JEE MAIN - Mathematics (2019 - 12th January Morning Slot - No. 11)
Let P = $$\left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$ and Q = [qij] be two 3 $$ \times $$ 3 matrices such that Q – P5 = I3.
Then $${{{q_{21}} + {q_{31}}} \over {{q_{32}}}}$$ is equal to :
Then $${{{q_{21}} + {q_{31}}} \over {{q_{32}}}}$$ is equal to :
15
9
135
10
Explanation
$$P = \left[ {\matrix{
1 & 0 & 0 \cr
3 & 1 & 0 \cr
9 & 3 & 1 \cr
} } \right]$$
$${P^2} = \left[ {\matrix{ 1 & 0 & 0 \cr {3 + 3} & 1 & 0 \cr {9 + 9 + 9} & {3 + 3} & 1 \cr } } \right]$$
$${P^3} = \left[ {\matrix{ 1 & 0 & 0 \cr {3 + 3 + 3} & 1 & 0 \cr {6.9} & {3 + 3 + 3} & 1 \cr } } \right]$$
$${P^n} = \left[ {\matrix{ 1 & 0 & 0 \cr {3n} & 1 & 0 \cr {{{n\left( {n + 1} \right)} \over 2}{3^2}} & {3n} & 1 \cr } } \right]$$
$${P^5} = \left[ {\matrix{ 1 & 0 & 0 \cr {5.3} & 1 & 0 \cr {15.9} & {5.3} & 1 \cr } } \right]$$
$$Q = {P^5} + {{\rm I}_3}$$
$$Q = \left[ {\matrix{ 2 & 0 & 0 \cr {15} & 2 & 0 \cr {135} & {15} & 2 \cr } } \right]$$
$${{{q_{21}} + {q_{31}}} \over {{q_{32}}}} = {{15 + 135} \over {15}} = 10$$
Aliter
$$P = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr 3 & 0 & 0 \cr 9 & 3 & 0 \cr } } \right)$$
$$P = {\rm I} + X$$
$$X = \left( {\matrix{ 0 & 0 & 0 \cr 3 & 0 & 0 \cr 9 & 3 & 0 \cr } } \right)$$
$${X^2} = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 9 & 0 & 0 \cr } } \right)$$
$${{X_3} = 0}$$
$${{P^5} = {\rm I} + 5X + 10{X^2}}$$
$${Q = {P^5} + {\rm I} = 2{\rm I} + 5X + 10{X^2}}$$
$$Q = \left( {\matrix{ 2 & 0 & 0 \cr 0 & 2 & 0 \cr 0 & 0 & 2 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr {15} & 0 & 0 \cr {15} & {15} & 0 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr {90} & 0 & 0 \cr } } \right)$$
$$ \Rightarrow \,\,Q = \left( {\matrix{ 2 & 0 & 0 \cr {15} & 2 & 0 \cr {135} & {15} & 2 \cr } } \right)$$
$${P^2} = \left[ {\matrix{ 1 & 0 & 0 \cr {3 + 3} & 1 & 0 \cr {9 + 9 + 9} & {3 + 3} & 1 \cr } } \right]$$
$${P^3} = \left[ {\matrix{ 1 & 0 & 0 \cr {3 + 3 + 3} & 1 & 0 \cr {6.9} & {3 + 3 + 3} & 1 \cr } } \right]$$
$${P^n} = \left[ {\matrix{ 1 & 0 & 0 \cr {3n} & 1 & 0 \cr {{{n\left( {n + 1} \right)} \over 2}{3^2}} & {3n} & 1 \cr } } \right]$$
$${P^5} = \left[ {\matrix{ 1 & 0 & 0 \cr {5.3} & 1 & 0 \cr {15.9} & {5.3} & 1 \cr } } \right]$$
$$Q = {P^5} + {{\rm I}_3}$$
$$Q = \left[ {\matrix{ 2 & 0 & 0 \cr {15} & 2 & 0 \cr {135} & {15} & 2 \cr } } \right]$$
$${{{q_{21}} + {q_{31}}} \over {{q_{32}}}} = {{15 + 135} \over {15}} = 10$$
Aliter
$$P = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr 3 & 0 & 0 \cr 9 & 3 & 0 \cr } } \right)$$
$$P = {\rm I} + X$$
$$X = \left( {\matrix{ 0 & 0 & 0 \cr 3 & 0 & 0 \cr 9 & 3 & 0 \cr } } \right)$$
$${X^2} = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 9 & 0 & 0 \cr } } \right)$$
$${{X_3} = 0}$$
$${{P^5} = {\rm I} + 5X + 10{X^2}}$$
$${Q = {P^5} + {\rm I} = 2{\rm I} + 5X + 10{X^2}}$$
$$Q = \left( {\matrix{ 2 & 0 & 0 \cr 0 & 2 & 0 \cr 0 & 0 & 2 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr {15} & 0 & 0 \cr {15} & {15} & 0 \cr } } \right) + \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr {90} & 0 & 0 \cr } } \right)$$
$$ \Rightarrow \,\,Q = \left( {\matrix{ 2 & 0 & 0 \cr {15} & 2 & 0 \cr {135} & {15} & 2 \cr } } \right)$$
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