Q219 / 270 · NDA 2022 (Phase 2) · Mathematics

What are the values of $x$ for which the angle between the vectors $2 x^2 \hat{i}+3 x \hat{j}+\hat{k}$ and $\hat{i}-2 \hat{j}+x^2 \hat{k}$ is obtuse?

  • Correct answer
    A
    $0 \lt x \lt 2$
  • B
    $x \lt 0$
  • C
    $x \gt 2$
  • D
    $0 \leq x \leq 2$

Solution

Given that angle between vectors is obtuse $\begin{array}{ll} \therefore & \vec{a} \cdot \vec{c} \lt 0 \\ \Rightarrow & 2 x^2-6 x+x^2 \lt 0 \quad \Rightarrow 3 x^2-6 x \lt 0 \\ \Rightarrow 3 x(x-2) \lt 0 \quad \Rightarrow 0 \lt x \lt 2 \end{array}$

Hence, the correct answer is option: 'A'.

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