Study questions platform-wide or filter by specific tests with correct answers revealed.
Let $\omega$ be a complex number with $2\omega+1=z$ where $z=\sqrt{-3}$. If $\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k$, find $k$.
Since $z=\sqrt{-3}=i\sqrt{3}$ and $2\omega+1=i\sqrt{3}$, so $\omega=\frac{-1+i\sqrt{3}}{2}=e^{2\pi i/3}$ (a primitive cube root of unity).
Properties: $\omega^3=1$, $1+\omega+\omega^2=0$, so $-\omega^2-1=\omega$ and $\omega^7=\omega$.
Matrix becomes: $\begin{vmatrix}1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega\end{vmatrix}$
This determinant $= 3(\omega-\omega^2)\cdot... $ After evaluation $= 3i\sqrt{3} = 3z$... so $k=z$.
In the figure, lines $m$ and $k$ are parallel (m โฅ k), cut by a transversal. The angle on line $m$ is labeled $sยฐ$ and the angle on line $k$ is labeled $tยฐ$ (co-interior / same-side interior angles). If $s = t + 30$, what is the value of $t$?
Since $m \parallel k$, the angles $sยฐ$ and $tยฐ$ are co-interior (same-side interior) angles, which means they are supplementary:
$s + t = 180ยฐ$
We are also given: $s = t + 30$
Substituting: $(t + 30) + t = 180$
$2t + 30 = 180$
$2t = 150$
$t = 75ยฐ$
Sign in to join the conversation and share your thoughts.
Log In to Comment