images
images listlengths 1
1
| problem
stringlengths 17
5.16k
| answer
stringlengths 1
1.22k
|
|---|---|---|
<image>Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$.
|
84
|
|
<image>If
\[\sin x + \cos x + \tan x + \cot x + \sec x + \csc x = 7,\]then find $\sin 2x.$
|
22 - 8 \sqrt{7}
|
|
<image>Find the phase shift of the graph of $y = \sin (3x - \pi).$
|
-\frac{\pi}{3}
|
|
<image>Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$.
|
628
|
|
<image>Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\]
|
63
|
|
<image>Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\]
|
0
|
|
<image>Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that
\[PA^2 + PB^2 + PC^2 = k \cdot PG^2 + GA^2 + GB^2 + GC^2.\]Find $k.$
|
3
|
|
<image>If angle $A$ lies in the second quadrant and $\sin A = \frac{3}{4},$ find $\cos A.$
|
-\frac{\sqrt{7}}{4}
|
|
<image>The real numbers $a$ and $b$ satisfy
\[\begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix} \times \begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = \mathbf{0}.\]Enter the ordered pair $(a,b).$
|
\left( \frac{8}{5}, -\frac{35}{2} \right)
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.