JEE MAIN - Mathematics (2019 - 10th January Morning Slot - No. 10)
Let d $$ \in $$ R, and
$$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\left( {\sin \theta } \right) + 2} & d \cr 5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr } } \right],$$
$$\theta \in \left[ {0,2\pi } \right]$$ If the minimum value of det(A) is 8, then a value of d is -
$$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\left( {\sin \theta } \right) + 2} & d \cr 5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr } } \right],$$
$$\theta \in \left[ {0,2\pi } \right]$$ If the minimum value of det(A) is 8, then a value of d is -
$$-$$ 7
$$2\left( {\sqrt 2 + 2} \right)$$
$$-$$ 5
$$2\left( {\sqrt 2 + 1} \right)$$
Explanation
$$\det A = \left| {\matrix{
{ - 2} & {4 + d} & {\sin \theta - 2} \cr
1 & {\sin \theta + 2} & d \cr
5 & {2\sin \theta - d} & { - \sin \theta + 2 + 2d} \cr
} } \right|$$
(R1 $$ \to $$ R1 + R3 $$-$$ 2R2)
$$ = \left| {\matrix{ 1 & 0 & 0 \cr 1 & {\sin \theta + 2} & d \cr 5 & {2\sin \theta - d} & {2 + 2d - \sin \theta } \cr } } \right|$$
= (2 + sin $$\theta $$) ( 2 + 2d $$-$$ sin$$\theta $$) $$-$$ d(2sin$$\theta $$ $$-$$ d)
= 4 + 4d $$-$$ 2sin$$\theta $$ + 2sin$$\theta $$ + 2dsin$$\theta $$ $$-$$ sin2$$\theta $$ $$-$$ 2dsin$$\theta $$ + d2
d2 + 4d + 4 $$-$$ sin2$$\theta $$
= (d + 2)2 $$-$$ sin2$$\theta $$
For a given d, minimum value of
det(A) = (d + 2)2 $$-$$ 1 = 8
$$ \Rightarrow $$ d = 1 or $$-$$ 5
(R1 $$ \to $$ R1 + R3 $$-$$ 2R2)
$$ = \left| {\matrix{ 1 & 0 & 0 \cr 1 & {\sin \theta + 2} & d \cr 5 & {2\sin \theta - d} & {2 + 2d - \sin \theta } \cr } } \right|$$
= (2 + sin $$\theta $$) ( 2 + 2d $$-$$ sin$$\theta $$) $$-$$ d(2sin$$\theta $$ $$-$$ d)
= 4 + 4d $$-$$ 2sin$$\theta $$ + 2sin$$\theta $$ + 2dsin$$\theta $$ $$-$$ sin2$$\theta $$ $$-$$ 2dsin$$\theta $$ + d2
d2 + 4d + 4 $$-$$ sin2$$\theta $$
= (d + 2)2 $$-$$ sin2$$\theta $$
For a given d, minimum value of
det(A) = (d + 2)2 $$-$$ 1 = 8
$$ \Rightarrow $$ d = 1 or $$-$$ 5
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