iCON Education - Mathematical Induction JEE Mains Test

Question 1: Palette 1/3

JEE Mains 2005 MCQ

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 $n \ge 1,$ by the principle of mathematical induction?

A

${A^n} = nA - \left( {n - 1} \right){\rm I}$

B

${A^n} = {2^{n - 1}}A - \left( {n - 1} \right){\rm I}$

C

${A^n} = nA + \left( {n - 1} \right){\rm I}$

D

${A^n} = {2^{n - 1}}A + \left( {n - 1} \right){\rm I}$

Question 2: Palette 2/3

JEE Mains 2004 MCQ

Let $S(K)$ $ = 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.$ Then which of the following is true

A

Principle of mathematical induction can be used to prove the formula

B

$S\left( K \right) \Rightarrow S\left( {K + 1} \right)$

C

$S\left( K \right) \ne S\left( {K + 1} \right)$

D

$S\left( 1 \right)$ is correct

Question 3: Palette 3/3

JEE Mains 2002 MCQ

If ${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } } $ having $n$ radical signs then by methods of mathematical induction which is true

A

${a_n} > 7\,\,\forall \,\,n \ge 1$

B

${a_n} < 7\,\,\forall \,\,n \ge 1$

C

${a_n} < 4\,\,\forall \,\,n \ge 1$

D

${a_n} > 3\,\,\forall \,\,n \ge 1$