For all real $x$, the vectors $C x \hat{i}-6 \hat{j}-3 \hat{k}$ and $x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2…

For all real $x$, the vectors $C x \hat{i}-6 \hat{j}-3 \hat{k}$ and $x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}$ make an obtuse angle with each other, then the value of C can be in
  1. $(0,1)$
  2. $\left(-2, \frac{-4}{3}\right)$
  3. $\left(\frac{-4}{3}, 0\right)$
  4. $\left(0, \frac{4}{3}\right)$

Solution

Let $\overline{\mathrm{a}}=C x \hat{\mathrm{i}}-6 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}$
Angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ is obtuse. $\begin{array}{ll} \therefore & \overline{\mathrm{a}} \cdot \overline{\mathrm{~b}} \lt \cos 180^{\circ} \\ \therefore & \overline{\mathrm{a}} \cdot \overline{\mathrm{~b}} \lt 0 \\ \therefore & \mathrm{C} x^2-12-6 \mathrm{C} x \lt 0 \end{array}$ $\begin{aligned} & \Rightarrow \mathrm{C} x^2-6 \mathrm{C} x-12 \lt 0 \\ & \Rightarrow \mathrm{C} \lt 0 \text { and } \mathrm{D} \lt 0 \\ & \Rightarrow \mathrm{C} \lt 0 \text { and } 36 \mathrm{C}^2+48 \mathrm{C} \lt 0 \\ & \Rightarrow \mathrm{C} \lt 0 \text { and } 3 \mathrm{C}^2+4 \mathrm{C} \lt 0 \\ & \Rightarrow \mathrm{C} \lt 0, \mathrm{C}(3 \mathrm{C}+4) \lt 0 \\ & \Rightarrow \mathrm{C} \lt 0,-\frac{4}{3} \lt \mathrm{C} \lt 0 \\ \therefore \quad \mathrm{C} & =\left(-\frac{4}{3}, 0\right)\end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 1)

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