Let A=[ … ] and B=[ … ] be two 2× 1matrices with real entries such that A=XB, where X= 1 √3[ … ], and k∈ R. If a_1^2+a_2^2= 2 3(b_1^2+b_2^2) and…
JEE Main 2021 — Mathematics Algebra
2021integerhard
Let A=[a1a2] and B=[b1b2] be two 2×1matrices with real entries such that A=XB, where X=31[11−1k], and k∈R. If a12+a22=32(b12+b22) and (k2+1)b22=−2b1b2, then the value of k is __________.
Official previous-year question
Held on 16 Mar 2021 · Verified 6 Jul 2026.
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Solution
A=XB
[a1a2]=31[11−1k][b1b2]
[3a13a2]=[b1−b2b1+kb2]
b1−b2=3a1…(1)
b1+kb2=3a2…(2)
Given, a12+a22=32(b12+b22)
(1)2+(2)2
(b1+b2)2+(b1+kb2)2=3(a12+a22)
a12+a22=32b12+3(1+k2)b22+32b1b2(k−1)
Given, a12+a22=32b12+32b22
On comparing we get
3k2+1=32⇒k2+1=2
⇒k=±1…(3)
32(k−1)=0⇒k=1…(4)
From both we get k=1
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