JEE Advanced 2016 · Paper 2 · Mathematics Q37

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MathematicsSingle correct+3 / -1Question 37 of 54

Let $P=\begin{bmatrix}1&0&0\\4&1&0\\16&4&1\end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$ , then $\dfrac{q_{31}+q_{32}}{q_{21}}$ equals

103

201

205

Answer and worked solution

Official answer: B

Nilpotent part. Write $P=I+N$ with $N=\begin{bmatrix}0&0&0\\4&0&0\\16&4&0\end{bmatrix}$ . Then $N^2=\begin{bmatrix}0&0&0\\0&0&0\\16&0&0\end{bmatrix}$ and $N^3=0$ .

Binomial expansion.
$P^{50}=I+50N+\binom{50}2N^2,\qquad Q=P^{50}-I=50N+1225N^2.$

Entries. $q_{21}=200$ , $q_{32}=200$ and $q_{31}=50(16)+1225(16)=20400$ .

Ratio. $\dfrac{20400+200}{200}=103$ .

(B)

Topics:Matrix algebra

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JEE Advanced 2016 · Paper 2, Mathematics question 37