−10\det(3\,\mathrm{adj}(2\,\mathrm{adj}((\det A)A)))=3^{-13}\cdot2^{-10} and…HardNumerical value
  7. 31 Jan 2024, Shift 2 · Q22Let A be a 3×33\times3 matrix and det⁡(A)=2\det(A)=2. If…MediumNumerical value

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 42 of the 174 JEE Main shifts held from 2017 to 2026.