Q61 / 150 · MHT CET 2024 (15 May Shift 1) · Mathematics

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are two unit vectors such that $5 \bar{a}+4 \bar{b}$ and $\bar{a}-2 \bar{b}$ are perpendicular to each other, then the angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ is

  • A
    $\frac{\pi}{3}$
  • Correct answer
    B
    $\frac{2 \pi}{3}$
  • C
    $\cos ^{-1}\left(\frac{2}{3}\right)$
  • D
    $\quad \cos ^{-1}\left(\frac{1}{3}\right)$

Solution

Let $\theta$ be the angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$. Since $5 \bar{a}+4 \bar{b}$ and $\bar{a}-2 \bar{b}$ are perpendicular to each other. $\begin{aligned} \therefore \quad & (5 \bar{a}+4 \overline{\mathrm{~b}}) \cdot(\overline{\mathrm{a}}-2 \overline{\mathrm{~b}})=0 \\ & \Rightarrow 5(\overline{\mathrm{a}} \cdot \overline{\mathrm{a}})-6(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}})-8(\overline{\mathrm{~b}} \cdot \overline{\mathrm{~b}})=0 \\ & \Rightarrow 5|\overline{\mathrm{a}}|^2-6|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta-8|\overline{\mathrm{~b}}|^2=0 \\ & \Rightarrow 5-6 \cos \theta-8=0 \\ & \Rightarrow \cos \theta=\frac{1}{2} \\ & \Rightarrow \theta=\frac{2 \pi}{3} \end{aligned}$

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

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