Embibe Experts Solutions for Chapter: Mathematical Methods, Exercise 1: MHT-CET 2020 (12-Oct Shift-1)

The unit vector $\left(a\hat{\mathrm{i}}+b\hat{\mathrm{j}}\right)$ is perpendicular to $\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right).$ The value of $b$ can be:

If $\overrightarrow{A}=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{B}=\hat{\mathrm{i}}-3\hat{\mathrm{ȷ}}+5\hat{\mathrm{k}}$ and $\overrightarrow{C}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-4\hat{\mathrm{k}}$ form a right angled triangle then out of the following which one is satisfied?

Figure shows three forces ${\overrightarrow{F}}_1,{\overrightarrow{F}}_2$ and ${\overrightarrow{F}}_3$ acting along the sides of an equilateral triangle. If the total torque acting at point $O$ (centre of the triangle) is zero then the magnitude of ${\overrightarrow{F}}_3$ is

Resultant of two vectors $\overrightarrow{\mathit{P}}$ and $\overrightarrow{\mathit{Q}}$ is of magnitude $R_1.$ If direction of $\overrightarrow{\mathit{Q}}$ is reversed, the resultant is of magnitude $R_2$. The value of $\left(R_1^2+R_2^2\right)$ is $\left[\cos \left(\pi -\theta \right)=-\cos \theta \right]$

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two non-zero vectors inclined at an angle $\theta$, $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ are unit vectors along $\overrightarrow{A}$ and $\overrightarrow{B}$ respectively. The component of $\overrightarrow{A}$ in the direction of $\overrightarrow{B}$ is