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JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Matrices

Let

A = [12 - 1 α]

and

B = [β 1 10], α, β \in R

. Let

α_{1}

be the value of

α

which satisfies

(A + B)^{2} = A^{2} + [2222]

and

α_{2}

be the value of

α

which satisfies

(A + B)^{2} = B^{2}

. Then

∣ α_{1} - α_{2} ∣

is equal to ______.

Answer: 2

Solution

First, compute

A + B = [β + 1 3 0 α]

.

For condition 1:

(A + B)^{2} = A^{2} + [2222]

.

Compute

(A + B)^{2} = [(β + 1)^{2} 3 (β + 1) + 3 α 0 α^{2}]

and

A^{2} = [- 1 2 + 2 α - 1 - α α^{2} - 2]

.

Adding the given matrix to

A^{2}

and equating entries gives

α_{1} = 1

.

For condition 2:

(A + B)^{2} = B^{2}

.

Compute

B^{2} = [β^{2} + 1 β β 1]

. Equating with

(A + B)^{2}

gives

β = 0

and

α_{2} = - 1

.

Therefore,

∣ α_{1} - α_{2} ∣ = ∣1 - (- 1) ∣ = 2

.

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Integer answer

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