This chapter is currently out of syllabus
1
AIEEE 2005
+4
-1
Out of Syllabus
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
$${A^n} = {2^{n - 1}}A + \left( {n - 1} \right){\rm I}$$
2
AIEEE 2004
+4
-1
Out of Syllabus
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
3
AIEEE 2002
+4
-1
Out of Syllabus
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
$${a_n} > 3\,\,\forall \,\,n \ge 1$$
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