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If \overrightarrow{x}\; \text{and}\; \overrightarrow{y} be two non-zero vectors such that \left | \overrightarrow{x}+\overrightarrow{y} \right |=\left |\overrightarrow{x} \right | and 2\overrightarrow{x}+\lambda \overrightarrow{y} is perpendicular to \overrightarrow{y}, then the value of \lambda is ________
Option: 1 0
Option: 2 1
Option: 3 -1
Option: 4 -2

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\\|\overrightarrow{\mathrm{x}}+\overrightarrow{\mathrm{y}}|=|\overrightarrow{\mathrm{x}}| \\ \\\sqrt{|\overrightarrow{\mathrm{x}}|^{2}+|\overrightarrow{\mathrm{y}}|^{2}+2 \overrightarrow{\mathrm{x}} \cdot \overrightarrow{\mathrm{y}}}=|\overrightarrow{\mathrm{x}}| \\ \\ |\overrightarrow{\mathrm{y}}|^{2}+2 \overrightarrow{\mathrm{x}} \cdot \overrightarrow{\mathrm{y}}=0\qquad\qquad\ldots(1)

\begin{array}{ll} \text { Now } & (2 \vec{x}+\lambda \vec{y}) \cdot \vec{y}=0 \\\\ & 2 \vec{x} \cdot \vec{y}+\lambda|\vec{y}|^{2}=0 \end{array}

\begin{aligned} &\text { From (1) }\\ &-|\vec{y}|^{2}+\lambda|\vec{y}|^{2}=0\\ &(\lambda-1)|\vec{y}|^{2}=0\\ &\text { given }|\vec{y}| \neq 0 \Rightarrow \lambda=1 \end{aligned}

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