JEE MAIN - Mathematics (2019 - 8th April Evening Slot - No. 8)
Let the number 2,b,c be in an A.P. and
A = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]$$. If det(A) $$ \in $$ [2, 16], then c lies in the interval :
A = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]$$. If det(A) $$ \in $$ [2, 16], then c lies in the interval :
[2, 3)
[4, 6]
(2 + 23/4, 4)
[3, 2 + 23/4]
Explanation
2, b, c are in AP.
Let common difference = d
$$ \therefore $$ b = 2 + d and c = 2 + 2d
|A| = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]$$
C2 = C2 - C1
C3 = C3 - C1
= $$\left| {\matrix{ 1 & 0 & 0 \cr 2 & {b - 2} & {c - 2} \cr 4 & {{b^2} - 4} & {{c^2} - 4} \cr } } \right|$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left| {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 1 \cr 4 & {b + 2} & {c + 2} \cr } } \right|$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left[ {c + 2 - b - 2} \right]$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left( {c - b} \right)$$
[ As b = 2 + d and c = 2 + 2d, then b - 2 = 4, c - 2 = 2d and c - b = d]
= (d) (2d) (d)
= 2d3
Given |A| $$ \in $$ [2, 16]
$$ \therefore $$ 2d3 $$ \in $$ [2, 16]
$$ \Rightarrow $$ d3 $$ \in $$ [1, 8]
$$ \Rightarrow $$ d $$ \in $$ [1, 2]
As c = 2 + 2d
then c $$ \in $$ [4, 6]
Let common difference = d
$$ \therefore $$ b = 2 + d and c = 2 + 2d
|A| = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]$$
C2 = C2 - C1
C3 = C3 - C1
= $$\left| {\matrix{ 1 & 0 & 0 \cr 2 & {b - 2} & {c - 2} \cr 4 & {{b^2} - 4} & {{c^2} - 4} \cr } } \right|$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left| {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 1 \cr 4 & {b + 2} & {c + 2} \cr } } \right|$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left[ {c + 2 - b - 2} \right]$$
= $$\left( {b - 2} \right)\left( {c - 2} \right)\left( {c - b} \right)$$
[ As b = 2 + d and c = 2 + 2d, then b - 2 = 4, c - 2 = 2d and c - b = d]
= (d) (2d) (d)
= 2d3
Given |A| $$ \in $$ [2, 16]
$$ \therefore $$ 2d3 $$ \in $$ [2, 16]
$$ \Rightarrow $$ d3 $$ \in $$ [1, 8]
$$ \Rightarrow $$ d $$ \in $$ [1, 2]
As c = 2 + 2d
then c $$ \in $$ [4, 6]
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