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JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Determinants

Let for

A = 1 a 1 231312

,

∣ A ∣ = 2

. If

∣2 adj (2 adj (2 A)) ∣ = 3 2^{n}

, then

3 n + α

is equal to

(A)10
(B)9
(C)12
(D)11✓ Correct

Answer: (D)

Solution

First, find

α

using

∣ A ∣ = 2

:

∣ A ∣ = 1 (6 - 1) - 2 (2 α - 1) + 3 (α - 3) = 5 - 4 α + 2 + 3 α - 9 = - α - 2 = 2

\Rightarrow α = - 4

Now simplify

∣2 adj (2 adj (2 A)) ∣

:

Using

∣ adj (B) ∣ = ∣ B ∣^{n - 1}

for an

n \times n

matrix:

∣2 adj (2 A) ∣ = 2^{3} ∣ adj (2 A) ∣ = 8 \cdot (2^{3} ∣ A ∣)^{2} = 8 \cdot 64 \cdot 4 = 2048 = 2^{11}

∣2 adj (2 adj (2 A)) ∣ = 2^{3} ∣ adj (2 adj (2 A)) ∣ = 8 \cdot (2^{11})^{2} = 8 \cdot 2^{22} = 2^{25} = (2^{5})^{5} = 3 2^{5}

So

n = 5

and

3 n + α = 15 - 4 = 11

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Single correct option

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