Mathematics - Vector Question with Solution | TestHub

(i) Solved the two equations, say, i.e. $x+y=|a|$ and $ax-y=1$, we get $x=\frac{|a|+1}{a+1}>0$ and $y=\frac{|a|-1}{a+1}>0$ when $a+1>0$; we get $a>1$ $\therefore a_0=1$ (ii) We have, $\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ $\Rightarrow \mathbf{a}\cdot \hat{\mathbf{k}}=\gamma$ Now, $\hat{\mathbf{k}}\times (\hat{\mathbf{k}}\times \mathbf{a})=(\hat{\mathbf{k}}\cdot \mathbf{a})-(\hat{\mathbf{k}}\cdot \hat{\mathbf{k}})\mathbf{a}=\gamma \hat{\mathbf{k}}-(\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}})$ $\Rightarrow \alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}=0\Rightarrow \alpha =\beta =0$ Also, $\alpha +\beta +\gamma =2\Rightarrow \gamma =2$
(iv) $\sin A\sin B\sin C+\cos A\cos B\le \sin A\sin B+\cos A\cos B=\cos (A-B)$ $$\begin{array}{lc}\Rightarrow & \cos (A-B)\ge 1\Rightarrow \cos (A-B)=1\\ \Rightarrow & \sin C=1\end{array}$$