JEE Main 2022 · 29 Jul · Shift 2 · Mathematics Q23

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MathematicsInteger+4 / -1Question 23 of 90

Let $X=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $A=\begin{bmatrix}-1&2&3\\0&1&6\\0&0&-1\end{bmatrix}$ . For k ∈ ℕ, if $X'A^kX=33$ , then k is equal to _______.

Your answer

Answer and worked solution

Official answer: 10

Square. $A^2=I+6E_{13}$ (only a 6 in the top-right corner), so $A^{2m}=I+6mE_{13}$ and $X'A^{2m}X=3+6m$ . Odd powers give $X'A^{2m+1}X=10-6m\le10$ .

Solve $3+6m=33$ : $m=5$ , $k=10$ .

Answer: 10

Topics:Matrix algebra

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JEE Main 2022 · 29 Jul · Shift 2, Mathematics question 23