JEE Main 2005 — Mathematical Induction Question with Solution
From: AIEEE 2005
Question
If A = \left[ {\matrix{
1 & 0 \cr
1 & 1 \cr
} } \right] and I = \left[ {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right], then which one of the following holds for all by the principle of mathematical induction?
Choose an option
Show full solutionCorrect option: A
Correct answer
A
Step-by-step explanation
Given A = \left[ {\matrix{
1 & 0 \cr
1 & 1 \cr
} } \right]
= = \left[ {\matrix{ 1 & 0 \cr 2 & 1 \cr } } \right]
and = = \left[ {\matrix{ 1 & 0 \cr 3 & 1 \cr } } \right]
So we can say = \left[ {\matrix{ 1 & 0 \cr n & 1 \cr } } \right]
Now
= \left[ {\matrix{ n & 0 \cr n & n \cr } } \right] - \left[ {\matrix{ {n - 1} & 0 \cr 0 & {n - 1} \cr } } \right]
= \left[ {\matrix{ 1 & 0 \cr n & 1 \cr } } \right] =
= = \left[ {\matrix{ 1 & 0 \cr 2 & 1 \cr } } \right]
and = = \left[ {\matrix{ 1 & 0 \cr 3 & 1 \cr } } \right]
So we can say = \left[ {\matrix{ 1 & 0 \cr n & 1 \cr } } \right]
Now
= \left[ {\matrix{ n & 0 \cr n & n \cr } } \right] - \left[ {\matrix{ {n - 1} & 0 \cr 0 & {n - 1} \cr } } \right]
= \left[ {\matrix{ 1 & 0 \cr n & 1 \cr } } \right] =
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This is a previous-year question from JEE Main 2005, covering the Mathematical Induction chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.