JEE Main 2023 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 12 April 2023, Shift 1, Maths Q3

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

Let

A=\begin{bmatrix}1&\frac{1}{51}\\0&1\end{bmatrix}

. If

B=\begin{bmatrix}1&2\\-1&-1\end{bmatrix}A\begin{bmatrix}-1&-2\\1&1\end{bmatrix}

, then the sum of all the elements of the matrix

\sum_{n=1}^{50}B^n

is equal to

(1) $50$
(2) $75$
(3) $100$ Official answer
(4) $125$

Official answer

Option 3

NTA final key.

Chapter: Matrices
Topic: Matrix Multiplication
Idea tested: Powers of Square Matrices

Same idea in other shifts

Asked 21× in all

4 Apr 2025, Shift 2 · Q4Let the matrix $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ satisfy $A^n=A^{n-2}+A^2-I$ for $n\ge 3$ . Then the sum of all the…HardSingle correct
4 Apr 2024, Shift 2 · Q3Let $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $B=I+\mathrm{adj}(A)+(\mathrm{adj}\,A)^2+\ldots+(\mathrm{adj}\,A)^{10}$ . Then, the sum of…MediumSingle correct
8 Apr 2024, Shift 1 · Q22Let $A=\begin{bmatrix}2&-1\\1&1\end{bmatrix}$ . If the sum of the diagonal elements of $A^{13}$ is $3^n$ , then $n$ is equal to ____________.MediumNumerical value
6 Apr 2023, Shift 2 · Q4Let $P$ be a square matrix such that $P^2=I-P$ . For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$ , if $P^\alpha+P^\beta=\gamma I-29P$ and…HardSingle correct
8 Apr 2023, Shift 1 · Q5Let $P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}$ ,…HardSingle correct
11 Apr 2023, Shift 1 · Q29Let $A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}$ , where $a, c \in \mathbb{R}$ . If $A^3 = A$ and the positive…HardNumerical value
25 Jul 2022, Shift 1 · Q21Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$ . If $\omega=\frac{\sqrt3\,i-1}{2}$ , then the number of elements in…HardNumerical value
25 Jul 2022, Shift 2 · Q23Let $A=\begin{bmatrix}1&a&a\\0&1&b\\0&0&1\end{bmatrix}$ , $a,b\in\mathbb{R}$ . If for some $n\in\mathbb{N}$ ,…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 108 of the 174 JEE Main shifts held from 2017 to 2026.