Consider the vectors x = i +2 j +3 k , y =2 i +3 j + k , and z =3 i + j +2 k For two distinct positive real numbers α and β , define X =α x +β y - z , Y =α y +β z - x , and Z =α z +β x - y If the vectors X , Y , and Z

Correct answer: 0 | array ccc α & β & -1 -1 & α & β β & -1 & α array | | array ccc 1 & 2 & 3 2 & 3 & 1 3 & 1 & 2 array | _ -0 =0 aligned & (α^3+β^3-1)-(-α β-α β-α β)=0 & α^3+β^3+3 α β=1 & α^3+β^3+(-1)^3=3(α)(β)(-1) & α+β-1=0 aligned So, α+β-3=-2

JEE Advanced 2025Paper 2

Exam: JEE Advanced
Year: 2025
Paper: Paper 2
Subject: Mathematics
Chapter: Vector Algebra

JEE Advanced 2025 · Paper 2 · Mathematics · Vector Algebra

Consider the vectors x = i +2 j +3 k , y =2 i +3 j + k , and z =3 i...

Consider the vectors

\vec{x}=\hat{i}+2 \hat{j}+3 \hat{k}, \quad \vec{y}=2 \hat{i}+3 \hat{j}+\hat{k}, \quad \text{ and } \quad \vec{z}=3 \hat{i}+\hat{j}+2 \hat{k}

For two distinct positive real numbers

\alpha

and

\beta

, define

\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text{ and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}

If the vectors

\vec{X}, \vec{Y}

, and

\vec{Z}

lie in a plane, the value of

\alpha+\beta-3

is ______ .

Correct answer

Step-by-step solution

\Rightarrow\left|

\begin{array}{ccc}\alpha & \beta & -1 \\ -1 & \alpha & \beta \\ \beta & -1 & \alpha\end{array}

\right| \underbrace{\left|

\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{array}

\right|}_{-0}=0

\begin{aligned} & \Rightarrow\left(\alpha^3+\beta^3-1\right)-(-\alpha \beta-\alpha \beta-\alpha \beta)=0 \\ & \Rightarrow \alpha^3+\beta^3+3 \alpha \beta=1 \\ & \Rightarrow \alpha^3+\beta^3+(-1)^3=3(\alpha)(\beta)(-1) \\ & \Rightarrow \alpha+\beta-1=0\end{aligned}

So,

\alpha+\beta-3=-2

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