MCQ
If $a\,.i\,=a\,.\,(i+j)=a\,.\,(i+j+k)$ , then $a = $
- ✓$i$
- B$k$
- C$j$
- D$i + j + k$
Then $a\,.\,i = (xi + yj + zk)\,.\,i = x$ and $a\,.\,(i + j) = x + y$ and $a\,.\,(i + j + k) = x + y + z$
Given that $x = x + y = x + y + z$
Now $x = x + y\,\,\, \Rightarrow y = 0$ and $x + y = x + y + z\,\, \Rightarrow \,\,z = 0$
Hence $x = 1$; $\therefore \,\,a = i$.
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$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$
where
$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$.
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?
$(A)$ $p \left(\frac{3+\sqrt{2}}{4}\right)<0$
$(B)$ $p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C)$ $p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D)$ $p \left(\frac{5-\sqrt{2}}{4}\right)<0$